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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.