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Bibliography of Wavelet and Time Series Titles

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[Abraham and Wei, 1984]

Bovus Abraham and William W. S. Wei. Inferences about the parameters of a time series model with changing variance. Metrika, 31:183-194, 1984.

[Abramovich and Benjamini, 1995]

Felix Abramovich and Y. Benjamini. Thresholding of wavelet coefficients as multiple hypotheses testing procedure. In [Antoniadis and Oppenheim, 1995], pages 5-14.

[Abramovich and Benjamini, 1996]

Felix Abramovich and Y. Benjamini. Adaptive thresholding of wavelet coefficients. Computational Statistics & Data Analysis, 22:351-361, 1996.

[Abramovich and Silverman, 1998]

F. Abramovich and B. W. Silverman. Wavelet decomposition approaches to statistical inverse problems. Biometrika, 85(1):115-129, 1998.

A wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise. Primary interest is in some function f(t) but data are accessible only about some linear transform corrupted by noise; The usual linear methods for such inverse problems do not perform satisfactorily when f(t) is spatially inhomogeneous. One existing nonlinear alternative is the wavelet-vaguelette decomposition method, based on the expansion of the unknown f(t) in wavelet series. In the vaguelette- wavelet decomposition method proposed here, the observed data are expanded directly in wavelet series. The performances of various methods are compared through exact risk calculations, in the context of the estimation of the derivative of a function observed subject to noise. A result is proved demonstrating that, with a suitable universal threshold somewhat larger than that used for standard denoising problems, both the wavelet-based approaches have an ideal spatial adaptivity property.

[Abramovich et al., 1996]

Felix Abramovich, T. Sapatinas, and Bernard Silverman. Wavelet thresholding via a Bayesian approach. Submitted, 1996.

[Abry and Flandrin, 1994]

P. Abry and P. Flandrin. On the initialization of the discrete wavelet transform algorithm. IEEE Signal Processing Letters, 1(2):32-34, 1994.

The authors show that making use of the discrete wavelet transform to analyse data implies performing a preliminary initialization of the fast pyramidal algorithm. An approximation enabling easy performance of such an initialization is proposed.

[Abry and Sellan, 1996]

P. Abry and F. Sellan. The wavelet-based synthesis for fractional Brownian motion - Proposed by F. Sellan and Y. Meyer: Remarks and fast implementation. Applied and Computational Harmonic Analysis, 3(4):377-383, 1996.

[Abry and Veitch, 1998]

P. Abry and D. Veitch. Wavelet analysis of long-range-dependent traffic. IEEE Transactions on Information Theory, 44(1):2-15, 1998.

A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced, The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as wed as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long-range dependence in Ethernet traffic traces, New features are found with important implications for the choice of valid models for performance evaluation, A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.

[Abry et al., 1993]

P. Abry, P. Gonclaves, and P. Flandrin. Wavelet-based spectral analysis of 1/f processes. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 3, pages 237-240, 1993. Minneapolis, MN, USA.

The authors attempt to show how and why a time-scale-based spectral estimation naturally suits the nature of 1/f processes, characterized by a power spectral density proportional to mod nu mod /sup - alpha /. They show that a time-scale approach allows an unbiased estimation of the spectral exponent alpha and interpret this result in terms of matched tilings of the time-frequency plane. They derive explicitly the probability density function of the estimated value of alpha. From this analysis, they find that there exists an optimum number of scales to use in a discrete wavelet scheme for obtaining a minimum variance estimator and that an improved procedure can be designed by making use of weighted least-squares in the estimation.

[Abry et al., 1995]

P. Abry, P. Gonclaves, and P. Flandrin. Wavelets, spectrum analysis and 1/f processes. In [Antoniadis and Oppenheim, 1995], pages 15-29.

The purpose of this paper is to evidence why wavelet-based estimators are naturally matched to the spectrum analysis of 1/f processes. It is shown how the revisiting of classical spectral estimators from a time-frequency perspective allows to define different wavelet-based generalizations which are proved to be statistically and computationally efficient. Discretization issues (in time and scale) are discussed in some detail, theoretical claims are supported by numerical experiments and the importance of the proposed approach in turbulence studies is underlined.

[Abry et al., 1998]

Patrice Abry, Darryl Veitch, and Patrick Flandrin. Long range dependence: Revisiting aggregation with wavelets. Journal of Time Series Analysis, 19(3):253-266, 1998.

The aggregation procedure is a natural way to analyse signals which exhibit long-range dependent features and has been used as a basis for estimation of the Hurst parameter, H. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly related to approximations of the signal in the sense of a Haar-multiresolution analysis. A natural wavelet based generalisation to traditional aggregation is then proposed: ``a-aggregation''. It is shown that a-aggregation cannot lead to good estimators of H, and so a new kind of aggregation, ``d-aggregation'', is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of H based on d-aggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data.

[Adorf, 1995]

H. M. Adorf. Interpolation of irregularly sampled data series -- A survey. In R. A. Shaw, H. E. Payne, and J. J. E. Hayes, editors, Astronomical Data Analysis Software and Systems IV, volume 77 of ASP Conference Series, pages 460-463, 1995.

Many astronomical observations, including spectra and time series, consist of irregularly sampled data series, the analysis of which is more complicated than that of regularly spaced data sets. Therefore a viable strategy consists of resampling a given irregularly sampled data series onto a regular grid, in order to use conventional tools for further analysis. Resampling always requires some form of interpolation, which permits the construction of an underlying continuous function representing the discrete data. This contribution surveys the methods used in astronomy for the interpolation of irregularly sampled one-dimensional data series.

[Aguilar, 1996]

Omar Aguilar. Wavelet and autoregressive decompositions for evaluating frequency compositions in time series. Technical report, Institute of Statisics and Decision Sciences, Duke University, 1996. Discussion Paper 96-22.

[Al-Mohimeed and Li, 1997]

Mohammed A. Al-Mohimeed and Ching-Chung Li. Application of shift-invariant wavelet transform to video coding. In Tzi cker Chiueh and Andrew G. Tescher, editors, Video Techniques and Software for Full-Service Networks, volume 2915 of Proceedings of the SPIE, pages 64-75, 1997.

The standard discrete wavelet transform lacks translation invariance in 1-D signals and 2-D images. The down-sampling at each coarser scale accentuates the undesirable effects of the shift-variance, in particular, on the motion estimation from decomposed subimages in video coding. In this paper, we present a study of applying the Chui-Shi shift-invariant wavelet transform using 'oversampling frames' to video compression. Further, we present an algorithm for approximating the motion fields at different scales and different frequency bands by utilizing the multiresolution structure of wavelet decomposition. Motion vectors at a higher resolution are predicted by the motion vectors at a lower resolution through a proper scaling. Experimental results on a salesman video sequence show that the use of the 2-D oversampling algorithm of a biorthogonal spline wavelet has reduced the required number of motion vectors while maintaining an acceptable prediction error when compared to the classical block matching technique using the standard wavelet transform. The proposed approach will advance the video compression methodology for applications to HDTV and video conferencing.

[Aldroubi and Feichtinger, 1997]

Akram Aldroubi and Hans Feichtinger. Complete iterative reconstruction algorithms for irregularly sampled data in spline-like spaces. BEIP, National Institute of Health, 1997.

We prove that the exact reconstruction of a function fv from its samples fv (x_i) on any `sufficiently dense' sampling set X_i in ind subset RR^n, where ind is a countable indexing set, can be obtained for a large class of spline-like spaces that belong to Lp (RR^n). Moreover, The reconstruction can be implemented using fast algorithms. Since, a special case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittacker sampling theorem on regular sampling and the Paley-Wiener theorem on nonuniform sampling.

[Aldroubi and Unser, 1996]

Akram Aldroubi and Michael Unser. Wavelets in Medicine and Biology. CRC Press Inc., Boca Raton, 1996.

Considerable attention from the international scientific community is currently focused on the wide ranging applications of wavelets. For the first time, the field's leading experts have come together to produce a complete guide to wavelet transform applications in medicine and biology. Wavelets in Medicine and Biology provides accessible, detailed, and comprehensive guidelines for all those interested in learning about wavelets and their applications to biomedical problems. The book consists of four main sections: Theory and Implementation of Wavelet Transforms, Wavelets in Medical Imaging and Tomography, Wavelets and Biomedical Signal Processing, Wavelets and Mathematical Models in Biology. The introductory material is written for non-experts and includes basic discussions of the theoretical and practical foundations of wavelet methods. The background and introduction is followed by contributions from the most prominent researchers in the field, giving the reader a complete survey of the use of wavelets in biomedical engineering. An international perspective is provided throughout the book, with contributions from experts from Germany, France, America, Belgium, Holland, Turkey, and Switzerland.

[Ali, 1989]

Mukhtar M. Ali. Tests for autocorrelation and randomness in multiple time series. Journal of the American Statistical Association, 84(406):533-540, 1989.

[Allan, 1966]

David W. Allan. Statistics of atomic frequency standards. Proceedings of the IEEE, 31:221-230, 1966.

[Allen and Tett, 1997]

M. R. Allen and S. F. B. Tett. Checking for model consistency in optimal fingerprinting. Technical Report RAL-TR-97-040, Council for the Central Laboratory of the Research Councils, 1997.

[Anderson and Walker, 1964]

T. W. Anderson and A. M. Walker. On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. The Annals of Mathematical Statistics, 35:1296-1303, 1964.

[Anderson and You, 1996]

T. W. Anderson and Linfeng You. Adequacy of asymptotic theory for goodness-of-fit criteria for spectral distributions. Journal of Time Series Analysis, 17(6):533-552, 1996.

Any of the Cramer-von Mises, Anderson-Darling, and Kolmogorov- Smirnov statistics can be used to test the null hypothesis that the standardized spectral distribution of a stationary stochastic process is a specified one. The asymptotic distributions of the criteria have been characterized (Anderson, 1993).They are the same as for probability distributions if the observations are independent (all autocorrelations zero), but are different when there is dependence. In this paper simulation with 10 000 replications has been used to determine the distributions of the criteria for samples of size 6, 10, 30 and 100 when the observations are independent. These empirical distributions have been compared with the asymptotic distributions in order to ascertain the sample sizes necessary for using the asymptotic tables. For practical purposes they are 30 for the Cramer-von Mises and Kolmogorov statistics and over 100 for Anderson-Darling.

[Anderson et al., 1984]

John R. Anderson, Duane E. Stevens, and Paul R. Julian. Temporal variations of the tropical 40-50 day oscillation. Monthly Weather Review, 112(12):2431-2438, 1984.

[Anderson, 1971]

T. W. Anderson. The Statistical Analysis of Time Series. John Wiley and Sons, Inc., New York, 1971.

[Anderson, 1993a]

James C. Anderson. A wavelet magnitude analysis theorem. IEEE Transactions on Signal Processing, 41(12):3541-3543, 1993.

Wavelet transform is the constant-Q special case of the generalized short time Fourier transform (GSTFT), and is useful for wavelet analysis. Scalograms are analyzed using specific types of filter/detector banks. GSTFT results are universally applicable to wavelet theory and are useful tools for scalogram sampling for computation and data reduction functions.

[Anderson, 1993b]

T. W. Anderson. Goodness of fit tests for spectral distributions. Applied Statistics, 21(2):830-847, 1993.

[Andreas and Treviño, 1997]

Edgar L. Andreas and George Treviño. Using wavelets to detect trends. Journal of Atmospheric and Oceanic Technology, 14(3):554-564, 1997.

Wavelets are a new class of basis functions that are finding wide use for analyzing and interpreting time series data. This paper describes a new use for wavelets-identifying trends in time series. The general signal considered has a quadratic trend. The inverted Haar wavelet and the elephant wavelet, respectively, provide estimates of the first-order and second-order coefficients in the trend polynomial. Unlike usual wavelet applications, however, this analysis requires only one wavelet dilation scale L, where L is the total length of the time series. Error analysis shows that wavelet trend detection is roughly half as accurate as least squares trend detection when accuracy is evaluated in terms of the mean-square error in estimates of the first-order and second-order trend coefficients. But wavelet detection is more than twice as efficient as least squares detection in the sense that it requires fewer than half the number of floating-point operations of least squares regression to yield the three coefficients of the quadratic trend polynomial. This paper demonstrates wavelet trend detection using artificial data and then various turbulence data collected in the atmospheric surface layer, and last, provides guidelines on when linear and quadratic trends are ``significant'' enough to require removal from a time series.

[Ansari et al., 1991]

R. Ansari, C. Guillemot, and J. F. Kaiser. Wavelet construction using lagrange halfband filters. IEEE Transactions on Circuits and Systems, 38(9):1116-1118, 1991.

Using the approach described by M.J.T. Smith and T.P. Barnwell (1986) for obtaining exact-reconstruction filter banks, the authors present conjugate-quadrature and linear-phase solutions for two-channel filter banks using Lagrange halfband filters. It is shown that the wavelet solutions obtained by I. Daubechies (1988) under certain regularity conditions are the same as the conjugate-quadrature solutions derived from Lagrange halfband filters using the above approach. The linear-phase solution that is described provides filters with simple coefficients.

[Antoniadis and Oppenheim, 1995]

Anestis Antoniadis and Georges Oppenheim, editors. Wavelets and Statistics, volume 103 of Lecture Notes in Statistics, New York, 1995. Springer-Verlag.

Wavelets theory has found applications in a remarkable diversity of disciplines. The volume presents the proceedings of a conference held at Villard de Lans, France in 1994. Both statistical results and practical contributions were presented. The material is wide in scope and ranges from the development of new tools for nonparametric curve estimation to applied problems such as detection of transients in signal processing and image segmentation.

[Antoniadis and Pham, 1996]

Anestis Antoniadis and Dinh Tuan Pham. Wavelet regression for random or irregular design. Technical report, IMAG - C.N.R.S. - I.N.R.I.A., 1996.

[Antoniadis et al., 1994]

A. Antoniadis, G. Grégoire, and I. W. McKeague. Wavelet methods for curve estimation. Journal of the American Statistical Association, 89(428):1340-1353, 1994.

[Antoniadis et al., 1997a]

A. Antoniadis, I. Gijbels, and G. Grégoire. Model selection using wavelet decomposition and applications. Biometrika, 84(4):751-763, 1997.

In this paper we discuss how to use wavelet decompositions to select a regression model. The methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients. Consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector. We then apply the selection rule to specific problems, including testing for pure white noise. The power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression.

[Antoniadis et al., 1997b]

Anestis Antoniadis, Gérard Grégoire, and Guy P. Nason. Density and hazard rate estimation for right censored data using wavelet methods. To appear in J. Roy. Statist. Soc., Series B, 1997.

[Ariño and Vidaković, 1995]

Miguel A. Ariño and Brani Vidaković. On wavelet scalograms and their applications in economic time series. Technical report, Institute of Statisics and Decision Sciences, Duke University, 1995.

[Aroian, 1947]

Leo A. Aroian. The probability function of the product of two normally distributed variables. The Annals of Mathematical Statistics, 18:265-271, 1947.

[Atkinson et al., 1994]

A. C. Atkinson, Siem Jan Koopman, and Neil Shephard. Outliers and switches in time series. In [Mandl and Huskova, 1994], pages 35-48.

[Bailey et al., 1998]

T. C. Bailey, T. Sapatinas, K. J. Powell, and W. J. Krzanowski. Signal detection in underwater sounds using wavelets. Journal of the American Statistical Association, 93:???--???, 1998.

[Bao and Erdol, 1994]

F. Bao and N. Erdol. The optimal wavelet transform and translation invariance. In IEEE International Conference on Acoustics, Speech and Signal Processing, volume 3, pages 13-16, 1994. 19-22 April 1994, Adelaide, SA, Australia.

Orthonormal wavelet representations are known to be time-variant. With shifting of the input signal, the energy distribution in time-scale plane also changes. We define the `separability' of a wavelet function both in the scale and translation domains as a measure of its localization with respect to translation. We derive a criterion for the optimal initial phase and then develop an algorithm for its choice in the case of stationary and nonstationary signals.

[Bao et al., 1995]

F. Bao, N. Erdol, and Z. Chen. Scale-translation filtering for wideband correlated noise attenuation. In [Szu, 1995], pages 652-660. 17-21, April 1994, Orlando, Florida.

A novel idea of scale-translation filtering based on the orthonormal wavelet transform is developed and demonstrated.

[Barnes and Allan, 1966]

James A. Barnes and David W. Allan. A statistical model of flicker noise. Proceedings of the IEEE, 31:176-179, 1966.

[Barnes, 1966]

James A. Barnes. Atomic timekeeping and the statistics of precision signal generators. Proceedings of the IEEE, 31:207-220, 1966.

[Bartlett, 1955]

Maurice S. Bartlett. An Introduction to Stochastic Processes, with Special Reference to Methods and Applications. Cambridge University Press, London, 1 edition, 1955.

[Bartlett, 1966]

Maurice S. Bartlett. An Introduction to Stochastic Processes, with Special Reference to Methods and Applications. Cambridge University Press, London, 2 edition, 1966.

[Basseville et al., 1992]

M. Basseville, A. Benveniste, K. C. Chou, S. A. Golden, R. Nikoukhah, and A. S. Willsky. Modeling and estimation of multiresolution stochastic processes. IEEE Transactions on Information Theory, 38(2):766-784, 1992.

An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. A natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it is shown how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time-like variable. The investigation of models on homogeneous trees is emphasized. The framework examined here allows for consideration, in a very natural way, of the fusion of data from sensors with differing resolutions. Also, thanks to the fact that wavelet transforms do an excellent job of 'compressing' large classes of covariance kernels, it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect.

[Bassingthwaighte et al., 1996]

J. B. Bassingthwaighte, D. A. Beard, D. B. Percival, and G. M. Raymond. Fractal structures and processes. In D. E. Herbert, editor, Chaos and the Changing Nature of Science and Medicine: An Introduction, pages 54-79, Woodbury, New York, 1996. AIP Press.

Fractals and chaos are closely related. Many chaotic systems have fractal features. Fractals are self-similar or self-affine structures, which means that they look much the same when magnified or reduced in scale over a reasonably large range of scales, at least two orders of magnitude and preferably more (Mandelbrot, 1983). The methods for estimating their fractal dimensions or their Hurst coefficients, which summarize the scaling relationships and their correlation structures, are going through a rapid evolutionary phase. Fractal measures can be regarded as providing a useful statistical measure of correlated random processes. They also provide a basis for analyzing recursive processes in biology such as the growth of arborizing networks in the circulatory system, airways, or glandular ducts.

[Bell and Percival, 1991]

B. M. Bell and D. B. Percival. A two step burg algorithm. IEEE Transactions on Signal Processing, 39(1):185-189, 1991.

The problem of estimating the parameters of a real-valued, stationary, nondeterministic, autoregressive process of order p from a time series of finite length is discussed. Burg's algorithm estimates these parameters indirectly by sequentially estimating one reflection coefficient at a time. The proposed approach is to sequentially estimate the reflection coefficients in pairs. The new algorithm has the same order of computational complexity as Burg's. It is guaranteed to generate parameter estimates that correspond to a stationary process (as does Burg's), and it produces estimates of the power spectral density that do not appear to suffer from spectral line splitting, in contrast to Burg's algorithm.

[Bell et al., 1993]

B. Bell, Donald B. Percival, and Andrew T. Walden. Calculating thomson's spectral multitapers by inverse iteration. Journal of Computational and Graphical Statistics, 2(1):119-130, 1993.

Spectral estimation using a set of orthogonal tapers is becoming widely used and appreciated in scientific research. It produces direct spectral estimates with more than 2 df at each Fourier frequency, resulting in spectral estimators with reduced variance. Computation of the orthogonal tapers from the basic defining equation is difficult, however, due to the instability of the calculations--the eigenproblem is very poorly conditioned. In this article the severe numerical instability problems are illustrated and then a technique for stable calculation of the tapers--namely, inverse iteration--is described. Each iteration involves the solution of a matrix equation. Because the matrix has Toeplitz form, the Levinson recursions are used to rapidly solve the matrix equation. FORTRAN code for this method is available through the Statlib archive. An alternative stable method is also briefly reviewed.

[Benedetto and Frazier, 1994]

John J. Benedetto and Michael W. Frazier, editors. Wavelets: Mathematics and Applications. CRC Press, Boca Raton, 1994.

[Benjamini and Hochberg, 1995]

Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society B, 57(1):289-300, 1995.

The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Terefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferroni-type procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples.

[Bentkus and Suvsinskas, 1982]

R. Ju. Bentkus and JU. V. Suvsinskas. On optimal statistical estimators of the spectral density. Soviet Math. Dokl., 25(2):415-419, 1982.

[Beran and Terrin, 1996]

J. Beran and N. Terrin. Testing for a change of the long-memory parameter. Biometrika, 83(3):627-638, 1996.

Long-range dependence is often observed in long time series. Correlations decay approximately like k(2H-2), With H epsilon(0.5, 1),as the lag k tends to infinity. The long-term features of the data are essentially characterised by the parameter H. Small changes of H have strong implications for the long-term behaviour of the process. In particular, rates of convergence of estimators for the mean, and for many other parameters of interest, differ for different values of H. For some data sets, H appears to change with time. In this paper we consider a simple test of the null hypothesis that H is constant. The test is based on a functional central limit theorem for quadratic forms. Critical values for the test statistic are given. Simulations confirm the validity of the test. A data example illustrates its practical application.

[Beran, 1992a]

Jan Beran. A goodness-of-fit test for time series with long range dependence. Journal of the Royal Statistical Society B, 54:749-760, 1992.

[Beran, 1992b]

Jan Beran. Statistical methods for data with long-range dependence. Statistical Science, 7(4):404-427, 1992.

[Beran, 1994]

Jan Beran. Statistics for Long-Memory Processes, volume 61 of Monographs on Statistics and Applied Probability. Chapman & Hall, New York, 1994.

[Beran, 1995]

Jan Beran. Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. Journal of the Royal Statistical Society B, 57(4):659-672, 1995.

In practical applications of Box-Jenkins autoregressive integrated moving average (ARIMA) models, the number of times that the observed time series must be differenced to achieve approximate stationarity is usually determined by careful, but mostly informal, analysis of the differenced series. For many time series, some differencing seems appropriate, but taking the first or the second difference may be too strong. As an alternative, Hosking, and Granger and Joyeux proposed the use of fractional differences. For -½ < d < ½ , the resulting fractional ARIMA processes are stationary. For 0 < d < ½ , the correlations are not summable. The parameter d can be estimated, for instance by maximum likelihood. Unfortunately, estimation methods known so far have been restricted to the stationary range -½ < d < ½ . In this paper, we show how any real d > -½ can be estimated by an approximate maximum likelihood method. We thus obtain a unified approach to fitting traditional Box-Jenkins ARIMA processes as well as stationary and non-stationary fractional ARIMA processes. A confidence interval for d can be given. Tests, such as for unit roots in the autoregressive parameter or for stationarity, follow immediately. The resulting confidence intervals for the ARMA parameters take into account the additional uncertainty due to estimation of d. A simple algorithm for calculating the estimate of d and the ARMA parameters is given. Simulations and two data examples illustrate the results.

[Beran, 1997]

Jan Beran. Estimating trends, long-range dependence adn nonstationarity. Department of Economics and Statistics, University of Konstanz, 1997.

[Beylkin and Saito, 1992]

Gregory Beylkin and Naoki Saito. Wavelets, their autocorrelation functions, and multiresolution representation of signals. In Intelligent Robots and Computer Vision XI: Biological, Neural Net and 3-D Methods, volume 1826 of Proceedings of the SPIE, pages 39-50, 1992.

We summarize the properties of the auto-correlation functions of compactly supported wavelets, their connection to iterative interpolation schemes, and the use of these functions for multiresolution analysis of signals. We briefly describe properties of representations using dilations and translations of these auto-correlation functions (the auto-correlation shell) which permit multiresolution analysis of signals.

[Bhargava and Kashyap, 1988]

U. K. Bhargava and R. L. Kashyap. Robust parametric approach for impulse response estimation. IEEE Transactions on Acoustics, Speech, and Signal Processing, 36(10):1592-1601, 1988.

A parametric technique for estimating the impulse response of a linear system using input-output observations in an outlier and distributionally uncertain environment is presented. The use of various cost functions for fitting the chosen output error model are discussed. By simulation, it is shown that the parametric approach based on the use of Huber's function as a criterion for fitting the model is robust. It is also shown that even though the parametric model for the impulse response is only an approximation to the true impulse response, the estimates from this approach still outperform the nonparametric approach in the presence of contaminated noise and low SNR.

[Bickel and Doksum, 1977]

Peter J. Bickel and Kjell A. Doksum. Mathematical Statistics: Basic Ideas and Selected Topics. Holden-Day, Inc., San Francisco, 1977.

[Bielza and Vidaković, 1996]

Concha Bielza and Brani Vidaković. Time adaptive wavelet denoising. Technical report, Institute of Statisics and Decision Sciences, Duke University, 1996.

[Bijaoui et al., 1994]

Albert Bijaoui, Jean-Luc Starck, and Fionn Murtagh. Restauration des images multi-echelles par l'Algorithme à trous. In French, 1994.

[Bijaoui et al., 1996]

A. Bijaoui, E. Slezak, F. Rue, and E. Lega. Wavelets and the study of the distant universe. Proceedings of the IEEE, 84(4):670-679, 1996.

The large-scale distribution of galaxies in the Universe exhibits structures at various scales, these so-called groups, clusters, and superclusters of galaxies being more or less hierarchically organized. A specific vision model is needed in order to detect, describe, and classify each component of this hierarchy. To do so rue have developed a multiscale vision model based on an unfolding into a scale space allowing us to detect structures of different sizes. A discrete wavelet transform is done by the a trous algorithm. The algorithm is implemented for astronomical images and also for lists of object positions, currently called catalogues in astronomical literature. Some applications on astrophysical data of cosmological interest are briefly described: 1) inventory procedures for galaxy counts on wide-field images, 2) processing of X-ray cluster images, leading to the analyses of the total matter distribution, and 3) detection of large-scale structures from galaxy counts. From the analyses of n-body simulations we show that the vision model from the wavelet transform provides a new statistical indicator on cosmological scenarios.

[Billingsley, 1968]

P. Billingsley. Convergence of Probability Measures. John Wiley & Sons, New York, 1968.

[Bingham et al., 1967]

Christopher Bingham, Michael D. Godfrey, and John W. Tukey. Modern techniques of power spectrum estimation. IEEE Transactions on Audio and Electroacoustics, 15(2):56-66, 1967.

[Bisaglia and Guégan, 1998]

Luisa Bisaglia and Dominique Guégan. A comparison of techniques of estimation in long-memory processes. Computational Statistics & Data Analysis, 27(1):61-81, 1998.

[Blackman and Tukey, 1958]

R. B. Blackman and J. W. Tukey. The Measurement of Power Spectra, from the Point of View of Communications Engineering. Dover Publications, Inc., New York, 1958. An unabridged and corrected republication of Part I and Part II of The measurement of power spectra from the point of view of communications engineering, which originally appeared in the January 1958 and March 1958 issues of volume XXXVII of the Bell system technical journal.

[Bloomfield, 1976]

Peter Bloomfield. Fourier Analysis of Time Series: An Introduction. John Wiley & Sons, New York, 1976.

[Booth and Smith, 1982]

N. B. Booth and A. F. M. Smith. A Bayesian approach to retrospective identification of change-points. Journal of Econometrics, 19:7-22, 1982.

[Box and Jenkins, 1976]

G. E. P. Box and G. M. Jenkins. Time Series Analysis: Forecasting and Control. Time Series Analysis and Digital Processing. Holden Day, San Francisco, 2 edition, 1976.

[Box and Pierce, 1970]

G. E. P. Box and David A. Pierce. Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association, 65(335):1509-1526, 1970.

[Bradshaw and McIntosh, 1994]

G. A. Bradshaw and B. A. McIntosh. Determining climate-induced patterns using wavelet analysis. Environmental Pollution, 83:133-142, 1994.

A method using wavelet analysis is introduced for the purpose of identifying and isolating inferred climatic components of the hydrologic record. This method affords an informed procedure for choosing filter dimensions for the purpose of signal decomposition.

[Bradshaw and Spies, 1992]

G. A. Bradshaw and Thomas A. Spies. Characterizing canopy gap structure in forests using wavelet analysis. Journal of Ecology, 80(2):205-215, 1992.

1. The wavelet transform is introduced as a technique to identify spatial structure in transect data. Its main advantages over other methods of spatial a nalysis are its ability to preserve and display hierarchical information while allowing for pattern decomposition. 2. Two applications are presented: simple one-dimensional simulations and a set of 200-m transect data of canopy opening measurements taken in 12 stands dominated by Pseudotsuga menziesii ranging over three developmental stages. 3. The calculation of the wavelet variance, derived from the transform, facilitates comparison based on dominant scale of pattern between multiple datase ts such as the stands described. 4. The results of the analysis indicate that while canopy pattern trends follow stand development, small to intermediate disturbances significantly influence canopy structure.

[Bretherton et al., 1998]

Christopher S. Bretherton, Martin Widmann, Valentin P. Dymnikov, John M. Wallace, and Ileana Bladé. Effective number of degrees of freedom of a spatial field. Submitted to Journal of Climate, 1998.

[Briggs and Henson, 1993]

William L. Briggs and Van Emden Henson. Wavelets and multigrid. SIAM Journal of Scientific Computing, 14(2):506-510, 1993.

[Briggs and Henson, 1995]

William L. Briggs and Van Emden Henson. The DFT: An Owner's Manual for the Discrete Fourier Transform. Society for Industrial and Applied Mathematics, Philadelphia, 1995.

Just as a prism separates white light into its component bands of colored light, so the discrete Fourier transform (DFT) is used to separate a signal into its constituent frequencies. Just as a pair of sunglasses reduces the glare of white light, permitting only the softer green light to pass, so the DFT may be used to modify a signal to achieve a desired effect. In fact, by analyzing the component frequencies of a signal or any system, the DFT can be used in an astonishing variety of problems. Among the applications of the DFT are digital signal processing, oil and gas exploration, medical imaging, aircraft and spacecraft guidance, and the solution of differential equations of physics and engineering. The DFT: An Owner's Manual for the Discrete Fourier Transform explores both the practical and theoretical aspects of the DFT, one of the most widely used tools in science, engineering, and computational mathematics. Designed to be accessible to an audience with diverse interests and mathematical backgrounds, the book is written in an informal style and is supported by many examples, figures, and problems. Conceived as an ``owner's'' manual, this comprehensive book covers such topics as the history of the DFT, derivations and properties of the DFT, comprehensive error analysis, issues concerning the implementation of the DFT in one and several dimensions, symmetric DFTs, a sample of DFT applications, and an overview of the FFT.

[Brillinger and Irizarry, 1998]

D. R. Brillinger and R. A. Irizarry. An investigation of the second- and higher-order spectra of music. Signal Processing, 65(2):161-179, 1998.

For a variety of musical pieces the following questions are addressed: Are the power spectra of 1/f form? Are the processes Gaussian? Are the higher-order spectra of 1/f form? Are the processes linear? Is long-range dependence present? Both score and acoustical signal representations of music are discussed and considered. Parametric forms are fit to sample spectra. Approximate distributions of the quantities computed are basic to drawing inferences. In summary, 1/f seems to be a reasonable approximation to the overall spectra of a number of pieces selected to be representative of a broad population. The checks for Gaussianity, really for bispectrum 0, in each case reject that hypothesis. The checks for linearity, really for constant bicoherence, reject that hypothesis in the case of the instantaneous power of the acoustical signal but not for the zero crossings of the signal or the score representation.

[Brillinger, 1969]

David R. Brillinger. Asymptotic properties of spectral estimates of second order. Biometrika, 56(2):375-389, 1969.

[Brillinger, 1974]

David R. Brillinger. Time Series: Data Analysis and Theory. Holt, Rinehart, and Winston, New York, 1974.

[Brillinger, 1978]

David R. Brillinger. Comparitive aspects of the study of ordinary time series and of point processes. In Developments in Statistics, volume 1, pages 34-133. Academic Press, Inc., 1978.

[Brillinger, 1979]

David R. Brillinger. Confidence intervals for the crosscovariance function. In Mathematical Statistics, volume 5 of Selecta Statistica Canadiana, pages 1-16. McMaster University Printing Services, Hamilton, Ontario, 1979.

[Brillinger, 1981]

David R. Brillinger. Time Series: Data Analysis and Theory. Holden-Day Series in Time Series Analysis. Holden-Day, San Francisco, 1981. Expanded edition.

[Brillinger, 1994]

David R. Brillinger. Trend analysis: Time series and point process problems. Environmetrics, 5:1-19, 1994.

[Brillinger, 1996]

David R. Brillinger. Some uses of cumulants in wavelet analysis. Nonparametric Statistics, 6:93-114, 1996.

[Brillinger, 1997]

David R. Brillinger. Some wavelet analysis of point process data. In Thirty-First Asilomar Conference on Signals, Systems and Computers, pages 93-114, 1997.

[Brockwell and Davis, 1991]

Peter J. Brockwell and Richard A. Davis. Time Series: Theory and Methods. Springer-Verlag, New York, 2 edition, 1991.

[Bronez, 1988]

Thomas P. Bronez. Spectral estimation of irregularly sampled multidimensional processes by generalized prolate spheroidal sequences. IEEE Transactions on Acoustics, Speech, and Signal Processing, 36(12):1862-1873, 1988.

A nonparametric spectral estimation method is presented for bandlimited random processes that have been sampled at arbitrary points in one or more dimensions. The method makes simultaneous use of several weight sequences that depend on the set of sampling point, the signal band, and the frequency band being analyzed. These sequences are solutions to a generalized matrix eigenvalue problem and are termed generalized prolate spheroidal sequences, being extensions of the familiar discrete prolate spheroidal sequences. Statistics of the estimator are derived, and the tradeoff among bias, variance, and resolution is quantified. The method avoids several problems typically associated with irregularly sampled data and multidimensional processes. A related method is suggested that has nearly as good performance while requiring significantly fewer computations

[Brown and Cai, 1997]

Lawrence D. Brown and T. Tony Cai. Wavelet shrinkage for nonequispaced samples. Technical Report 97-06, Department of Statistics, Purdue University, 1997.

[Brown, 1986]

Robert H. Brown. The distribution function of positive definite quadratic forms in normal random variables. SIAM Journal on Scientific and Statistical Computing, 7:689-695, 1986.

[Bruce and Gao, 1996a]

Andrew Bruce and Hong-Ye Gao. Applied Wavelet Analysis with S-PLUS. Springer, New York, 1996.

This book introduces applied wavelet analysis through the S-PLUS software system. Using a visual data analysis approach, wavelet concepts are explained in a way that is intuitive and easy to understand. In addition to wavelets, a whole range of related signal processing techniques such as wavelet packets, local cosine analysis, and matching pursuits are covered. Applications of wavelet analysis are illustrated, including nonparametric function estimation, digital image compression, and time-frequency signal analysis. The book and software is intended for a broad range of data analysts, scientists, and engineers. While most textbooks on wavelet analysis require advanced training in mathematics, this book minimizes reliance on formal mathematical methods. Readers should be familiar with calculus and linear algebra at the undergraduate level.

[Bruce and Gao, 1996b]

Andrew Bruce and Hong-Ye Gao. Understanding WaveShrink: Variance and bias estimation. Biometrika, 83(4), 1996.

Donoho and Johnstone's WaveShrink procedure has proven valuable for signal de-noising and non-parametric regression. WaveShrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise. WaveShrink has very broad asymptotic near-optimality properties. In this paper, we derive computationally efficient formulas for computing the exact bias, variance and L_2 risk of WaveShrink estimates in finite sample situations. These formulas provide a new way of understanding how WaveShrink works, what its limitations are, and the pros and cons of the shrinkage schemes: soft shrink vs. hard shrink. It complements the tools of simulation and asymptotic analysis. We use these formulas to estimate the bias, the variance and the L_2 risk for WaveShrink. Variance estimates are used to construct approximate pointwise confidence intervals and applied to synthetic and real examples. We also address the problem of threshold selection, computing minimax thresholds and ideal thresholds for both hard and soft shrinkage.

[Bruce et al., 1996]

Andrew Bruce, David Donoho, and Hong-Ye Gao. Wavelet analysis [for signal processing]. IEEE Spectrum, 33(10):26-35, 1996.

As every engineering student knows, any signal can be portrayed as an overlay of sinusoidal waveforms of assorted frequencies. But while classical analysis copes superbly with naturally occurring sinusoidal behavior-the kind seen in speech signals-it is ill-suited to representing signals with discontinuities, such as the edges of features in images. Latterly, another powerful concept has swept applied mathematics and engineering research: wavelet analysis. In contrast to a Fourier sinusoid, which oscillates forever, a wavelet is localized in time-it lasts for only a few cycles. Like Fourier analysis, however, wavelet analysis uses an algorithm to decompose a signal into simpler elements. Here, the authors describe how localized waveforms are powerful building blocks for signal analysis and rapid prototyping-and how they are now available in software toolkits.

[Burn et al., 1997]

J. F. Burn, A. M. Wilson, and G. P. Nason. Impact during equine locomotion: Techniques for measurement and analysis. Equine Veterinary Journal, 23:9-12, 1997.

[Burns et al., 1996]

T. J. Burns, S. K. Rogers, M. E. Oxley, and D. W. Ruck. A wavelet multiresolution analysis for spatio-temporal signals. IEEE Transactions on Aerospace and Electronic Systems, 32(2):628-649, 1996.

The wavelet filters of the conventional 3D multiresolution analysis possess homogeneous spatial and temporal frequency characteristics which Limits one's ability to match filter frequency characteristics to signal frequency behavior. Also, the conventional 3D multiresolution analysis employs an oct-tree decomposition structure which restricts the analysis of signal details to identical resolutions in space and time. This paper presents a 3D wavelet multiresolution analysis constructed from nonhomogeneous spatial and temporal filters, and an orthogonal sub-band coding scheme that decouples the spatial and temporal decomposition processes.

[Caccia et al., 1997]

D. C. Caccia, D. Percival, Cannon M. J., G. Raymond, and J. B. Bassingthwaighte. Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods. Physica A, 246(3-4):609-632, 1997.

Precise reference signals are required to evaluate methods for characterizing a fractal time series. Here we use fGp (fractional Gaussian process) to generate exact fractional Gaussian noise (fGn) reference signals for one-dimensional time series. The average autocorrelation of multiple realizations of fGn converges to the theoretically expected autocorrelation. Two methods commonly used to generate fractal time series, an approximate spectral synthesis (SSM) method and the successive random addition (SRA) method, do not give the correct correlation structures and should be abandoned. Time series from fGp were used to test how well several versions of rescaled range analysis (RIS) and dispersional analysis (Disp) estimate the Hurst coefficient(0 < H < 1.0). Disp is unbiased for H < 0.9 and series length N greater than or equal to 1024, but underestimates H when H > 0.9. R/S-detrended overestimates H for time series with H < 0.7 and underestimates H for H > 0.7. Estimates of H((H) over cap)) from all versions of Disp usually have lower bias and variance than those from R/S. All versions of dispersional analysis, Disp, now tested on fGp, are better than we previously thought and are recommended for evaluating time series as long-memory processes.

[Cai and Brown, 1998]

T. Tony Cai and Lawrence D. Brown. Wavelet shrinkage for nonequispaced samples. Annals of Statistics, to appear, 1998.

[Cai and Silverman, 1998]

T. Tony Cai and Bernard W. Silverman. Incorporating information on neighboring coefficients into wavelet estimation. Technical Report 98-13, Department of Statistics, Purdue University, 1998.

[Cai et al., 1998]

Z. W. Cai, C. M. Hurvich, and C. L. Tsai. Score tests for heteroscedasticity in wavelet regression. Biometrika, 85(1):229-234, 1998.

We consider two Score tests for heteroscedasticity in the errors of a signal;plus-noise model, where the signal is estimated;by wavelet thresholding methods. The error variances are assumed to depend on observed covariates, through a parametric relationship of known form. The tests are based on the approaches of Breusch & Pagan (1979) and Koenker (1981). We establish the asymptotic validity of the tests and examine their performance in a simulation study. The Koenker test is found to perform well, in terms of both size and power.

[Cai, 1996]

T. Tony Cai. Minimax wavelet estimation via block thresholding. Technical Report 96-41, Department of Statistics, Purdue University, 1996.

[Cai, 1997]

T. Tony Cai. On adaptivity of BlockShrink wavelet estimator over Besov spaces. Technical Report 97-05, Department of Statistics, Purdue University, 1997.

[Cambanis and Masry, 1994]

S. Cambanis and Elias Masry. Wavelet approximation of deterministic and random signals: convergence properties and rates. IEEE Transactions on Information Theory, 40(4):1013-1029, 1994.

The multiresolution decomposition of deterministic and random signals and the resulting approximation at increasingly finer resolution is examined. Specifically, an nth-order expansion is developed for the error in the wavelet approximation at resolution 2^-l of deterministic and random signals. The deterministic signals are assumed to have n continuous derivatives, while the random signals are only assumed to have a correlation function with continuous nth-order derivatives off the diagonal-a very mild assumption. For deterministic signals square integrable over the entire real line, for stationary random signals over finite intervals, and for nonstationary random signals with finite mean energy over the entire real line, the smoothness of the scale function can be matched with the signal smoothness to substantially improve the quality of the approximation. In sharp contrast, this is feasible only in special cases for nonstationary random signals over finite intervals and for deterministic signals which are only locally square integrable.

[Cannon et al., 1997]

M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, and J. B. Bassingthwaighte. Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series. Physica A, 241(3-4), 1997.

Three scaled windowed variance methods (standard, linear regression detrended, and bridge detrended) for evaluating the Hurst coefficient (H) are evaluated. The Hurst coefficient, with 0 < H < 1, characterizes self-similar decay in the time series autocorrelation function. The scaled windowed variance methods estimate H for fractional Brownian motion (fBm) signals which are cumulative sums of fractional Gaussian noise (fGn) signals. For all three methods both the bias and standard deviation of estimates are less than 0.05 for series have 512 points or more. Estimates for short series (less than 256 points) are unreliable. To have a 95% probability of distinguishing between two signals with true H differing by 0.1, more than 32,768 points are needed. All three methods proved more reliable (based on bias and variance of estimates) than Hurst's rescaled range analysis, periodogram analysis, and autocorrelation analysis, and as reliable as dispersional analysis. These latter methods can only be applied to fGn or differences of fBm, while the scaled windowed variance methods must be applied to fBm or cumulative sums of fGn.

[Carmona and Hudgins, 1994]

Réne A. Carmona and Lonnie H. Hudgins. Wavelet denoising of EEG signals and identification of evokedresponse potentials. In [Laine and Unser, 1994], pages 91-104. 24-29 July, 1994, San Diego, California.

The purpose of this study is to apply a recently developed wavelet based de-noising filter to the analysis of human electroencephalogram (EEG) signals, and measure its performance. The data used contained subject EEG responses to two different stimuli using the `odd-ball' paradigm. Electrical signals measured at standard locations on the scalp were processed to detect and identify the Evoked Response Potentials (ERP's). First, electrical artifacts emitting from the eyes were identified and removed. Second, the mean signature for each type of response was extracted and used as a matched filter to define baseline detector performance for the noisy data. Third, a nonlinear filtering procedure based on the wavelet extrema representation was used to de-noise the signals. Overall detection rates for the de-noised signals were then compared to the baseline performance. It was found that while the filtered signals have significantly lower noise than the raw signals, detector performance remains comparable. We therefore conclude that all of the information that is important to matched filter detection is preserved by the filter. The implication is that the wavelet based filter eliminates much of the noise while retaining ERP's.

[Carmona and Wang, 1996]

R. A. Carmona and A. Wang. Comparison tests for the spectra of dependent multivariate time series. In Robert J. Adler, Peter Müller, and Boris Rozovskii, editors, Stochastic Modelling in Physical Oceanography, volume 39 of Progress in Probability, pages 69-88. Birkhauser, Boston, 1996.

[Carmona et al., 1997]

René A. Carmona, Wen L. Hwang, and Brun Torrésani. Characterization of signals by the ridges of their wavelet transforms. IEEE Transactions on Signal Processing, 45(10):2586-2590, 1997.

We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional (1-D) signals, These detection procedures are shown to be robust to additive white noise, We also derive and test a new reconstruction procedure, The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides a very efficient way to code the information contained in the signal.

[Carmona, 1993]

René A. Carmona. Wavelet identification of transients in noisy time series. In [Laine, 1993], pages 392-400. 11-16 July, 1993, San Diego, California.

The detection of transients in noisy time series is an important part of modern signal analysis because of the importance of its civil and military applications. The author presents a new denoising procedure, the output of which gives a very reasonable guess for the component of the input signal which was buried in noise. The algorithm has two main components. The first one concerns the identification of the main characteristics of the noise component and of the typical effects it has on the wavelet transform of the input signal. This information is used to identify the points in the time-scale space which cannot be extrema of the wavelet transform, unless something else than noise was present in the input signal. This is done by bootstrap in general but direct Monte Carlo simulations can be used when parametric knowledge on the distribution of the noise is available. The second part deals with the actual reconstruction of what is believed to be the component of the input which is to be identified. This part of the algorithm uses the reconstruction procedure of Mallat and Zhong (1992) as revised by the author (1992) the main novelty being the fact that this procedure is fed with the set of points in the time-scale plane which passed the trimming test of the extrema of the wavelet transform. The author illustrates the efficiency of the reconstruction algorithm using the examples of transients used previously by the author (1992).

[Chan and Ho, 1996]

Y. T. Chan and K. C. Ho. Multiresolution analysis, its link to the discrete parameter wavelet transform, and its initialization. IEEE Transactions on Signal Processing, 44(4):1001-1007, 1996.

Two-scale wavelet equations are derived for equivalent multiresolution analysis (MRA) detail parameters and the discrete parameter (DP) wavelet transform coefficients for a signal s(t). MRA initialization by prefiltering its input signal s(n) obtains the equivalence between the DP and MRA coefficients. MRA gives the DP of a signal s(t) when s(n) are samples of the inner product of s(t) and the scaling function. A simulation example is presented to discuss the prefiltering procedure's effectiveness.

[Chan et al., 1996]

Ngai Hang Chan, Joseph B. Kadane, Robert N. Miller, and Wilfredo Palma. Estimation of tropical sea level anomaly by an improved kalman filter. Journal of Physical Oceanography, 26(7):1286-1303, 1996.

Kalman filler theory and autoregressive time series are used to map sea level height anomalies in the tropical Pacific. Our Kalman filters are implemented with a linear state space model consisting of evolution equations for the amplitudes of baroclinic Kelvin and Rossby waves and data from the Pacific tide gauge network. Ln this study, three versions of the Kalman filter are evaluated through examination of the innovation sequences, that is, the time series of differences between the observations and the model predictions before updating. In a properly tuned Kalman filter, one expects the innovation sequence to be white (uncorrelated, with zero mean). A white innovation sequence can thus be taken as an indication that there is no further information to be extracted from the sequence of observations. This is the basis for the frequent use of whiteness, that is, lack of autocorrelation, in the innovation sequence as a performance diagnostic for the Kalman filter. Our long-wave model embodies the conceptual basis of current understanding of the large-scale behavior of the tropical ocean. When the Kalman filter was used to assimilate sea level anomaly data, we found the resulting innovation sequence to be temporally correlated, that is, nonwhite and well fitted by an autoregressive process with a lag of one month. A simple modification of the way in which sea level height anomaly is represented in terms of the state vector for comparison to observation results in a slight reduction in the temporal correlation of the innovation sequences and closer fits of the model to the observations, but significant autoregressive structure remains in the innovation sequence. This autoregressive structure represents either a deficiency in the model or some source of inconsistency in the data. When an explicit first-order autoregressive model of the innovation sequence is incorporated into the filter, the new innovation sequence is white. In an experiment with the modified filter in which some data were held back from the assimilation process, the sequences of residuals at the withheld stations were also white. To our knowledge, this has not been achieved before in an ocean data assimilation scheme with real data. Implications of our results for improved estimates of model error statistics and evaluation of adequacy of models are discussed in detail.

[Chan et al., 1997]

N. H. Chan, J. B. Kadane, and T. Jiang. Time series analysis of diurnal cycles in small-scale turbulence. To appear in Environmetrics, 1997.

[Chan, 1995]

Y. T. Chan. Wavelet Basics. Kluwer Academic Publishers, Boston, 1995.

Since the study of wavelets is a relatively new area, much of the research coming from mathematicians, most of the literature uses terminology, concepts and proofs that may, at times, be difficult and intimidating for the engineer. Wavelet Basics has therefore been written as an introductory book for scientists and engineers. The mathematical presentation has been kept simple, the concepts being presented in elaborate detail in a terminology that engineers will find familiar. Difficult ideas are illustrated with examples which will also aid in the development of an intuitive insight. Chapter 1 reviews the basics of signal transformation and discusses the concepts of duals and frames. Chapter 2 introduces the wavelet transform, contrasts it with the short-time Fourier transform and clarifies the names of the different types of wavelet transforms. Chapter 3 links multiresolution analysis, orthonormal wavelets and the design of digital filters. Chapter 4 gives a tour d'horizon of topics of current interest: wavelet packets and discrete time wavelet transforms, and concludes with applications in signal processing.

[Chen and An, 1997]

Min Chen and Hong Zhi An. A kolomogorov-smirnov type test for conditional heteroskedasticity in time series. Statistics & Probability Letters, 33(3):321-331, 1997.

[Chen and Gupta, 1997]

Jie Chen and A. K. Gupta. Testing and locating variance changepoints with application to stock prices. Journal of the American Statistical Association, 92(438):739-747, 1997.

[Chen et al., 1996]

Shuyi S. Chen, Robert A. Houze Jr., and Brian E. Mapes. Multiscale variability of deep convection in relation to large-scale circulation in TOGA COARE (Tropical Ocean Global Atmosphere Coupled Ocean-Atmosphere Response Experiment). Journal of Atmospheric Science, 53(10):1380-1409, 1996.

Deep convection over the Indo-Pacific oceanic warm pool in the Tropical Ocean Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA COARE) occurred in cloud clusters, which grouped together in regions favoring their occurrence. These large groups of cloud clusters produced large-scale regions of satellite-observed cold cloud-top temperature. This paper investigates the manner in which the cloud clusters were organized on time and space scales ranging from the seasonal mean pattern over the whole warm-pool region to the scale of individual cloud clusters and their relationship to the large-scale circulation and sea surface temperature (SST).

[Chen, 1997]

Ying Chen. Wavelet analysis and statistics of CN tower current waveforms. Master's thesis, Department of Electrical and Computer Engineering, University of Western Ontario, 1997.

[Chiann and Morettin, 1996]

Chang Chiann and Pedro A. Morettin. A wavelet analysis for stationary processes. University of São Paulo, São Paulo, Brazil, 1996.

Short abstract: In this paper a wavelet analysis for stationary time series is proposed. A wavelet spectrum (with respect to a given wavelet family) is defined and asymptotic properties of the finite wavelet transform, the periodogram and scalegram are derived.

[Chipman et al., 1997]

Hugh A. Chipman, Eric D. Kolaczyk, and Robert E. McCulloch. Adaptive bayesian wavelet shrinkage. Journal of the American Statistical Association, 92(440):1413-1421, 1997.

[Chui et al., 1994]

Charles K. Chui, Laura Montefusco, and Luigia Puccio, editors. Wavelets: Theory, Algorithms, and Applications, volume 5 of Wavelet Analysis and its Applications. Academic Press, Inc., 1994.

Wavelets: Theory, Algorithms, and Applications is the fifth volume in the highly respected series, WAVELET ANALYSIS AND ITS APPLICATIONS. This volume shows why wavelet analysis has become a tool of choice in fields ranging from image compression, to signal detection and analysis in electrical engineering and geophysics, to analysis of turbulent or intermittent processes. The 28 papers comprising this volume are organized into seven subject areas: multiresolution analysis, wavelet transforms, tools for time-frequency analysis, wavelets and fractals, numerical methods and algorithms, and applications. More than 135 figures supplement the text.

[Chui, 1992a]

C. K. Chui. An Introduction to Wavelets, volume 1 of Wavelet Analysis and its Applications. Academic Press, Inc., 1992.

This is the first volume in the series WAVELET ANALYSIS AND ITS APPLICATIONS. It is an introductory treatise on wavelet analysis, with an emphasis on spline wavelets and and time-frequency analysis. Among the basic topics covered are time frequency localization, intergral wavelet transforms, dyadic wavelets, frames, spine wavelets, orthonormal wavelet bases, and wavelet packets. Is is suitable as a textbook for a beginning course on wavelet analysis and is directed toward both mathematicians and engineers who wish to learn about the subject.

[Chui, 1992b]

C. K. Chui. Wavelets: A Tutorial in Theory and Applications, volume 2 of Wavelet Analysis and its Applications. Academic Press, Inc., 1992.

Wavelets: A Tutorial in Theory and Applications is the second volume in the new series WAVELET ANALYSIS AND ITS APPLICATIONS. As a companion to the first volume in this series, this volume covers several of the most important areas in wavelets, ranging from the development of the basic theory such as construction and analysis of wavelet bases to an introduction of some of the key applictions, including Mallat's local wavelet maxima technique in second generagion image coding.

[Chui, 1997]

Charles K. Chui. Wavelets: A Mathematical Tool for Signal Analysis. SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, Philadelphia, 1997.

Wavelets continue to be powerful mathematical tools that can be used to solve problems for which the Fourier (spectral) method does not perform well or cannot handle. This book is for engineers, applied mathematicians, and other scientists who want to learn about using wavelets to analyze, process, and synthesize images and signals. Applications are described in detail and there are step-by-step instructions about how to construct and apply wavelets. The only mathematically rigorous monograph written by a mathematician specifically for nonspecialists, it describes the basic concepts of these mathematical techniques, outlines the procedures for using them, compares the performance of various approaches, and provides information for problem solving, putting the reader at the forefront of current research.

[Ciarlini et al., 1994]

P. Ciarlini, M. Cox, R. Monaco, and F. Pavese, editors. Advanced Mathematical Tools in Metrology, volume 16 of Advances in Mathematics for Applied Sciences, Singapore, 1994. World Scientific. Proceedings of the International Workshop.

[Clark et al., 1980]

A. P. Clark, C. P. Kwong, and F. McVerry. Estimation of the sampled impulse-response of a channel. Signal Processing, 2(1):39-53, 1980.

Describes various techniques for estimating the sampled impulse-response of a noise linear channel. The estimators are suitable for use with maximum-likelihood detection processes such as the Viterbi-algorithm detector, in applications where a digital data signal is transmitted over a channel introducing severe intersymbol interference and where the receiver may or may not have some prior knowledge of the channel. Results of computer simulation tests are presented, showing, for each estimator, the magnitude of the error in the channel estimate over the reception of a typical data signal. Both time-invariant and time-varying channels are used in the tests and the performances of the estimators are compared for the different cases where the receiver initially has some or no knowledge of the channel and where the detected data symbols are all correct or contain some errors. It is shown that, even under quite unfavourable conditions, a surprisingly good estimate of the channel can be obtained by means of a relatively simple estimator.

[Clyde et al., 1998]

M. Clyde, G. Parmigiani, and B. Vidakovic. Multiple shrinkage and subset selection in wavelets. Biometrika, 85(2):391-401, 1998.

[Coates and Diggle, 1986]

D. S. Coates and P. J. Diggle. Tests for comparing two estimated spectral densities. Journal of Time Series Analysis, 7:7-20, 1986.

[Cohen and Ryan, 1995]

A. Cohen and R. D. Ryan. Wavelets and Multiscale Signal Processing. Chapman & Hall, 1995.

Since their appearance in the mid-1980s, wavelets and, more generally, multiscale methods have become powerful tools in mathematical analysis and in applications to numerical analysis and signal processing. This book is based on Ondelettes et Traitement Numerique du Signal by Albert Cohen. It has been translated from French by Robert D. Ryan and extensively updated by both Cohen and Ryan. It studies the existing relations between filter banks and wavelet decompositions and shows how these relations can be exploited in the context of digital signal processing. Throughout, the book concentrates on the fundamentals. It begins with a chapter on the concept of multiresolution analysis, which contains complete proofs of the basic results. The description of filter banks that are related to wavelet bases is elaborated in both the orthogonal case (Chapter 2), and in the biorthogonal case (Chapter 4). The regularity of wavelets, how this is related to the properties of the filters, and the importance of regularity for the algorithms are the subjects of Chapter 3. Chapter 5 looks at multiscale decomposition as it applies to stochastic processing, in particular to signal and image processing. Wavelets and Multiscale Signal Processing will be of particular interest to mathematicians working in analysis, academic and research electrical engineers, and researchers who need to analyse time series, in areas such as hydrodynamics, aeronautics, meteorology, geophysics, statistics and economics.

[Cohen et al., 1993]

A. Cohen, I. Daubechies, and P. Vial. Wavelets on the interval and fast wavelet transforms. Applied and Computational Harmonic Analysis, 1(1):54-81, 1993.

The authors discuss several constructions of orthonormal wavelet bases on the interval, and they introduce a new construction that avoids some of the disadvantages of earlier constructions.

[Cohen et al., 1997]

Israel Cohen, Shalom Raz, and David Malah. Orthonormal shift-invariant wavelet packet decomposition and representation. To appear in Signal Processing, 57(3), 1997.

In this work, a shifted wavelet packet (SWP) library, containing all the time shifted wavelet packet bases, is defined. A corresponding shift-invariant wavelet packet decomposition (SIWPD) search algorithm for a ``best basis'' is introduced. The search algorithm is representable by a binary tree, in which a node symbolizes an appropriate subspace of the original signal. We prove that the resultant ``best basis'' is orthonormal and the associated expansion, characterized by the lowest information cost, is shift-invariant. The shift-invariance stems from an additional degree of freedom, generated at the decomposition stage and incorporated into the search algorithm. The added dimension is a relative shift between a given parent-node and its respective children-nodes. We prove that for any subspace it suffices to consider one of two alternative decompositions, made feasible by the SWP library. These decompositions correspond to a zero shift and a 2^-ell relative shift where ell denotes the resolution level. The optimal relative shifts, which minimize the information cost, are estimated using finite depth subtrees. By adjusting their depth, the quadratic computational complexity associated with SIWPD may be controlled at the expense of the attained information cost down to O(N log_2 N).

[Cohen, 1994]

Leon Cohen. Time Frequency Analysis: Theory and Applications. Prentice Hall, Inc., New Jersey, 1994.

Featuring traditional coverage as well as new research results that, until now, have been scattered throughout the professional literature, this book brings together --- in simple language --- the basic ideas and methods that have been developed to study natural and man-made signals whose frequency content changes with time; e.g., speech, sonar and radar, optical images, mechanical vibrations, acoustic signals, biological/biomedical and geophysical signals. Covers time analysis, frequency analysis, and scale analysis; time-bandwidth relations; instantaneous frequency; densities and local quantities; the short time Fourier Transform; time-frequency analysis; the Wigner representation; time-frequency representations; computation methods; the synthesis problem; spatial-spatial/frequency representations; time-scale representations; operators; general joint representations; stochastic signals; and higher order time-frequency distributions. Illustrates each concept with examples and shows how the methods have been extended to other variables, such as scale.

[Coifman and Donoho, 1995]

Ronald R. Coifman and David Donoho. Time-invariant wavelet denoising. In [Antoniadis and Oppenheim, 1995], pages 125-150.

[Coifman and Wickerhauser, 1992]

Ronald R. Coifman and Mladen Victor Wickerhauser. Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory, 38(2):713-718, 1992.

Adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals, such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets and localized trigonometric functions, and have reasonably well-controlled time-frequency localization properties. The idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost. The method relies heavily on the remarkable orthogonality properties of the new libraries: all expansions in a given library conserve energy and are thus comparable. Several cost functionals are useful; one of the most attractive is Shannon entropy, which has a geometric interpretation in this context.

[Coifman et al., 1992a]

Ronald R. Coifman, Yves Meyer, and Mladen Victor Wickerhauser. Size properties of wavelet packets. In [Ruskai et al., 1992], pages 453-470.

[Coifman et al., 1992b]

Ronald R. Coifman, Yves Meyer, and Mladen Victor Wickerhauser. Wavelet analysis and signal processing. In [Ruskai et al., 1992], pages 153-178.

This describes the use of wavelet analysis for various tasks in signal processing.

[Combes et al., 1989]

Jean-Michel Combes, Alexander Grossman, and Philippe Tchamitchian, editors. Wavelets: Time-Frequency Methods and Phase Space, Inverse Problems and Theoretical Imaging, Berlin, 1989. Springer-Verlag. Proceedings of the International Converence, Marseille, France, December 14-18, 1987.

[Craig, 1936]

Cecil C. Craig. On the frequency function of xy. The Annals of Mathematical Statistics, 7:1-15, 1936.

[Creusere and Hewer, 1994]

C. D. Creusere and G. Hewer. A wavelet-based method of nearest neighbor pattern classification using scale sequential matching. In A. Singh, editor, Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers, volume 2, pages 1123-1127, 1994.

In this method of pattern classification a wavelet transform is used to extract features from the input signal which are then compared in a scale sequential manner (from coarse to fine) to a trained nearest neighbor codebook. At each scale, possible classification categories are eliminated until only one class is left. We apply this pattern classifier to the problem of fingerprinting post-detection radar pulses and analyze its performance in noise using Monte Carlo simulations. To make our classifier shift invariant, we process the input with an undecimated wavelet transform until the pulse edge is sensed and then start decimating the wavelet coefficients as appropriate to each scale.

[Croisier et al., 1976]

A. Croisier, D. Esteban, and C. Galand. Perfect channel splitting by use of interpolation/decimation/tree decomposition techniques. In Int. Conf. on Inform. Sciences and Systems, pages 443-446, 1976. Patras, Greece.

[Crouse et al., 1998]

Matthew S. Crouse, Robert D. Nowak, and Richard G. Baraniuk. Wavelet-based statistical signal processing using hidden markov models. IEEE Transactions on Signal Processing, 46(4), 1998.

[D'Agostino and Stephens, 1986]

Ralph B. D'Agostino and Michael A. Stephens, editors. Goodness-of-Fit Techniques, volume 68 of STATISTICS: Textbooks and Monographs. Marcel Dekker, New York, 1986.

[Daubechies and Lagarias, 1991]

Ingrid Daubechies and J. Lagarias. Two-scale difference equations, I. SIAM Journal of Mathematical Analysis, 22:1388-1410, 1991.

[Daubechies and Lagarias, 1992]

Ingrid Daubechies and J. Lagarias. Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals. SIAM Journal of Mathematical Analysis, 23:1031-1079, 1992.

We study solutions of the functional equation f(x)=sumsp Nsb n=0csb nf(kx-n), where kgeq 2 is an integer, and sumsp Nsb n=0csb n=k. Part I showed [SIAM J. Math. Anal. 22 (1991), no. 5, 1388-1410; MR 92d:39001] that equations of this type have at most one Lsp 1-solution up to a multiplicative constant, which necessarily has compact support in [0,N/k-1]. This paper gives a time-domain representation for such a function f(x) (if it exists) in terms of infinite products of matrices (that vary as x varies). Sufficient conditions are given on csb n for a continuous nonzero Lsp 1-solution to exist. Additional conditions sufficient to guarantee fin Csp r are also given. The infinite matrix product representations are used to bound from below the degree of regularity of such an Lsp 1-solution and to estimate the Holder exponent of continuity of the highest-order well-defined derivative of f(x). Such solutions f(x) are often smoother at some points than others. For certain f(x) a hierarchy of fractal sets in bold R corresponding to different Holder exponents of continuity for f(x) is described.

[Daubechies and Sweldens, 1996]

I. Daubechies and W. Sweldens. Factoring wavelet transforms into lifting steps. Technical report, Bell Laboratories, Lucent Technologies, 1996.

The lifting scheme is a new flexible tool for constructing wavelets and wavelet transforms. In this paper, we use the Euclidean algorithm to show how any discrete wavelet transform or two band subband transform with finite filters can be obtained with a finite number of lifting steps starting from the Lazy wavelet (or polyphase transform). We show a bound on the number of lifting steps which is proportional to the length of the filters. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal (non-unitary) case. The lifting factorization asymptotically reduces the computational complexity of the transform by a factor of two and allows for wavelet transforms that map integers to integers.

[Daubechies, 1988]

Ingrid Daubechies. Orthonormal bases of compactly supported wavelets. Communications in Pure and Applied Mathematics, 41:909-996, 1988.

[Daubechies, 1989]

Ingrid Daubechies. Orthonormal bases of wavelets with finite support -- connection with discrete filters. In [Combes et al., 1989], pages 38-66. Proceedings of the International Converence, Marseille, France, December 14-18, 1987.

[Daubechies, 1990]

I. Daubechies. The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information Theory, 36(5):961-1005, 1990.

Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems.

[Daubechies, 1991]

Ingrid Daubechies. The wavelet transform: A method for time-frequency localization. In [Haykin, 1991], pages 366-417.

[Daubechies, 1992]

Ingrid Daubechies. Ten Lectures on Wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, 1992.

Wavelets are a mathematical development that may revolutionize the world of information storage and retrieval according to many experts. They are a fairly simple mathematical tool now being applied to the compression of data-such as fingerprints, weather satellite photographs, and medical x-rays-that were previously thought to be impossible to condense without losing crucial details. This monograph contains 10 lectures presented by Dr. Daubechies as the principal speaker at the 1990 CBMS-NSF Conference on Wavelets and Applications. The author has worked on several aspects of the wavelet transform and has developed a collection of wavelets that are remarkably efficient. The opening chapter provides an overview of the main problems presented in the book. Following chapters discuss the theoretical and practical aspects of wavelet theory, including wavelet transforms, orthonormal bases of wavelets, and characterization of functional spaces by means of wavelets. The last chapter presents several topics under active research, as multidimensional wavelets, wavelet packet bases, and a construction of wavelets tailored to decompose functions defined in a finite interval. Because of their interdisciplinary origins, wavelets appeal to scientists and engineers of many different backgrounds.

[David, 1966]

F. N. David. Tables of the correlation coefficient. In E. S. Pearson and H. O. Hartley, editors, Biometrika Tables for Statisticians, volume 1. Cambridge University Press, Cambridge, 3 edition, 1966.

[Davies and Harte, 1987]

R. B. Davies and D. S. Harte. Tests for Hurst effect. Biometrika, 74:95-101, 1987.

[Davies, 1980]

Robert B. Davies. The distribution of a linear combination of chi^2 random variables. Applied Statistics, 29:323-333, 1980.

[Davis et al., 1994]

Anthony Davis, Alexander Marshak, and Warren Wiscombe. Wavelet-based multifractal analysis of non-stationary and/or intermittent geophysical signals. In [Foufoula-Georgiou and Kumar, 1994], pages 249-298.

[Davis, 1979]

William W. Davis. Robust methods for detection of shifts of the innovation variance of a time series. Technometrics, 21(3):313-320, 1979.

[Dejak et al., 1990]

C. Dejak, D. Franco, R. Pastres, and G. Pecenik. Irregular environmental historical series: Software for statistical and periodic analyses. In P. Zannetti, editor, Computer Techniques in Environmental Studies III, pages 489-500, 1990. Proceedings of the Third International Conference on Development and Application of Computer Techniques to Environmental Studies. Montreal, Que., Canada. 11-13 Sept. 1990.

When dealing with historical time series of environmental water quality parameters, irregular and sparse data sets are frequently met, particularly when data refer to multiannual surveys. Since common statistical methods for handling time series require equispaced data sets, program is described, which, by including different alternatives, permits one to regularize the series. Techniques include linear interpolations and parabolic best fits. After regularization, the data sets are analyzed for detecting and removing the long term trend, with extrapolation of missing values at both tails, and the seasonal component, leaving the stochastic fluctuations. Testing for Gaussian behaviour is performed to the former, while the latter are examined through Fourier series, which are optimized through variance analysis, and, as a general approach, with the negentropy method, in order to avoid data overfitting or underfitting.

[del Marco and Weiss, 1994]

Stephen del Marco and John Weiss. M-band wavepacket-based transient signal detector using a translation-invariant wavelet transform. Optical Engineering, 33(7):2175-2182, 1994.

This paper develops a two-dimensional M-band translation-invariant wavelet transform (2-D MTI). Use of the MTI overcomes the shift-variance of the wavelet transform by applying a cost function over M shifts of the input signal. The new transform is proven to be translation-invariant. Use of M-band wavelets enables a finer frequency partitioning and greater energy compaction in the transform representation. Examples are presented which show that the translation-invariant transforms provide superior energy concentration compared to the corresponding nominal wavelet transforms. Examples are also presented comparing the energy concentration capability of M-band wavelets and the modulated lapped transform (MLT). We explored the MTI as a tool for image processing by using it to represent several different images.

[del Marco and Weiss, 1997]

Stephen del Marco and John Weiss. Improved transient signal detection using a wavepacket-based detector with an extended translation-invariant wavelet transform. IEEE Transactions on Signal Processing, 45(4):841-850, 1997.

This paper presents the theory of M-band, extended translation-invariant (ETI) wavelet transforms. The ETI generalizes the translation-invariant wavelet transform of Weiss. It is shown that iteration of the ETI, in a tree structure, provides a signal decomposition into an orthonormal wavepacket basis, Other properties such as translation invariance and invertibility of the transform are proven, The theory is then applied to transient signal detection through development of a family of translation-invariant wavepacket-based detectors. This family of detectors provides improved performance over previously defined wavepacket-based detectors, A performance analysis is conducted. ROC curves generated by Monte-Carlo simulation are presented, indicating detector performance, Detector performance is demonstrated to be independent of the signal translation.

[Delgado and Robinson, 1996]

Miguel A. Delgado and Peter M. Robinson. Optimal spectral bandwidth for long memory. Statistica Sinica, 6:97-112, 1996.

[Delgado, 1996]

Miguel A. Delgado. Testing serial independence using the sample distribution function. Journal of Time Series Analysis, 17(3):271-285, 1996.

This paper presents and discusses a nonparametric test for detecting serial dependence. We consider a Cramèr-von Mises statistic based on the difference between the joint sample distribution and the product of the marginals. Exact critical values can be approximated from the asymptotic null distribution, or by resampling, randomly permuting the original series. A Monte Carlo experiment illustrates the test performance with small sample sizes. The paper also includes an application, testing the random walk hypothesis of exchange rate returns for several currencies.

[Delyon and Juditsky, 1995]

Bernard Delyon and Anatoli Juditsky. Estimating wavelet coefficients. In [Antoniadis and Oppenheim, 1995], pages 151-168.

[Delyon and Juditsky, 1997]

B. Delyon and A. Juditsky. On the computation of wavelet coefficients. Journal of Approximation Theory, 88(1):47-79, 1997.

We consider fast algorithms of wavelet decomposition of a function f when discrete observations of f (supp f subset of or equal to[0,1](d)) are available. The properties of the algorithms are studied for three types of observation design which for d=1 can be described as follows: the regular design, when the observations f(xi) are taken on the regular grid x(i)=i/N, i=1, ..., N; the case of a jittered regular grid, when it is only known that for all 1 less than or equal to i less than or equal to N, i/N less than or equal to x(i)<i+1)/N; and the random design case; in which x(i), i=1, ..., N, are independent and identically distributed random variables on [0,1]. We show that these algorithms are in a certain sense efficient when the accuracy of the approximation is concerned. The proposed algorithms are computationally straightforward; the whole effort to compute the decomposition is order N for the sample size N.

[Denison et al., 1998]

D. G. T. Denison, A. T. Walden, A. Balogh, and R. J. Forsyth. Multitaper testing of spectral lines and the detection of the solar rotation frequency and its harmonics. Technical Report 98-04, Department of Mathematics, Imperial College of Science, Technology & Medicine, 1998.

[DeRose et al., 1993]

Tony D. DeRose, Michael Lounsbery, and Joe Warren. Multiresolution analysis for sufaces of arbitrary topological type. Technical Report 93-10-05, Department of Computer Science and Engineering, University of Washington, 1993.

[Diaz, 1982]

Joaquin Diaz. Bayesian detection of a change of scale parameter in sequences of independent gamma random variables. Journal of Econometrics, 19(1):23-29, 1982.

[Diggle and Fisher, 1991]

Peter J. Diggle and Nicholas I. Fisher. Nonparametric comparison of cumulative periodograms. Applied Statistics, 40(3):423-434, 1991.

Motivated by a problem in the analysis of hormonal time series data, this paper proposes a simple graphical method for comparing two periodograms and describes a new nonparametric approach to testing the hypothesis that the two underlying spectra are the same. Simulation studies show that the new test has power characteristics that are competitive with existing procedures. The relative merits of nonparametric and semiparametric tests are discussed.

[Diggle, 1990]

Peter J. Diggle. Time Sereis: A Biostatistical Introduction. Oxford Statistical Science Series 5. Clarendon Press, Oxford, 1990.

[Dijkerman and Mazumdar, 1994a]

R. W. Dijkerman and R. R. Mazumdar. On the correlation structure of the wavelet coefficients of fractional Brownian motion. IEEE Transactions on Information Theory, 40(5):1609-1612, 1994.

Shows that the interdependence of the discrete wavelet coefficients of fractional Brownian motion, defined by normalized correlation, decays exponentially fast across scales and hyperbolically fast along time.

[Dijkerman and Mazumdar, 1994b]

R. W. Dijkerman and R. R. Mazumdar. Wavelet representations of stochastic processes and multiresolution stochastic models. IEEE Transactions on Signal Processing, 42(7):1640-1652, 1994.

Deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years. In the context of stochastic processes, the use of wavelet bases has not yet been fully investigated. We use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in L/sup 2/ defined in the time domain. We derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales. We study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by Basseville et al. (see IEEE Trans. Inform. Theory, vol.38, p.766-784, Mar. 1992). We propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process. These models are simple due to the strong decorrelation of the wavelet transform. Experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated.

[Dijkerman et al., 1995]

R. W. Dijkerman, R. R. Mazumdar, and A. Bagchi. Reciprocal processes on a tree-modeling and estimation issues. IEEE Transactions on Automatic Control, 40(2):330-335, 1995.

Motivated by multiresolution decomposition methods such as the discrete wavelet transformation, the authors introduce reciprocal processes on truncated N-ary trees. The authors discuss the relationship between such processes and nearest neighbor models. The authors show that they can derive a recursive description of the process, and that all reciprocal processes on N-ary trees reduce to autoregressive processes in the case of zero-valued boundary values at the bottom of the tree, corresponding to truncation of the tree. The authors then study the smoothing equations associated with such models.

[Donoho and Johnstone, 1993]

David L. Donoho and Iain M. Johnstone. Adapting to unknown smoothness by wavelet shrinkage. Technical report, Department of Statistics, Stanford University, 1993. Technical Report 425.

[Donoho and Johnstone, 1994]

David L. Donoho and Iain M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3):425-455, 1994.

[Donoho and Johnstone, 1996]

David L. Donoho and Iain M. Johnstone. Neo-classical minimax problems, thresholding and adaptive function estimation. Bernoulli, 2(1):39-62, 1996.

We study the problem of estimating θ from data Y&126;N(θ, σ2) under squared-error loss. We define three new scalar minimax problems in which the risk is weighted by the size of θ. Simple thresholding gives asymptotically minimax estimates in all three problems. We indicate the relationships of the new problems to each other and to two other neo-classical problems: the problems of the bounded normal mean and of the risk-constrained normal mean. Via the wavelet transform, these results have implications for adaptive function estimation in two settings: estimating functions of unknown type and degree of smoothness in a global l2 norm; and estimating a function of unknown degree of local Hölder smoothness at a fixed point. In the latter setting, the scalar minimax results imply: Lepskii's results that it is not possible fully toadapt the unknown degree of smoothness without incurring a performance cost; and that simple thresholding of the empirical wavelet transform gives an estimate of a function at a fixed point which is, to within constants, optimally adaptive to unknown degree of smoothness.

[Donoho and Johnstone, 1997]

David L. Donoho and Iain M. Johnstone. Minimax estimation via wavelet shrinkage. To appear in Annals of Statistics, 1997.

[Donoho et al., 1995]

David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, and Dominique Picard. Wavelet shrinkage: Asymptopia? (with discussion). Journal of the Royal Statistical Society B, 57(2):301-369, 1995.

Much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects - curves, densities, spectral densities, images - from noisy data. A now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems. Unfortunately, the results have rarely moved into practice, for a variety of reasons - among them being similarity to known methods, computational intractability and lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data: translate the empirical wavelet coefficients towards the origin by an amount [sq.root](2 log n) [sigma]/[sq.root]n. The proposal differs from those in current use, is computationally practical and is spatially adaptive; it thus avoids several of the previous objections. Further, the method is nearly minimax both for a wide variety of loss functions - pointwise error, global error measured in Lp-norms, pointwise and global errors in estimation of derivatives - and for a wide range of smoothness classes, including standard Hölder and Sobolev classes, and bounded variation. This is a much broader near optimality than anything previously proposed: we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity.

[Donoho et al., 1997]

David L. Donoho, Stepháne Mallat, and Rainer von Sachs. Estimating covariances of locally stationary processes: Rates of convergence of best basis methods. 1997.

[Donoho, 1992]

David L. Donoho. Interpolating wavelet transforms. Technical report, Technical Report 408, Department of Statistics, Stanford University, 1992.

[Donoho, 1993a]

David L. Donoho. Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data. In Proceedings of Symposia in Applied Mathematics, volume 47, pages 173-205. American Mathematical Society, 1993.

Wavelet methods for the recovery of objects from noisy and incomplete data are described. The common themes: (a) the new methods use nonlinear operations in the wavelet domain; (b) they accomplish tasks which are not possible by traditional linear/Fourier approaches to such problems. An attempt is made to indicate the heuristic principles, theoretical foundations and possible application areas for these methods, i.e. wavelet de-noising, wavelet approaches to linear inverse problems, wavelet packet de-noising, segmented multiresolutions, and nonlinear multi-resolutions.

[Donoho, 1993b]

David L. Donoho. Smooth wavelet decompositions with blocky coefficient kernels. In [Schumaker and Webb, 1993], pages 1-43.

[Donoho, 1995]

David L. Donoho. De-noising by soft-thresholding. IEEE Transactions on Information Theory, 41(3):613-627, 1995.

Donoho and Johnstone (1994) proposed a method for reconstructing an unknown function f on (0,1) from noisy data d/sub i/=f(t/sub i/)+ sigma z/sub i/, i=0, ..., n-1,t/sub i/=i/n, where the z/sub i/ are independent and identically distributed standard Gaussian random variables. The reconstruction f*/sub n/ is defined in the wavelet domain by translating all the empirical wavelet coefficients of d toward 0 by an amount sigma . square root (2log (n)/n). The authors prove two results about this type of estimator. (Smooth): with high probability f*/sub n/ is at least as smooth as f, in any of a wide variety of smoothness measures. (Adapt): the estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. The present proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model.

[Doroslovavcki, 1998]

Milovs L. Doroslovavcki. On the least asymmetric wavelets. IEEE Transactions on Signal Processing, 46(4):1125-1130, 1998.

The asymmetry of Daubechies' (1988, 1992) scaling functions and wavelets can be diminished by minimizing a special second moment in time for the wavelet-generating discrete-time filter. The moment is involved in an uncertainty relation for discrete-time signals. Other measures of asymmetry are addressed as well, and corresponding results are compared.

[Downie and Silverman, 1996]

T. R. Downie and B. W. Silverman. The discrete multiple wavelet transform and thresholding methods. Department of Mathematics, University of Bristol, 1996.

[Dutilleux, 1989]

P. Dutilleux. An implementation of the ``algorithme à trous'' to compute the wavelet transform. In [Combes et al., 1989], pages 298-304. Proceedings of the International Converence, Marseille, France, December 14-18, 1987.

[Edwards, 1991]

Tim Edwards. Discrete wavelet transforms: Theory and implementation. Deptartment of Statistics, Stanford University, 1991.

This includes a brief introduction to wavelets in general and the discrete wavelet transform in particular, covering a number of implementation issues that are often missed in the literature. A hardware implementation on a commercially available DSP system is described along with a program listing to show how such an implementation can be simulated.

[Efron and Morris, 1975]

Bradley Efron and Carl Morris. Data analysis using Stein's estimator and its generalizations. Journal of the American Statistical Association, 70(350):311-319, 1975.

[Einstein, 1914]

A. Einstein. Method for the determination of the statistical values of observations concerning quantities subject to irregular fluctuations. Archives de Sciences Physiques et Naturalles, 37:254-256, 1914.

[Erdol and Feng, 1994]

N. Erdol and Bao. Feng. Use of shift variance of the wavelet transform for signal detection. In Sixth IEEE Digital Signal Processing Workshop, 1994. 2-5 Oct. 1994, Yosemite National Park, CA, USA.

Characterizes signals according to the degree with which a time shift affects their wavelet series coefficients and develops a measure called the `shift index' to quantify that effect. The authors argue that the shift index can be used to locate, separate and cluster and/or detect pulse like signals with random arrival times. Examples are given to verify the established theory.

[Erdol et al., 1995]

Nurgun Erdol, Feng Bao, and Zajing Chen. Wavelet interpolation: From orthonomal to the oversampled wavelet transform. In International Conference on Acoustics, Speech, and Signal Processing, volume 2, pages 1093-1096, 1995. 9-12 May 1995, Detroit, MI, USA.

The orthonormal wavelet transform is an efficient way for signal representation since there is no redundancy in its expression, but due to aliasing in the decimation stage it lacks the often desired property of shift invariance. On the other hand, the oversampled or nonorthogonal wavelet offers a finer resolution in translation; thus reducing the effect of shift of origin, it becomes more robust to changes in the initial phase of the signal. In some areas of signal processing, such as wideband correlation processing, sensitivity to time alignment necessitates the use of the nonorthogonal wavelet transform. The price paid for the advantage of robustness to shifting is the introduction of redundancy in the expression. In many applications, both of these two properties are needed in different stages of signal processing. Thus there is a need to know the conditions under which the redundant and nonorthonormal wavelet transform coefficients can be derived from the orthonormal wavelet transform coefficients. The answer provides us with a convenient way to switch between these two forms: the orthonormal wavelet for efficient expression, and the nonorthogonal one whenever it is necessary for feature extraction.

[Eskridge et al., 1997]

Robert E. Eskridge, Jia-Yeong Ku, S. Trivikrama Rao, P. Steven Porter, and Igor G. Zurbenko. Separating different scales of motion in time series of meteorological variables. Bulletin of the American Meteorological Society, 78(7):1473-1484, 1997.

The removal of synoptic and seasonal signals from time series of meteorological variables leaves datasets amenable to the study of trends, climate change, and the reasons for such trends and changes. In this paper, four techniques for separating different scales of motion are examined and their effectiveness compared. These techniques are PEST, anomalies, wavelet transform, and the Kolmogorov-Zurbenko (KZ) filter. It is shown that PEST and anomalies do not cleanly separate the synoptic and seasonal signals from the data as well as the other two methods. The KZ filter method is shown to have the same level of accuracy as the wavelet transform method. However, the KZ filter method can be applied to datasets with missing observations and is much easier to use than the wavelet transform method.

[Fan et al., 1996]

J. Q. Fan, P. Hall, M. A. Martin, and P. Patil. On local smoothing of nonparametric curve estimators. Journal of the American Statistical Association, 91(433):258-266, 1996.

We develop new local versions of familiar smoothing methods; such as cross-validation and smoothed cross-validation, in the contexts of density estimation and regression. These new methods are locally adaptive in the sense that they capture smooth local fluctuations in the curve by using smoothly varying bandwidths that change as the character of the curve changes. Moreover, the new methods are accurate, easy to apply, and computationally expedient.

[Fan, 1996]

J. Q. Fan. Test of significance based on wavelet thresholding and Neyman's truncation. Journal of the American Statistical Association, 91(434):674-688, 1996.

Traditional nonparametric tests, such as the Kolomogorov-Smirnov test and the Cramer-Von Mises test, are based on the empirical distribution functions. Although these tests possess root-n consistency, they effectively use only information contained in the low frequencies. This leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high-frequency alternations. The drawback can be repaired via smoothing-based test statistics. In this article we propose two such kind of test statistics based on the wavelet thresholding and the Neyman truncation. We provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations, while maintaining the same capability in detecting smooth alternative densities as the traditional tests. Similar conclusions can be made for two-sample nonparametric tests of distribution functions. In that case, the traditional linear rank tests such as the Wilcoxon test and the Fisher-Yates test have low power in detecting two nearby densities where one has local features or contains high-frequency components, because these procedures are essentially testing the uniform distribution based on the sample mean of rank statistics. In contrast, the proposed tests use more fully the sampling information and have better ability in detecting subtle features.

[Farebrother, 1985]

R. W. Farebrother. Eigenvalue-Free methods for computing the distribution of a quadratic form in normal variables. Statistische Hefte, 26:287-302, 1985.

[Farebrother, 1990]

R. W. Farebrother. The distribution of a quadratic form in normal variables. Applied Statistics, 39(2):294-309, 1990.

[Farge et al., 1993]

M. Farge, Julian C. R. Hunt, and J. C Vassilicos, editors. Wavelets, fractals, and Fourier transforms, volume 43 of Institute of Mathematics and Its Applications conference series, New York, 1993. Clarendon Press. Based on the proceedings of a conference on wavelets, fractals, and Fourier transforms held at Newnham College, Cambridge in December 1990.

[Farge et al., 1996]

M. Farge, N. Kevlahan, V. Perrier, and U. Goirand. Wavelets and turbulence. Proceedings of the IEEE, 84(4):639-669, 1996.

We have used wavelet transform techniques to analyze, model, and compute turbulent flows. The theory and open questions encountered in turbulence are presented The wavelet-based techniques that we have applied to turbulence problems are explained and the main results obtained are summarized.

[Farge, 1992]

M. Farge. Wavelet transforms and their applications to turbulence. Annual Review of Fluid Mechanics, 24:395-457, 1992.

[Fargues and Brooks, 1993]

Monique P. Fargues and William A. Brooks. Applications of time-frequency and time-scale transforms to ultra-wideband radar transient signal detection. In Franklin T. Luk, editor, Advanced Signal Processing Algorithms, Architectures, and Implementations IV, volume 2027, pages 180-193, San Diego, California, 1993. The International Society for Optical Engineering.

Compared to conventional radars, ultra-wideband (UWB) radars are characterized by very large bandwidth and fine range resolution. Potential applications of this type of radar include terrain mapping, and target identification/classification. In this paper we use a non- stationary approach and analyze UWB radar data using time- frequency and time-scale transformations. The time-frequency transformations considered are the Short-Time Fourier Transform (STFT), the Wigner-Ville Distribution (WD), the Instantaneous Power Spectrum (IPS), and the ZAM transform. Two discrete implementations of the Wavelet Transform (DWT) are also investigated: the decimated A-trous algorithm proposed by Holschneider et al, which uses non-orthogonal wavelets; and the Mallat algorithm, which employs orthogonal wavelets. The transients under study are UWB radar returns from a boat (with and without corner reflector) in the presence of sea clutter, multipath, and radio frequency interferences (RFI). Results show that all time-frequency and time-scale transforms clearly detect the transient radar returns corresponding to the boat with a corner reflector. However, as the radar cross section of the target decreases (boat without a corner reflector), results change drastically as the RFI component dominates the signal. Simulations show that the Instantaneous Power Spectrum may be better adapted for localizing the transient among the time-frequency techniques studied. The decimated A-trous algorithm has the best time resolution of the techniques studied as the return appears better localized in the scalogram.

[Feng and Erdol, 1993]

Bao. Feng and N. Erdol. On the discrete wavelet transform and shiftability. In A. Singh, editor, Conference Record of the Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, volume 2, pages 1442-1445, 1993. 1-3 Nov. 1993, Pacific Grove, CA, USA.

We analyze the relationship between the change that is observed in the wavelet coefficients when a signal is time shifted and the time and frequency distributions of the wavelet functions. We address the effects of shift variance and show how it can be useful.

[Fernández et al., 1996]

G. Fernández, S. Periaswamy, and Wim Sweldens. LIFTPACK: A software package for wavelet transforms using lifting. In [Unser et al., 1996], page 1044. 4-9 August, 1996, Denver, Colorado.

We present LIFTPACK: A software package written in C for fast calculation of 2D biorthogonal wavelet transforms using the lifting scheme. The lifting scheme is a new approach for the construction of biorthogonal wavelets entirely in the spatial domain, i.e., independent of the Fourier Transform. Constructing wavelets using lifting consists of three simple phases: the first step or Lazy wavelet splits the data into two subsets, even and odd, the second step calculates the wavelet coefficients (high pass) as the failure to predict the odd set based on the even, and finally the third step updates the even set using the wavelet coefficients to compute the scaling function coefficients (low pass). The predict phase ensures polynomial cancelation in the high pass (vanishing moments of the dual wavelet) and the update phase ensures preservation of moments in the low pass (vanishing moments of the primal wavelet). By varying the order, an entire family of transforms can be built. The lifting scheme ensures fast calculation of the forward and inverse wavelet transforms that only involve FIR filters. The transform works for images of arbitrary size with correct treatment of the boundaries. Also, all computations can be done in-place.

[Fisher, 1915]

R. A. Fisher. Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10:507-521, 1915.

[Fisher, 1929]

R. A. Fisher. Tests of significance in harmonic analysis. Proceedings of the Royal Society of London, Series A, 125:54-59, 1929.

[Flandrin, 1989]

Patrick Flandrin. On the spectrum of fractional Brownian motions. IEEE Transactions on Information Theory, 35(1):197-199, 1989.

Fractional Brownian motions (FBMs) provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order. However, due to the nonstationary nature of these processes, the precise meaning of such spectra remains generally unclear. Two complementary approaches are proposed which are intended to clarify this point. The first one, based on a time-frequency analysis, takes into account the nonstationary nature of FBM and puts emphasis on time-averaged measurements; the second one, based on a time-scale analysis, is matched to self-similarity properties of FBM and reveals an underlying stationary structure relative to each time-scaling.

[Flandrin, 1992]

Patrick Flandrin. Wavelet analysis and synthesis of fractional Brownian motion. IEEE Transactions on Information Theory, 38(2):910-917, 1992.

Fractional Brownian motion (FBM) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and 1/f-type spectral behavior over wide ranges of frequencies. Statistical self-similarity is an essential feature of FBM and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of FBM. It reveals a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of FBM can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered, and some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out.

[Foster, 1996]

Grant Foster. Wavelets for period analysis of unevenly sampled time series. The Astronomical Journal, 112(4):1709-1729, 1996.

[Foufoula-Georgiou and Kumar, 1994]

Efi Foufoula-Georgiou and Praveen Kumar, editors. Wavelets in Geophysics, volume 4 of Wavelet Analysis and its Applications. Academic Press, Inc, San Diego, 1994.

Applications of wavelet analysis to the geophysical sciences grew from Jean Morlet's work on seismic signals in the 1980s. Used to detect signals against noise, wavelet analysis excels for transients or for spatially localized phenomena. In this fourth volume in the renown WAVELET ANALYSIS AND ITS APPLICATIONS Series, Efi Foufoula-Georgiou and Praveen Kumar begin with a self-contained overview of the nature, power, and scope of wavelet transforms. The eleven original papers that follow in this edited treatise show how geophysical researchers are using wavelets to analyze such diverse phenomena as intermittent atmospheric turbulence, seafloor bathymetry, marine and other seismic data, and flow in aquifiers. Wavelets in Geophysics will make informative reading for geophysicists seeking an up-to-date account of how these tools are being used as well as for wavelet researchers searching for ideas for applications, or even new points of departure.

[Fournier, 1996a]

Aimé Fournier. Wavelet analysis of observed geopotential and wind: Blocking and local energy coupling across scales. In [Unser et al., 1996], page 1044. 4-9 August, 1996, Denver, Colorado.

Atmospheric blocking during three unusual winter months is studied by multiresolution analysis and a wavelet based adaptation of traditional Fourier series based energetics. We demonstrate that blocking, in part a large and localized meteorological phenomenon, is largely described by just the largest scale part of the multiresolution analysis. New forms of the transfer functions of kinetic energy with the mean and eddy parts of the atmospheric circulation are introduced. These quantify the spatially localized conversion of energy between scales. A new accounting method for wavelet indexed transfers permits the introduction of a physically meaningful localized scale flux function. These techniques are applied to the data, and some support is found for the hypothesis that blocking is partially maintained by an inverse cascade.

[Fournier, 1996b]

Aimé Fournier. Wavelet multiresolution analysis of numerically sinulated 3D radiative convection. In [Szu, 1996], pages 642-653. 8-12 April 1996, Orlando, Florida.

A wavelet multiresolution analysis is performed on atmospheric fields simulated by a multilevel 3-dimensional atmospheric boundary layer model. Wavelet cospectra of the vertical wind and potential temperature are calculated and compared with radial Fourier cospectra. The former indicate most of the field variance to have horizontal scales roughly equal to the vertical scale, as should be the case for convectively driven turbulence. Fourier spectra exhibit a -3 power law, suggesting that the statistics may depend only on a quantity with units of time. Observations of time-and scale-dependent structures suggest certain physical mechanisms at work. The multiresolution analysis analogue of turbulent energy equations are formulated. This framework supports the proposed physical mechanisms.

[Fournier, 1996c]

Aimé Fournier. Wavelet representation of lower-atmospheric long nonlinear wave dynamics, governed by the Benjamin-Davis-Ono-Burgers equation. Department of Physics, Yale University, 1996.

[Fox and Taqqu, 1986]

Robert Fox and Murad S. Taqqu. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Applied Statistics, 14(2):517-532, 1986.

[Fox and Taqqu, 1987]

Robert Fox and Murad S. Taqqu. Central limit theorems for quadratic forms in random variables having long-range dependence. Probability Theory and Related Fields, 74:213-240, 1987.

[Fuller, 1976]

Wayne A. Fuller. Introduction to Statistical Time Series. John Wiley and Sons, Inc., New York, 1976.

[Fuller, 1996]

Wayne A. Fuller. Introduction to Statistical Time Series. Wiley-Interscience, New York, 2 edition, 1996.

Retaining its theorem-proof format, this updated edition incorporates new results in such areas as nonstationary, multivariate and nonlinear models and empirical model identification. Features additional sections on the Wold decomposition, partial autocorrelation and the Kalman filter. Most of the homework problems can be worked with any number of statistical packages.

[Gabor, 1946]

D. Gabor. Theory of communication. Journal of the IEE, 93:429-457, 1946.

The purpose of these studies is an inquiry into the essence of the ``information'' conveyed by channels of communication, and the application of the result of this inquiry to the practical problem of optimum utilization of frequency bands. In Part 1, a new method of analysing signals is presented in which time and frequency play symmetrical parts, and which contains ``time analysis'' and ``frequency analysis'' as special cases. It is shown that the information conveyed by a frequency band in a given time-interval can be analysed in various ways into the same number of elementary ``quanta of information,'' each quantum conveying one numerical datum. In Part 2, this method is applied to the analysis of hearing sensations. It is shown on the basis of existing experimental material that in the band between 60 an 1000 c/s the human ear can discriminate very nearly every second datum of information, and that this efficiency of nearly 50 percent is independent of the duration of the signals in a remarkably wide interval. This fact, which cannot be explained by any mechanism in the inner ear, suggests a new phenomenon in nerve conduction. At frequencies above 1000 c/s the efficiency of discrimination falls off sharply, proving that sound reproductions which are far from faithful may be perceived by the ear as perfect, and that ``condensed'' methods of transmission and reproduction with improved waveband economy are possible in principle. In Part 3, suggestions are discussed for compresse transmission and reproduction of speech or music, and the first experimental results obtained with one of these methods are described.

[Gallant and Hutchinson, 1997]

J. C. Gallant and M. F. Hutchinson. Scale dependence in terrain analysis. Mathematics and Computers in Simulation, 43(3-6):313-321, March 1997.

Topographic attributes computed from digital elevation models are dependent on the resolution of the elevation data from which they are computed. A regular rectangular grid is not an ideal representation of topographic surfaces for the study of scale effects. Spectral and wavelet techniques are obvious alternatives but have several deficiencies, particularly in their use of oscillatory basis functions. The positive wavelet representation has very attractive properties of localisation and feature representation. Preliminary application to one-dimensional topographic data (profiles) yields useful results, including the identification of changes in topographic structure with scale. Extension to two-dimensional analysis will allow quantification of characteristic shapes, scales and orientations in the landscape.

[Gao and Bruce, 1996]

Hong-Ye Gao and Andrew Bruce. WaveShrink with firm shrinkage. Technical report, Research Report 39, Statistical Sciences Division, MathSoft, Inc, 1996. To appear in Statistica Sinica.

Donoho and Johnstone's WaveShrink procedure has proven valuable for signal de-noising and non-parametric regression. WaveShrink has very broad asymptotic near-optimality properties. In this paper, we introduce a new shrinkage scheme, firm, which generalizes the hard and soft shrinkage proposed by Donoho and Johnstone. We derive minimax thresholds and provide formulas for computing the pointwise variance, bias, and risk for WaveShrink with firm shrinkage. We study the properties of the shrinkage functions, and demonstrate that firm shrinkage offers advantages over both hard shrinkage (uniformly smaller risk and less sensitivity to small perturbations in the data) and soft shrinkage (smaller bias and overall L_2 risk). Software is provided to reproduce all results in this paper.

[Gao and Li, 1993]

W. Gao and BL. Li. Wavelet analysis of coherent structures at the atmosphere- forest interface. Journal of Applied Meteorology, 32(11):1717-1725, 1993.

Wavelet studies were used for the turbulent data obtained inside and over a deciduous forest to investigate spatial and scale properties of a coherent structure in the area. Discrete warm and cool centers are linked to organized updrafts and downdrafts. Their patterns are alike, but the magnitudes vary at various heights. Temperature structures over the canopy possess a shorter duration, but the rate of reduction in the time scale with increasing height seems proportional to the rise in mean wind speed.

[Gao, 199]

Hong-Ye Gao. Wavelets and isotonic regression. Statistical Sciences Division, MathSoft, Inc, 199?

Consider the following isotonic regression model: [ y_i = f(t_i) + z_i] where f is only known to be a decreasing function and z_i are iid Gaussian with mean zero and variance sigma^2. We propose a simple thresholding procedure based on the fact that the wavelet coefficients for f, under Haar basis, are non-negative. We show that our estimator is competitive with the Grenander estimator both theoretically and numerically (in the sense of mean-square-error).

[Gao, 1996]

Hong-Ye Gao. Spectral density estimation via wavelet shrinkage. Statistical Sciences Division, MathSoft, Inc, 1996.

We study the problem of estimating the spectral density of a stationary Gaussian time series. We use an orthogonal wavelet system whose members are periodic functions and have a finite number of non-zero Fourier coefficients -- periodized Meyer wavelets. We apply shrinkage rules to the empirical wavelet coefficients. We show that estimates based on thresholds t_j,n = lm_jlog n for certain lm_j, with n the sample size, have near-optimal L_2 convergence rates, over any Besov class in a wide range. In some cases, which includes the Bump Algebra, wavelet shrinkage procedures significantly outperform classical linear procedures, such as window methods and AR approximation methods.

[Gao, 1997a]

Hong-Ye Gao. Choice of thresholds for wavelet shrinkage estimate of the spectrum. Journal of Time Series Analysis, 18(3), 1997.

We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t_j,n depending on sample size n, wavelet basis and resolution level j. At fine resolution levels (j=1, 2,...), we propose [ t_j,n = A_jlog n, ] where A_j are level-dependent constants and at coarse levels (j>>1), [ t_j,n = fracpisqrt3sqrtlog n. ] The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties.

[Gao, 1997b]

Hong-Ye Gao. Threshold selection in WaveShrink. Statistical Sciences Division, MathSoft, Inc, 1997.

Donoho and Johnstone's wavelet shrinkage denoising technique (known as WaveShrink) consists three steps: (1) transform data into wavelet domain; (2) shrink the wavelet coefficients; and (3) transform the shrunk coefficients back. The choice of shrinkage function and thresholds in step (2) plays an important role for WaveShrink both theoretically and in practice. In this paper, we discuss the issue of threshold selection in WaveShrink. We first review the threshold selection procedure based minimax thresholds and Stein's Unbiased Risk Estimate (SURE). We then propose a new threshold selection procedure based on combining Coifman and Donoho's cycle-spinning and SURE. We call our new procedure SPINSURE. We use examples to show that SPINSURE is numerically more stable than SURE: smaller standard deviation and smaller range. Various comparisons with the ideal and minimax thresholds are also presented.

[Gao, 1997c]

Hong-Ye Gao. Wavelet shrinkage denoising using non-negative garrote. Statistical Sciences Division, MathSoft, Inc, 1997.

In this paper, we combine Donoho and Johnstone's Wavelet Shrinkage denoising technique (known as WaveShrink) with Breiman's non-negative garrote. We show that the non-negative garrote shrinkage estimate enjoys the same asymptotic convergence rate as the hard and the soft shrinkage estimates. For finite sample simulations, non-negative garrote performs better (smaller mean-square-error) than both hard and soft, and comparable to the firm shrinkage. We derive the minimax thresholds for the non-negative garrote. We study the threshold selection procedure based on Stein's Unbiased Risk Estimate (SURE) for both non-negative garrote and soft shrinkages. We propose a new threshold selection procedure based on combining Coifman and Donoho's cycle-spinning and SURE. We call our new procedure SPINSURE. We use examples to show that SPINSURE is more stable than SURE: smaller standard deviation and smaller range.

[Gao, 1997d]

Hong-Ye Gao. Wavelet shrinkage estimates for heteroscedastic regression models. Statistical Sciences Division, MathSoft, Inc, 1997.

We extend Donoho and Johnstone's wavelet shrinkage smoothing technique (known as WaveShrink) to handle data with heteroscedastic noise. We first show that if the noise variances are known, WaveShrink estimate achieves the same near-optimal convergence rate as in the white noise case. We then propose a procedure for estimating the noise variances. Our procedure is based on applying running MAD (Median Absolute Deviation from the median) to the non-decimated finest level wavelet coefficients. We apply our technique to various numerical examples.

[Gardner, 1987]

W. A. Gardner. Common pitfalls in the application of stationary process theory to time-sampled and modulated signals. IEEE Transactions on Communications, COM-35(5):529-534, 1987.

The common practice of applying the theory of stationary stochastic processes to a cyclostationary process by introducing random phase(s) into the probabilistic model in order to stationarize the process can lead to erroneous results, such as incorrect formulas for power spectral density. This is illustrated by showing that commonly used formulas for signals that have undergone frequency conversion or time sampling can be incorrect. The source of error is shown to be inappropriate phase-randomization procedures. The correct procedure is described, and corrected formulas are given. The problem is further illustrated by showing that commonly used resolution and reliability (mean and variance) formulas for spectrum analyzers must be corrected for cyclostationary signals. It is explained that all corrections to formulas reflect the effects of spectrum correlation. These effects are inappropriately averaged out by inappropriate phase-randomization procedures. It is further explained that these inappropriate procedures destroy the important property of ergodicity of the probabilistic model.

[Gardner, 1988]

William A. Gardner. Statistical Spectral Analysis: A Nonprobabilistic Theory. Prentice Hall, New Jersey, 1988.

[Geronimo et al., 1994]

Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust. Fractal functions and wavelet expansions based on several scaling functions. Journal of Approximation Theory, 78(3):373-401, 1994.

[Geweke and Porter-Hudak, 1983]

John Geweke and Susan Porter-Hudak. The estimation and application of long memory time series models. Journal of Time Series Analysis, 4(4):221-238, 1983.

[Giraitis and Leipus, 1995]

L. Giraitis and R. Leipus. A generalized fractionally differencing approach in long-memory modeling. Lietuvos Matematikos Rinkinys, 35(1):65-81, 1995.

[Goodman, 1957]

N. R. Goodman. On the joint estiamtion of spectra, cospectrum and quadrature spectrum of a two-dimensional stationary Gaussian process. Sci. Paper No. 10. Engrng. Statist. Lab., New York Univ., New York, 1957.

[Goodman, 1963]

N. R. Goodman. Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). The Annals of Mathematical Statistics, 34:152-177, 1963.

[Goupil et al., 1991]

M. J. Goupil, M. Auvergne, and A. Baglin. Wavelet analysis of pulsating white dwarfs. Astronomy and Astrophysics, 250(1):89-98, 1991.

Parts of light curves of two variable white dwarfs, Giclas 191-16 (BR Cam) and PG 1351+489 (EM UMa), are investigated by means of a wavelet analysis. This time-frequency analysis decomposes the light curves into their different oscillating components whose temporal behaviors are then individually studied. In addition to an oscillation of large amplitude, small amplitude oscillations are thereby clearly emphasized for both stars. Amplitude variations are found for most detected oscillations with periods of modulation as long or greater than the time intervals of the corresponding runs. A wavelet analysis of a comparison star gives the quality of the night in localizing perturbative atmospheric events.

[Graf, 1983]

Hans-Peter Graf. Long-range correlations and estimation of the self-similarity parameter. PhD thesis, Eidgenössische Technische Hochschule, Zürich, 1983.

[Granger and Joyeux, 1980]

C. W. J. Granger and Roselyne Joyeux. An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis, 1:15-29, 1980.

[Granger, 1963]

C. W. J. Granger. A quick test for serial correlation suitable for use with non-stationary time series. Journal of the American Statistical Association, 58:728-736, 1963.

[Graps, 1995]

Amara Graps. An introduction to wavelets. IEEE Computational Science and Engineering, 2(2):50-61, 1995.

Wavelets were developed independently by mathematicians, quantum physicists, electrical engineers and geologists, but collaborations among these fields during the last decade have led to new and varied applications. What are wavelets, and why might they be useful to you? The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers feel that using wavelets means adopting a whole new mind-set or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. Most of the basic wavelet theory has now been done. The mathematics have been worked out in excruciating detail, and wavelet theory is now in the refinement stage. This involves generalizing and extending wavelets, such as in extending wavelet packet techniques. The future of wavelets lies in the as-yet uncharted territory of applications. Wavelet techniques have not been thoroughly worked out in such applications as practical data analysis, where, for example, discretely sampled time-series data might need to be analyzed. Such applications offer exciting avenues for exploration.

[Gray, 1988]

B. M. Gray. Seasonal frequency variations of the 40-50 day oscillation. Journal of Climatology, 8:511-519, 1988.

[Greenblatt, 1994]

Seth A. Greenblatt. Wavelets in econometrics: An application to outlier testing. University of Reading, 1994.

[Greenhall, 1990]

Charles A. Greenhall. Orthogonal sets of data windows constructed from trigonometric polynomials. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(5):870-872, 1990.

Suboptimal, easily computable substitutes for the discrete prolate spheroidal windows used by D.J. Thomson (Proc. IEEE, vol.70, p.1055-1096, 1982) for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated.

[Greenhall, 1991]

Charles A. Greenhall. Recipes for degrees of freedom of frequency stability estimators. IEEE Transactions on Instrumentation and Measurement, 40(6):994-999, 1991.

The Allan variance for an averaging time tau can be estimated either from all available phase samples or from a subgrid of samples with spacing tau . The author gives a set of computational recipes that yield the variance of both estimators, with less than 2% error, for the five power-law components of the classical continuous-time clock noise model.

[Greenhall, 1997]

Charles A. Greenhall. The third-difference approach to modified allan variance. IEEE Transactions on Instrumentation and Measurement, 46(3):696-703, 1997.

This study gives strategies for estimating the modified Allan variance (MVAR), and formulas for computing the equivalent degrees of freedom (edf) of the estimators. A third-difference formulation of MVAR leads to a tractable formula for edf in the presence of power-law phase noise. The effect of estimation stride on edf is shown, First-degree rational-function approximations for edf are derived, and their errors tabulated. A theorem allowing conservative estimates of edf in the presence of compound noise processes is given.

[Greenshields and Rosiene, 1998]

I. R. Greenshields and J. A. Rosiene. A fast wavelet-based karhunen-loeve transform. Pattern Recognition, 31(7):839-845, 1998.

The paper describes the role of the standard wavelet decomposition in computing a fast Karhunen-Loeve transform. The standard wavelet decomposition (which we show is different from the conventional wavelet transform) leads to a highly sparse and simply structured transformed version of the correlation matrix which can be easily subsetted (with little loss of Frobenius norm). The eigenstructure of this smaller matrix can be efficiently computed using standard algorithms such as QL. Finally, we provide an example of the use of the efficient transform by classifying a 219-channel AVIRIS image with respect to its eigensystem.

[Gregg et al., 1993]

M. C. Gregg, H. E. Seim, and D. B. Percival. Statistics of shear and turbulent dissipation profiles in random internal wave fields. Journal of Physical Oceanography, 23(8):1777-1799, 1993.

Because breaking internal waves produces most of the turbulence in the thermocline, the statistics of epsilon , the rate of turbulent dissipation, cannot be understood apart from the statistics of internal wave shear. The statistics of epsilon shear are compared for two sets of profiles from the northeast Pacific. One set, PATCHEX, has internal wave shear close to the Garrett and Munk model, but the other set, PATCHEX north, has average 10-m shear squared, (S/sub 10//sup 2/), about four times larger than the model. The 10-m shear components, S/sub x/ and S/sub y/, were measured between 1 and 9 MPa and referenced to a common stratification by WKB scaling. The scaled components, S/sub x/ and S/sub y/, are found to be independent and normally distributed with zero means, as assumed by Garrett and Munk. This readily leads to analytic forms for the probability densities of S/sub 10//sup 2/ and S/sub 10//sup 4/. The observed probability densities of S/sub 10//sup 2/ and S/sub 10//sup 4/ are close to the predicted forms, and both are strongly skewed. Moreover, sigma /sub InS//sub 10//sup 2/ and sigma /sub InS//sub 10//sup 4/ are constants, independent of the standard deviations of S/sub x/ and S/sub y/. The probability density of the inverse Richardson number is a scaled version of the probability density of S/sub 10//sup 2/. The PATCHEX distribution cuts off near Ri/sub 10//sup -1/=4, as found by Eriksen, but the PATCHEX north distribution extends to higher values, as predicted analytically. Consequently, a cutoff at Ri/sub 10//sup -1/=4 is not a universal constraint. Over depths where (N/sup 2/) is nearly uniform, the probability density of 0.5-m epsilon can be approximated, to varying degrees of accuracy, as the sum of a noise variate with an empirically determined distribution and a lognormally distributed variate whose parameters can be estimated using a minimum chi-square fitting procedure.

[Greiner et al., 1997]

M. Greiner, J. Giesemann, and P. Lipa. Translational invariance in turbulent cascade models. Physical Review E, 56(4):4263-4274, 1997.

Due to the underlying hierarchical structure, spatial correlation functions calculated from multiplicative cascade models are not translationally invariant. A scheme is presented that restores translational invariance by averaging over the experimentally unknown spatial location of cascade realizations with respect to the observation window. The impact of this scheme on multiplier distributions for the energy dissipation field in fully developed turbulence is analyzed; only the experimental multiplier distribution is found to be invariant under a wide range of scales.

[Grenander and Rosenblatt, 1957]

Ulf Grenander and Murray Rosenblatt. Statistical analysis of stationary time series. Wiley, New York, 1957.

[Grossmann and Morlet, 1984]

A. Grossmann and J. Morlet. Decomposition of hardy functions into square integrable wavelets of constant shape. SIAM Journal of Mathematical Analysis, 15(4):723-736, 1984.

[Grossmann et al., 1989]

A. Grossmann, R. Kronland-Martinet, and J. Morlet. Reading and understanding continuous wavelet transforms. In [Combes et al., 1989], pages 2-20. Proceedings of the International Converence, Marseille, France, December 14-18, 1987.

An introduction to continuous wavelet transforms and a description of the representation methods that have evolved. Also discusses the influence of the choice of the wavelet in the interpretation of wavelet transforms.

[Grubb and Walden, 1997]

H. J. Grubb and A. T. Walden. Characterizing seismic time series using the discrete wavelet transform. Geophysical Prospecting, 45(2):183-205, 1997.

The discrete wavelet transform (DWT) has potential as a tool for supplying discriminatory attributes with which to characterize or cluster groups of seismic traces in reservoir studies. The wavelet transform has the great advantage over the Fourier transform in being able to better localize changes. The multiscale nature and structure of the DWT leads to a method of display which highlights this and allows comparison of changes in the transform with changing data. Many different sorts of wavelet exist and it is found that the quality of reconstruction of a seismic trace segment, using some of the coefficients, is dependent on the choice of wavelet, which leads us to consider choosing a wavelet under a 'best reconstruction' criterion. Location shifts, time zero uncertainties, are also shown to affect the transform, as do truncations, resampling, etc. Using real data, examples of utilizing the DWT coefficients as attributes for whole trace segments or fractional trace segments are given. Provided the DWT is applied consistently, for example with a fixed wavelet, and non-truncated data, the transform produces useful results. Care must be exercised if it is applied to data of different lengths. However, as the algorithm is refined and improved in the future, the DWT should prove increasingly useful.

[Guo, 1995]

Haitao Guo. Theory and applications of the shift-invariant, time-varying and undercimated wavelet transforms. Master's thesis, Electrical and Computer Engineering Department, Rice University, 1995.

In this thesis, we generalize the classical discrete wavelet transform, and construct wavelet transforms that are shift-invariant, time-varying, undecimated, and signal dependent. The result is a set of powerful and efficient algorithms suitable for a wide variety of signal processing tasks, e.g., data compression, signal analysis, noise reduction, statistical estimation, and detection. These algorithms are comparable and often superior to traditional methods. In this sense, we put wavelets in action.

[Haar, 1910]

Alfred Haar. Zur Theorie der orthogonalen Funktionen-Systeme. Mathematische Annalen, 69:331-371, 1910. In German.

[Hall and Nason, 1996]

Peter Hall and Guy P. Nason. On choosing a non-integer resolution level when using wavelet methods. Technical report, Centre for Mathematics and its Applications, Australian National University, 1996.

[Hall and Patil, 1995]

Peter Hall and Prakash Patil. On wavelet methods for estimating smooth functions. Bernoulli, 1(1):41-58, 1995.

Without assuming any prior knowledge of wavelet methods, we develop theory describing their performance as estimators of smooth functions. The linear part of the wavelet estimator is discussed by analogy with classical kernel methods. Concise formulae are developed for its bias, variance and mean square error. These quantities oscillate somewhat erratically on a wavelength that is equivalent to the bandwidth, reflecting the irregular numerical fluctuations that are observed in practice. Nevertheless, the contributions of these oscillations to mean integrated square error tend to dampen one another out, even over very small intervals, with the result that mean integrated square error properties of linear wavelet methods are much closer to those of kernel methods than is perhaps reasonable, given the local behaviour. We illustrate the adaptive qualities of the nonlinear component of a wavelet estimator by describing its performance when the target function is smooth but has high-frequency oscillations. It is shown that the nonlinear component automatically adapts to changing local conditions, to the extent of achieving (except for a logarithmic factor) the same convergence rate as the optimal linear estimator, but without a need to adjust the underlying bandwidth. This makes explicitly clear the way in which the linear part of the estimator takes care of the ‘average’ characteristics of the unknown curve, while the nonlinear part corrects for more erratic fluctuations, in a manner which is virtually independent of the construction of the linear part.

[Hall and Patil, 1996a]

P. Hall and P. Patil. On the choice of smoothing parameter, threshold and truncation in nonparametric regression by non-linear wavelet methods. Journal of the Royal Statistical Society B, 58(2):361-377, 1996.

Concise asymptotic theory is developed for non-linear wavelet estimators of regression means, in the context of general error distributions, general designs, general normalizations in the case of stochastic design, and non-structural assumptions about the mean. The influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters. Mainly, the tail weight is described in an extremely simple way, by a moment condition; previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal. Different approaches to correction for stochastic design are suggested. These include conventional kernel estimation of the design density, in which case the interaction between the smoothing parameters of the non-linear wavelet estimator and the linear kernel method is described.

[Hall and Patil, 1996b]

Peter Hall and Prakash Patil. Effect of thresholding rules on performance of wavelet-based curve estimators. Statistica Sinica, 6:331-345, 1996.

[Hall and Turlach, 1995]

Peter Hall and Berwin A. Turlach. Convolution and interpolation: Competitors with local polynomial smoothing. Technical Report SRR95-037, Centre for Mathematics and its Applications, Australian National University, 1995.

Local polynomial smoothing enjoys a variety of very attractive features. It is often viewed as superior to convolution and interpolation methods, which offer greater numerical stability but inferior theoretical performance. In this paper we show that modifications to convolution and interpolation techniques produce effective competitors with local polynomial smoothing, enjoying similar bias, variance and mean squared error properties but without the downside of numerical instability. The methods suggested here may be employed as the basis for empirical wavelet transforms of ungridded data.

[Hall and Turlach, 1997]

Peter Hall and Berwin A. Turlach. Enhancing convolution and interpolation methods for nonparametric regression. Biometrika, 84(4):779-790, 1997.

[Hall et al., 1996]

Peter Hall, Ian McKay, and Berwin Turlach. Performance of wavelet methods for functions with many discontinuities. Annals of Statistics, 24(6):???--???, 1996.

[Hall et al., 1997]

Peter Hall, Spiridon Penev, Gérard Kerkyacharian, and Dominique Picard. Numerical performance of block thresholded wavelet estimators. Statistics and Computing, 7(2):115-124, 1997.

Usually, methods for thresholding wavelet estimators are implemented term by term, with empirical coefficients included or excluded depending on whether their absolute values exceed a level that reflects plausible moderate deviations of the noise. We argue that performance may be improved by pooling coefficients into groups and thresholding them together. This procedure exploits the information that coefficients convey about the sizes of their neighbours. In the present paper we show that in the context of moderate to low signal-to-noise ratios, this `block thresholding' approach does indeed improve performance, by allowing greater adaptivity and reducing mean squared error. Block thresholded estimators are less biased than term-by-term thresholded ones, and so react more rapidly to sudden changes in the frequency of the underlying signal. They also suffer less from spurious aberrations of Gibbs type, produced by excessive bias. On the other hand, they are more susceptible to spurious features produced by noise, and are more sensitive to selection of the truncation parameter.

[Hannan, 1970]

E. J. Hannan. Multiple Time Series. John Wiley and Sons, Inc., New York, 1970.

[Hannan, 1976]

E. J. Hannan. The asymptotic distribution of serial covariances. Applied Statistics, 4:396-399, 1976.

[Haslett and Raftery, 1989]

John Haslett and Adrian E. Raftery. Space-time modelling with long-memory dependence: Assessing ireland's wind power resource. Applied Statistics, 38(1):1-50, 1989.

[Hawkins, 1988]

D. L. Hawkins. Retrospective and sequential tests for a change in distribution based on Kolmogorov-Smirnov-type statistics. Sequential Analysis, 7(1):23-51, 1988.

[Haykin, 1991]

Simon Haykin, editor. Advances in Spectrum Analysis and Array Processing. Prentice Hall, Englewood, Cliffs, N.J., 1991.

[Helson and Sarason, 1967]

Henry Helson and Donald Sarason. Past and future. Mathematica Scandanavica, 21(1):5-16, 1967.

[Hendon and Salby, 1993]

Harry H. Hendon and Murry L. Salby. The life cycle of the madden-julian oscillation. Center for Atmospheric Theory and Analysis, University of Colorado, 1993.

[Heneghan et al., 1994]

Conor Heneghan, Shyam Khanna, Åke Flock, Mats Ulfendahl, Lou Brundin, and Malvin C. Teich. Investigating the nonlinear dynamics of cellular motion in the inner ear using the short-time Fourier and continuous wavelet transforms. IEEE Transactions on Signal Processing, 42(12):3335-3352, 1994.

The short-time Fourier transform (STFT) and the continuous wavelet transform (CWT) are used to analyze the time course of cellular motion in the inner ear. The velocity responses of individual outer hair cells and Hensen's cells to sinusoidal and amplitude modulated (AM) acoustical signals applied at the ear canal display characteristics typical of nonlinear systems, including the generation of harmonic and half-harmonic components. The STFT proves to be valuable for following the time course of the frequency components generated using sinusoidal and ARM input signals. The CWT is also useful for analyzing these signals; however, it is generally not as effective as the STFT when octave-band-based CWT's are used. For the transient response, the spectrogram (which is the squared magnitude of the STFT) and the octave-band-based scalogram (which is the squared magnitude of the CWT) prove equally valuable, and the authors have used both to study the responses of these cells to step-onset tones of different frequencies. Such analyses reveal information about the preferred vibration frequencies of cells in the inner ear and are useful for deciding among alternative mathematical models of nonlinear cellular dynamics. A modified Duffing oscillator model yields results that bear some similarity to the data.

[Hernández and Weiss, 1996]

Eugenio Hernández and Guido Weiss. A First Course on Wavelets. CRC Press Inc., Boca Raton, 1996.

Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets. The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets. Some of the topics treated include: Several bases generated by a single function via translations and dilations; Multiresolution analysis, compactly supported wavelets, and spline wavelets; Band-limited wavelets; Unconditionality of wavelet bases; Characterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functions. The authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented. Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets.

[Hess-Nielsen and Wickerhauser, 1996]

Nikolaj Hess-Nielsen and Mladen Victor Wickerhauser. Wavelets and time-frequency analysis. Proceedings of the IEEE, 84(4):523-540, 1996.

We present a selective overview of time-frequency analysis and some of its key problems. In particular we motivate the introduction of wavelet and wavelet packet analysis. Different types of decompositions of an idealized time-frequency plane provide the basis for understanding the performance of the numerical algorithms and their corresponding interpretations within the continuous models. As examples we show how to control the frequency spreading of wavelet packets at high frequencies using nonstationary filtering and study some properties of periodic wavelet packets. Furthermore we derive a formula to compute the time localization of a wavelet packet from its indexes which is exact for linear phase filters, and show how this estimate deteriorates with deviation from linear phase.

[Hidalgo and Robinson, 1996]

J. Hidalgo and P. M. Robinson. Testing for structural change in a long-memory environment. Journal of Econometrics, 70(1):159-174, 1996.

Long-memory time-series analysis is apt to be applied to economic time series which extend over many years, in which circumstances the possibility of structural breaks is likely to be entertained. Tests for a change in parameter values at a given time point are proposed in linear regression models with long-memory errors. Existing tests based on the assumption of serially independent or weakly dependent errors will typically be invalid in such an environment. The tests are derived in case of certain nonstochastic and stochastic regressors, and are given large-sample justification. A small Monte Carlo study of finite-sample behaviour is included.

[Hidalgo, 1996]

Javier Hidalgo. Spectral analysis for bivariate time series with long memory. Econometric Theory, 12(5):773-792, 1996.

This paper provides limit theorems for spectral density matrix estimators and functionals of it for a bivariate covariance stationary process whose spectral density matrix has singularities not only at the origin but possibly at some other frequencies and, thus, applies to time series exhibiting long memory. In particular, we show that the consistency and asymptotic normality of the spectral density matrix estimator at a frequency, say lambda, which hold for weakly dependent time series, continue to hold for long memory processes when lambda lies outside any arbitrary neighborhood of the singularities. Specifically, we show that for the standard properties of spectral density matrix estimators to hold, only local smoothness of the spectral density matrix is required in a neighborhood of the frequency in which we are interested. Therefore, we are able to relax, among other conditions, the absolute summability of the autocovariance function and of the fourth-order cumulants or summability conditions on mixing coefficients, assumed in much of the literature, which imply that the spectral density matrix is globally smooth and bounded.

[Hidalgo, 1997]

Javier Hidalgo. Non-parametric estimation with strongly dependent multivariate time series. Journal of Time Series Analysis, 18(2):95-122, 1997.

Smooth non-parametric kernel density and regression estimators are studied when the data are strongly dependent. In particular, we derive central (and non-central) limit theorems for the kernel density estimator of a multivariate Gaussian process and an infinite-order moving average of an independent identically distributed process, as well as the estimator's consistency for other types of data, such as non-linear functions of a Gaussian process. We find that the kernel density estimator at two different points, under certain conditions, is not only perfectly correlated but may converge to the same random variable. Also, central (and non-central) limit theorems of the non-parametric kernel regression estimator are studied.One important and surprising characteristic found is that its asymptotic variance does not depend on the point at which the regression function is estimated and also that its asymptotic properties are the same whether or not regressors are strongly dependent. Finally, a Monte Carlo experiment is reported to assess the behaviour of the estimators in finite samples.

[Hilton et al., 1996]

M. Hilton, Ogden T., D. Hattery, and G. Jawerth B. Eden. Wavelet denoising of functional MRI data. In [Aldroubi and Unser, 1996], pages 93-114.

[Hirchoren and DAttellis, 1998]

G. A. Hirchoren and C. E. DAttellis. Estimation of fractal signals using wavelets and filter banks. IEEE Transactions on Signal Processing, 46(6):1624-1630, 1998.

A filter bank design based on orthonormal wavelets and equipped with a multiscale Wiener filter mas recently proposed for signal restoration and for signal smoothing of 1/f family of fractal signals corrupted by external noise. The conclusions obtained in these papers are based on the following simplificative hypotheses: 1) The wavelet transformation is a whitening filter, and 2) the approximation term of the wavelet expansion can be avoided when the number of octaves in the multiresolution analysis is large enough. In this paper, we shelf that the estimation of 1/f processes in noise can be improved avoiding these two hypotheses. Explicit expressions of the mean-square error are given, and numerical comparisons with previous results are shown.

[Hirji, 1998]

Karim F. Hirji. Assessing fast Fourier transform algorithms. Computational Statistics & Data Analysis, 27:1-9, 1998.

[Holschneider et al., 1989]

M. Holschneider, R. Kronland-Martinet, J. Morlet, and Ph. Tchamitchian. A real-time algorithm for signal analysis with the help of the wavelet transform. In [Combes et al., 1989], pages 286-297. Proceedings of the International Converence, Marseille, France, December 14-18, 1987.

[Hondré, 1994]

C. Hondré. Wavelets, probability and statistics: some bridges. In [Benedetto and Frazier, 1994].

[Horne and Baliunas, 1986]

James H. Horne and Sallie L. Baliunas. A prescription for period analysis of unevenly sampled time series. The Astrophysical Journal, 302:757-763, 1986.

[Hosking, 1981]

J. R. M. Hosking. Fractional differencing. Biometrika, 68(1):165-176, 1981.

The family of autoregressive integrated moving-average processes, widely used in time series analysis, is generalized by permitting the degree of differencing to take fractional values. The fractional differencing operator is defined as an infinite binomial series expansion in powers of the backward-shift operator. Fractionally differenced processes exhibit long-term persistence and antipersistence; the dependence between observations a long time span apart decays much more slowly with time span than is the case with the more commonly studied time series models. Long-term persistent processes have applications in economics and hydrology; compared to existing models of long-term persistence, the family of models introduced here offers much greater flexibility in the simultaneous modelling of the short-term and long-term behaviour of a time series.

[Hosking, 1984]

J. R. M. Hosking. Modeling persistence in hydrological time series using fractional differencing. Water Resources Research, 20(12):1898-1908, 1984.

The class of autoregressive integrated moving average (ARIMA) time series models may be generalized by permitting the degree of differencing d to take fractional values. Models including fractional differencing are capable of representing persistent series (d>0) or short-memory series (d=0). The class of fractionally differences ARIMA processes provides a more flexible way than has hitherto been available of simultaneously modeling the long-term and short-term behavior of a time series. In this paper some fundamental properties of fractionally differenced ARIMA processes are presented. Methods of simulating these processes are described.

[Hosking, 1996]

J. R. M. Hosking. Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series. Journal of Econometrics, 73(1):261-284, 1996.

We derive the asymptotic distributions of the sample mean, autocovariances, and autocorrelations for a time series whose autocovariance function gamma(k) has the powerlaw decay gamma(k) similar to gamma k(-alpha), lambda > 0, 0 < alpha < 1, as k --> infinity. The results differ in important respects from the corresponding results for short-memory processes, whose autocovariance functions are absolutely summable. For long-memory processes the variances of the sample mean, and of the sample autocovariances and autocorrelations for 0 < alpha less than or equal to 1/2, are not of asymptotic order n(-1). When 0 < alpha < 1/2 the asymptotic distributions of the sample autocovariances and autocorrelations are not Normal.

[Howe and Percival, 1995]

David A. Howe and Donald B. Percival. Wavelet variance, Allan variance, and leakage. IEEE Transactions on Instrumentation and Measurement, 44(2):94-97, 1995.

Wavelets have recently been a subject of great interest in geophysics, mathematics and signal processing. The discrete wavelet transform can be used to decompose a time series with respect to a set of basis functions, each one of which is associated with a particular scale. The properties of a time series at different scales can then be summarized by the wavelet variance, which decomposes the variance of a time series on a scale by scale basis. The wavelet variance corresponding to some of the recently discovered wavelets can provide a more accurate conversion between the time and frequency domains than can be accomplished using the Allan variance. This increase in accuracy is due to the fat that these wavelet variances give better protection against leakage than does the Allan variance.

[Hsu et al., 1974]

Der-Ann Hsu, Robert B. Miller, and Dean W. Wichern. On the stable Paretian behavior of stock-market prices. Journal of the American Statistical Association, 69:108-113, 1974.

[Hsu, 1977]

Der-Ann Hsu. Tests for variance shift at an unknown time point. Applied Statistics, 26(3):279-284, 1977.

[Hsu, 1979]

Der-Aan Hsu. Detecting shifts of parameter in gamma sequences with applications to stock price and air traffic flow analysis. Journal of the American Statistical Association, 74:31-40, 1979.

[Huang and Cressie, 1997]

Hsin-Cheng Huang and Noel Cressie. Deterministic/stochastic wavelet decomposition for recovery of signal from noise data. Department of Statistics, Iowa State University, 1997.

[Hubbard, 1996]

Barbara Burke Hubbard. The World According to Wavelets: The Story of a Mathematical Technique in the Making. A K Peters, Wellesley, Massachusetts, 1996.

This book, lovingly written and highly accessible, embraces the often unheralded notion that mathematics contains ideas that can, and deserve, to be communicated to a wider public ­p; even if what is communicated is at the level of appreciation rather than practical knowledge. Put simply, it is a book about the wavelet transform, that strange and scientifically intriguing new method of encoding information with an abundance of practical applications. This book is a wonderfully successful attempt to entice the non-mathematical reader into formerly uncharted territory without sacrificing precision. The material is masterfully organized so mathematical details can be assimilated at one's own pace; the main text is devoid of formulas and relates a story of people and ideas, while separate boxes and appendices contain intricate discussions for the more mathematically adventurous. This book is a rarity in mathematics books in that it recognizes that both mathematicians and readers interested in mathematics have a human side.

[Hudgins et al., 1993]

Lonnie Hudgins, Carl. A. Friehe, and Meinhard E. Mayer. Wavelet transforms and atmospheric turbulence. Physical Review Letters, 71(20):3279-3282, 1993.

Wavelet cross spectra and cross scalograms are used to analyze the time-scale structure of bivariate turbulence data from the boundary layer over the ocean. The cross scalogram for the streamwise and vertical turbulent velocity components shows a highly intermittent pattern with significant contributions of opposite signs appearing at two specific scales, approximately 60 m and approximately 2 km, believed to be related to small- scale turbulent mixing and large-scale secondary flow in the boundary layer.

[Hudgins, 1992]

Lonnie H. Hudgins. Wavelet Analysis of Atmospheric Turbulence. PhD thesis, University of California, Irvine, 1992.

[Ibragimov and Rozanov, 1978]

I. A. Ibragimov and Yu. A. Rozanov. Gaussian Random Processes, volume 9 of Applications of Mathematics. Springer-Verlag, New York, 1978.

[Ibragimov, 1965]

I. A. Ibragimov. On the spectrum of stationary gaussian sequences satisfying the strong mixing condition I. necessary conditions. Theory of Probability and Its Applications, 10(1):85-106, 1965.

[Ibragimov, 1975]

I. A. Ibragimov. A note on the central limit theorem for dependent random variables. Theory of Probability and Its Applications, 20(1):135-141, 1975.

[Imhof, 1961]

J. P. Imhof. Computing the distribution of a quadratic form in normal variables. Biometrika, 48:419-426, 1961.

[Inclán and Tiao, 1994]

Carla Inclán and George C. Tiao. Use of cumulative sums of squares for retrospective detection of changes of variance. Journal of the American Statistical Association, 89(427):913-923, 1994.

[Inclán, 1992]

Carla Inclán. Effect of a change in variance on the Portmanteau statistic. Estadistica, 44(142):149-170, 1992.

[Inclán, 1993]

Carla Inclán. Detection of multiple changes of variance using posterieor odds. Journal of Business and Economic Statistics, 11(3):289-300, 1993.

[Isserlis, 1918]

L. Isserlis. On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika, 12:134-139, 1918.

[Istas, 1992]

Jacques Istas. Wavelet coefficients of a Gaussian process and applications. Annales de l'Institut Henri Poincare, Section B, Calcul des Probabilities et Statistique, 28:537-556, 1992. In French.

The author gives the relations between the covariance functions and the spectral densities of the approximation and the details of a Gaussian stationary process at different resolutions. He studies the rate of convergence of the square error between the process and its wavelet transform. Then he shows the convergence in distribution of the projection of the process to the original process. Finally, proposes a choice of the regularity of the wavelet in order to minimize the correlation between the approximation and the details.

[Jaruvsková and Antoch, 1994]

Daniela Jaruvsková and Jarom´ir Antoch. Detection of change in variance. In [Mandl and Huskova, 1994], pages 297-302.

[Jawerth and Sweldens, 1994]

B. Jawerth and Wim Sweldens. An overview of wavelet based multiresolution analyses. SIAM Rev., 36(3):377-412, 1994.

Wavelet-based multiresolution analysis helps in data compression, operator analysis and developing a periodic fast wavelet transform algorithm. The analysis requires definition of a multiresolution analysis and investigation of the method in which wavelets fit into the multiresolution analysis. The fitting process requires a consideration of the semiorthogonal, orthogonal and biorthogonal wavelets. The application process requires an understanding of the wavelets on an interval, wavelet packets, multidimensional waves and fast wavelet transforms.

[Jenkins and Watts, 1968]

Gwilym M. Jenkins and Donald G. Watts. Spectral Analysis and Its Applications. Holden-Day, San Francisco, 1968.

[Jensen, 1994]

Mark J. Jensen. Wavelet analysis of fractionally integrated processes. Technical Report ewp-em/9405001, Department of Economics, Washington University, 1994.

In this paper we apply wavelet analysis to the class of fractionally integrated processes to show that this class is a member of the 1/f family of processes as defined by Wornell (1993) and to produce an alternative method of estimating the fractional differencing parameter. Currently the method by Geweke and Porter-Hudak (1983) is used most often to estimate and test the fractional differencing parameter. The GPH approach, however, has been shown to have poor statistical properties and suffers from subjective decisions that the users must make. The wavelet analysis estimate of the fractional differencing parameter is shown to be more straightforward and to provide results that are more robust than the GPH method.

[Jensen, 1995]

Mark J. Jensen. OLS estimate of fractional differencing parameter using wavelets derived from smoothing kernels. Technical Report 95-12, Department of Economics, Southern Illinois University at Carbondale, 1995.

This paper develops a consistent OLS estimate of a fractionally integrated processes' differencing parameter, using continuous wavelet theory as constructed from smoothing kernels. We show that a log-log linear relationship exists between the variance of the wavelet coefficient and the level at which the fractionally integrated processes is smoothed. This linear relationship occurs because the self-simularity property of the fractionally integrated process and the self-similarity of the wavelet causes the smoothing level to continually appear in the wavelet transformation. Since the wavelet coefficient can be interpreted as the k-th order details of the series at some level of smoothing, we also show that the above log-log relationship can be derived from the variance of the 1-st order derivative of the time series smoothed by a kernel that is well localized in both time and frequency space. Lastly, we derive the asymptotic biasness and variance of the OLS estimate and test our consistent estimate with a number of Monte Carlo experiments.

[Jensen, 1998]

Mark J. Jensen. An approsimate wavelet MLE of short and long memory parameters. Department of Economics, University of Missouri, 1998.

[Johnson and Kotz, 1970]

Norman L. Johnson and Samuel Kotz. Continuous Univariate Distributions. Houghton Mifflin, New York, 1970.

[Johnson and Kotz, 1972]

Norman L. Johnson and Samuel Kotz. Continuous Multivariate Distributions. John Wiley & Sons, Inc., New York, 1972.

[Johnstone and Silverman, 1997]

I. M. Johnstone and B. W. Silverman. Wavelet threshold estimators for data with correlated noise. Journal of the Royal Statistical Society B, 59(2):319-351, 1997.

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the coefficients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean-squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable 'bench-mark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an 'oracle' that provides information that is not actually available in the data. It is shown that the level-dependent threshold estimator performs well relative to the bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short-and long-range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.

[Jones et al., 1996]

C. L. Jones, G. T. Lonergan, and D. E. Mainwaring. Wavelet packet computation of the Hurst exponent. Journal of Physics A, 29(10):2509-2527, 1996.

Wavelet packet analysis was used to measure the global scaling behaviour of homogeneous fractal signals from the slope of decay for discrete wavelet coefficients belonging to the adapted wavelet best basis. A new scaling function for the size distribution correlation between wavelet coefficient energy magnitude and position in a sorted vector listing is described in terms of a power law to estimate the Hurst exponent. Profile irregularity and long-range correlations in self-affine systems can be identified and indexed with the Hurst exponent, and synthetic one-dimensional fractional Brownian motion (fBm) type profiles are used to illustrate and test the proposed wavelet packet expansion. We also demonstrate an initial application to a biological problem concerning the spatial distribution of local enzyme concentration in fungal colonies which can be modelled as a self-affine trace or an `enzyme walk'. The robustness of the wavelet approach applied to this stochastic system is presented, and comparison is made between the wavelet packet method and the root-mean-square roughness and second-moment approaches for both examples. The wavelet packet method to estimate the global Hurst exponent appears to have similar accuracy compared with other methods, but its main advantage is the extensive choice of available analysing wavelet filter functions for characterizing periodic and oscillatory signals.

[Jones, 1980]

Richard H. Jones. Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics, 22(3):389-395, 1980.

[Jones, 1985]

Richard H. Jones. Time series analysis with unequally spaced data. In Edward James Hannan, Paruchuri R. Krishnaiah, and Malempati Madhusudana Rao, editors, Time Series in the Time Domain, volume 5 of Handbook of Statistics, pages 157-177. North Holland Press, 1985.

[Kadambe, 1992]

Shubha Kadambe. On the choice of a wavelet, and the energy and the phase distributions of the wavelet transform. In Time-Frequency and Time-Scale Analysis, pages 379-382, Victoria, B.C., Canada, 1992. IEEE Signal Processing Society.

[Kaiser, 1994]

Gerald Kaiser. A Friendly Guide to Wavelets. Springer-Verlag, New York, 1994.

This volume consists of two parts. The first part, forming chapters 1-8, is designed as a textbook for an introductory one-semester course on wavelet analysis and time-frequency analysis aimed at graduate students or advanced undergraduates in science and engineering. Each of the first eight chapters ends with a set of straightforward exercises designed to drive home the concepts just covered, and the graphics should further facilitate absorption. The second part, form-ing chapters 9-11, represents original research and is written in a more advanced style. This section can be used as a textbook for a second-semester course or, when combined with chapters 1 & 3, as a reference for an advanced research-level seminar.

[Kaiser, 1996a]

Gerald Kaiser. Physical wavelets and radar: A variational approach to remote sensing. IEEE Antennas and Propagation Magazine, 38(1):15-24, 1996.

Physical wavelets are acoustic or electromagnetic waves, resulting from the emission of a time signal by a localized acoustic or electromagnetic source moving along an arbitrary trajectory in space. Thus, they are localized solutions of the wave equation or Maxwell`s equations. Under suitable conditions, such wavelets can be used as ``basis'' functions, to construct general acoustic or electromagnetic waves. This gives a local alternative to the construction of such waves in terms of (nonlocal) plane waves, via Fourier transforms. We give a brief, self-contained introduction to physical wavelets, and apply them to remote sensing. We define the ambiguity functional, generalization of the radar and sonar ambiguity functions, which applies not only to wideband signals, but also to targets and radar platforms executing arbitrary nonlinear motions.

[Kaiser, 1996b]

Gerald Kaiser. Wavelet filtering in the scale domain. In [Szu, 1996], pages 51-54. 8-12 April 1996, Orlando, Florida.

It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform establishes a one-to-one correspondence between frequency filters (system or transfer functions) and scale filters, which are defined as multiplication operators in the scale domain, subject to the convergence of the defining integrals. Applications to the denoising of random signals are proposed. We argue that the present method is more suitable for removing the effects of atmospheric turbulence than the conventional procedures based on Fourier analysis because it is ideally suited for resolving spectral power laws.

[Kaiser, 1996c]

Gerald Kaiser. Wavelet filtering with the Mellin transform. Applied Mathematical Letters, 9(5):69-74, 1996.

It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform gives a one-to-one correspondence between frequency filters (system functions) and scale filters (multiplication operators in the scale domain), subject to the convergence of the defining integrals. Applications to the denoising of random signals are proposed. It is argued that the present method is more suitable for removing the effects of atmospheric turbulence than the conventional procedures because it is ideally suited for resolving spectral power laws.

[Kaplan and Kuo, 1993]

Lance M. Kaplan and C.-C. Jay Kuo. Fractal estimation from noisy data via discrete fractional gaussian noise (DFGN) and the haar basis. IEEE Transactions on Signal Processing, 41(12):3554-3562, 1993.

The authors show that the application of the discrete wavelet transform (DWT) using the Haar basis to the increments of fractional Brownian motion (fBm), also known as discrete fractional Gaussian noise (DFGN), yields coefficients which are weakly correlated and have a variance that is exponentially related to scale. Similar results were derived by Flandrin (1989), Tewfik, and Kim for a continuous-time fBm going through a continuous wavelet transform (CWT). The new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by Wornell and Oppenheim. The performance of the new algorithm is compared with that of Wornell and Oppenheim's (see IEEE Trans. Signal Processing, vol. 40, p. 611-623, Mar. 1992) algorithm in numerical simulation.

[Karl et al., 1996]

Thomas R. Karl, Philip D. Jones, and Richard W. Knight. Testing for bias in the climate record. Science, 271(5257):1879-1883, 1996.

The method climatologists use to calculate trends on monthly and annual time series do not introduce significant bias as has been suggested. Perihelion calender shifts were used to test for bias because they have no impact on annual mean temperature trends. Monthly differences were insignificant.

[Kawata and Arimoto, 1996]

Kouzou Kawata and Suguru Arimoto. Signal matching using wavelet correlation. Electronics and Communications in Japan 3, 79(9):23-34, 1996. Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. 78-A, No. 12, December 1995, pp. 1655-1664.

The problem of detecting corresponding points is studied in the case in which local deformations exist and a new method named ``wavelet correlation'' is proposed. There is a difficulty in that a reasonable window width cannot be designed in local correlation, which is one of the methods for a corresponding problem. The wavelet correlation is derived by extending the notion of local correlation and is considered to overcome difficulty. The fundamental concept is derived by the belief that a signal can be decomposed to several (sinusoidal) components and the window width can be varied according to each component. It is claimed that any algorithm using local correlation can be replaced by the one using wavelet correlation. In this paper, the wavelet correlation derived from local correlation is compared with the Laplacian distance and the local correlation itself by experiments. Further, a matching method that uses a narrow-band property of a wavelet correlation function is proposed and the matching error is evaluated through experiments using one-dimensional signals. Finally, an absolute measure of matching by using normalized wavelet correlation is introduced and applied for detecting discontinuities of local deformations.

[Kay, 1981]

S. M. Kay. Efficient generation of colored noise. Proceedings of the IEEE, 69(4):480-481, 1981.

A new technique is presented for efficiently generating colored noise. Instead of discarding initial samples to account for the transient the approach proposed here is to set the initial conditions of the filter so that the output process will be stationary. It is shown that the Levinson-Durbin algorithm provides an efficient means for determining these initial conditions.

[Keqin, 1993]

Xu Keqin. Two updated methods for impulse response function estimation. Mechanical Systems and Signal Processing, 7(5):451-460, 1993.

The impulse response function (IRF) is a complete description of the dynamic behaviour of a linear structure. The study of effectively identifying this function is still far from complete. The conventional method, i.e., the inverse discrete Fourier transform (IDFT) of the frequency response function (FRF) has been proved to be inaccurate. After investigation of some features of the FRF, two revised approaches for IRF estimation are advocated. One is called the imaginary part transform and the other compensatory amendment. The former exploits the energy concentration property of the imaginary part of the FRF and the feasibility of obtaining the IRF from it. The latter is a simple revision of the conventional method by taking the frequency truncation into consideration. A numerical example is provided to demonstrate that the two revised methods given produce much more accurate IRF estimations than the conventional solution.

[Kerkyacharian et al., 1996]

Gérard Kerkyacharian, Dominique Picard, and Karine Tribouley. Lp adaptive density estimation. Bernoulli, 2(3):229-247, 1996.

We provide global adaptive wavelet-type density estimates. Our procedures illustrate the refinement which can be obtained by replacing the Fourier basis by the wavelet basis in estimation methods. The main argument consists in observing that the estimated total energy of the details of a specified level j will be smaller or greater than some known threshold if precisely j is above or below the theoretical optimal level calculated with the a priori knowledge of the regularity of the density. This balancing effect leads directly to an adaptation procedure, and some natural extensions. We investigate the minimax properties of these procedures and explain their evolution for different global error measures.

[Kikkawa and Ishida, 1988]

S. Kikkawa and M. Ishida. Number of degrees of freedom, correlation times, and equivalent bandwidths of a random process. IEEE Transactions on Information Theory, 34(1):151-155, 1988.

New definitions for the number of degrees of freedom (NDF) of a stationary process are proposed and their general form derived for Gaussian processes. Correlation times and equivalent bandwidths, which have been important in random processes and some fields in physics, are deduced from the first-order and second-order NDF and studied.

[Kikkawa, 1994]

S. Kikkawa. Number of degrees of freedom, Fisher information, and frequency-time products of a random process. Electronics and Communications in Japan 3, 77(3):28-39, 1994.

The number of nth order degrees of freedom (nth order NDF) of stationary random process is defined in terms of the sample moment of order n of a finite length sample. Correlation times and equivalent bandwidths are derived from the NDFs. It is shown that the 2nd order NDF asymptotically approaches the degrees of freedom of a gamma distribution, by which the distribution of the sample variance is approximated. It is also shown for the case of a Gaussian process that the 1st order NDF is the same as the standardized Fisher information about the mean; and, furthermore, the 2nd order NDF of a Gaussian autoregressive process is the same as the standardized Fisher information about the variance. A problem with the Fisher information is that it cannot always be calculated. Since the NDF defined in this paper is based on sample moments, it can easily be calculated from the observed data. Further, useful features can be derived from the NDF, such as the correlation time and the equivalent bandwidth. Finally, for a Gaussian process, the significance of the approximation of the NDF by 2WT, where W denotes the equivalent bandwidth and T is the time duration, is discussed. It is shown, in particular, that the approximation error for the 2nd order NDF decreases only slowly, depending on the logarithm of 2WT. This result is interesting in examining the dimension of the signal space for each moment of a random process, since the situation is the same as in the case where the dimension of the signal space for the deterministic signal is approximated by 2WT.

[Kolaczyk, 1996a]

E. D. Kolaczyk. A wavelet shrinkage approach to tomographic image reconstruction. Journal of the American Statistical Association, 91(435):1079-1090, 1996.

A method is proposed for reconstructing images from tomographic data with respect to a two-dimensional wavelet basis. The Wavelet-vaguelette decomposition (WVD) is used as a framework within which expressions for the necessary wavelet coefficients may be derived. These coefficients are calculated using a version of the filtered back-projection algorithm as a computational tool, in a multiresolution fashion. The necessary filters are defined in terms of the underlying wavelets. Denoising is accomplished through an adaptation of the wavelet shrinkage (WS) approach of Donoho et al. and amounts to a form of regularization. Combining these two steps yields the proposed WVD/WS reconstruction algorithm, which is compared to the traditional filtered backprojection method in a small study.

[Kolaczyk, 1996b]

Eric D. Kolaczyk. An application of wavelet shrinkage to tomography. In [Aldroubi and Unser, 1996], pages 77-92.

[Kolaczyk, 1997a]

Eric D. Kolaczyk. Estimation of intensities of burst-like poisson processes using haar wavelets. Submitted to the Journal of the Royal Statistical Society, Series B, 1997.

I present a method for producing estimates of the intensity function of certain `burst-like' inhomogeneous Poisson processes, based on the shrinkage of Haar wavelet coefficients of the observed counts. The Haar basis is a natural wavelet basis in which to work in this context, and I derive thresholds for shrinkage estimation based on the distribution of the coefficients. The translation-invariant de-noising approach of Donoho and Coifman (1995) is used in conjunction with these level-dependent thresholds to yield smooth estimates, without the usual `staircase' structure associated with Haar wavelets. This work is motivated by recent efforts in astronomy to model the intensity functions underlying gamma-ray bursts. It is demonstrated that the method proposed herein (TIPSH) yields sharper estimates of the detail structure in these signals than those obtained through an analogous version of the standard adaptation of wavelet shrinkage for Poisson counts, based on the square-root transformation.

[Kolaczyk, 1997b]

Eric D. Kolaczyk. A method for wavelet shrinkage estimation of certain poisson intensity signals using corrected thresholds. To appear in Statistica Sinica, 1997.

Wavelet shrinkage estimation has been found to be a powerful tool for the non-parametric estimation of spatially variable phenomena. Most work in this area to date has concentrated primarily on the use of wavelet shrinkage techniques in contexts where the data are modeled as observations of a signal plus additive, Gaussian noise. When the data instead take the form of Poisson counts, a common procedure is to first pre-process the data using Anscombe's square root transformation, thereby normalizing the data and stabilizing the variance. However, this approach has a tendency to smooth away sharp, brief structure in the underlying intensity function, especially in situations involving very low levels of counts. In this paper, I introduce an alternative approach to estimating intensity functions for a certain class of `burst-like' Poisson processes using wavelet shrinkage. The proposed method is based on the shrinkage of wavelet coefficients of the original, un-transformed count data. `Corrected' versions of the usual Gaussian-based shrinkage thresholds are used. The corrections explicitly account for effects of the first few cumulants of the Poisson distribution on the tails of the coefficient distributions. A large deviations argument is used to justify these corrections. The performance of the new method is examined, and compared to that of the pre-processing approach, in the context of an application to an astronomical gamma-ray burst signal.

[Kolaczyk, 1997c]

Eric D. Kolaczyk. Non-parametric estimation of gamma-ray burst intensities using haar wavelets. The Astrophysical Journal, 483(1):340-349, 1997.

In this article, I present a method for the non-parametric (model-free) estimation of intensity profiles underlying gamma-ray bursts. The method, TIPSH, is based on applying specially calibrated thresholds to the Haar wavelet coefficients of binned counts gathered from such bursts. As functions well-localized with respect to both time and scale, wavelets are an ideal tool for working with the often sharp, abrupt nature of gamma-ray burst signals. When applied to an idealized signal in a small simulation study and a selection of actual gamma-ray bursts, the TIPSH algorithm is found to be well capable of simultaneously estimating the smooth, uniform background and the pulse-like structure of gamma-ray burst signals.

[Koopmans, 1974]

Lambert Herman Koopmans. The Spectral Analysis of Time Series. New York. Academic Press, 1974.

[Kotz et al., 1982]

Samuel Kotz, Norman L. Johnson, and Campbell B. Read, editors. Encyclopedia of Statistical Sciences. Wiley, New York, 1982.

[Krim and Pesquet, 1995]

H. Krim and J. C. Pesquet. Multiresolution analysis of a class of nonstationary processes. IEEE Transactions on Information Theory, 41(4):1010-1020, 1995.

Processing nonstationary signals is an important and challenging problem. We focus on the class of nonstationary processes with stationary increments of an arbitrary order, and place them in a multiscale framework. Unlike other related studies, we concentrate on the discrete-time analysis and derive a number of new results in addition to placing the related existing ones in the same framework. We extend the study to various parametric models for which we derive the resulting multiresolution description. We show that wide-sense stationarity may be achieved by adequately selecting the analysis wavelet. After generalizing the study to wavelet packet analysis, we show that the latter possesses additional properties which are useful in the presence of other types of nonstationarities.

[Krim et al., 1992]

H. Krim, K. Drouiche, and J. C. Pesquet. Multiscale detection of nonstationary signals. In Time-Frequency and Time-Scale Analysis, pages 105-108, Victoria, B.C., Canada, 1992. IEEE Signal Processing Society.

A statistical method for detecting and/or localizing nonstationarities in a process observed over a time interval T is presented. Stationarity is induced by taking a wavelet transform of the process. A parametric model is fitted to the result. The error incurred in fitting the model is shown to preserve the singularity manifested in the transform. The error is then used to establish a statistical detection test that is free of any prior knowledge about the class of signals being analyzed, and of any user input.

[Krim et al., 1994]

H. Krim, J. C. Pesquet, and A. S. Willsky. Robust multiscale representation of processes and optimal signal reconstruction. In Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, pages 1-4, 1994. 25-28 Oct. 1994, Philadelphia, PA, USA.

We propose a statistical approach to obtain a ``best basis'' representation of an observed random process. We derive statistical properties of a criterion first proposed to determine the best wavelet packet basis, and, proceed to use it in constructing a statistically sound algorithm. For signal enhancement, this best basis algorithm is followed by a nonlinear filter based on the minimum description length (MDL) criterion. We show that it is equivalent to a min-max based algorithm proposed by Donoho and Johnstone (1992).

[Krogstad, 1989]

Harald E. Krogstad. Simulation of multivariate gaussian time series. Communications in Statistics B, 18(3):929-941, 1989.

[Kuhnel, 1989]

Ivan Kuhnel. Spatial and temporal variations in Australo--Indonesian region cloudiness. International Journal of Climatology, 9(4):395-405, 1989.

[Kumar and Foufoula-Georgiou, 1993]

Praveen Kumar and Efi Foufoula-Georgiou. A new look at rainfall fluctuations and scaling properties of spatail rainfall using orthogonal wavelets. Journal of Applied Meteorology, 32:209-222, 1993.

Orthogonal wavelet transforms of the rainfall fields are analyzed. Results show that wavelet multiresolution analysis provides methods for the study of nonhomogeneous anisotropic processes and for defining fluctuations in two dimensions. Moreover, orthogonal wavelet transforms segregate large-scale features from small-scale features by providing convenient orthogonal decompositions with directionality. Lastly, orthogonal wavelet analysis is applied to a squall-line storm.

[Kumar and Foufoula-Georgiou, 1997]

Praveen Kumar and Efi Foufoula-Georgiou. Wavelet analysis for geophysical applications. Review of Geophysics, 35(4):385-412, 1997.

Wavelet transforms originated in geophysics in the early 1980s for the analysis of seismic signals. Since then, significant mathematical advances in wavelet theory have enabled a suite of applications in diverse fields. In geophysics the power of wavelets for analysis of nonstationary processes that contain multiscale features, detection of singularities, analysis of transient phenomena, fractal and multifractal processes, and signal compression is now being exploited for the study of several processes including space-time precipitation, remotely sensed hydrologic fluxes, atmospheric turbulence, canopy cover, land surface topography, seafloor bathymetry, and ocean wind waves. It is anticipated that in the near future, significant further advances in understanding and modeling geophysical processes will result from the use of wavelet analysis. In this paper we review the basic properties of wavelets that make them such an attractive and powerful tool for geophysical applications, We discuss continuous, discrete, orthogonal wavelets and wavelet packets and present applications to geophysical processes.

[Kumar, 1995]

Praveen Kumar. A wavelet based methodology for scale-space anisotropic analysis. Geophysical Research Letters, 22(20):2777-2780, 1995.

It is well known that several geophysical fields exhibit characteristic features at different scales. For some such fields scale-space anisotropy is also present, that is, features contributing a significant fraction of energy are oriented in different directions at different scales. Examples of such fields include clouds, rainfall, hurricanes etc. A technique based on wavelet transforms (with two-dimensional directionally oriented Morlet wavelet) is developed to analyze such random fields. This methodology has significant advantage over Fourier transform based techniques and is demonstrated using the analysis of a spatial rainfall field.

[Kumar, 1996]

Praveen Kumar. Role of coherent structures in the stochastic-dynamic variability of precipitation. Journal of Geophysical Research-Atmospheres, 101(D21):26393-26404, 1996.

Using time-frequency-scale elements obtained from wavelet packets as a basis, we describe a broad framework of analysis which can be used to reveal the essential dynamics, identified as coherent structures, of precipitation. We show that the matching pursuits algorithm with nearly symmetric orthogonal wavelets provides an optimal representation of the inner structure of rainfall time series and can describe features that range from scales of isolated singularity to synoptically forced large-scale features. We describe the analysis of time series of several storms and show that there exist distinct scales of variation identifiable with rain cell and synoptic-scale activity, which is in contradistinction to the scale invariance hypothesis.

[L. et al., 1997]

Starck J. L., Siebenmorgen R., and Gredel R. Spectral analysis using the wavelet transform. The Astrophysical Journal, 482(2):1011-1020, 1997.

We introduce a new signal processing technique to analyze noisy spectra. The method is based on the wavelet transform and employs the a trous algorithm. Noise determination and detection criteria are discussed in detail, together with pitfalls related to the use of wavelets in the analysis of spectra. Simulations are presented to demonstrate the power and the shortcomings of our method. We apply our technique to the case of continuum sources that show superposed interstellar or circumstellar absorption or emission bands that are shallow and broad. In particular, we analyze an L-band spectrum of the Herbig-Haro energy source HH 100 IRS. The analysis indicates the presence of a shallow emission band near 3.51 mu m that is tentatively assigned to arise from aliphatic (CH2) vibrations.

[Laine and Unser, 1994]

Andrew F. Laine and Michael A. Unser, editors. Wavelet applications in signal and image processing II, volume 2303 of Proceedings of SPIE, 1994. 24-29 July, 1994, San Diego, California.

[Laine et al., 1995]

Andrew F. Laine, Michael A. Unser, and Mladen V. Wickerhauser, editors. Wavelet applications in signal and image processing III, volume 2569 of Proceedings of SPIE, 1995. 12-14 July, 1995, San Diego, California.

[Laine, 1993]

Andrew F. Laine, editor. Mathematical Imaging: Wavelet Applications in Signal and Image Processing, volume 2034 of Proceedings of the SPIE, 1993. 11-16 July, 1993, San Diego, California.

[Lamoureux and Lastrapes, 1990]

Christopher G. Lamoureux and William D. Lastrapes. Persistence in variance, structural change, and the GARCH model. Journal of Business and Economic Statistics, 8(2):225-234, 1990.

[Lang et al., 1995]

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells. Nonlinear processing of a shift invariant DWT for noise reduction. In [Szu, 1995], pages 640-651. 17-21, April 1994, Orlando, Florida.

A novel approach for noise reduction is presented. Similar to Donoho, we employ thresholding in some wavelet transform domain but use a nondecimated and consequently redundant wavelet transform instead of the usual orthogonal one. Another difference is the shift invariance as opposed to the traditional orthogonal wavelet transform. We show that this new approach can be interpreted as a repeated application of Donoho`s original method. The main feature is, however, a dramatically improved noise reduction compared to Donoho`s approach, both in terms of the l/sub 2/ error and visually, for a large class of signals. This is shown by theoretical and experimental results, including synthetic aperture radar (SAR) images.

[Lang et al., 1996]

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells. Noise reduction using an undecimated discrete wavelet transform. IEEE Signal Processing Letters, 3(1):10-12, 1996.

A new nonlinear noise reduction method is presented that uses the discrete wavelet transform. Similar to Donoho (1995) and Donoho and Johnstone (1994, 1995), the authors employ thresholding in the wavelet transform domain but, following a suggestion by Coifman, they use an undecimated, shift-invariant, nonorthogonal wavelet transform instead of the usual orthogonal one. This new approach can be interpreted as a repeated application of the original Donoho and Johnstone method for different shifts. The main feature of the new algorithm is a significantly improved noise reduction compared to the original wavelet based approach. This holds for a large class of signals, both visually and in the l/sub 2/ sense, and is shown theoretically as well as by experimental results.

[Lari and Zakhor, 1992]

Francesco Lari and Avideh Zakhor. Automatic classification of active sonar data using time-frequency transforms. In Time-Frequency and Time-Scale Analysis, pages 21-24, Victoria, B.C., Canada, 1992. IEEE Signal Processing Society.

Automatic classification of active sonar signals using the Wigner-Ville transform (WVT), the wavelet transform (WT) and the scalogram is addressed. Features are extracted by integrating over regions in the time-frequency (TF) distribution, and are classified by a decision tree. Experimental results show classification and detection rates of up to 92% at -4 dB of SNR. The WT outperforms the WVT and the scalogram, particularly at high noise levels. This can be partially attributed to the absence of cross terms in the WT.

[Lau and Weng, 1995]

K. M. Lau and Hengyi Weng. Climate signal detection using wavelet transform: How to make a time series sing. Bulletin of the American Meteorological Society, 76(12):23-41, 1995.

In this paper, the application of the wavelet transform (WT) to climate time series analyses is introduced. A tutorial description of the basic concept of WT, compared with similar concepts used in music, is also provided. Using an analogy between WT representation of a time series and a music score, the authors illustrate the importance of local versus global information in the time-frequency localization of climate signals. Examples of WT applied to climate data analysis are demonstrated using analytic signals as well as real climate time series. Results of WT applied to two climate time series - that is, a proxy paleoclimate time series with a 2.5-Myr deep-sea sediment record of [[Delta].sup.18]O and a 140-yr monthly record of Northern Hemisphere surface temperature - are presented. The former shows the presence of a 40-kyr and a 100-kyr oscillation and an abrupt transition in the oscillation regime at 0.7 Myr before the present, consistent with previous studies. The latter possesses a myriad of oscillatory modes from interannual (2-5 yr), interdecadal (10-12 yr, 20-25 yr, and 40-60 yr), and century ([approximately]180 yr) scales at different periods of the data record. In spite of the large difference in timescales, common features in time-frequency characteristics of these two time series have been identified. These features suggest that the variations of the earth's climate are consistent with those exhibited by a nonlinear dynamical system under external forcings.

[Lawrence and Kottegoda, 1977]

A. J. Lawrence and N. T. Kottegoda. Stochastic modelling of riverflow time series. Journal of the Royal Statistical Society A, 140(1):1-47, 1977.

[Lebrun and Vetterli, 1998]

Jér^ome Lebrun and Martin Vetterli. Balanced multiwavelets theory and design. IEEE Transactions on Signal Processing, 46(4):1119-1125, 1998.

This article deals with multiwavelets, which are a generalization of wavelets in the context of time-varying filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multichannel signals. This is the main issue in which we are interested. First, we review material on multiwavelets and their links with multifilter banks and, especially, time-varying filter banks. Then, we have a close look at the problems encountered when using multiwavelets in applications, and we propose new solutions for the design of multiwavelets filter banks by introducing the so-called balanced multiwavelets

[Leduc et al., 1997]

Jean-Pierre Leduc, Fernando Mujica, Romain Murenzi, and Mark Smith. Spatio-temporal wavelet transforms for motion tracking. Georgia Institute of Technology, 1997.

[Leduc, 1997]

Jean-Pierre Leduc. Spatio-temporal wavelet transforms for digital signal analysis. Signal Processing, 60(1):23-41, 1997.

The goal of this paper is to investigate spatio-temporal continuous wavelet transforms. A new wavelet family called the Galilean wavelet has been designed to tune to four main parameters namely the scale, the spatio-temporal position, the spatial orientation, and the velocity. The paper starts with the theory of motion-compensated wavelet filtering in the discrete realm of image processing. As a major difference to multi-dimensional homogeneous spaces, the spatio-temporal signals involve motions that warp the signal along the temporal dimension. Modeling motion with 2-D affine transformations leads to spatio-temporal generalizations. Decomposition in to elementary operators lead to developing transformation groups and exploiting the related representation theory. The construction of continuous spatio-temporal wavelets in R^n times R spaces is then handled with classical techniques of calculation. Close connections may then be established among all the spatio-temp oral wavelet transforms through different sets of transformations. This approachgenerates a general framework for the study of future tools. Frames of wavelets are thereafter investigated to revisit discrete wavelet transforms in a more general way. Eventually illustrations demonstrate the ability of the Galilean wavelet transforms to analyze spatio-temporal contents.

[Lee et al., 1996]

GeungHee Lee, Jeffrey D. Hart, and F. Michael Speed. Automated smoothing of tides data using wavelets. Technical Report 268, Department of Statistics, Texas A&M University, 1996.

[Lees and Park, 1995]

Jonathan M. Lees and Jeffrey Park. Multiple-taper spectral analysis: A stand-alone C-subroutine. Computers &amp; Geosciences, 21(2):199-236, 1995.

A simple set of subroutines in ANSI-C are presented for multiple taper spectrum estimation. The multitaper approach provides an optimal spectrum estimate by minimizing spectral leakage while reducing the variance of the estimate by averaging orthogonal eigenspectrum estimates. The orthogonal tapers are Slepian n pi prolate functions used as tapers on the windowed time series. Because the taper functions are orthogonal, combining them to achieve an average spectrum does not introduce spurious correlations as standard smoothed single-taper estimates do. Furthermore, estimates of the degrees of freedom and F-test values at each frequency provide diagnostics for determining levels of confidence in narrow band (single frequency) periodicities. The program provided is portable and has been tested on both Unix and Macintosh systems.

[Lehmann, 1983]

E. L. Lehmann. Theory of Point Estimation. Wiley, New York, 1983.

[Lehmann, 1986]

E. L. Lehmann. Testing Statistical Hypotheses. Wiley, New York, 2 edition, 1986.

[Leipus and Viano, 199]

Remigijus Leipus and Marie-Claude Viano. Modelling long-memory time series with finite or infinite variance: A general approach. Department of Mathematics, Vilnius University, 199?

[Leipus, 1994]

R. Leipus. A posteriori and sequential methods of change-point detection in FARIMA-type time series. In B. Grigelionis, J. Kubilius, H. Pragarauskas, and V. Statulevicius, editors, Probability Theory and Mathematical Statistics, pages 485-496, Netherlands, 1994. VSP. Proceedings of the 6th Vilnius Conference, Vilnius, Lithuania.

[Li and Nozaki, 1997]

Hui Li and Tsutomu Nozaki. Application of wavelet cross-correlation analysis to a plane turbulent jet. Japanese Society of Mechanical Engineers International Journal, Series B, 40(1):58-66, 1997.

A new cross-correlation method, which is called wavelet cross-correlation analysis and is used to express the statistical cross-correlation of two arbitrary signals in terms of scale and time delay, is proposed and its main properties are presented, analyzing two test signals, it is shown that wavelet cross-correlation does not have the limitations of classical cross-correlation. As a practical application to fluid mechanics, wavelet cross-correlation is employed to determine the cross-correlation relationships between the x-components of the fluctuation velocities at two points on opposite sides of the centerline and along the centerline of a plane turbulent jet in terms of period and time delay. In the distributions of the wavelet cross-correlation coefficients, similar structures with various scales are observed instantaneously, and the period of eddy and apparent flapping motions can be determined easily in terms of period and time delay. It is found that the apparent flapping behavior appears first in region with an intermediate period. It is also revealed that a similar structure with a high period consists of similar structures with a low period, i.e., a large eddy contains small eddies.

[Liang and Parks, 1994]

Jie Liang and Thomas W. Parks. A two-dimensional translation invariant wavelet representation and its applications. In Proceedings ICIP-94, volume 1, pages 66-70, 1994.

Addresses the problem of the sensitivity of wavelet representations to translations for two-dimensional signals. The authors describe a fast algorithm to calculate the two-dimensional wavelet transforms for all the circular translates of an input image. They select the optimal translate for the decomposition using a quadtree search algorithm. The resulted wavelet representation is invariant under translations measured by an additive cost criterion. The complexity of the whole algorithm is O(N/sup 2/ log N) for a N*N input block. They apply this translation invariant wavelet transform to data compression. The results show that by taking into account the effect of translations, additional compression can be achieved beyond that achieved by a standard wavelet transform.

[Liang and Parks, 1996]

Jie Liang and Thomas W. Parks. A translation-invariant wavelet representation algorithm with applications. IEEE Transactions on Signal Processing, 44(2):225-232, 1996.

We address the time-varying problem of wavelet transforms, and a new translation-invariant wavelet representation algorithm is proposed. Using the algorithm introduced by Beylkin (see SIAM J. Numer. Anal., vol. 29, p.1716-1740, 1992), we compute the wavelet transform for all the circular time shifts of a length- N signal in O(N log N) operations. The wavelet coefficients of the time shift with minimal cost are selected as the best representation of the signal using a binary tree search algorithm with an appropriate cost function. We apply the translation-invariant representation algorithm to a geoacoustic data compression application. The results show that the new algorithm can reduce the distortion (the squared error in our case) substantially, if the input signals are transients that are sensitive to time shifts.

[Liang and Parks, 1998]

Jie Liang and Thomas W. Parks. Image coding using translation invariant wavelet transforms with symmetric extensions. IEEE Transactions on Image Processing, 7(5):762-769, 1998.

In this correspondence, we address the problem of translation sensitivity of conventional wavelet transforms for two-dimensional (2-D) signals. We propose wavelet transform algorithms that achieve the following desirable properties simultaneously: i) translation invariance, ii) reduced edge effects, and iii) sice-limitedness. We apply this translation invariant biorthogonal wavelet transform with symmetric extensions to image coding applications with good results.

[Liang et al., 1996]

Kai-Chieh Liang, Jin Li, and C.-C. Jay Kuo. Image compression with embedded multiwavelet coding. In [Szu, 1996]. 8-12 April 1996, Orlando, Florida.

[Liebmann and Hendon, 1990]

Brant Liebmann and Harry H. Hendon. Synoptic-scale disturbances near the equator. Journal of Atmospheric Science, 47(12):1463-1479, 1990.

[Lilly and Park, 1995]

J. M. Lilly and J. Park. Multiwavelet spectral and polarization analyses of seismic records. Geophysical Journal International, 122(3):1001-1021, 1995.

Presents an algorithm, based on the wavelet transform and multiple taper spectral analysis, for providing a low-variance spectrum estimate of a non-stationary data process. The `multiwavelet' algorithm uses, within each frequency band, a number of mutually orthogonal Slepian wavelets, optimally concentrated in frequency. The sum of squared wavelet transforms with the Slepian wavelets results in a spectrum estimate that is both low-variance and resistant to broad-band bias. The multiwavelet algorithm is used to estimate the time-varying spectral density matrix S(f,t) for two or more time series, in particular for three-component seismic data. Coherent three-component motion is described by motion along a single trajectory, with appropriate projections onto the three component axes. This trajectory is found by applying a singular value decomposition (SVD) to a matrix M(f,t) of wavelet transform values. The normalized first singular value of the SVD determines whether a correlation among the three components of the seismogram is statistically significant. Where significant, coherent particle motion is reconstructed by a linear combination of the wavelets with coefficients specified by the first left-singular vector. The polarization of this motion with respect to the coordinate axes is given by the first right-singular vector. Where the wavelets are real-valued, the usefulness of this method is limited to cases in which the three components of the seismic record oscillate in phase with each other, as is often the case for seismic body waves. Elliptical polarization is handled by pairing even and odd Slepian wavelets into complex-valued wavelets, capable of detecting phase shifts between components. The authors demonstrate the multiwavelet spectrum and polarization estimators on seismic data from a large shallow earthquake in the Solomon Islands, and from the deep earthquakes beneath Fiji (1994 March 9) and Bolivia (1994 June 9).

[Lindsay et al., 1996]

Ronald W. Lindsay, Donald B. Percival, and D. Andrew Rothrock. The discrete wavelet transform and the scale analysis of the surface properties of sea ice. IEEE Transactions on Geoscience and Remote Sensing, 34(3):771-787, 1996.

The formalism of the one-dimensional discrete wavelet transform (DWT) based on Daubechies wavelet filters is outlined in terms of finite vectors and matrices. Both the scale-dependent wavelet variance and wavelet covariance are considered and confidence intervals for each are determined. The variance estimates are more accurately determined with a maximal-overlap version of the wavelet transform. The properties of several Daubechies wavelet filters and the associated basis vectors are discussed. Both the Mallat orthogonal-pyramid algorithm for determining the DWT and a pyramid algorithm for determining the maximal-overlap version of the transform are presented in terms of finite vectors. As an example, the authors investigate the scales of variability of the surface temperature and albedo of spring pack ice in the Beaufort Sea. The data analyzed are from individual lines of a Landsat TM image (25-m sample interval) and include both reflective (channel 3, 30-m resolution) and thermal (channel 6, 120-m resolution) data. The wavelet variance and covariance estimates are presented and more than half of the variance is accounted for by scales of less than 800 m. A wavelet-based technique for enhancing the lower-resolution thermal data using the reflected data is introduced. The simulated effects of poor instrument resolution on the estimated lead number density and the mean lead width are investigated using a wavelet-based smooth of the observations.

[Lindsay, 1998]

Ronald W. Lindsay. Temporal variability of the energy balance of thick arctic pack ice. Journal of Climate, 11(3):313-333, 1998.

The temporal variability of the six terms of the energy balance equation for a slab of ice 3 m thick is calculated based on 45 yr of surface meteorological observations from the drifting ice stations of the former Soviet Union. The equation includes net radiation, sensible heat flux, latent heat flux, bottom heat flux, heat storage, and energy available for melting. The energy balance is determined with a time-dependent 10-layer thermodynamic model of the ice slab that determines the surface temperature and the ice temperature profile using 3-h forcing values. The observations used for the forcing values are the 2-m air temperature, relative humidity and wind speed, the cloud fraction, the snow depth and density, and the albedo of the nonponded ice. The downwelling radiative fluxes are estimated with parameterizations based on the cloud cover, the air temperature and humidity, and the solar angle. The linear relationship between the air temperature and both the cloud fraction and the wind speed is also determined for each month of the year. The annual cycles of the mean values of the terms of the energy balance equation are all nearly equal to those calculated by others based on mean climatological forcing values. The short-term variability, from 3 h to 16 days, of both the forcings and the fluxes, is investigated on a seasonal basis with the discreet wavelet transform. Significant diurnal cycles are found in the net radiation, storage, and melt, but not in the sensible or latent heat fluxes. The total annual ice-melt averages 0.67 m, ranges between 0.29 and 1.09 m, and exhibits large variations from year to year. It is closely correlated with the albedo and, to a lesser extent, with the latitude and the length of the melt season.

[Liu, 1994]

Paul C. Liu. Wavelet spectrum analysis and ocean wind waves. In [Foufoula-Georgiou and Kumar, 1994], pages 151-166.

[Ljung and Box, 1978]

G. M. Ljung and G. E. P. Box. On a measure of lack of fit in time series models. Biometrika, 65(2):297-304, 1978.

[Lo, 1991]

Andrew W. Lo. Long-term memory in stock market prices. Econometrica, 59(5):1279-1313, 1991.

[Lobato and Savin, 1998]

I. N. Lobato and N. E. Savin. Real and spurious long-memory properties of stock-market data. Journal of Business and Economic Statistics, 16(3), 1998.

We test for the presence of long memory in daily stock returns and their squares using a robust semiparametric procedure. Spurious results can be produced by nonstationarity and aggregation. We address these problems by analyzing subperiods of returns and using individual stocks. The test results show no evidence of long memory in the returns. By contrast, there is strong evidence in the squared returns.

[Lobato, 1997]

Ignacio N. Lobato. Consistency of the averaged cross-periodogram in long memory series. Journal of Time Series Analysis, 18(2):137-155, 1997.

Several aspects of inference with long memory series in a multivariate framework are examined. The main result of this paper is to prove the consistency of the averaged cross-periodogram evaluated in a degenerating neighbourhood of zero frequency. We also illustrate several applications of that result and consider some specification issues.

[Ma et al., 1997]

Yanyuan Ma, Gilbert Strang, and Brani Vidakovic. The first moment of wavelet random variables. Technical Report 97-10, Institute of Statistics and Decision Sciences, Duke University, 1997.

[Madden and Julian, 1971]

Roland A. Madden and Paul R. Julian. Detection of a 40-50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Science, 28:702-708, 1971.

[Madden and Julian, 1972]

Roland A. Madden and Paul R. Julian. Description of global-scale circulation cells in the tropics with a 40-50 day period. Journal of Atmospheric Science, 29:1109-1123, 1972.

[Madden and Julian, 1994]

Roland A. Madden and Paul R. Julian. Observations of the 40-50 day tropical oscillation: A review. Monthly Weather Review, 122(5):814-837, 1994.

[Madden, 1986]

Roland A. Madden. Seasonal variation of the 40-50 day oscillation in the tropics. Journal of Atmospheric Science, 43(24):3138-3158, 1986.

Daily rawinsonde data from 19 near-equatorial stations are examined to learn more about annual variations of the 40-50 day oscillations. Lengths of the available time series range from 5 to 28 years. A technique is devised to isolate spectral and cross-spectral quantities as a function of season. It is determined that a variance of the zonal wind in a relatively broad band centered on 47-day periods generally exceeds that in adjacent lower and higher frequency bands by the largest amount during December, January and February (DJF) and at stations in the Indian and western Pacific Oceans during all seasons. The coherence between lower- and upper-tropospheric zonal winds tends to be largest in the summer hemisphere for stations located in the Indian and western Pacific Oceans. Upper tropospheric zonal and meridional winds are coherent and out of (in) phase at several stations there during DJF (June, July and August (JJA)). These results, coupled with composited wind and outgoing longwave radiation data, lead the authors to conclude that in the Indian and western Pacific Oceans the eastward- moving regions of enhanced convection associated with the 40-50 day oscillation force a Kelvin-like wave to the east and anticyclonic, Rossby-like waves to the west.

[Maejima, 1989]

Makoto Maejima. Self-similar processes and limit theorems. Sugaku Expositions, 2(1):103-123, 1989.

[Mahrt, 1991]

L. Mahrt. Eddy asymmetry in the sheared heated boundary layer. Journal of Atmospheric Science, 48(3):472-482, 1991.

Statistical measures are developed to study the influence of mean shear on the asymmetry of eddy updrafts as observed from low-level aircraft flights in HAPEX, FIFE, and SESAME. This asymmetry involves formation of microfronts between updrafts with slow horizontal motion and downdrafts with faster horizontal motion. The variance of the Haar-wavelet transform (step-function basis) is found to be a superior indicator of the dominant scales of such eddies compared to the structure function. For those analyses where scale dependence is not of interest, the simpler structure function is applied. The coherent structures at the dominant scale are examined by computing eigenvectors of the lagged correlation matrix based on conditionally sampled events.

[Mak, 1995]

Mankin Mak. Orthogonal wavelet analysis: Interannual variability in the sea surface temperature. Bulletin of the American Meteorological Society, 76:2179-2186, 1995.

The unique capability of orthogonal wavelets, which have attractive time-frequency localization properties as exemplified by the Meyer wavelet, is demonstrated in a diagnosis of the interannual variability using a 44-year dataset of the sea surface temperature (SST). This wavelet analysis is performed in conjunction with an empirical orthogonal function analysis and a Fourier analysis to illustrate their complementary capability. The focus of this article is on the equatorial Pacific SST, which is known to have far-reaching impacts on short-term climate variability. The Meyer spectrum brings to light intriguing episodic characteristics of three separate sequences of El Niño (abnormally warm) and La Niña (abnormally cold) events during the past 42 years. It quantifies the relative contributions to the variability associated with different frequency ranges at different times. Through a wavelet cross-spectral analysis with the SST at an equatorial location and at a midlatitude location in the Pacific Ocean, the planetary character of the SST field associated with such events is also illustrated.

[Mallat and Hwang, 1992]

S. G. Mallat and W. L. Hwang. Singularity detection and processing with wavelets. IEEE Transactions on Information Theory, 38(2):617-643, 1992.

The mathematical characterization of singularities with Lipschitz exponents is reviewed. Theorems that estimate local Lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed. It is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately. The local frequency of such oscillations is measured from the wavelet transform modulus maxima. It has been shown numerically that one- and two-dimensional signals can be reconstructed, with a good approximation, from the local maxima of their wavelet transform modulus. As an application, an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales. In two dimensions, the wavelet transform maxima indicate the location of edges in images.

[Mallat and Zhong, 1992]

S. Mallat and S. Zhong. Characterization of signals from multiscale edges. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(7):710-732, 1992.

A multiscale Canny edge detection is equivalent to finding the local maxima of a wavelet transform. The authors study the properties of multiscale edges through the wavelet theory. For pattern recognition, one often needs to discriminate different types of edges. They show that the evolution of wavelet local maxima across scales characterize the local shape of irregular structures. Numerical descriptors of edge types are derived. The completeness of a multiscale edge representation is also studied. The authors describe an algorithm that reconstructs a close approximation of 1-D and 2-D signals from their multiscale edges. For images, the reconstruction errors are below visual sensitivity. As an application, a compact image coding algorithm that selects important edges and compresses the image data by factors over 30 has been implemented.

[Mallat, 1989]

Stéphane G. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674-693, 1989.

Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed.

[Mallat, 1991]

S. G. Mallat. Zero-crossings of a wavelet transform. IEEE Transactions on Information Theory, 37(4):1019-1033, 1991.

The completeness, stability, and application to pattern recognition of a multiscale representation based on zero-crossings is discussed. An alternative projection algorithm is described that reconstructs a signal from a zero-crossing representation, which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero-crossings. The reconstruction algorithm has a fast convergence and each iteration requires O(N log/sup 2/ (N)) computation for a signal of N samples. The zero-crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems. As an example, the implementation and results of a coarse-to-fine stereo-matching algorithm are described.

[Mallat, 1996]

S. Mallat. Wavelets for a vision. Proceedings of the IEEE, 84(4):604-614, 1996.

Early on, computer vision researchers have realized that multiscale transforms are important to analyze the information content of images. The wavelet theory gives a stable mathematical foundation to understand the properties of such multiscale algorithms. This tutorial describes major applications to multiresolution search, multiscale edge detection, and texture discrimination.

[Mandelbrot and van Ness, 1968]

Benoit B. Mandelbrot and John W. van Ness. Fractional Brownian motions, fractional noises and applications. SIAM Review, 10(4):422-437, 1968.

[Mandl and Huskova, 1994]

Petr Mandl and Marie Huskova, editors. Asymptotic Statistics: Proceedings of the fifth Prague Symposium, Contributions to Statistics, Heidelberg, 1994. Physica-Verlag.

[Mann and Lees, 1996]

Michael E. Mann and Jonathan M. Lees. Robust estimation of background noise and signal detection in climatic time series. Climate Change, 33:409-445, 1996.

We present a new technique for isolating climate signals in time series with a characteristic 'red' noise background which arises from temporal persistence. This background is estimated by a 'robust' procedure that, unlike conventional techniques, is largely unbiased by the presence of signals immersed in the noise. Making use of multiple-taper spectral analysis methods, the technique further provides for a distinction between purely harmonic (periodic) signals, and broader-band ('quasiperiodic') signals. The effectiveness of our signal detection procedure is demonstrated with synthetic examples that simulate a variety of possible periodic and quasiperiodic signals immersed in red noise. We apply our methodology to historical climate and paleoclimate time series examples. Analysis of a approximate to 3 million year sediment core reveals significant periodic components at known astronomical forcing periodicities and a significant quasiperiodic 100 year peak. Analysis of a roughly 1500 year tree-ring reconstruction of Scandinavian summer temperatures suggests significant quasiperiodic signals on a near-century timescale, an interdecadal 16-18 year timescale, within the interannual El Ninio/Southem Oscillation (ENSO) band, and on a quasibiennial timescale. Analysis of the 144 year record of Great Salt Lake monthly volume change reveals a significant broad band of significant interdecadal variability, ENSO-timescale peaks, an annual cycle and its harmonics. Focusing in detail on the historical estimated global-average surface temperature record, we find a highly significant secular trend relative to the estimated red noise background, and weakly significant quasiperiodic signals within the ENSO band. Decadal and quasibiennial signals are marginally significant in this series.

[Mann and Wald, 1943]

H. B. Mann and A. Wald. On stochastic limit and order relationships. The Annals of Mathematical Statistics, 14:217-226, 1943.

[Marron et al., 1996]

S. J. Marron, S. Adak, Iain Johnstone, Michael H. Neumann, and P. Patil. Exact risk analysis of wavelet regression. To appear in Journal of Computational and Graphical Statistics, 1996.

[Masry, 1991]

Elias Masry. Flicker noise and the estimation of the allan variance. IEEE Transactions on Information Theory, 37(4):1173-1177, 1991.

Flicker noise is a random process observed in a variety of contexts, including current fluctuations in metal film and semiconductor devices, loudness fluctuations in speech and music, and neurological patterns. The quadratic-mean convergence of appropriate estimates of the Allan variance for flicker noise is established when the latter is modeled as a stochastic process with stationary increments. A precise asymptotic expression of the mean-square error is given along with the rate of convergence.

[Masry, 1993]

Elias Masry. The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion. IEEE Transactions on Information Theory, 39(1):260-264, 1993.

The wavelet transform of random processes with wide-sense stationary increments is shown to be a wide-sense stationary process whose correlation function and spectral distribution are determined. The second-order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained. Applications to the spectral analysis and to the synthesis of fractional Brownian motion are given.

[Masry, 1996]

Elias Masry. Convergence properties of wavelet series expansions of fractional Brownian motion. Applied and Computational Harmonic Analysis, 3(3):239-253, 1996.

We consider the approximation of a fractional Brownian motion by a wavelet series expansion at resolution 2^-l. The approximation error is measured in the integrated mean squared sense over finite intervals and we obtain its expansion as a sum of terms with increasing rates of convergence. The dependence of the coefficients in the expansion of the error on the scale function is explicitly determined.

[McCoy and Walden, 1996]

Emma J. McCoy and Andrew T. Walden. Wavelet analysis and synthesis of stationary long-memory processes. Journal of Computational and Graphical Statistics, 5(1):26-56, 1996.

The discrete wavelet transform (DWT) can be interpreted as a filtering of a time series by a set of octave band filters such that the width of each band as a proportion of its center frequency is constant. A long-memory process having a power spectrum that plots as a straight line on log-frequency/log-power scales over many octaves of frequency is intrinsically related to such a structure. As an example of such processes, we focus on one class of discrete-time, stationary, long-memory processes, the fractionally differenced Gaussian white noise processes (fdGn). We show how the DWT breaks down a fdGn, and show the exact correlation structure of the resulting coefficients for different wavelets (Daubechies' minimum-phase and least-asymmetric and Haar). The DWT is an impressive ``whitening filter.'' A discrete wavelet-based scheme for simulating fdGn's is discussed and is shown to be equivalent to a spectral decomposition of the covariance matrix of the process; however, it can be carried out using only information on the nature of the spectrum of the process --- that is, time-domain information is not required. It produces results comparable with theexact Hosking method. We then show that, using wavelet methods, the spectral slope parameter d can be estimated as well, or better, than when using the best Fourier-based method known to us, namely regression on multitaper spectral ordinates. Since wavelet analysis and synthesis methods can be applied to a much wider variety of empirical or theoretical long-memory processes, wavelet methods could prove a valuable tool in the future in the analysis and synthesis of stochastic processes.

[McCoy et al., 1995]

Emma J. McCoy, Donald B. Percival, and Andrew T. Walden. On the phase of least-asymmetric scaling and wavelet filters. Technical Report TR-95-15, Dept. of Mathematics, Imperial College of Science, Technology and Medicine, 1995. Submitted to IEEE Transactions on Signal Processing.

The advance applied to Daubechies' least-asymmetric wavelet filters at each scale, in order to obtain near zero phase, is derived. The appropriate advance depends on whether half the length of each of the original quadrature mirror filters is even or odd. The departures from zero phase are illustrated.

[McCoy et al., 1998]

Emma J. McCoy, Andrew T. Walden, and Donald B. Percival. Multitaper spectral estimation of power law processes. IEEE Transactions on Signal Processing, 46(3):655-668, 1998.

In many branches of science, particularly astronomy and geophysics, power spectra of the form f(-beta), where beta is a positive, power-law exponent, are common, This form of spectrum is characterized by a sharp increase in the spectral density as the frequency f decreases toward zero, A power spectrum analysis method that has proven very powerful wherever the spectrum of interest is detailed and/or varies rapidly with a large dynamic range is the multitaper method, With multitaper spectral estimation, a set of orthogonal tapers are applied to the time series, and the resulting direct spectral estimators (``eigenspectra'') are averaged, thus, reducing the variance. One class of processes with spectra of the power-law type are fractionally differenced Gaussian processes that are stationary and can model certain types of long-range persistence, Spectral decay f(-beta) can be modeled for 0 < beta < 1. Estimation of the spectral slope parameter by regression on multitaper spectral ordinates is examined for this class of processes, It is shown that multitapering, or using sine or Slepian tapers, produces much better results than using the periodogram and is attractive compared with other competing methods, The technique is applied to a geophysical estimation problem.

[McCoy, 1994]

Emma J. McCoy. Some New Statistical Approaches to the Analysis of Long Memory Processes. PhD thesis, Imperial College, UK, Deptartment of Mathematics, 1994.

[McCulloch and Tsay, 1993]

Robert E. McCulloch and Ruey S. Tsay. Bayesian inference and prediction for mean and variance shifts in autoregressive time series. Journal of the American Statistical Association, 88(423):968-978, 1993.

[Meeker and Escobar, 1994]

William Q. Meeker and Luis A. Escobar. An algorithm to compute the CDF of the product of two normal random variables. Communications in Statistics A, 23(1):271-280, 1994.

[Mehrabi et al., 1997]

A. R. Mehrabi, H. Rassamdana, and M. Sahimi. Characterization of long-range correlations in complex distributions and profiles. Physical Review E, 56(1):712-722, 1997.

Characterizing long-range correlations in complex distributions, such as the porosity logs of field-scale porous media, and profiles, such as the fracture surfaces of rock and materials, is an important problem. We carry out an extensive analysis of such distributions represented by synthetic and real data to determine which method provides the most efficient and accurate tool for characterizing them. The synthetic data and profiles are generated by a fractional Brownian motion (FBM) and the real data analyzed are a porosity log of an oil reservoir and time variations of the pressure fluctuations in three-phase flow in a fluidized bed. The FBM is generated by three different numerical methods and the data are analyzed by seven different techniques. Our analysis indicates that the size of the data array greatly influences the accuracy of characterization of its long-range correlations. We also find that if the size of the data array is large enough, the commonly used rescaled-range (R/S) method of analyzing FBM series fails to provide accurate estimates of the Hurst exponent, although it can provide a reasonably accurate analysis of a data array that is generated by a fractional Gaussian noise. In contrast, the maximum entropy and wavelet decomposition methods offer highly accurate and efficient tools of characterizing long-range correlations in complex distributions and profiles. New methods that an somewhat similar to the R/S method are also suggested.

[Meneveau, 1991]

C. Meneveau. Analysis of turbulence in the orthonormal wavelet representation. Journal of Fluid Mechanics, 232:469-520, 1991.

A decomposition of turbulent velocity fields into modes that exhibit both localization in wavenumber and physical space is performed. The author reviews some basic properties of such a decomposition, the wavelet transform. The wavelet-transformed Navier-Stokes equations are derived, and he defines new quantities such as e(r,x), t(r,x) and pi (r,x) which are the kinetic energy, the transfer of kinetic energy and the flux of kinetic energy through scale r at position x. The discrete version of e(r,x) is computed from laboratory one-dimensional velocity signals in a boundary layer and in a turbulent wake behind a circular cylinder. The author also computes (r,x), t(r,x) and pi (r,x) from three-dimensional velocity fields obtained from direct numerical simulations. His findings are that the localized kinetic energies become very intermittent in x at small scales and exhibit multifractal scaling. The transfer and flux of kinetic energy are found to fluctuate greatly in physical space for scales between the energy containing scale and the dissipative scale.

[Meneveau, 1993]

C. Meneveau. Wavelet analysis of turbulence: The mixed energy cascade. In [Farge et al., 1993], pages 251-264. Based on the proceedings of a conference on wavelets, fractals, and Fourier transforms held at Newnham College, Cambridge in December 1990.

The wavelet-transformed Navier-Stokes equations are used to define quantities such as the transfer of kinetic energy and the flux of kinetic energy by scale and position. Direct numerical simulations are performed which show large spatial variability at every scale and non-Gaussian statistics. The local energy flux exhibits large spatial intermittency and is often negative, indicating local inverse cascades.

[Meyer, 1992]

Yves Meyer. Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge University Press, 1992. Translated to English by D. H. Salinger.

The first book in English to provide a comprehensive account of the mathematical theory of wavelets which has proved to be a powerful tool for harmonic analysts, and an alternative to the standard theory of Fourier analysis

[Meyer, 1993]

Yves Meyer. Wavelets: Algorithms & Applications. Society for Industrial and Applied Mathematics, Philadelphia, 1993. Translated and revised by Robert D. Ryan.

[Meyers and O'Brien, 1994]

Steven D. Meyers and James J. O'Brien. Spatial and temporal 26-day SST variations in the equatorial Indian Ocean using wavelet analysis. Geophysical Research Letters, 21(9):777-780, 1994.

Two-year sea-surface temperature time series of satellite data at two sites in the equatorial Indian Ocean are examined for oscillations with periods 2-70 days. The wavelet transform of the signals reveals a changing wavelet spectrum between August 1987 and November 1987 in the 10-30 day range, whereas the same time period in 1988 shows a relatively fixed spectrum. At 3 degrees latitude and 563 degrees E the 1987 wavelet coefficients with scales 10-30 days have about twice the amplitude they have in 1988. At 3 degrees latitude and 56 degrees E the 1987 waves have roughly half the amplitude of the 1988 waves. Activity with wavelet spectral peaks at periods near 12 days often procedes these waves.

[Mohr, 1981]

Donna L. Mohr. Modeling Data as a Fractional Gaussian Noise. PhD thesis, Princeton University, 1981.

[Morettin, 1996]

Pedro A. Morettin. From fourier to wavelet analysis of time series. In A. Prat, editor, Proceedings in Computational Statistics, pages 111-122, 1996.

[Moulin, 1994]

Pierre Moulin. Wavelet thresholding techniques for power spectrum estimation. IEEE Transactions on Signal Processing, 42(11):3126-3136, 1994.

Estimation of the power spectrum S(f) of a stationary random process can be viewed as a nonparametric statistical estimation problem. We introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown S(f). This approach offers the ability to capture statistically significant components of ln S(f) at different resolution levels and guarantees nonnegativity of the spectrum estimator. The spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non-Gaussian noise. We propose a wavelet thresholding technique to solve this problem under specified noise/resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables. The thresholds are computed using a saddle-point approximation to the distribution of the noise coefficients.

[Müller and Vidakovic, 1995]

Peter Müller and Brani Vidakovic. Bayesian inference with wavelets: Density estimation. Technical Report 95-34, Institute of Statisics and Decision Sciences, Duke University, 1995.

[Murtagh and Aussem, 1996]

Fionn Murtagh and Alex Aussem. Using the wavelet transform for multivariate data analysis and time series forecasting. Proc. IFCS'96, Kobe, Springer-Verlag, accepted (subject to minor revision), 1996.

[Murtagh, 1996]

Fionn Murtagh. Wedding the wavelet transform and multivariate data analysis. To appear Journal of Classification, 1996.

[Myers, 1990]

Raymond H. Myers. Classical and Modern Regression with Applications. The Duxbury Advanced Series in Statistics and Decision Sciences. PWS-KENT, Boston, 2 edition, 1990.

[Nason and Silverman, 1994]

Guy P. Nason and Bernard W. Silverman. The discrete wavelet transform in S. Journal of Computational and Graphical Statistics, 3(2):163-191, 1994.

The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one- and two-dimensional discrete wavelet transforms. The transforms and their inverses can be computed using any particular wavelet selected from a range of different families of wavelets. Pictures can be drawn of any of the one- or two-dimensional wavelets available in the package. The wavelet coefficients can be presented in a variety of ways to aid in the interpretation of data. The package's wavelet transform ``engine'' is written in C for speed and the object-oriented functionality of S makes wavethresh easy to use. We provide a tutorial introduction to wavelets and the wavethresh software. We also discuss how the software may be used to carry out nonlinear regression and image compression. In particular, thresholding of wavelet coefficients is a method for attempting to extract signal from noise and wavethresh includes functions to perform thresholding according to methods in the literature.

[Nason and Silverman, 1995]

Guy P. Nason and Bernard W. Silverman. The stationary wavelet transform and some statistical applications. In [Antoniadis and Oppenheim, 1995], pages 281-300.

[Nason and Silverman, 1997]

Guy P. Nason and Bernard W. Silverman. Wavelets for regression and other statistical problems. In M. G. Schimek, editor, Smoothing and Regression: Approaches, Computation and Application. Wiley, 1997.

[Nason et al., 1997a]

G. P. Nason, T. Sapatinas, and A. Sawczenko. Statistical modelling of time series using non-decimated wavelet representations. Technical report, Department of Mathematics, University of Bristol, Bristol, 1997.

[Nason et al., 1997b]

Guy P. Nason, Rainer von Sachs, and Gerald Kroisandt. Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Technical Report 516, Deptartment of Statistics, Stanford University, 1997.

[Nason, 1994]

Guy P. Nason. Wavelet regression by cross-validation. Technical report, Deptartment of Mathematics, University of Bristol, 1994.

This paper is about using wavelets for regression. The main aim is to introduce and develop a cross-validation method for selecting a wavelet regression threshold that produces good estimates with respect to L_2 error. The selected threshold determines which coefficients to keep in an orthogonal wavelet expansion of noisy data and acts in a similar way to a smoothing parameter in non-parametric regression.

[Nason, 1995]

Guy P. Nason. Choice of the threshold parameter in wavelet function estimation. In [Antoniadis and Oppenheim, 1995], pages 261-280.

[Nason, 1996]

Guy P. Nason. Wavelet shrinkage by cross-validation. Journal of the Royal Statistical Society B, 58:463-479, 1996.

Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage. A modified form of twofold cross-validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2. The cross-validation algorithm is then extended to data sets of any length and to multidimensional data sets.The algorithms are compared with established threshold choosers by using simulation. An application to a real data set arising from anaesthesia is presented.

[Nason, 1997]

Guy P. Nason. Wavelets. New Electronics, April 1997.

[Neumann and von Sachs, 1995]

Michael H. Neumann and Ranier von Sachs. Wavelet thresholding: Beyond the Gaussian I.I.D situation. In [Antoniadis and Oppenheim, 1995], pages 301-329.

[Neumann and von Sachs, 1997]

Michael H. Neumann and Ranier von Sachs. Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Annals of Statistics, 25(1):???--???, 1997.

[Neumann, 1994]

Michael H. Neumann. Spectral density estimation via nonlinear wavelet methods for stationary non-gaussian time series. Technical report, Statistics Research Report SRR 028-94, CMA, Australian National University, Canberra, 1994.

[Neumann, 1996]

Michael H. Neumann. Spectral density estimation via nonlinear wavelet methods for stationary non-gaussian time series. Journal of Time Series Analysis, 17(6):601-633, 1996.

In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis. The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram.Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression. It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their L 2 risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.

[Newland, 1993a]

D. E. Newland. Harmonic wavelet analysis. Proceedings of the Royal Society of London, Series A, 443(1917):203-225, 1993.

A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x)=(exp(i4 pi x)-exp(i2 pi x))/i2 pi x. This function w(x) is concentrated locally around x=0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.

[Newland, 1993b]

David Edward Newland. An Introduction to Random Vibrations, Spectral & Wavelet Analysis. Longman Scientific & Technical, New York, 3 edition, 1993.

[Newland, 1994a]

D. E. Newland. Some properties of discrete wavelet maps. Probability Engineering Mechanics, 9(1):59-69, 1994.

[Newland, 1994b]

D. E. Newland. Wavelet analysis of vibration, Part 2: wavelet maps. Transactions of the ASME. Journal of Vibration and Acoustics, 116(4):417-25, 1994.

For pt. 1, see ibid., vol. 116, p. 409-16, (1994). Wavelet maps provide a graphical picture of the frequency composition of a vibration signal. This paper, which is Part 2 of a pair, describes their construction and properties. In the case of harmonic wavelets, there are close similarities between wavelet maps and sonograms. A range of practical examples illustrate how the wavelet method may be applied to vibration analysis and some of its advantages.

[Newland, 1994c]

D. E. Newland. Wavelet analysis of vibration, Part I: theory. Transactions of the ASME. Journal of Vibration and Acoustics, 116(4):409-416, 1994.

Wavelets provide a new tool for the analysis of vibration records. They allow the changing spectral composition of a nonstationary signal to be measured and presented in the form of a time-frequency map. The purpose of this paper, which is Part I of a pair, is to introduce and review the theory of orthogonal wavelets and their application to signal analysis. It includes the theory of dilation wavelets, which have been developed over a period of about ten years, and of harmonic wavelets which have been proposed recently by the author. Part II is about presenting the results on wavelet maps and gives a selection of examples. The papers will interest those who work in the field of vibration measurement and analysis and who are in positions where it is necessary to understand and interpret vibration data.

[Ninness, 1998]

B. Ninness. Estimation of 1/f noise. IEEE Transactions on Information Theory, 44(1):32-46, 1998.

Several models have emerged for describing 1/f(gamma) noise processes. Based on these, various techniques for estimating the properties of such processes have been developed. This paper provides theoretical analysis of a new wavelet-based approach which has the advantages of having low computational complexity and being able to handle the case where the 1/f(gamma) noise might be embedded in a further white-noise process. However, the analysis conducted here shows that these advantages are balanced by the fact that the wavelet-based scheme is only consistent for spectral exponents gamma in the range gamma is an element of (0, 1). This is in contradiction to the results suggested in previous empirical studies. When gamma is an element of (0, 1) this paper also establishes that wavelet-based maximum-likelihood methods are asymptotically Gaussian and efficient. Finally, the asymptotic rate of mean-square convergence of the parameter estimates Is established and is shown to slow as gamma approaches one. Combined with a survey of non-wavelet-based methods, these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1/f(gamma) noise processes

[Nuri and Herbst, 1969]

W. A. Nuri and L. J. Herbst. Fourier methods in the study of variance fluctuations in time series analysis. Technometrics, 11(1):103-113, 1969.

[Odegard and Burrus, 1996]

Jan E. Odegard and C. Signey Burrus. New class of wavelets for signal approximation. Department of Electrical and Computer Engineering, Rice University, 1996.

[Ogden and Cheng, 1997]

R. Todd Ogden and Cheng Cheng. Testing for abrupt jumps with wavelets. In Proceedings of the 1997 Conference on the Interface of Statistics and Computer Science, 1997.

[Ogden and Hilton, 1997]

R. Todd Ogden and M. Hilton. Data analytic wavelet threshold selection in 2-D signal denoising. IEEE Transactions on Signal Processing, 45(2):496-500, 1997.

A data adaptive scheme for wavelet shrinkage-based noise removal is developed. The method involves a statistical test of hypothesis that takes into account the wavelet coefficients' magnitudes and relative positions. The amount of smoothing performed during noise removal is controlled by the user-supplied confidence level of the tests.

[Ogden and Parzen, 1996a]

R. Todd Ogden and Emanuel Parzen. Change-point approach to data analytic wavelet thresholding. Statistics and Computing, 6(2):93-99, 1996.

[Ogden and Parzen, 1996b]

R. Todd Ogden and Emanuel Parzen. Data dependent wavelet thresholding in nonparametric regression with change-point applications. Computational Statistics & Data Analysis, 22:53-70, 1996.

[Ogden, 1994]

R. Todd Ogden. Wavelet Thresholding in Nonparametric Regression with Change-Point Applications. PhD thesis, Texas A&M University, 1994. (PostScript)

[Ogden, 1996a]

R. Todd Ogden. On preconditioning the data for the wavelet transform when the sample size is not a power of two. Technical report, Department of Statistics, University of South Carolina, 1996.

[Ogden, 1996b]

R. Todd Ogden. Wavelets in Bayesian change-point analysis. Department of Statistics, University of South Carolina, 1996.

[Ogden, 1997]

R. Todd Ogden. Essential Wavelets for Statistical Applications and Data Analysis. Birkhauser, Boston, 1997.

Exciting new developments in wavelet theory have attracted much attention and sparked new research in many fields of applied mathematics. New tools are available for efficient data compression, image analysis, and signal processing, and there is a great deal of activity in developing new wavelet methods. The same features that make wavelets useful in these fields also make wavelets a natural and attractive choice in many areas of statistical data analysis. Essential Wavelets for Statistical Applications and Data Analysis presents an accesible, introductory survey for new wavelet analysis tools and how they can be applied to fundamental data analysis problems. A variety of problems in statistics are discussed in a non-theoretical style, with an emphasis on understanding of wavelet methods. The only technical prerequisite is basic knowledge of undergraduate calculus, linear algebra, and basic statistical theory.

[Pando and Fang, 1998]

Jesús Pando and Li-Zhi Fang. Discrete wavelet transform power spectrum estimator. Physical Review E, 57(3):3593-3601, 1998.

A method for measuring the spectrum of a density field by the discrete wavelet transform (DWT) is studied. We show how the Fourier power spectrum can be detected by using the wavelet function coefficients (WFC) of the DWT. This method can successfully measure the power spectrum in samples for which traditional methods often fail because the samples are finite sized, have a complex geometry, or are varyingly sampled. We demonstrate that the spectrum features, such as the power law index, the magnitude, and the typical scales can be determined by the DWT reconstructed spectrum. We apply this method to analyze the power spectrum of the spatial distribution of the Ly-alpha clouds. The two popular data sets used for the spectrum detection have quite different geometries and samplings, yet the one-dimensional (1D) power spectra and their 3D reconstruction given by the DWT estimator show the same features. The analysis makes clear that the DWT estimator is a sensitive tool in revealing common and physical properties from diverse data sets.

[Parzen, 1984]

Emanuel Parzen, editor. Time Series Analysis of Irregularly Observed Data, volume 25 of Lecture Notes in Statistics, New York, 1984. Springer-Verlag. Proceedings of a symposium held at Texas A&M University, College Station, Texas, February 10-13, 1983.

[Parzen, 1992]

Emanual Parzen. Comparison change analysis. In A. K. Md. E. Saleh, editor, Nonparametric Statistics and Related Topics, pages 3-15, Amsterdam, 1992. North Holland. Proceedings of the International Symposium on Nonparametric Statistics adn Related Topics.

[Pensky and Vidakovic, 1998]

Marianna Pensky and Brani Vidakovic. On non-equally spaced wavelet regression. Technical Report 98-06, Institute of Statistics and Decision Sciences, Duke University, 1998.

Wavelet-based regression analysis is widely used mostly for equally-spaced designs. For such designs wavelets are superior to other traditional orthonormal bases because of their versatility and ability to parsimoniously describe irregular functions. If the regression design is random, an automatic solution is not available. Given the observations (X_i, Y_i), i = 1,..., n, we estimate the regression function m(x)=E(Y|X=x) as a series sum_k hat c_jk phi_jk(x) where phi_jk(x),  k in Z are scaling functions spanning the multiresolution subspace V_j. We propose a method that utilizes a probabilistic model on X_i's in defining the empirical coefficients hat c_jk. The paper deals with both theoretical and practical aspects of the proposed estimator. We explore MSE convergence rates of the estimator. The performance of the estimator is compared to that of some traditional regression methods.

[Percival and Bruce, 1997]

Donald B. Percival and Andrew G. Bruce. Estimation of long memory processes with missing data. Technical Report 64, MathSoft, Inc., 1700 Westlake Avenue N., Seattle, WA 98109-9891, 1997.

[Percival and Guttorp, 1994a]

Donald B. Percival and Peter Guttorp. An introduction to spectral analysis and wavelets. In [Ciarlini et al., 1994], pages 175-186. Proceedings of the International Workshop.

[Percival and Guttorp, 1994b]

Donald B. Percival and Peter Guttorp. Long-memory processes, the Allan variance and wavelets. In [Foufoula-Georgiou and Kumar, 1994], pages 325-344.

[Percival and Mofjeld, 1997]

Donald B. Percival and Harold O. Mofjeld. Analysis of subtidal coastal sea level fluctuations using wavelets. Journal of the American Statistical Association, 92(439):868-880, 1997.

Subtidal coastal sea level fluctuations affect coastal ecosystems and the consequences of destructive events such as tsunamis. We analyze a time series of subtidal fluctuations at Crescent City, California, during 1980-1991 using the maximal overlap discrete wavelet transform (MODWT). Our analysis shows that the variability in these fluctuations depends on the season for scales of 32 days and less. We show how the MODWT characterizes nonstationary behavior succinctly and how this characterization can be used to improve forecasts of inundation during tsunamis and storm surges. Pie provide pseudocode and enough details so that data analysts in other disciplines can readily apply MODWT analysis to other nonstationary time series.

[Percival and Walden, 1993]

Donald B. Percival and Andrew T. Walden. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press, Cambridge, 1993.

This up-to-date introduction to univariate spectral analysis at the graduate level reflects a new scientific awareness of its complexity, as well as its widespread usage on digital computers with considerable computational power.

[Percival and Walden, 1999]

Donald B. Percival and Andrew T. Walden. Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge, 1999. Forthcoming.

[Percival, 1983a]

Donald B. Percival. On the sample mean and variance of a long memory process. Technical report, Department of Statistics, University of Washington, 1983.

[Percival, 1983b]

Donald Bame Percival. The Statistics of Long Memory Processes. PhD thesis, Department of Statistics, University of Washington, 1983.

[Percival, 1991]

Donald B. Percival. Characterization of frequency stability: frequency-domain estimation of stability measures. Proceedings of the IEEE, 79(7):961-972, 1991.

The author focuses on the frequency domain approach, which provides a complete characterization of frequency. The standard characterization of frequency stability in the frequency domain is the spectral density function (SDF). The author describes SDFs that model sampled frequency stability data and that are related to the SDFs of the standard characterization. On the basis of standard techniques in spectral analysis, he outlines a systematic way of estimating SDFs typical of frequency stability data. The recommended procedure is to check for broadband bias in the periodogram using a sequence of data tapers and, if bias is in evidence, to design an autoregressive prewhitening filter to prewhiten the data. The author considers the relationship between the Allan variance and the SDF and outlines two nonparametric ways of translating stability measures between the two domains-one based upon pilot analysis and the other upon J. Rutman's bandpass variance (1978).

[Percival, 1992]

Donald B. Percival. Simulating Gaussian random processes with a specified spectra. Computing Science and Statistics, 24:534-538, 1992.

We discuss the problem of generating realizations of length N from a Gaussian stationary process Y_t with a specified spectral density function S_Y(f). We review three methods for generating the required realizations and consider their relative merits. In particular, we discuss an approximate frequency domain technique that is evidently used frequently in practice, but that has some potential pitfalls. We discuss extensions to this technique that allow it to be used to generate realizations from a power-law process with spectral density function similar to S(f) = |f|^alpha for alpha < 0.

[Percival, 1993]

Donald B. Percival. Three curious properties of the sample variance and autocovariance for stationary processes with unknown mean. The American Statistician, 47(4):274-276, 1993.

In most books on time series analysis, estimators of the variance and autocovariance for a stationary process are discussed under the assumption that the process mean is known. Here we illustrate that, if the process mean is unknown and hence is estimated by the sample mean, these estimators have some surprising properties.

[Percival, 1994]

Donald B. Percival. Spectral analysis of univariate and bivariate time series. In [Stanford and Vardeman, 1994], pages 313-348.

[Percival, 1995]

Donald B. Percival. On estimation of the wavelet variance. Biometrika, 82(3):619-631, 1995.

Thw wavelet variance decomposes the variance of a time series into components associated with differen scales. We consider two estimators of the wavelet variance: the first based upon the discrete wavelet transform, and the second, called the maximal-overlap estimator, based upon a filtering interpretation of wavelets. We determine the large sample distribution for both estimatorsand show that the maximal-overlap estimator ismore efficient for a class of processes of interest in the physical sciences. We discuss methods for determining an approximate confidence interval for the wavelet variance. We demonstrate through Monte Carlo experiments that the large sample distribution for the maximal-overlap estimator is a reasonable approximation even for the moderate sample size of 128 observations. We apply our proposed methodology to a series of observations related to vertical shear in the ocean.

[Perrier et al., 1995]

Valérie Perrier, Thierry Philipovitch, and Claude Basdevant. Wavelet spectra compared to Fourier spectra. Journal of Mathematical Physics, 36(3):1506-1519, 1995.

The relation between Fourier spectra and spectra obtained from wavelet analysis is established. Small scale asymptotic analysis shows that the wavelet spectrum is meaningful only when the analyzing wavelet has enough vanishing moments. These results are related to regularity theorems in Besov spaces. For the analysis of infinitely regular signals, a new wavelet, with an infinite number of cancellations is proposed.

[Pesquet et al., 1996a]

J. C. Pesquet, H. Krim, D. Leporini, and E. Hamman. Bayesian approach to best basis selection. In IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 5, pages 2634-2637, 1996. 7-10 May 1996, Atlanta, GA, USA.

Wavelet packets and local trigonometric bases provide an efficient framework and fast algorithms to obtain a `best basis' or `best representation' of deterministic signals. Applying these deterministic techniques to stochastic processes may, however, lead to variable results. We revisit this problem and introduce a prior model on the underlying signal in noise and account for the contaminating noise model as well. We thus develop a Bayesian-based approach to the best basis problem, while preserving the classical tree search efficiency.

[Pesquet et al., 1996b]

Jean-Christophe Pesquet, Hamid Krim, and Hervé Carfantan. Time-invariant orthonormal wavelet representations. IEEE Transactions on Signal Processing, 44(8):1964-1970, 1996.

A simple construction of an orthonormal basis starting with a so-called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptanceand generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. We address the time-invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it,is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation.

[Petit and Bendjoya, 1996]

J. M. Petit and Ph. Bendjoya. A new insight in Uranus rings: A wavelet analysis of the Voyager 2 data. In Terrence W. Rettig and Joseph M. Hahn, editors, Completing the Inventory of the Solar System, volume 107 of Astronomical Society of the Pacific Conference Proceedings, pages 137-146, 1996.

A new signal processing analysis, based on the wavelet transform has been developed. It allows the detection and the reconstruction of fine structures in a very noisy signal. It removes the noise and gives a quantified level of detection of the structures against chance fluctuations. This powerful method has been applied on the PPS Voyager 2 data on the Uranus rings. A preliminary catalog of structures found in the sigma Sagitarii occultation experiment, is proposed here.

[Petris, 1997a]

Giovanni Petris. Bayesian Analysis of Long Memory Time Series. PhD thesis, Institute of Statistics and Decision Sciences, Duke University, 1997. (PostScript)

[Petris, 1997b]

Giovanni Petris. Bayesian spectral analysis of long memory time series. Technical Report 97-08, Institute of Statistics and Decision Sciences, Duke University, 1997.

[Pettitt and Stephens, 1982]

A. N. Pettitt and M. A. Stephens. EDF statistics for testing for the Gamma distribution. Technical Report 323, Department of Statistics, Stanford University, 1982.

[Pinheiro and Vidakovic, 1997]

A. Pinheiro and B. Vidakovic. Estimating the square root of a density via compactly supported wavelets. Computational Statistics & Data Analysis, 25(4):399-415, 1997.

[Plonka and Strela, 1998]

Gerlind Plonka and Vasily Strela. From wavelets to multiwavelets. In M. Dahlem, T. Lyche, and L. Shumaker, editors, Mathamatical Methods for Curves and Surfaces II. Vanderbilt University Press, 1998.

[Porter-Hudak, 1982]

Susan Porter-Hudak. Long-Term Memory Modelling -- A Simplified Spectral Approach. PhD thesis, University of Wisconsin, 1982.

[Press et al., 1992]

William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2 edition, 1992.

[Priestley, 1981]

M. B. Priestley. Spectral Analysis and Time Series. Academic Press, Inc., London, 1981.

[Priestley, 1996]

M. B. Priestley. Wavelets and time-dependent spectral analysis. Journal of Time Series Analysis, 17(1):85-104, 1996.

One of the key features of wavelet analysis is its potential use for effecting time-frequency decompositions of non-stationary signals. The relationship between wavelet analysis and timedependent spectral analysis has so far rested mainly on heuristic reasoning: in this paper we examine the relationship in a more precise mathematical form. A crucial feature of this analysis is the need to define carefully the notion of `frequency' when applied to non-stationary signals.

[Qiu and Er, 1995]

Lunji Qiu and Meng Hwa Er. Wavelet spectrogram of noisy signals. International Journal of Electronics, 79(5):665-677, 1995.

The wavelet transform is of interest for analysing non- stationary signals. The squared modulus of the wavelet transform leads to the wavelet spectrogram or scalogram. When signals are embedded in additive noise, it is important to study the estimation accuracy in terms of bias and variance. The mean and variance statistical properties of the wavelet spectrogram of a signal embedded in additive gaussian white noise are derived in this paper. Examples and simulation results are also presented.

[Ramanathan and Zeitouni, 1991]

J. Ramanathan and O. Zeitouni. On the wavelet transform of fractional Brownian motion. IEEE Transactions on Information Theory, 37(4):1156-1158, 1991.

A theorem characterizing fractional Brownian motion by the covariance structure of its wavelet transform is established. The authors examine whether there are alternate Gaussian processes whose wavelet transforms have a natural covariance structure. In addition, the authors examine if there are any Gaussian processes whose wavelet transform is stationary with respect to the affine group (i.e. the statistics of the wavelet transform do not depend on translations and dilations of the process).

[Ramsey and Lampart, 1998]

J. B. Ramsey and C. Lampart. Decomposition of economic relationships by timescale using wavelets - Money and income. Macroeconomic Dynamics, 2(1):49-71, 1998.

Economists have long known that timescale matters in that the structure of decisions as to the relevant time horizon, degree of time aggregation, strength of relationship, and even the relevant variables differ by timescale. Unfortunately, until recently it was difficult to decompose economic time series into orthogonal timescale components except for the shea or long run in which the former is dominated by noise. Wavelets are used to produce an orthogonal decomposition of some economic variables by timescale over six different timescales. The relationship of interest is that between money and income, i.e., velocity. We confirm that timescale decomposition is very important for analyzing economic relationships. The analysis indicates the importance of recognizing variations in phase between variables when investigating the relationships between them and throws considerable light on the conflicting results that have been obtained in the literature using Granger causality tests.

[Rao et al., 1997]

T. Subba Rao, M. B. Priestly, and O. Lessi, editors. Applications of Time Series Analysis in Astronomy and Meteorology. Chapman & Hall, London, 1997.

Statistical techniques, in particular time series techniques, are widely used in astronomy and meteorology. Despite this, until recently there have been few attempts to bring researchers from the fields of statistics, astronomy and meteorology together to discuss and formalize important problems. Applications of Time Series Analysis in Astronomy and Meteorology brings together a series of papers by experts in these fields evenly devoted to the theory and methodology of time series and to its applications to astronomy, meteorology and climatology. The topics covered include detection of periodicities, spectral analysis of unequally spaced data, detection of change points and higher order spectral methods of non-linear and non-Gaussian signals. Estimation of fractal dimension and applications of wavelet methods to astronomy are also considered. In addition, this book includes a floppy disc containing data sets to serve as a benchmark series. Applications of Time Series Analysis in Astronomy and Meteorology is of interest to statisticians, astronomers, meteorologists and climatologists alike.

[Ray and Tsay, 1997]

B. K. Ray and R. S. Tsay. Bandwidth selection for kernel regression with long-range dependent errors. Biometrika, 84(4):791-802, 1997.

We investigate the effect of long-range dependence on bandwidth selection for kernel regression with the plug-in method of Herrmann, Gasser & Kneip (1992). A new bandwidth estimator is proposed to allow for long-range dependence. Properties of the proposed estimator are investigated theoretically and via simulation. We find that the proposed estimator performs well in terms of integrated squared error of the estimated trend, allowing us to incorporate both deterministic nonlinear features having an unknown structure and long-range dependence into a single model. The method is illustrated using biweekly measurements of the volume of the Great Salt Lake.

[Reschenhofer, 1989]

E. Reschenhofer. Adaptive test for white noise. Biometrika, 76(3):629-632, 1989.

[Rice, 1945]

S. O. Rice. Mathematical analysis of random noise, part III: Statistical properties of random noise currents. Bell Systems Technical Journal, 24:46-156, 1945.

[Rice, 1980]

S. O. Rice. Distribution of quadratic forms in normal random variables -- Evaluation by numerical integration. SIAM Journal on Scientific and Statistical Computing, 1(4):438-448, 1980.

[Riedel and Sidorenko, 1995]

Kurt S. Riedel and Alexander Sidorenko. Minimum bias multiple taper spectral estimation. IEEE Transactions on Signal Processing, 43(1):188-195, 1995.

Two families of orthonormal tapers are proposed for multitaper spectral analysis: minimum bias tapers, and sinusoidal tapers ( upsilon /sup (k/)), where upsilon /sub n//sup (k/)= square root (2/(N+1))sin( pi kn/N+1), and N is the number of points. The resulting sinusoidal multitaper spectral estimate is S(f)=(1/2K(N+1)) Sigma /sub j=1//sup K/ mod y(f+j/(2N+2))-y(f- j/(2N+2)) mod /sup 2/, where y(f) is the Fourier transform of the stationary time series, S(f) is the spectral density, and K is the number of tapers. For fixed j, the sinusoidal tapers converge to the minimum bias tapers like 1/N. Since the sinusoidal tapers have analytic expressions, no numerical eigenvalue decomposition is necessary. Both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth. The bandwidth of the jth taper is simply 1/N centered about the frequencies (+or-j)/(2N+2). Thus, the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers. The band limited spectral concentration, integral /sub -w//sup w/ mod V(f) mod /sup 2/df of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the Slepian (1978) tapers. In contrast, the Slepian tapers can have the local bias, integral /sub - 1/2 //sup 1/2 /f/sup 2/ mod V(f) mod /sup 2/df, much larger than of the minimum bias tapers and the sinusoidal tapers.

[Riedel and Sidorenko, 1996]

Kurt S. Riedel and Alexander Sidorenko. Adaptive smoothing of the log-spectrum with multiple tapering. IEEE Transactions on Signal Processing, 44(7):1794-1800, 1996.

A hybrid estimator of the log-spectral density of a stationary time series is proposed, First, a multiple taper estimate is performed, followed by kernel smoothing the log-multiple taper estimate, This procedure reduces the expected mean square error by (pi(2)/4)(4/5) over simply smoothing the log tapered periodogram, A data-adaptive implementation of a variable-bandwidth kernel smoother is given.

[Rios and Vidaković, 1997]

David Insua Rios and Brani Vidaković. Wavelet-based random densities. Technical report, Institute of Statisics and Decision Sciences, Duke University, 1997. Discussion Paper 97-05.

[Rioul and Flandrin, 1992]

Olivier Rioul and Patrick Flandrin. Time-scale energy distributions: A general class extending wavelet transforms. IEEE Transactions on Signal Processing, 40(7):1746-1757, 1992.

A proposed theoretical framework for time-scale energy representation is based on local frequency which is covariant under modulations and time scaling which is covariant under dilations or contractions. The frameworks seeks to illustrate the relationship between scalograms and spectrograms. Results show that, from the Wigner-Ville distribution, it is possible to shift continuously to either a scalogram or a spectrogram. The approach simultaneously maintains a balance between time-frequency resolution and cross-terms reduction in both time-scale and time-frequency representations.

[Rioul and Vetterli, 1991]

Olivier Rioul and Martin Vetterli. Wavelets and signal processing. IEEE Signal Processing Magazine, 8(4):14-38, 1991.

A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes. The discussion includes nonstationary signal analysis, scale versus frequency, wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing. The main definitions and properties of wavelet transforms are covered, and connections among the various fields where results have been developed are shown.

[Risager, 1980]

Folmer Risager. Simple correlated autoregressive processes. Scandanavian Journal of Statistics, 7(1):49-60, 1980.

[Risager, 1981]

Folmer Risager. Model checking of simple correlated autoregressive processes. Scandanavian Journal of Statistics, 8(3):137-153, 1981.

[Robertson et al., 1998a]

A. N. Robertson, K. C. Park, and K. F. Alvin. Extraction of impulse response data via wavelet transform for structural system identification. Journal of Vibration and Acoustics, 120(1):252-260, 1998.

This paper presents a wavelet transform-based method of extracting the impulse response characteristics from the measured disturbances and response histories of linear structural dynamic systems. The proposed method is found to be effective in determining the impulse response functions for systems subjected to harmonic (narrow frequency-band) input signals and signals with sharp discontinuities, thus alleviating the Gibbs phenomenon encountered in FFT methods. When the system is subjected to random burst input signals for which the FFT methods are known to perform well, the proposed wavelet method performs equally well with a fewer number of ensembles than FFT-based methods. For completely random input signals, both the wavelet and FFT methods experience difficulties, although the wavelet method appears to perform somewhat better in tracing the fundamental response modes.

[Robertson et al., 1998b]

A. N. Robertson, K. C. Park, and K. F. Alvin. Identification of structural dynamics models using wavelet-generated impulse response data. Journal of Vibration and Acoustics, 120(1):261-266, 1998.

[Rozanov, 1967]

Yu. A. Rozanov. Stationary Random Processes. Holden-Day, San Francisco, 1967.

[Ruskai et al., 1992]

Mary Beth Ruskai, Gregory Beylkin, Ronald Coifman, Ingrid Daubechies, Stephane Mallat, Yves Meyer, and Louise Raphael, editors. Wavelets and Their Applications. Jones and Bartlett Publishers, 1992.

[Saito and Beylkin, 1993]

Naoki Saito and Gregory Beylkin. Multiresolution representations using the autocorrelation functions of compactly supported wavelets. IEEE Transactions on Signal Processing, 41(12):3584-3590, 1993.

Proposes a shift-invariant multiresolution representation of signals or images using dilations and translations of the autocorrelation functions of compactly supported wavelets. Although these functions do not form an orthonormal basis, their properties make them useful for signal and image analysis. Unlike wavelet-based orthonormal representations, the present representation has (1) symmetric analyzing functions, (2) shift-invariance, (3) associated iterative interpolation schemes, and (4) a simple algorithm for finding the locations of the multiscale edges as zero-crossings. The authors also develop a noniterative method for reconstructing signals from their zero-crossings (and slopes at these zero-crossings) in their representation. This method reduces the reconstruction problem to that of solving a system of linear algebraic equations.

[Saito, 1994]

Naoki Saito. Local Feature Extraction and Its Applications Using a Library of Bases. PhD thesis, Yale University, 1994.

[Sakaguchi, 1995]

F. Sakaguchi. Pseudodiagonalization of the autocorrelation of a stochastic process by an over-complete wavelet system. Electronics and Communications in Japan 3, 78(4):16-27, 1995. Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. 77-A, No. 8, August 1994, pp. 1065-1074.

If a stochastic process can be regarded as a superposition of the wavelets which arise randomly and independently of one another, the random-wavelet picture of a stochastic process is intuitive and convenient. This paper investigates theoretically in what case the picture can be used; i.e., in what case the autocorrelation of the stochastic process can be diagonalized by using the over-complete wavelet system. A general method for calculating the pseudodiagonal form from an arbitrarily given autocorrelation function using the operator algebra is proposed. Next, some properties of stationary wavelet-diagonal processes are investigated where it is shown that the power spectra of these processes are related to the spectral estimates under the circumstances in which the number of the time points are constrained to a constant finite number.

[Salazar, 1982]

Diego Salazar. Structural changes in time series models. Journal of Econometrics, 19(1):147-163, 1982.

[Salby and Hendon, 1994]

Murry L. Salby and Harry H. Hendon. Intraseasonal behavior of clouds, temperature, and motion in the tropics. Journal of Atmospheric Science, 51(15):2207-2224, 1994.

The spectral character of tropical convection is investigated in an 1 1-yr record of outgoing longwave radiation from the Advanced Very High Res olution Radiometer to identify interaction with the tropical circulation. Alo ng the equator in the eastern hemisphere, the space-time spectrum of convect ion possesses a broad peak at wavenumbers 1-3 and eastward periods of 35- 95 days. Significantly broader than the dynamical signal of the Madden-Julian oscillation (MJO), this quasi-discrete convective signal is associate d with a large- scale anomaly that propagates across and modulates time mean or 'climatological convection' over the equatorial Indian Ocean and west ern Pacific. Outside that region the convective signal is small, even tho ugh, under amplified conditions, coherence can be found east of the date l ine and in the subtropics. Having a zonal scale of approximately wavenumber 2 , anomalous convection propagates eastward at some 5 m |s.sup.-1 and suppresses as well as reinforces climatological convection in the eas tern hemisphere. The convective signal amplifies to a seasonal maximum nea r vernal equinox and, to a weaker degree, again near autumnal equinox, when climatological convection and warm SST cross the equator. Contemporaneous records of motion from ECMWF analyses and tropospheri c-mean temperature from Microwave Sounding Unit reveal an anomalous componen t of the tropical circulation that coexists with the convective signal and emb odies many of the established properties of the MJO. Unlike anomalous conve ction, that dynamical signal extends globally around the Tropics. The anomal ous circulation differs fundamentally between the eastern and western hemispheres. In the eastern hemisphere, subtropical Rossby gyres and zonal Kelvin structure along the equator flank the convective anomaly as it tracks eastward, giving the anomalous circulation the form of a 'forced resp onse.' In the western hemisphere, the dynamical signal is composed chiefly o f wavenumber-1 Kelvin structure, which has the form of a 'propagating r esponse' that is excited in and radiates away from anomalous convection at som e 10 m |s.sup.-1 . Kelvin structure comprising the propagating response appe ars in 850-mb and 200-mb zonal winds even when the convective signal is abse nt, albeit with much smaller amplitude. In contrast, the signal in 1000-m b convergence appears only when accompanied by anomalous convection, wh ich suggests that convergence in the boundary layer is instrumental in ac hieving strong interaction with the convective pattern.

[Sardy et al., 1997]

Sylvain Sardy, Donald B. Percival, Andrew G. Bruce, Hong-Ye Gao, and Werner Stuetzle. Wavelet denoising for unequally spaced data. Technical report, StatSci Division of MathSoft, Inc., 1700 Westlake Ave. N., Seattle, WA 98109-9891, 1997.

Wavelet shrinkage (WaveShrink) is a relatively new technique for nonparametric function estimation that has been shown to have asymptotic near-optimality properties over a wide class of functions. As originally formulated by Donoho and Johnstone, WaveShrink assumes equally spaced data. Because so many statistical applications (e.g., scatterplot smoothing) naturally involve unequally spaced data, we investigate in this paper how WaveShrink can be adapted to handle such data. Focusing on the Haar wavelet, we propose four approaches that extend the Haar wavelet transform to the unequally spaced case. Each approach is formulated in terms of continuous wavelet basis functions applied to a piecewise constant interpolation of the observed data, and each approach leads to wavelet coefficients that can be computed via a matrix transform of the original data. For each approach, we propose a practical way of adapting WaveShrink. We compare the four approaches in a Monte Carlo study and find them to be quite comparable in performance. The computationally simplest approach (isometric wavelets) has an appealing justification in terms of a weighted mean square error criterion and readily generalizes to wavelets of higher order than the Haar.

[Sari-Sarraf and Brzakovic, 1997]

Hamed Sari-Sarraf and Dragana Brzakovic. A shift-invariant discrete wavelet transform. IEEE Transactions on Signal Processing, 45(10):2621-2626, 1997.

This correspondence presents a unifying approach to the derivation and implementation of a shift-invariant wavelet transform of one-and two-dimensional (1-D and 2-D) discrete signals, Starting with Mallat's multiresolution wavelet representation (MRWAR), the correspondence presents an analytical process through which a shift-invariant, orthogonal, discrete wavelet transform called the multiscale wavelet representation (MSWAR) is obtained, The coefficients in MSWAR are shown to be inclusive of those in MRWAR with the implication that the derived representation is invertible. The computational complexity of MSWAR is quantified in terms of the required convolutions, and its implementation is shown to be equivalent to the filter upsampling technique.

[Scargle et al., 1993]

J. D. Scargle, T. Steiman, Cameron, K. Young, D. L. Donoho, J. P. Crutchfield, and J. Imamura. The quasi-periodic oscillations and very low frequency noise of Scorpius X-1 as transient chaos: a dripping handrail? Astrophysical Journal Letters, 411(2):91-94, 1993.

The authors present evidence that the quasi-periodic oscillations (QPO) and very low frequency noise (VLFN) characteristic of many accretion sources are different aspects of the same physical process. They analyzed a long, high time resolution EXOSAT observation of the low-mass X-ray binary (LMXB) Sco X-1. The X-ray luminosity varies stochastically on time scales from milliseconds to hours. The nature of this variability-as quantified with both power spectrum analysis and a new wavelet technique, the scalegram-agrees well with the dripping handrail accretion model, a simple dynamical system which exhibits transient chaos. In this method both the QPO and VLFN are produced by radiation from blobs with a wide size distribution, resulting from accretion and subsequent diffusion of hot gas, the density of which is limited by an unspecified instability to lie below a threshold.

[Scargle, 1982]

Jeffrey D. Scargle. Studies in astronomical time series analysis. II. statistical aspects of spectral analysis of unevenly spaced data. The Astrophysical Journal, 263:835-853, 1982.

For pt.I see Astrophys. J. Suppl. Ser., vol.45, no.1, p.1-71 (1981). Detection of a periodic signal hidden in noise is frequently a goal in astronomical data analysis. This paper does not introduce a new detection technique, but instead studies the reliability and efficiency of detection with the most commonly used technique, the periodogram, in the case where the observation times are unevenly spaced. This choice was made because, of the methods in current use, it appears to have the simplest statistical behavior. A modification of the classical definition of the periodogram is necessary in order to retain the simple statistical behavior of the evenly spaced case. With this modification, periodogram analysis and least-squares fitting of sine waves to the data are exactly equivalent. Certain difficulties with the use of the periodogram are less important than commonly believed in the case of detection of strictly periodic signals. In addition, the standard method for mitigating these difficulties (tapering) can be used just as well if the sampling is uneven. An analysis of the statistical significance of signal detections is presented, with examples.

[Scargle, 1989]

Jeffrey D. Scargle. Studies in astronomical time series analysis. III. fourier transforms, autocorrelation functions, and cross-correlation functions of unevenly spaced data. The Astrophysical Journal, 343(2):874-887, 1989.

For pt.II see ibid., vol.263, no.2, p.835-53 (1982). The paper develops techniques to evaluate the discrete Fourier transform (DFT), the autocorrelation function, and the cross-correlation function (CCF) of time series which are not evenly sampled. The series may consist of quantized point data (e.g. yes/no processes such as photon arrival). The DFT, which can be inverted to recover the original data and the sampling, is used to compute correlation functions by means of a procedure which is effectively, but not explicitly, an interpolation. The CCF can be computed for two time series not even sampled at the same set of times. Techniques for removing the distortion of the correlation functions caused by the sampling, determining the value of a constant component to the data, and treating unequally weighted data are also discussed. A FORTRAN code for the Fourier transform algorithm and numerical examples of the techniques are given.

[Scargle, 1994]

Jeffrey D. Scargle. Detection and modeling of chaotic dynamics using wavelet techniques. In [Szu, 1994], page 994.

Powerful new data analysis techniques based on wavelets are proving extremely useful in the reduction and interpretation of time series data. The goals of these methods include denoising, characterizing, modeling, and compressing of time series data. The multi-scale nature of wavelet analysis makes it especially useful for detection and characterization of self-similar or 'scaling' behavior, such as is common for chaotic processes. This paper describes how wavelet techniques led to a transient-chaos model for a rapidly fluctuating celestial X-ray source. The methods described here are freely available in a new software package called TeachWave, developed by David Donoho and Iain Johnstone of Stanford University (anonymous ftp to playfair.stanford.edu; the software is in directory /pub/software/wavelets, and a number of related technical papers are in /pub/reports).

[Scargle, 1996]

Jeffrey D. Scargle. Astronomical time series analysis: New methods for studying periodic and aperiodic systems. In The Weise Observatory 25th Anniversary Symposium: Astronomical Time Series, 1996.

[Scargle, 1997]

Jeffrey D. Scargle. Wavelet methods in astronomical time series analysis. In [Rao et al., 1997], pages 226-248.

[Scholl, 1998]

D. J. Scholl. Translation-invariant data visualization with orthogonal discrete wavelets. IEEE Transactions on Signal Processing, 46(7):2031-2034, 1998.

Orthogonal discrete wavelet transforms, can be made translation-invariant by adding redundant wavelet coefficients through repeated shifting operations. Othogonality is lost, but isometry and compact time support can be preserved. The practical application to data visualization of scalograms based on such transforms is discussed and illustrated with measured transient signals.

[Schröder and Sweldens, 1995]

Peter Schröder and Wim Sweldens. Spherical wavelets: Efficiently representing functions on the sphere. In Computer Graphics Proceedings (SIGGRAPH 95), pages 161-172. ACM Siggraph, 1995.

Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allofw fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bi-directional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

[Schumaker and Webb, 1993]

Larry L. Schumaker and Glenn Webb, editors. Recent Advances in Wavelet Analysis, volume 3 of Wavelet Analysis and its Applications. Academic Press, Inc., 1993.

Recent Advances in Wavelet Analysis is the third volume in the WAVELET ANALYSIS AND ITS APPLICATIONS series. This edited volume features ten timely and important articles authored by various experts in their respective fields, including such notable contributors as David L. Donoho, Ingrid Daubechies (MacArthur grant awardees in ‘91 and ‘92, respectively), Phillippe Tchamitchian, Patrick Flandrin (both featured speakers at the ‘92 International Wavelets Conference in Toulouse), Charles Chui, and Bjorn Jawerth (one of the editors of the Wavelet Digest). This book covers recent advances in wavelet analysis and applications in areas including wavelets on bounded intervals, wavelet decomposition of special interest to statisticians, wavelets approach to differential and integral equations, analysis of subdivision operators, and wavelets related to problems in engineering and physics. Anyone interested in the ever-evolving field of wavelets will find this book an excellent addition to the series and to the literature overall.

[Schuster, 1898]

A. Schuster. On the investigation of hidden periodicities with application to a supposed 26-day period of meterological phenomena. Terrestrial Magnetism, 3:13-41, 1898.

[Schwarzenberg-Czerny, 1996]

A. Schwarzenberg-Czerny. Fast and statistically optimal period search in uneven sampled observations. The Astrophysical Journal, 460(2):107-110, 1996.

The classical methods for searching for a periodicity in uneven sampled observations suffer from a poor match of the model and true signals and/or use of a statistic with poor properties. We present a new method employing periodic orthogonal polynomials to fit the observations and the analysis of variance (ANOVA) statistic to evaluate the quality of the fit. The orthogonal polynomials constitute a flexible and numerically efficient model of the observations. Among all popular statistics, ANOVA has optimum detection properties as the uniformly most powerful test. Our recurrence algorithm for expansion of the observations into the orthogonal polynomials is fast and numerically stable. The expansion is equivalent to an expansion into Fourier series. Aside from its use of an inefficient statistic, the Lomb-Scargle power spectrum can be considered a special case of our method. Tests of our new method on simulated and real light curves of nonsinusoidal pulsators demonstrate its excellent performance. In particular, dramatic improvements are gained in detection sensitivity and in the damping of alias periods.

[Serroukh and Walden, 1998]

Abdeslam Serroukh and Andrew T. Walden. The scale analysis of bivariate non-Gaussian time series via wavelet cross-covariance. Technical Report 98-02, Department of Mathematics, Imperial College of Science, Technology & Medicine, 1998.

[Serroukh et al., 1998]

Abdeslam Serroukh, Andrew T. Walden, and Donald B. Percival. Statistical properties of the wavelet variance estimator for non-Gaussian/non-linear time series. Technical Report 98-03, Department of Mathematics, Imperial College of Science, Technology & Medicine, 1998.

[Seshadri et al., 1969]

V. Seshadri, M. CsörgH o, and M. A. Stephens. Tests for the exponential distribution using Kolmogorov-type statistics. Journal of the Royal Statistical Society B, 31(3):499-509, 1969.

[Shen and Strang, 1998]

J. H. Shen and G. Strang. Asymptotics of Daubechies filters, scaling functions, and wavelets. Applied and Computational Harmonic Analysis, 5(3):312-331, 1998.

We study the asymptotic form as p --> infinity of the Daubechies orthogonal minimum phase filter h(p)[n], scaling function phi(p)(t), and wavelet w(p)(t). Kateb and Lemarie calculated the leading term in the phase of the frequency response H-p(omega). The infinite product <(phi)over cap>(p)(omega) = Pi H-p(omega/2(k)) leads us to a problem in stationary phase, for an oscillatory integral with parameter t. The leading terms change form with tau = t/p and we find three regions for phi(p)(tau): (1) An Airy function up to near tau(0): root 42 pi/p Ai(-root 42 pi p(2)(tau - tau(0))) + o(p(-1/3)) (2) An oscillating region root 2/pi pG'(omega(tau))cos [p(G((-1))(omega(tau)) - G(omega(tau))omega(tau)) + pi/4] + o(p(-1/2)) (3) A rapid decay after tau(1): (1/p pi)(1/(tau - tau(1)))sin[p(G((-1))(pi) - tau pi)] + o(p(-1)) The numbers tau(0) similar or equal to 0.1817 and tau(1) similar or equal to 0.3515 are known constants. The function G and its integral G((-1)) are independent of p. Regions 1 and 2 are matched over the interval p(-2/3) much less than tau - tau(0) much less than 1. The wavelets have a simpler asymptotic expression because the Airy wavefront is removed by the highpass filter. We also find the asymptotics of the impulse response h(p)[n] -a different function g(omega) controls the three regions. The difficulty throughout is to estimate the phase.

[Sheng et al., 1992]

Yunlong Sheng, Donny Roberge, and Harold H. Szu. Optical wavelet transform. Optical Engineering, 31(9):1840-1845, 1992.

The wavelet transform is implemented using an optical multichannel correlator with a bank of wavelet transform filters. This approach provides a shift-invariant wavelet transform with continuous translation and discrete dilation parameters. The wavelet transform filters can be in many cases simply optical transmittance masks. Experimental results show detection of the frequency transition of the input signal by the optical wavelet transform.

[Shensa, 1992]

Mark J. Shensa. The discrete wavelet transform: Wedding the à trous and Mallat algorithms. IEEE Transactions on Signal Processing, 40(10):2464-2482, 1992.

Two separately motivated implementations of the wavelet transform are brought together. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by the choice of filters. In fact, the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown that the commonly used Lagrange a trous filters are in one-to-one correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Relevant signal processing parameters are examined, and it is observed that orthonormality is balanced by restrictions on resolution.

[Shensa, 1996]

M. J. Shensa. Discrete inverses for nonorthogonal wavelet transforms. IEEE Transactions on Signal Processing, 44(4):798-807, 1996.

Discrete nonorthogonal wavelet transforms play an important role in signal processing by offering finer resolution in time and scale than their orthogonal counterparts. The standard inversion procedure for such transforms is a finite expansion in terms of the analyzing wavelet. While this approximation works quite well for many signals, it fails to achieve good accuracy or requires an excessive number of scales for others. This paper proposes several algorithms that provide more adequate inversion and compares them in the case of Morlet wavelets. In the process, both practical and theoretical issues for the inversion of nonorthogonal wavelet transforms are discussed.

[Shusterman and Feder, 1998]

E. Shusterman and M. Feder. Analysis and synthesis of 1/f processes via shannon wavelets. IEEE Transactions on Signal Processing, 46(6):1698-1702, 1998.

1/f processes can he very useful in modeling processes with long-term correlation. We propose analysis and synthesis procedures to express these processes in terms of the Shannon wavelet. Unlike previous techniques, our analysis procedure generates uncorrelated decomposition coefficients for the 1/f process. This is done hy taking onto account, and then removing, the residual correlation between the wavelet components. The analysis procedure is the major contribution of this work. The proposed synthesis algorithm, which is a byproduct of the proposed analysis algorithm, is competitive with other techniques.

[Siegel, 1979]

Andrew F. Siegel. The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity. Biometrika, 66(2):381-386, 1979.

[Simoncelli and Freeman, 1995]

Eero P. Simoncelli and William T. Freeman. The steerable pyramid: A flexible architecture for multi-scale derivative computation. In International Conference on Image Processing, volume 3, pages 444-447, 23-26 Oct. 1995, Washington, DC, USA, October 1995.

We describe an architecture for efficient and accurate linear decomposition of an image into scale and orientation subbands. The basis functions of this decomposition are directional derivative operators of any desired order. We describe the construction and implementation of the transform.

[Simoncelli et al., 1992]

Eero P. Simoncelli, William T. Freeman, Edward H. Adelson, and David J. Heeger. Shiftable multiscale transforms. IEEE Transactions on Information Theory, 38(2):587-607, 1992.

One of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance: the content of wavelet subbands is unstable under translations of the input signal. Wavelet transforms are also unstable with respect to dilations of the input signal and, in two dimensions, rotations of the input signal. The authors formalize these problems by defining a type of translation invariance called shiftability. In the spatial domain, shiftability corresponds to a lack of aliasing; thus, the conditions under which the property holds are specified by the sampling theorem. Shiftability may also be applied in the context of other domains, particularly orientation and scale. Jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored. Two examples of jointly shiftable transforms are designed and implemented: a 1-D transform that is jointly shiftable in position and scale, and a 2-D transform that is jointly shiftable in position and orientation. The usefulness of these image representations for scale-space analysis, stereo disparity measurement, and image enhancement is demonstrated.

[Skaug and Tjøstheim, 1993]

Hans Julius Skaug and Dag Tjøstheim. A nonparametric test of serial independence based on the empirical distribution function. Biometrika, 80(3):591-602, 1993.

[Slepian, 1978]

D. Slepian. Prolate spheroidal wave functions, Fourier analysis, and uncetainty -- V: The discrete case. Bell System Technical Journal, 57:1371-1430, 1978.

[Slingo et al., 1995]

J. M. Slingo, K. R. Sperber, J. S. Boyle, J.-P. Ceron, M. Dix, B. Dugas, W. Ebisuzaki, J. Fyfe, D. Gregory, J.-F. Gueremy, J. Hack, A. Harzallah, P. Inness, A. Kitoh, W. K.-M. Lau, B. McAvaney, R. Madden, A. Matthews, T. N. Palmer, C.-K. Park, D. Randell, and N. Renno. Intraseasonal oscillations in 15 atmospheric general circulation models (results from an AMIP diagnostic subproject). Technical Report 661, World Meteorological Organization, 1995.

[Spokoiny, 1996]

V. G. Spokoiny. Adaptive hypothesis testing using wavelets. Annals of Statistics, 24(6):??--??, 1996.

[Srivastava, 1993]

M. S. Srivastava. Comparison of CUMSUM and EWMA procedures for detecting a shift in the mean or an increase in the variance. Journal of Applied Statistical Science, 1(4):445-468, 1993.

[Stanford and Vardeman, 1994]

John L. Stanford and Stephen B. Vardeman, editors. Statistical Methods for Physical Science, volume 28 of Methods of Experimental Physics. Academic Press, Inc., Boston, 1994.

[Stark et al., 1995]

J. L. Stark, F. Murtagh, and A. Bijaoui. Multiresolution and astronomical image processing. In R. A. Shaw, H. E. Payne, and J. J. E. Hayes, editors, Astronomical Data Analysis Software and Systems IV, volume 77 of ASP Conference Series, pages 279-288, 1995.

We present several wavelet transform algorithms and their applications in astronomical image processing (restoration, object detection, compression, etc.).

[Stein, 1981]

Charles M. Stein. Estimation of the mean of a multivariate normal distribution. Annals of Statistics, 9(6):1135-1151, 1981.

[Stephens, 1970]

Michael A. Stephens. Use of the Kolmogorov--Smirnov, Cramér-von Mises and related statistics without extensive tables. Journal of the Royal Statistical Society B, 32(1):115-122, 1970.

[Stephens, 1974]

M. A. Stephens. EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69(347):730-737, 1974.

[Stephens, 1986a]

Michael A. Stephens. Tests based on EDF statistics. In [D'Agostino and Stephens, 1986], pages 97-193.

[Stephens, 1986b]

Michael A. Stephens. Tests for the exponential distribution. In [D'Agostino and Stephens, 1986], pages 421-459.

[Stigler, 1986]

Stephen M. Stigler. Estimating serial correlation by visual inspection of diagnostic plots. The American Statistician, 40(2):111-116, 1986.

[Stoffel, 1998]

Alexander Stoffel. Remarks on the unsubsampled wavelet transform and the lifting scheme. Submitted to Signal Processing, 1998.

[Strang and Nguyen, 1996]

Gilbert Strang and Truong Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, MA, 1996.

This new textbook by Gilbert Strang and Truong Nguyen offers a clear and easy-to-understand introduction to two central ideas -- filter banks for discrete signals, and wavelets. The connections are fully explained -- the wavelet is determined by a choice of filter coefficients. All important wavelet properties (orthogonality or biorthogonality, symmetry, accuracy of approximation, and smoothness) come from specific properties of the filters. The text shows how to construct those filters and wavelets. The applications are very widespread -- and they continue to develop rapidly. The book gives a direct approach to signal processing and image processing through filter banks that iterate on the lowpass filter (this is the wavelet idea). Blocking and ringing artifacts are analyzed, along with many MATLAB applications. Wavelets and Filter Banks is written for the very broad audience that uses these ideas: Digital Signal Processing and Speech Processing, Image Processing including Medical Imaging, Scientific and Engineering Applications, Students and Professionals (wanting to understand wavelets!)

[Strang and Strela, 1994]

Gilbert Strang and Vasily Strela. Orthogonal multiwavelets with vanishing moments. Optical Engineering, 33(7):2104-2107, 1994.

A scaling function is the solution to a dilation equation Phi(t)= Sigma c/sub k/ Phi (2t-k), in which the coefficients come from a low-pass filter. The coefficients in the wavelet W(t)= Sigma d/sub k/ Phi (2t-k) come from a high-pass filter. When these coefficients are matrices, Phi and W are vectors: there are two or more scaling functions and an equal number of wavelets. By dilation and translation of the wavelets, we have an orthogonal basis W/sub ijk/=W/sub i/(2/sup j/t-k) for all functions of finite energy. These ``multiwavelets'' open new possibilities. They can be shorter, with more vanishing moments, than single wavelets. They can be symmetric, which is impossible for scalar wavelets (except for Haar's). We determine the conditions to impose on the matrix coefficients c/sub k/ in the design of multiwavelets, and we construct a new pair of piecewise linear orthogonal wavelets with two vanishing moments.

[Strang, 1989]

Gilbert Strang. Wavelets and dilation equations: A brief introduction. SIAM Review, 31(4):614-627, 1989.

This is an introduction to the construction of wavelets from the solution to a dilation equation. It discusses the approximation and orthogonal properties of wavelets and describes the recursive algorithms that decompose and reconstruct a function. The object of wavelets is to localize as far as possible in both time and frequency, with efficient algorithms.

[Strang, 1993]

Gilbert Strang. Wavelet transforms versus Fourier transforms. Bulletin of the American Mathematical Society (N.S.), 28(2):288-305, 1993.

An orthogonal basis for piecewise constant functions is constructed by dilation and translation. The wavelength transform maps each function to its coefficients with respect to this basis. The approximation is found to be poor and is improved by dilation equations. Higher-order wavelets are constructed and indirect and recursive methods are used to compute them. The practicality of the wavelet transform and Fourier transform in signal processing are discussed.

[Strang, 1994]

G. Strang. Wavelets. American Scientist, 82:250-255, 1994.

The transformation of signals into a sum of small, overlapping waves offers a new method for analyzing, storing and transmitting information. The author discusses: Fourier and wavelet transforms; choosing the best basis; higher dimensions; fast wavelet transform; Daubechies wavelets; high-definition television; the future of fingerprints.

[Strang, 1996]

Gilbert Strang. Creating and comparing wavelets. Department of Mathematics, Massachusetts Institute of Technology, 1996.

[Strela and Walden, 1998a]

Vasily Strela and Andrew T. Walden. Orthogonal and biorthogonal multiwavelets for signal denoising and image compression. In [Szu, 1998]. 14-16 April 1998, Orlando, Florida.

[Strela and Walden, 1998b]

Vasily Strela and Andrew T. Walden. Signal and image denoising via wavelet thresholding: Orthogonal and biorthogonal, scalar and multiple wavelet transforms. Technical Report TR-98-01, Statistics Section, Department of Mathematics, Imperial College of Science, Technology & Medicine, 1998.

[Strela et al., 1995]

V. Strela, P. N. Heller, G. Strang, P. Topiwala, and C. Heil. The application of multiwavelet filter banks to image processing. Submitted to IEEE Transactions on Image Processing, 1995.

[Strela, 1996]

Vasily Strela. Multiwavelets: Theory and Applications. PhD thesis, Massachusetts Institute of Technology, 1996.

[Strichartz, 1993]

Robert S. Strichartz. How to make wavelets. American Mathematical Monthly, 100(6):539-557, 1993.

Wavelet bases where Haar functions are constructed from a single function by dyadic dilations and integer translations are considered as approximate definitions of a wavelet expansion. First, a scaling function and associated multiresolution analysis are chosen. The orthonormality conditions should be satisfied by generation of a multiresolution analysis of the function. The wavelets are then constructed by solving two algebraic identities and establishing the properties of the wavelet functions.

[Strickland and Hahn, 1996]

Robin N. Strickland and Hee Il Hahn. Wavelet transforms for detecting microcalcifications in mammograms. IEEE Transactions on Medical Imaging, 15(2):218-229, 1996.

Clusters of fine, granular microcalcifications in mammograms may be an early sign of disease. Individual grains are difficult to detect and segment due to size and shape variability and because the background mammogram texture is typically inhomogeneous. We develop a two-stage method based on wavelet transforms for detecting and segmenting calcifications. The first stage is based on an undecimated wavelet transform, which is simply the conventional filter bank implementation without downsampling, so that the low-low (LL), low-high (LH), high-low (HL), and high-high (HH) sub-bands remain at full size. Detection takes place in HH and the combination LH+HL. Four octaves are computed with two inter- octave voices for finer scale resolution. By appropriate selection of the wavelet basis the detection of microcalcifications in the relevant size range can be nearly optimized. In fact, the filters which transform the input image into HH and LH+HL are closely related to prewhitening matched filters for detecting Gaussian objects (idealized microcalcifications) in two common forms of Markov (background) noise. The second stage is designed to overcome the limitations of the simplistic Gaussian assumption and provides an accurate segmentation of calcification boundaries. Detected pixel sites in HH and LH+HL are dilated then weighted before computing the inverse wavelet transform. Individual microcalcifications are greatly enhanced in the output image, to the point where straightforward thresholding can be applied to segment them. FROC curves are computed from tests using a freely distributed database of digitized mammograms.

[Sweldens and Schröder, 1996]

Wim Sweldens and Peter Schröder. Building your own wavelets at home. In ``Wavelets in Computer Graphics'', ACM SIGGRAPH Course Notes, 1996.

We give an practical overview of three simple techniques to construct wavelets under general circumstances: interpolating subdivision, average interpolation, and lifting. We include examples concerning the construction of wavelets on an interval, weighted wavelets, and wavelets adapted to irregular samples.

[Sweldens, 1995a]

W. Sweldens. The lifting scheme: A construction of second generation wavelets. Technical Report 1995:6, Department of Mathematics, University of South Carolina, 1995.

We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included.

[Sweldens, 1995b]

W. Sweldens. The lifting scheme: A new philosophy in biorthogonal wavelet constructions. In [Laine et al., 1995], pages 68-79. 12-14 July, 1995, San Diego, California.

In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in a later stage. We show how lifting leads to a faster, fully in-place implementation of the wavelet transform. Moreover, it can be used in the construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one function. A typical example of the latter are wavelets on the sphere.

[Sweldens, 1996a]

W. Sweldens. The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal., 3(2):186-200, 1996.

We present the lifting scheme, a new idea of constructing compactly supported wavelets with compactly supported duals. The lifting scheme provides a simple relationship between all multiresolution analyses with the same scaling function. It isolates the degrees of freedom remaining after fixing the biorthogonality relations. Then one has full control over these degrees of freedom to custom-design the wavelet for a particular application. It also leads to a faster implementation of the fast wavelet transform. We illustrate the use of the lifting scheme in the construction of wavelets with interpolating scaling functions.

[Sweldens, 1996b]

W. Sweldens. Wavelets: What next?. Proc. IEEE, 84(4):680-685, 1996.

In this concluding article, we want to look ahead and see what the future can bring to wavelet research. We try to find a common denominator for ``wavelets'' and identify promising research directions and challenging problems.

[Szatmary et al., 1994]

K. Szatmary, J. Vinko, and J. Gal. Application of wavelet analysis in variable star research. I. Properties of the wavelet map of simulated variable star light curves. AASS, 108(2):377-394, 1994.

A type of the relatively new time-frequency method, the wavelet analysis is studied. Some results of testing this method are presented. The test data series were defined so that they show similarities with the light variations of variable stars. The effects of observational noise and irregularities in data sampling are pointed out. The wavelet analysis seems to be a suitable method for detecting the local behaviour of the light curves, e.g. phase jump or mode switching. The investigation of time-dependent phenomena, e.g. amplitude or frequency modulation, is more available than in the case of standard Fourier analysis. In order to interpret the real wavelet maps of variable stars it is necessary to take into account the properties of the method presented by similar tests.

[Szatmary et al., 1996]

K. Szatmary, J. Gal, and L. L. Kiss. Application of wavelet analysis in variable star research. II. The semiregular star V Bootis. Astronomy and Astrophysics, 308(3):791-8, 1996.

For pt.I see Astron. Astrophys. Suppl. Ser., vol.108, no.2, p.377-94 (1994). Light curve analysis of the SRa-type variable V Boo is presented and discussed. The periods are determined and the stability of these periods as well as their amplitudes are investigated with wavelet analysis. The amplitude decrease is studied with the so-called ridge procedure, which shows that the amplitude of the longer period strongly decreased while the amplitude of the shorter period seems to remain stable. The possible interpretations of this effect are discussed. Using theoretical models and observational relations physical parameters and pulsational modes of V Boo are also estimated.

[Szilagyi et al., 1996]

Jozsef Szilagyi, Gabriel G. Katul, Marc B. Parlange, John D. Albertson, and Anthony T. Cahill. The local effect of intermittency on the inertial subrange energy spectrum of the atmospheric surface layer. Boundary-Layer Meteorology, 79(1-2):35-50, 1996.

Orthonormal wavelet expansions are applied to atmospheric surface layer velocity measurements. The effect of intermittent events on the energy spectrum of the inertial subrange is investigated through analysis of wavelet coefficients. The local nature of the orthonormal wavelet transform in physical space makes it possible to identify a relationship between the inertial subrange slope of the local wavelet spectrum and a simple indicator (i.e. the local variance of the signal) of local intermittency buildup. The slope of the local wavelet energy spectrum in the inertial subrange is shown to be sensitive to the presence of intermittent events. During well-developed intermittent events (coherent structures), the slope of the energy spectrum is somewhat steeper than -5/3, while in less active regions the slope is found to be flatter than -5/3. When the slopes of local wavelet spectra are ensemble averaged, a slope of -5/3 is recovered for the inertial subrange.

[Szu, 1994]

Harold H. Szu, editor. Wavelet Applications, volume 2242 of Proceedings of SPIE, 1994. 4-8, April 1994, Orlando, Florida.

[Szu, 1995]

Harold H. Szu, editor. Wavelet Applications II, volume 2491 of Proceedings of SPIE, 1995. 17-21, April 1994, Orlando, Florida.

[Szu, 1996]

Harold H. Szu, editor. Wavelet Applications III, volume 2762 of Proceedings of SPIE, 1996. 8-12 April 1996, Orlando, Florida.

[Szu, 1998]

Harold H. Szu, editor. Wavelet Applications V, volume 3391 of Proceedings of SPIE, 1998. 14-16 April 1998, Orlando, Florida.

[Tachibana, 1998]

Y. Tachibana. The differentiation by a wavelet and its application to the estimation of a transfer function. IEICE Transactions on Fundamentals of Electronics Communications and Computer Science, E81A(6):1194-1200, 1998.

This paper deals with a set of differential operators for calculating the differentials of an observed signal by the Daubechies wavelet and its application for the estimation of the transfer function of a linear system by using non-stationary step-like signals. The differential operators are constructed by iterative projections of the differential of the scaling function for a multiresolution analysis into a dilation subspace. By the proposed differential operators we can extract the arbitrary order differentials of a signal. We propose a set of identifiable filters constructed by the sum of multiple filters with the first order lag characteristics. Using the above differentials and the identifiable filters we propose an identification method for the transfer function of a linear system. In order to ensure the appropriateness and effectiveness of the proposed method some numerical simulations are presented.

[Teolis, 1997]

A. Teolis. Computational Signal Processing with Wavelets. Springer-Verlag, 1997.

Computational Signal Processing with Wavelets examines both theoretical and practical aspects of computational signal processing using wavelets. Theoretically, an emphasis is placed on balancing the accessibility of the material with the level of mathematical rigor which sacrifices as little as possible of both. Computationally, wavelet signal processing algorithms are presented and applied to signal compression, noise suppression, and signal identification. Numerical illustrations of these computational techniques are further provided with interactive software (MATLAB) via an internet accessible WEB site. Starting from basic principles of signal representation with atomic functions, a mathematically well founded theory of the discretization of analog signals is developed. General families are specialized to wavelet families and the discrete representation are specialized to generally non-orthogonal wavelet transforms. The theory leads naturally to the computer implementation of the non-orthogonal wavelet transform. Specific topics covered include general signal representation, continuous and discrete Fourier transforms, orthonormal and biorthogonal bases, frames, wavelet frames and frame reconstruction, discrete wavelet transform, multi-resolution analysis, orthonormal wavelets, continuous wavelet transform, non-orthogonal wavelet transform, and wavelet based signal processing algorithms for compression, noise suppression, and identification. The discussion is at the level of a senior or beginning graduate student level and is accessible to signal processing professionals and practicioners. Dissemination of the material is provided by a hybrid combination of traditional (text) and non-traditional (internet and electronic) media.

[Teti and Kritikos, 1992]

Joseph G. Teti and H. N. Kritikos. SAR ocean image representation using wavelets. IEEE Transactions on Geoscience and Remote Sensing, 30(5):1089-1094, 1992.

The utility of wavelet analysis as a tool for geophysical research is examined using both continuous and discrete versions of the wavelet transform. In both cases, waveform decomposition and reconstruction is possible using somewhat different computational procedures. The theoretical background of each procedure is briefly described and applied using a specific 'wavelet'. The wavelet used is based on a Gaussian function, and provides simultaneous time-frequency (or space-wavenumber) localization that meets the lower limit of the uncertainty principle. A representation of this type is ideally suited for the analysis of waveforms that arise from nonstationary processes. The properties of wavelet analysis are examined by expanding an FM-chirp waveform and azimuth cuts taken from two different SAR ocean images. The performance and ease of implementation are compared for the continuous and discrete formulations, and the effects of filtering in wavelet phase space using the discrete case are also examined.

[Teverovsky and Taqqu, 1997]

Vadim Teverovsky and Murad Taqqu. Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator. Journal of Time Series Analysis, 18(3):279-304, 1997.

[Tewfik and Kim, 1992]

A. H. Tewfik and M. Kim. Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Transactions on Information Theory, 38(2):904-909, 1992.

It is shown that the discrete wavelet coefficients of fractional Brownian motion at different scales are correlated and that their auto- and cross-correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional Brownian motion itself. The rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet.

[Thomson and Chave, 1991]

David J. Thomson and Alan D. Chave. Jackknifed error estimates for spectra, coherences, and transfer functions. In [Haykin, 1991], pages 58-113.

[Thomson, 1982]

David J. Thomson. Spectrum estimation and harmonic analysis. IEEE Proceedings, 70(9):1055-1096, 1982.

In the case of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or 'smoothing', are dominant. The author presents a new method based on a 'local' eigen-expansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weighted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. The author shows relations of this estimate to maximum-likelihood estimates, shows that the estimation capacity of the estimate is high, and shows applications to coherence and polyspectrum estimates.

[Thomson, 1995]

David J. Thomson. The seasons, global temperature, and precession. Science, 268(5207):59-68, 1995.

Analysis of instrumental temperature records beginning in 1659 shows that in much of the world the dominant frequency of the seasons is one cycle per anomalistic year (the time from perihelion to perihelion, 365.25964 days), not one cycle per tropical year (the time from equinox to equinox, 365.24220 days), and that the timing of the annual temperature cycle is controlled by perihelion. The assumption that the seasons were timed by the equinoxes has caused many statistical analyses of climate data to be badly biased. Coherence between changes in the amplitude of the annual cycle and those in the average temperature show that between 1854 and 1922 there were small temperature variations, probably of solar origin. Since 1922, the phase of the Northern Hemisphere coherence between these quantities switched from 0[degree] to 180[degrees] and implies that solar variability cannot be the sole cause of the increasing temperature over the last century. About 1940, the phase patterns of the previous 300 years began to change and now appear to be changing at an unprecedented rate. The average change in phase is now coherent with the logarithm of atmospheric C[O.sub.2] concentration.

[Tillman et al., 1993]

J. E. Tillman, N. C. Johnson, P. Guttorp, and D. B. Percival. The Martian annual atmospheric pressure cycle: years without great dust storms. Journal of Geophysical Research, 98(E6):10963-10971, 1993.

A model of the annual cycle of pressure on Mars has been developed for a 2-year period chosen to include 1 year at Lander 2 and to minimize the effect of great dust storms at the 22 degrees N Lander 1 site. The model was developed by weighted least squares fitting of the Viking Lander pressure measurements to an annual mean, and fundamental and the first four harmonics of the annual cycle. The very close agreement between the two years suggests that an accurate representation of the annual CO/sub 2/ condensation-sublimation cycle can be established for such years. The two annual mean pressures are identical to 0.006 mbar out of 7.9 mbar, and the differences in amplitudes for the first five periodic components between the two years range from 0.017 to 0.001 mbar.

[Titchmarsh, 1939]

E. C. Titchmarsh. The Theory of Functions. Oxford University Press, Oxford, 2 edition, 1939.

[Torrence and Compo, 1998]

Christopher Torrence and Gilbert P. Compo. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1):61-78, 1998.

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino-Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmöller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of changes in ENSO variance on interdecadal timescales. Using new datasets that extend back to 1871, the Nino3 sea surface temperature and the Southern Oscillation index show significantly higher power during 1880-1920 and 1960-90, and lower power during 1920-60, as well as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in 2-8-yr wavelet power in both longitude and time.

[Toussoun, 1925]

Omar Toussoun. Mémoire sur l'histoire du nil. In Mémoires a l'Institut d'Egypte, volume 18, pages 366-404. 1925.

[Treviño and Andreas, 1996]

Beorge Treviño and Edgar L. Andreas. On wavelet analysis of nonstationary turbulence. Boundary-Layer Meteorology, 81(3-4):271-288, 1996.

Wavelets are new tools for turbulence analysis that are yielding important insights into boundary-layer processes. Wavelet analysis, however, has some as yet undiscussed limitations: failure to recognize these can lead to misinterpretation of wavelet analysis results. Here we discuss some limitations of wavelet analysis when applied to nonstationary turbulence. Our main point is that the analysis wavelet must be carefully matched to the phenomenon of interest, because wavelet coefficients obscure significant information in the signal being analyzed. For example, a wavelet that is a second-difference operator can provide no information on the linear trend in a turbulence signal. Wavelet analysis also yields no meaningful information about nonlinear behavior in a signal - contrary to claims in the literature - because, at any instant, a wavelet is a single-scale operator, while nonlinearity involves instantaneous interactions among many scales.

[Tribouley, 1995a]

K. Tribouley. Adaptive density estimation. In [Antoniadis and Oppenheim, 1995], pages 385-395.

[Tribouley, 1995b]

K. Tribouley. Practical estimation of multivariate densities using wavelet methods. Statistica Neerlandica, 49(1):41-62, 1995.

This paper describes a practical method for estimating multivariate densities using wavelets. As in kernel methods, wavelet methods depend on two types of parameters. On the one hand we have a functional parameter: the wavelet [phi] (comparable to the kernel K) and on the other hand we have a smoothing parameter: the resolution index (comparable to the bandwidth h). Classically, we determine the resolution index with a cross-validation method. The advantage of wavelet methods compared to kernel methods is that we have a technique for choosing the wavelet [phi] among a fixed family. Moreover, the wavelets method simplifies significantly both the theoretical and the practical computations.

[Tsay, 1988]

Ruey S. Tsay. Outliers, level shifts, and variance changes in time series. Journal of Forecasting, 7:1-20, 1988.

[Tsonis et al., 1997]

A. A. Tsonis, P. Kumar, J. B. Elsner, and P. A. Tsonis. Wavelet analysis of DNA sequences. Physical Review E, 53(2):1828-1834, 1997.

In this paper we use wavelet analysis in order to probe the localized structure of DNA sequences. We demonstrate that, unlike other conventional approaches, wavelets are able to decompose seemingly homogeneous regions in noncoding sequences into smaller distinct regions that obey their own repetition and construction rules. The significance of this result to gene evolution is discussed.

[Tukey, 1949]

John. W. Tukey. The sampling theory of power spectrum estimates. In Symposium on Applications of Autocorrelation Analysis to Physical Problems, pages 47-67. Office of Naval Research, Department of the Navy, Washington, U.S.A., 1949.

[Turlach and Hall, 1997]

Berwin A. Turlach and Peter Hall. Interpolation methods for nonlinear wavelet regression with irregularly spaced design. AS, 25(5), 1997.

We suggest and discuss interpolation methods that enable nonlinear wavelet estimators to be employed with stochastic design, or non-dyadic regular design, in problems of nonparametric regression. This approach allows relatively rapid computation, involving dyadic approximations to wavelet-after-interpolation techniques. New types of interpolation are described, enabling first-order variance reduction at the expense of second-order increases in bias. The effect of interpolation on threshold choice is addressed, and appropriate thresholds are suggested for error distributions with as few as four finite moments. A concise account of mean squared error properties is given for interpolation-based wavelet estimators applied to piecewise-smooth functions.

[Unser and Aldroubi, 1996]

M. Unser and A. Aldroubi. A review of wavelets in biomedical applications. Proceedings of the IEEE, 84(4):626-638, 1996.

In this paper we present an overview of the various uses of the wavelet transform (WT) in medicine and biology. We start by describing the wavelet properties that are the most important for biomedical applications. In particular, we provide an interpretation of the continuous wavelet transform (CWT) as a prewhitening multiscale matched filter. Me also briefly indicate the analogy between the WT and some of the biological processing that occurs in the early components of the auditory and visual system. We then review the rises of the WT for the analysis of 1-D physiological signals obtained by phonocardiography, electrocardiography (ECC), and electroencephalography (EEG), including evoked response Next, we provide a survey of recent wavelet developments in medical imaging. These include biomedical image processing algorithms (e.g., noise reduction, image enhancement, and detection of microcalcifications in mammograms), image reconstruction and acquisition schemes (tomography, and magnetic resonance imaging (MRI)), and multiresolution methods for the registration and statistical analysis of functional images of the brain (positron emission tomography (PET) and functional MRI (fMRI)). In each case, we provide the reader with some general background information and a brief explanation of how the methods work.

[Unser et al., 1996]

Michael A. Unser, Akram Aldroubi, and Andrew F. Laine, editors. Wavelet applications in signal and image processing IV, volume 2825 of Proceedings of SPIE, 1996. 4-9 August, 1996, Denver, Colorado.

[Unser et al., 1998]

M. Unser, P. Thevenaz, and A. Aldroubi. Shift-orthogonal wavelet bases. IEEE Transactions on Signal Processing, 46(7):1827-1836, 1998.

Shift-orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation (or shifts) within one level but not with respect to dilations across scales. In this paper, we characterize these wavelets and investigate their main properties by considering two general construction methods. In the first approach, we start by specifying the analysis and synthesis function spaces and obtain the corresponding shift-orthogonal basis functions by suitable orthogonalization. In the second approach, we take the complementary view and start from the digital filterbank. We present several illustrative examples, including a hybrid version of the Battle-Lemarie spline wavelets. We also provide filterbank formulas for the fast wavelet algorithm. A shift-orthogonal wavelet transform is closely related to an orthogonal transform that uses the same primary scaling function; both transforms have essentially the same approximation properties. One experimentally confirmed benefit of relaxing the interscale orthogonality requirement is that we can design wavelets that decay faster than their orthogonal counterpart.

[Uosaki and Kawagoe, 1988]

K. Uosaki and M. Kawagoe. Backward SPRT failure detection system for detection of innovation variance change. In Han-Fu Chen, editor, Identification and System Parameter Estimation, volume 2, pages 1153-1157, 1988.

It is known that the failure detection system based on the Wald's sequential probability ratio test (SPRT) suffers an extra time delay in detecting system degradation characterized by the presence of a systematic non-zero mean. Chien and Adams (1976) developed a modified Wald's SPRT system by utilizing a feedback of the logarithm of likelihood ratio function (LLR) for improvement of the detection system. Uosaki (1986) proposed a backward SPRT failure detection system to improve the characteristics in detecting the above system degradation. The system uses the LLR evaluated in reverse from the current observation to the past ones. Recognizing the fact that system parameter change causes the change in innovation variance rather than in innovation mean, the authors apply the idea of the backward SPRT to detect degradation characterized by the increase of innovation variance. The mean detection time is derived using the theory of the absorbing Markov chain. This quantity can be used to determine the decision boundary in the system.

[Vaidyanathan, 1993]

P. P. Vaidyanathan. Multirate Systems And Filter Banks. Prentice-Hall, Inc., New Jersey, 1993.

KEY BENEFIT: Presenting general principles without bias towards specific application-oriented detail, this text offers a thorough, unified, and in-depth treatment of the techniques of multirate signal processing. KEY TOPICS: It is the first book to cover the topics of digital filter banks, multidimensional multirate systems, and wavelet representations under one cover. MARKET: This manual will be valuable to engineers working with applications of speech and image compression, digital audio, and statistical and adaptive signal processing.

[Vannucci and Corradi, 1997]

Marina Vannucci and Fabio Corradi. Some findings on the covariance structure of wavelet coefficients: Theory and models in a bayesian perspective. Technical report, Institute of Mathematics and Statistics, University of Kent at Canterbury, 1997. UKC/IMS/97/05.

[Vannucci and Vidaković, 1995]

Marina Vannucci and Brani Vidaković. Preventing the dirac disaster: Wavelet based density estimation. Technical report, Institute of Statisics and Decision Sciences, Duke University, 1995. Discussion Paper 95-27.

[Vannucci, 1996]

Marina Vannucci. On the Application of Wavelets in Statistics. PhD thesis, Dipartimento di Statistica ``G. Parenti'', University of Florence, Italy, 1996. In Italian. (PostScript)

[Velis and Ulrych, 1996]

D. R. Velis and T. J. Ulrych. Simulated annealing wavelet estimation via fourth-order cumulant matching. Geophysics, 61(6):1939-1948, 1996.

The fourth-order cumulant matching method has been developed recently for estimating a mixed-phase wavelet from a convolutional process. Matching between the trace cumulant and the wavelet moment is done in a minimum mean-squared error sense under the assumption of a non-Gaussian, stationary, and statistically independent reflectivity series. This leads to a highly nonlinear optimization problem, usually solved by techniques that require a certain degree of linearization, and that invariably converge to the minimum closest to the initial model. Alternatively, we propose a hybrid strategy that makes use of a simulated annealing algorithm to provide reliability of the numerical solutions by reducing the risk of being trapped in local minima. Beyond the numerical aspect, the reliability of the derived wavelets depends strongly on the amount of data available. However, by using a multidimensional taper to smooth the trace cumulant, we show that the method can be used even in a trace-by-trace implementation, which is very important from the point of view of stationarity and consistency. We demonstrate the viability of the method under several reflectivity models. Finally, we illustrate the hybrid strategy using marine and held real data examples. The consistency of the results is very encouraging because the improved cumulant matching strategy we describe can be effectively used with a limited amount of data.

[Vetterli and Kovavcević, 1995]

Martin Vetterli and Jelena Kovavcević. Wavelets and Subband Coding. Prentice Hall PTR, New Jersey, 1995.

[Vidaković and Müller, 1994]

Brani Vidaković and Peter Müller. Wavelets for kids: Tutorial introduction. Institute of Statisics and Decision Sciences, Duke University, 1994.

[Vidakovic, 1994]

Brani Vidakovic. Nonlinear wavelet shrinkage with Bayes rules and Bayes factors. Technical Report 94-24, Institute of Statisics and Decision Sciences, Duke University, 1994.

[von Sachs and Neumann, 1997]

Rainier von Sachs and Michael H. Neumann. A wavelet-based test for stationarity. Technical Report AGTM 182, Department of Mathematics, University of Kaiserslautern, 1997.

[von Sachs et al., 1996]

Rainier von Sachs, Guy P. Nason, and Gerald Kroisandt. Spectral representation and estimation for locally stationary wavelet processes. FB Mathematik, Universität Kaiserslautern, D-67653 Kaiserslautern, Germany, 1996.

[von Sachs et al., 1997]

Rainier von Sachs, Guy P. Nason, and Gerald Kroisandt. Adaptive estimation of the evolutionary wavelet spectrum. Technical Report 516, Department of Statistics, Stanford University, 1997.

[von Sachs, 1996]

Rainier von Sachs. Modelling and estimation of the time-varying structure of nonstationary time series. Technical report, Department of Statistics, Stanford University, 1996.

[Wahba, 1968]

Grace Wahba. On the distribution of some statistics useful in the analysis of jointly stationary time series. The Annals of Mathematical Statistics, 39(6):1849-1862, 1968.

[Wahba, 1971]

Grace Wahba. Some tests of independence for stationary multivariate time series. Journal of the Royal Statistical Society B, 33(1):153-166, 1971.

[Wahba, 1980]

Grace Wahba. Automatic smoothing of the log periodogram. Journal of the American Statistical Association, 75(36):122-132, 1980.

[Walden and Cristan, 1996]

Andrew T. Walden and A. Contreras Cristan. Matching pursuit by undecimated discrete wavelet transform for arbitrary-length time series. Technical Report TR-96-02, Imperial College of Science, Technology and Medicine, Statistics Section, 1996.

[Walden and Cristan, 1997]

Andrew T. Walden and Alberto Contreras Cristan. The phase-corrected undecimated discrete wavelet packet transform and the recurrence of high latitude interplanetary shock waves. Technical Report TR-97-03, Imperial College of Science, Technology and Medicine, Statistics Section, 1997.

We develop and apply advanced time-frequency methodology to examine the recurrence time between shock waves identified in a non-stationary time series of hourly-averaged southern hemisphere solar magnetic field magnitude data acquired by the Ulysses spacecraft. The discrete cyclic filtering steps of the maximal overlap discrete wavelet packet transform (MODWPT) are fully explained. Energy preservation is proven. With filter coefficients chosen from Daubechies least asymmetric class, the optimum time shifts to apply to ensure approximate zero phase filtering at every level of the MODWPT are studied, and applied to the wavelet packet coefficients to give phase-corrections which ensure alignment with the original time series. Also the time series values at each time are decomposed into details associated with each frequency band, and these line up perfectly with features in the original time series since the details are shown to arise through exact zero phase filtering. We carry out a level 4 phase-corrected MODWPT of the Ulysses magnetic field data, and show that the recurrence times of the shock waves previously determined via manual pattern-matching on the raw data match those times in the time-frequency plot where a broadband spectrum is obtained; in other words, the phase-corrected MODWPT provides an approach to picking the location of complicated events. Furthermore, the phase-corrected MODWPT time-frequency plot strongly suggests that the first shock wave is a composite of two events, possibly one associated with a corotating interaction region, and one due to a coronal mass ejection. This might explain why the first shock wave has been differently classified in recent studies.

[Walden and Prescott, 1983]

A. T. Walden and P. Prescott. Statistical distributions for tidal elevations. Geophysical Journal of the Royal Astronomical Society, 72(1):223-36, 1983.

Recent developments in the analysis of extreme sea-levels using the joint density function of surges and tides has generated interest in statistical modelling of the distributions of tidal elevations as encountered in British coastal waters. In this paper some relatively simple stochastic models are proposed and their suitability examined and compared using tidal heights determined for Newlyn and Portsmouth. Particular attention has been paid to the computational procedures involved in order to illustrate the theoretical and practical difficulties which may be encountered.

[Walden and White, 1984]

A. T. Walden and R. E. White. On errors of fit and accuracy in matching synthetic seismograms and seismic traces. Geophysical Prospecting, 32(5):871-891, 1984.

A synthetic seismogram that closely resembles a seismic trace recorded at a well may not be at all reliable for, say, stratigraphic interpretation around the well. The most accurate synthetic seismogram is, in general, not the one that displays the smallest errors to fit to the trace but the one that best estimates the noise on the trace. If the match is confined to a short interval of interest or if the seismic reflection wavelet is allowed to be unduly long, there is considerable danger of forcing a spurious fit that treats the noise on the trace as part of the seismic reflection signal instead of making a genuine match with the signal itself. This paper outlines tests that allow an objective and quantitative evaluation of the accuracy of any match and illustrates their application with practical examples.

[Walden and White, 1990]

A. T. Walden and R. E. White. Estimating the statistical bandwidth of a time series. Biometrika, 77:699-707, 1990.

[Walden and White, 1998]

A. T. Walden and R. E. White. Seismic wavelet estimation: A frequency domain solution to a geophysical noisy input-output problem. IEEE Transactions on Geoscience and Remote Sensing, 36(1):287-297, 1998.

In seismic reflection prospecting for oil and gas a key step is the ability to estimate the seismic wavelet (impulse response) traveling through the earth, Such estimation enables filters to be designed to deblur the recorded seismic time series and allows the integration of ``downhole'' and surface seismic data for seismic interpretation purposes. An appropriate model for the seismic time series is a noisy-input/noisy-output linear model, We tackle the estimation of the impulse response in the frequency domain by estimating its frequency response function. We use a novel approach where multiple coherence analysis is applied to the replicated observed output series to estimate the output signal-to-noise ratio (SNR) at each frequency. This, combined with an estimate of the ordinary coherence between observed input and observed output, and with the spectrum of the observed input and cross-spectrum of the observed input and output, enables estimation of the frequency response function. The methodology is seen to work well on real and synthetic data.

[Walden and Williams, 1993]

Andrew T. Walden and Mark L. Williams. Deconvolution, bandwidth, and the trispectrum. Journal of the American Statistical Association, 88(424):1323-1330, 1993.

Three important applications of the time series analysis are composed of deconvolution, bandwidth and the trispectrum. In geophysical exploration, non-Gaussian and non-invertible deconvolution was done to estimate the single shift parameter of the series. However, the limitations posed by deconvolution requires the study of the trispectrum, which establishes the use of kurtosis in phase correction during instances of band limitation. Analysis of the discrete-parameter trispectrum under standard linear models showed non-zero trispectrums in the inner and outer volumes.

[Walden et al., 1994]

A. T. Walden, E. McCoy, and D. B. Percival. The variance of multitaper spectrum estimates for real gaussian processes. IEEE Transactions on Signal Processing, 42(2):479-482, 1994.

Multitaper spectral estimation has proven very powerful as a spectral analysis method wherever the spectrum of interest is detailed and/or varies rapidly with a large dynamic range. In his original paper D.J. Thomson (1982) gave a simple approximation for the variance of a multitaper spectral estimate which is generally adequate when the spectrum is slowly varying over the taper bandwidth. The authors show that near zero or Nyquist frequency this approximation is poor even for white noise and derive the exact expression of the variance in the general case of a stationary real-valued time series. This expression is illustrated on an autoregressive time series and a convenient computational approach outlined. It is shown that this multitaper variance expression for real-valued processes is not derivable as a special case of the multitaper variance for complex-valued, circularly symmetric processes, as previously suggested in the literature.

[Walden et al., 1995a]

A. T. Walden, E. J. McCoy, and D. B. Percival. The effective bandwidth of a multitaper spectral estimator. Biometrika, 82(1):201-214, 1995.

[Walden et al., 1995b]

Andrew T. Walden, Donald B. Percival, and Emma J. McCoy. Spectrum estimation by wavelet thresholding of multitaper estimators. Technical report, Dept. of Mathematics, Imperial College of Science, Technology and Medicine,, 1995.

Current methods for producing a power spectrum estimate by wavelet thresholding apply thresholding to the empirical wavelet coefficients derived from the log periodogram. Unfortunately, the periodogram is a very poor spectrum estimation method when the true spectrum has a high dynamic range and/or is rapidly varying. Such spectra are common in science. Also, because of the form of the distribution of the log periodogram, complicated wavelet-dependent thresholding schemes are needed to try to force the problem into the `ideal' wavelet shrinkage strait-jacket. Instead, we start with a robust and powerful multitaper spectrum estimator. The logarithm of this estimator is close to Gaussian distributed provided at least five tapers are used, and this enables the computation of the correlation of the log spectrum estimator. For scale-independent `ideal' thresholding the correlation acts in an ideal way to strongly suppress `noise spikes' while leaving informative coarse-scale coefficients relatively unattentuated. This apparently rather crude aproach is seen to work very well in practice. By way of contrast, the progression of the variance of wavelet coefficients with scale can be accurately calculated so that it is also possible to compute scale-dependent `ideal' thresholds; in fact these do not lead to appealing spectrum smooths, undoubtedly due to the finite-sample size sensitivity of the various finely-tuned asymptotic `ideal' thresholds.

[Walden, 1982]

Andrew T. Walden. The Statistical Analysis of Extreme High Sea Levels Utilizing Data from the Solent Area. PhD thesis, University of Southampton, 1982.

[Walden, 1989]

A. T. Walden. Accurate approximation of a 0th order discrete prolate spheroidal sequence for filtering and data tapering. Signal Processing, 18(3):341-8, 1989.

The index-limited 0th order discrete prolate spheroidal sequence (DPSS) is very useful as both a data taper for spectral analysis and as a FIR filter since its frequency response has very low sidelobes. However its calculation is not straightforward. Kaiser (1966) produced a Bessel approximation to the continuous prolate spheroidal wave function. The author discusses how to sample the continuous Bessel expression in order to approximate the 0th order DPSS. The obvious approximation, with end points which involve the modified Bessel function evaluated at zero, is not the best. A much better result is obtained by slightly altering the sampling positions. A range of values for sample size and bandwidth are used to compare the recommended approximation with the actual 0th order DPSS. Differences are expressed in terms of sum of squared errors, by crossplotting corresponding sequence values, and by comparing magnitude squared transfer functions. In all cases the recommended approximation performs well.

[Walden, 1990a]

A. T. Walden. Improved low-frequency decay estimation using the multitaper spectral analysis method. Geophysical Prospecting, 38(1):61-86, 1990.

Seismic spectra exhibit very large dynamic ranges particularly at low frequencies. Estimation of low-frequency decay is very important for accurate modelling. However, when using traditional spectral estimates incorporating smoothing windows, too much sidelobe energy leaks from high power into low power areas. The multitaper method of spectral analysis, which uses a set of orthogonal data tapers, yields much less sidelobe contamination, while maintaining a stable estimate. The trace is tapered by each of a subset of the orthogonal tapers, and a raw spectral estimate produced in each case. These are combined to produce a final spectral estimate. The technique can be made adaptive by applying different weights to the different raw spectra at different frequencies. A comparison of seismic spectral estimation using this multitaper technique with a traditional approach having the same analysis bandwidth and stability demonstrates the very different estimates of spectral decay in the areas of high dynamic range.

[Walden, 1990b]

A. T. Walden. Maximum likelihood estimation of magnitude-squared multiple and ordinary coherence. Signal Processing, 19(1):75-82, 1990.

It is shown that the first and second derivatives of the probability density function can be written very simply in terms of the probability density function itself. As a result the iterative scheme for maximum likelihood estimation can be expressed in much simpler terms than done heretofore. A way of treating small maximum likelihood ordinary coherence estimates stated in the recent study is shown to be false. Illustrative plots are given showing the relationship between the standard multiple and ordinary coherence estimates, and maximum likelihood counterparts. An expression for the mean of the standard multiple coherence estimate is given in terms of generalized hypergeometric series and is shown to reduce to the correct form for ordinary coherence. A simple recursive formula is given for computing the mean.

[Walden, 1990c]

A. T. Walden. Variance and degrees of freedom of a spectral estimator following data tapering and spectral smoothing. Signal Processing, 20(1):67-79, 1990.

The equivalent degrees of freedom of spectral estimators resulting from data tapering combined with smoothing in the frequency domain are calculated using digital techniques for six different tapers and two types of spectral smoothing. The tapers have a 'kurtosis' ranging from 1 to 4 (i.e. from flat to quite spikey). The degrees of freedom calculated are compared to those expected using the standard approximate correction for tapering and an additional approximation. Both these approximations become less accurate as the kurtosis of the taper increases, but are still found to be quite satisfactory for practical purposes.

[Walden, 1991]

A. T. Walden. Wavelet estimation using the multitaper method. Geophysical Prospecting, 39(5):625-42, 1991.

An accurate estimate of the seismic wavelet on a seismic section is extremely important for interpretation of fine details on the section and for estimation of acoustic impedance. Thomson's (1982) multitaper method of cross-spectral estimation, which suffers little from side-lobe leakage, is applied and is compared with the result of using frequency smoothing with the Papoulis (1973) window. The multitaper method seems much less prone to estimating spuriously high coherences at very low frequencies. The wavelet estimated by the multitaper approach from the data used here is equivalent to imposing a low-frequency roll-off of some 48 dB/oct (below 3.91 Hz) on the amplitude spectrum. Using Papoulis smoothing the equivalent roll-off is only about 36 dB/oct. Thus the multitaper method gives a low-frequency decay rate of the amplitude spectrum which is some four times greater than for Papoulis smoothing. It also gives more consistent results across the section.

[Walden, 1994]

A. T. Walden. Interpretation of geophysical borehole data via interpolation of fractionally differenced white noise. Applied Statistics, 43(2):335-345, 1994.

[Walden, 1995]

A. T. Walden. Multitaper estimation of the innovation variance of a stationary time series. IEEE Transactions on Signal Processing, 43(1):181-187, 1995.

Accurate computation of the innovation variance of a stationary time series by a nonparametric method provides useful information to judge the quality of fit of parametric models for the time series. Previous estimators of the innovation variance have made use of raw periodogram ordinates, smoothed periodogram ordinates, or periodogram ordinates following tapering. Smoothing provides more degrees of freedom at each frequency but fewer independent estimates, whereas tapering reduces side-lobe leakage if the dynamic range of the spectrum is high but produces only two degrees of freedom at each frequency. Here, we investigate estimation of innovation variance from finite sample sizes by the use of multiple tapering. The tapers are designed to reduce side-lobe leakage and produce increased degrees of freedom at each frequency. It is demonstrated that the multiple tapering approach produces much better estimates of the innovations variance than the other methods when the spectrum has a high dynamic range and/or is rapidly varying. The multitaper bandwidth parameter W may be selected using an obvious heuristic approach or by an automatic method. The multitaper method is hence an attractive alternative to conventional techniques.

[Walden, 1997]

A. T. Walden. Estimated cross spectrum matrices and their inverses. Imperial College of Science, Technology and Medicine, Statistics Section, 1997.

[Walker, 1928]

Gilbert T. Walker. World weather. Monthly Weather Review, 56:167-170, 1928.

[Walter, 1994]

Gilbert G. Walter. Wavelets and Other Orthogonal Systems with Applications. CRC Press Inc., Boca Raton, 1994.

This book makes accessible to both mathematicians and engineers important elements of the theory, construction, and application of orthogonal wavelets. It is integrated with more traditional orthogonal series, such as Fourier series and orthogonal polynomials. It treats the interaction of both with generalized functions (delta functions), which have played an important part in engineering theory but whose rules are often vaguely presented. Unlike most other books that are excessively technical, this text/reference presents the basic concepts and examples in a readable form. Much of the material on wavelets has not appeared previously in book form. Applications to statistics, sampling theorems, and stochastic processes are given. In particular, the close affinity between wavelets and sampling theorems is explained and developed.

[Wang et al., 1997]

Yazhen Wang, Joseph E. Cavanaugh, and Changyong Song. Self-similarity index estimation via wavelets for locally self-similar processes. Department of Statistics, University of Missouri, 1997.

[Wang, 1995]

Yazhen Wang. Jump and sharp cusp detection by wavelets. Biometrika, 82(2):385-397, 1995.

A method proposed to detect jumps and sharp cusps in a function which is observed with noise, by checking if the wavelet transformation of the data has significantly large absolute values across fine scales. Asymptotic theory is established and practical implementation is discussed. The method is tested on simulated examples, and applied to stock market return data.

[Wang, 1996]

Yazhen Wang. Function estimation via wavelet shrinkage for long-memory data. Annals of Statistics, 24(2):466-484, 1996.

[Wang, 1997]

Yazhen Wang. Change curve estimation via wavelets. Journal of the American Statistical Association, to be published in 1998, 1997.

[Wei and Bovik, 1998]

D. Wei and A. C. Bovik. Enhancement of compressed images by optimal shift-invariant wavelet packet basis. Journal of Visual Communication and Image Representation, 9(1):15-24, 1998.

A novel postprocessing method based on the optimal shift-invariant wavelet packet (SIWP) representation and wavelet shrinkage is proposed to enhance compressed images. At the encoder, the optimal (in the mean square error sense) SIWP basis is searched using a fast optimization algorithm and the location of the best basis in the entire SIWP library is transmitted as overhead information to the decoder. The selected basis is jointly optimal in terms of both the time-frequency tiling and the relative time-domain offset (or shift) between a signal and its wavelet packet representation. After the decoder reconstructs the compressed image, the postprocessor performs wavelet shrinkage using the optimal basis. Due to its powerful adaptability, the method is shown to achieve a better trade-off between enhancement performance and decoder complexity than both the orthonormal wavelet transform and the undecimated wavelet transform-based methods.

[Weiss and Dixon, 1997]

L. G. Weiss and T. L. Dixon. Wavelet-based denoising of underwater acoustic signals. Journal of the Acoustical Society of America, 101(1):377-383, 1997.

Underwater environmental measurements of the ocean require signals that are free from unwanted backscatter and clutter. Removing these unwanted signal components usually amounts to applying some form of filtering technique such as a high pass filter, a bandpass filter, a Wiener filter, etc. These approaches however are limited in their abilities to remove acoustic returns that vary spectrally. This paper presents a multiresolution approach to removing unwanted backscatter from high-frequency underwater acoustic signals and compares it to high pass filtering of the same signals. The filtering approach presented applies wavelet transforms for signal recovery and denoising of high-frequency acoustic signals. It is shown that by computing a wavelet transform of the returned signals, applying a denoising technique, and then reconstructing the signal, additional unwanted backscatter can be removed.

[Weiss, 1993]

John Weiss. Translation invariance and the wavelet transform. Applied Mathematics Group, 1993.

A translation invariant wavelet transform algorithm is defined. The algorithm is an extension of the best basis approach and can be used to define translation invariant best bases and wavelet transforms. The computational cost is a factor of m greater than the usual algorithms, where m is the multiplier of the wavelet system. Some applications to transient detection are presented. A general form of an invariant wavelet transform is presented. This transform is shown to be invariant under a large group of symmetries described, most naturally, by the g-circulant transformations. The symmetries include translation and time-reversal of a periodic data vector. In our construction the expansion coefficients of g-circulant transformations of a data vector areshown to be simply related by periodic shifts of their expansion coefficients. Therefore, under g-circulant transformations the numerical values and ordering are invariant.

[West et al., 1997]

Mike West, Raquel Prado, and Andrew Krystal. Latent structure in non-stationary time series with application to studies of EEG traces. Technical Report 97-14, Institute of Statistics and Decision Sciences, Duke University, 1997.

[Weyrich and Warhola, 1998]

N. Weyrich and G. T. Warhola. Wavelet shrinkage and generalized cross validation for image denoising. IEEE Transactions on Image Processing, 7(1):82-90, 1998.

We present a denoising method based on wavelets and generalized cross validation and apply these methods to image denoising, We describe the method of modified wavelet reconstruction and show that the related shrinkage parameter vector can be chosen without prior knowledge of the noise variance by using the method of generalized cross validation, By doing so, we obtain an estimate of the shrinkage parameter vector and, hence, the image, which is very close to the best achievable mean-squared error result-that given by complete knowledge of the underlying clean image.

[Whitcher et al., 1998]

Brandon Whitcher, Simon D. Byers, Peter Guttorp, and Donald B. Percival. Testing for homogeneity of variance in time series: Long memory, wavelets and the Nile River. Submitted to Biometrika, 1998.

We consider the problem of testing for homogeneity of variance in a time series that has long memory structure. We demonstrate that a test whose null hypothesis is designed to be white noise can in fact be applied, on a scale by scale basis, to the discrete wavelet transform of long memory processes. In particular, we show that evaluating a normalized cumulative sum of squares test statistic using critical levels appropriate for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractional difference process. The point at which the test statistic, using the maximal overlap discrete wavelet transform, achieves its maximum value can be used to estimate the time of the unknown variance change. We apply our proposed test statistic on a time series of Nile River yearly minimum water levels covering the years 622 to 1284 AD. The test confirms an inhomogeneity of variance at short scales and identifies the change point around 720 AD, which coincides closely with the construction of a new device in 715 AD for measuring Nile River water levels.

[Whitcher, 1998]

Brandon Whitcher. Assessing Nonstationary Events in Time Series. PhD thesis, University of Washington, 1998.

[Wichern et al., 1976]

Dean W. Wichern, Robert B. Miller, and Der-Ann Hsu. Changes of variance in first-order autoregressive time series models -- with an application. Applied Statistics, 25(3):248-256, 1976.

[Wickerhauser, 1994]

Mladen Victor Wickerhauser. Adapted Wavelet Analysis from Theory to Software. A K Peters, Ltd., Wellesley, Massachusetts, 1994.

This detail-oriented text is intended for engineers and applied mathematicians who must write computer programs to perform wavelet and related analyses on real data. It should also be useful to the pure mathematician with questions about wavelet theory applications and to the instructor or student as a textbook in the mathematics and latest techniques in transient signal analysis and processing. Beginning with an overview of mathematical prerequisites, successive chapters rigorously examine the properties of waveforms used in adapted wavelet analysis: discrete ``fast'' Fourier transforms, orthogonal and biorthogonal wavelets, wavelet packets, and localized trigonometric or lapped orthogonal functions. Other chapters discuss the ``best-basis'' method,time-frequency analysis, and combinations of these algorithms useful for signal analysis, de-noising, and data compression. Each chapter discusses the technical aspects of implementation giving examples in pseudocode backed up with a Standard C source code (on optional diskette) and closes with a list of worked exercises.

[Wong et al., 1996]

Heung Wong, Wai-Cheung Ip, Yihui Luan, and Zhongjie Xie. Wavelet detection of jump points and an application to exchange rates. The Hong Kong Polytechnic University, Hong Kong, 1996.

[Wong, 1993]

P. W. Wong. Wavelet decomposition of harmonizable random processes. IEEE Transactions on Information Theory, 39(1):7-18, 1993.

The discrete wavelet decomposition of second-order harmonizable random processes is considered. The deterministic wavelet decomposition of a complex exponential function is examined, where its pointwise and bounded convergence to the function is proved. This result is then used for establishing the stochastic wavelet decomposition of harmonizable processes. The similarities and differences between the wavelet decompositions of general harmonizable processes and a subclass of processes having no spectral mass at zero frequency, e.g., those that are wide-sense stationary and have continuous power spectral densities, are also investigated. The relationships between the harmonization of a process and that of its wavelet decomposition are examined. Finally, certain linear operations such as addition, differentiation, and linear filtering on stochastic wavelet decompositions are considered. It is shown that certain linear operations can be performed term by term with the decomposition.

[Wood and Chan, 1994]

Andrew T. A. Wood and Grace Chan. Simulation of stationary Gaussian processes in [0,1]^d. Journal of Computational and Graphical Statistics, 3(4):409-432, 1994.

A method for simulating a stationary Gaussian process on a fine rectangular grid in [0,1]^d subset IR^d is described. It is assumed that the process is stationary with respect to translations of IR^d, but the method does not require the process to be isotropic. As with some other approaches to this simulation problem, our procedure uses discrete Fourier methods and exploits the efficiency of the fast Fourier transform. However, the introduction of a novel feature leads to a procedure that is exact in principle when it can be applied. It is established that sufficient conditions for it to be possible to apply the procedure are (1) the covariance function is summable on IR^d, and (2) a certain spectral density on the d-dimensional torus, which is determined by the covariance function on IR^d, is strictly positive. The procedure can cope with more than 50,000 grid points in many cases, even on a relatively modest computer. An approximate procedure is also proposed to cover cases where it is not feasible to apply the procedure in its exact form.

[Wornell and Oppenheim, 1992]

Gregory W. Wornell and Alan V. Oppenheim. Wavelet-based representations for a class of self-similar signals with application to fractal modulation. IEEE Transactions on Information Theory, 38(2):785-800, 1992.

A potentially important family of self-similar signals based upon a deterministic scale-invariance characterization is introduced. These signals, which are referred to as 'dy-homogeneous' signals because they generalize the well-known homogeneous functions, have highly convenient representations in terms of orthonormal wavelet bases. In particular, wavelet representations can be exploited to construct orthonormal self-similar bases for these signals. The spectral and fractal characteristics of dy-homogeneous signals make them appealing candidates for use in a number of applications. As one potential example, their use in a communications-based context is considered. Specifically, a strategy for embedding information into a dy-homogeneous waveform on multiple time-scales is developed. This multirate modulation strategy, called fractal modulation, is potentially well-suited for use with noisy channels of simultaneously unknown duration and bandwidth.

[Wornell, 1990]

Gregory W. Wornell. A Karhunen-Loéve-like expansion for 1/f processes via wavelets. IEEE Transactions on Information Theory, 36(4):859-861, 1990.

While so-called 1/f or scaling processes emerge regularly in modeling a wide range of natural phenomena, as yet no entirely satisfactory framework has been described for the analysis of such processes. Orthonormal wavelet bases are used to provide a new construction for nearly 1/f processes from a set of uncorrelated random variables.

[Wornell, 1993]

G. W. Wornell. Wavelet-based representations for the 1/f family of fractal processes. Proceedings of the IEEE, 81(10):1428-1450, 1993.

It is demonstrated that 1/f fractal processes are, in a broad sense, optimally represented in terms of orthonormal wavelet bases. Specifically, via a useful frequency-domain characterization for 1/f processes, the wavelet expansion's role as a Karhunen-Loeve-type expansion for 1/f processes is developed. As an illustration of potential, it is shown that wavelet-based representations naturally lead to highly efficient solutions to some fundamental detection and estimation problems involving 1/f processes.

[Wornell, 1996]

Gregory W. Wornell. Signal Processing with Fractals: A Wavelet Based Approach. Prentice Hall, New Jersey, 1996.

Fractal signals, derived from wavelet theory, are ideally suited for use in many engineering applications, ranging from communications to remote sensing. This book provides an introduction to wavelet theory from a signal processing perspective, and details fractal signals and a collection of practical wavelet-based techniques for representing and manipulating fractal signals in various applications.

[Wright, 1998]

Jonathan H. Wright. Testing for a structural break at unknown date with long-memory errors. Journal of Time Series Analysis, 19(3):369-376, 1998.

We derive the null limiting distributions of the usual sup-Wald and CUSUM tests for structural stability with an unknown potential break date in a polynomial regression model where the errors are I(d), -0.50, both tests diverge under the null so that the asymptotic size of either test is unity. For d<0, both tests converge to zero under the null so that the asymptotic size of either test is zero.

[Wu and Su, 1996]

Bing-Fei Wu and Yu-Lin Su. On the relationship between the self-similarities of fractal signals and wavelet transforms. In B. Boashash, N. Harle, and A. A. Zoubir, editors, International Symposium on Signal Processing and its Applications, pages 736-739, 1996.

Since many natural phenomena are occasionally defined as stochastic processes and the corresponding fractal characteristics are hidden from their correlation functions or power spectra, the topic would be of interest in signal processing. In this paper, we summarize the fractal dimensions and the relationship of the fractal in probability measure, variance, time series, time-averaging autocorrelation, ensemble-averaging autocorrelation, time-averaging power spectrum, average power spectrum and distribution functions for stationary and nonstationary processes. We also propose that the preservation of the one-dimensional self-similarity of a fractal signal is obtained by using the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT) with the perfect reconstruction quadrature mirror filter structure. Moreover, we extend the results to the two-dimensional case and point out the relationship of the self-similarities between the CWT and DWT of the fractal signals. A fractional Brownian motion process is provided as an example to show the results of this paper.

[Xia et al., 1996]

Xiang-Gen Xia, Jeffrey S. Geronimo, Douglas P. Hardin, and Bruce W. Suter. Design of prefilters for discrete multiwavelet transforms. IEEE Transactions on Signal Processing, 44(1):25-35, 1996.

The pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. The authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be thought of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete-time signals. The authors then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms.

[Xiang et al., 1994]

Gen Xia Xiang, C. C. J. Kuo, and Zhang Zhen. Wavelet coefficient computation with optimal prefiltering. IEEE Transactions on Signal Processing, 42(8):2191-7, 1994.

Discrete wavelet transform (DWT) is often used to approximate wavelet series transform (WST) and continuous wavelet transform (CWT), since it can be computed numerically. In this research, we first study the accuracy of the computed DWT coefficients obtained from the Shensa (see ibid., vol.40, no.10, p.2464-2482, 1992) algorithm as an approximate of the WST coefficients. Based on the accuracy analysis, we then propose a procedure to design optimal FIR prefilters used in the Shensa algorithm to reduce the approximation error. Finally, numerical examples are presented to demonstrate the performance of the optimal FIR prefilters.

[Yang et al., 1997]

Zi-Jiang Yang, Setsuo Sagara, and Teruo Tsuji. System impulse response identification using a multiresolution neural network. Automatica, 33(7)