Past Lectures

• lecture 1 (Monday 1/7):
  • what is spectral analysis? [overhead 1:1, i.e., Chapter 1, first overhead (PDF)]
  • time series & time series analysis [1:2 to 1:7]
  • lagged scatter plots [1:8 to 1:13]
  • sample autocorrelation sequence [1:14 to 1:18]
  • modeling of time series [1:19]
  • frequency domain modeling [1:20 to 1:25]
  • the spectrum [1:26 to 1:39]
  • nonparametric estimation of the spectrum [1:40 to 1:46]
  • parametric estimation of the spectrum [1:47 to 1:51]
  • `industrial strength' theory of spectral analysis [1:52 to 1:55]
  • uses of spectral analysis [1:56]
• lecture 2 (Wednesday 1/9, first hour):
  • definition of a stochastic process [2:1 (LaTeX; PostScript; PDF)]
  • basic theory for stochastic processes [2:2]
  • definition of a stationary process [2:3]
  • basic theory for stationary processes [2:4 to 2:6]
• lecture 3 (Wednesday 1/9, second hour):
  • white noise processes [2:7]
  • moving average processes [2:8]
  • autoregressive processes [2:9]
  • harmonic processes [2:10 and 2:11]
  • stationary processes as models [2:12]
• lecture 4 (Friday 1/11):
  • overview of Fourier theory[3:1 (LaTeX; PostScript; PDF)]
  • four "flavors" of Fourier theory [3:2]
  • Fourier theory for periodic functions (continuous time/discrete frequency) [3:3]
  • Parseval's theorem [3:4]
  • discrete power spectrum [3:5]
  • Fourier theory for square integrable functions (continuous time/continuous frequency) [3:6]
  • energy spectral density function [3:7 and 3:8]
• lecture 5 (Monday 1/14):
  • time-limited and band-limited functions [3:9]
  • similarity theorem [3:10]
  • equivalent width [3:11]
  • fundamental uncertainty relationship [3:12]
  • definition of convolution and the convolution theorem [3:13]
  • convolution as a smoothing operation [3:14]
  • example of smoothing via convolution [3:15]
  • cross correlations and autocorrelations [3:16]
• lecture 6 (Wednesday 1/16, first hour):
  • Fourier theory for square summable sequences (discrete time/continuous frequency) [3:17 to 3:19]
  • approximating a Fourier transform from a finite sample (loss of resolution, leakage and the Gibbs phenomenon) [3:20]
  • Cesàro sums [3:21]
  • tradeoffs in using convergence factors (windows) [3:22]
• lecture 7 (Wednesday 1/16, second hour):
  • aliasing [3:23]
  • discrete/continuous concentration problem [3:24 and 3:25]
  • Fourier theory for finite sequences (discrete time/discrete frequency) [3:26]
  • zero padding [3:27]
• lecture 8 (Friday 1/18; audio (32 megabytes)):
  • spectral representation theorem for stationary processes [4:1 to 4:4 (LaTeX; PostScript; PDF)]
  • relationships between integrated spectrum, autocovariance sequence (acvs), and spectral density function (sdf) [4:5]
  • spectral representation theorem for continuous parameter and complex-valued stationary processes [4:6]
  • basic properties of integrated spectrum and sdf, and Wold's theorem [4:7]
• lecture 9 (Wednesday 1/23, first hour):
  • spectrum for white noise [4:8]
  • spectrum for a harmonic process [4:9]
  • classification of spectra (Lebesgue decomposition theorem) [4:10]
  • purely continuous, purely discrete, discrete and mixed spectra [4:11]
  • sampling and aliasing [4:12]
  • comparison of sdf's and acvs's as informative characterizations
• lecture 10 (Wednesday 1/23, second hour):
  • convolution and energy spectra [5:1 (LaTeX; PostScript; PDF)]
  • definition of linear time invariant (LTI) analog filter [5:2 and 5:3]
  • eigenfunctions and eigenvalues of LTI analog filter [5:3]
  • spectral density function (sdf) of a filtered stationary process [5:4]
  • transfer, gain and phase functions [5:5]
  • definition of linear time invariant (LTI) digital filter [5:6]
  • impulse response sequence [5:7]
  • sdf for a moving average process [5:8]
  • sdf for an autoregressive process [5:9]
  • low-pass, high-pass, band-pass and cascaded filters [5:10]
  • filtering interpretation of sdf [5:11]
• lecture 11 (Friday 1/25; audio (30 megabytes)):
  • simple examples of LTI filters [5:12 and 5:13]
  • basic ideas behind nonparametric spectral estimation [6:1 (LaTeX; PostScript; PDF)]
  • sample mean as estimator of process mean [6:2]
  • best linear unbiased estimator of process mean [6:3] [SKIPPED]
  • unbiased estimator of autocovariance sequence (acvs) [6:4]
  • biased estimator of acvs [6:5]
• lecture 12 (Monday 1/28):
  • definition of the periodogram [6:6]
  • inverse Fourier transform of the periodogram [6:7]
  • deserata for sampling properties of the periodogram [6:8]
  • expected value of the periodogram [6:9]
  • Fejer's kernel [6:10]
  • bias and the periodogram [6:11]
• lecture 13 (Wednesday 1/30, first hour; audio (26 megabytes)):
  • bias reduction via tapering [6:12]
  • spectral windows [6:13]
  • examples of data tapers and spectral windows [6:14]
  • examples of leakage suppression [6:15]
  • connection between leakage and mismatch between ends of time series
• lecture 14 (Wednesday 1/30, second hour; audio (31 megabytes)):
  • autocovariance sequence for direct spectral estimators [6:16]
  • normalization of data tapers [6:17 and 6:18]
  • tapering and estimation of process mean [6:19]
  • bias reduction via prewhitening [6:20 and 6:21]
  • bias reduction via rejection filtration [6:22]
  • statistical properties of direct spectral estimators
    • Gaussian white noise, general taper [6:23]
    • Gaussian white noise, default taper [6:24]
  • role of chi-square distribution [6:24 and 6:25]
• lecture 15 (Friday 2/1; audio (133 megabytes)):
  • statistical properties of direct spectral estimators [continued]
    • colored noise, default taper [6:26]
    • colored noise, general taper [6:27]
  • inconsistency of direct spectral estimators [6:28]
  • probability density function of direct spectral estimators [6:29]
  • periodogram-based test for white noise [6:30]
  • smoothing direct spectral estimators [6:31]
• lecture 16 (Monday 2/4; audio (103 megabytes)):
  • discretely smoothed direct spectral estimators [6:32]
  • smoothing windows and lag windows [6:33]
  • lag window spectral estimators [6:34]
  • conditions on smoothing windows [6:35]
  • smoothing window bandwidth [6:36]
  • expected value of lag window spectral estimators; spectral window [6:37 and 6:38]
• lecture 17 (Wednesday 2/6, first hour; audio (129 megabytes)):
  • smoothing window bias [6:39]
  • reshaped lag windows [6:40] [SKIPPED]
  • large sample variance/covariance for lag window estimators [6:41 to 6:43]
  • large sample distribution of lag window estimators via chi-square approximation [6:44]
• lecture 18 (Wednesday 2/6, second hour):
  • confidence intervals for the true spectral density function [6:45]
  • percentages points for the chi-square distribution [6:46]
  • confidence intervals for log of spectral density function [6:47]
  • examples of lag windows and smoothing windows [6:48]
    • Bartlett lag window [6:49]
    • Daniell lag window [6:50 amd 6:51]
• lecture 19 (Friday 2/8):
  • examples of lag windows and smoothing windows [6:48]
    • Parzen lag window [6:52]
    • Papoulis lag window [6:53]
  • example of reshaped lag window [6:54] [SKIPPED]
  • criteria for choosing lag window [6:55 and 6:56]
  • criteria for choosing lag window parameter; spectral bandwidth [6:57]
  • estimation of spectral bandwidth [6:58 and 6:59]
• lecture 20 (Monday 2/11; audio (137 megabytes)):
  • automatic smoothing of log spectral estimators [6:60]
  • computational pathways [6:61 and 6:62]
  • Welch's overlapped segment averaging (WOSA) spectral estimation: motivation [6:63]
  • definition of WOSA [6:64]
  • first moment properties of WOSA estimator [6:65]
  • variance of WOSA estimator [6:66]
  • distribution of WOSA estimator [6:67]
  • advantages and disadvantages of the WOSA estimator [6:68]
• lecture 21 (Wednesday 2/13, first hour; audio (152 megabytes)):
  • examples of nonparametric spectral estimation
    • ocean wave data [6:69 to 6:72]
    • ice profile data [6:73 and 6:74]
• lecture 22 (Wednesday 2/13, second hour; audio (130 megabytes)):
  • intermezzo: Einstein (1914) and spectral estimation
  • motivation for multitaper spectral estimation [7:1 (LaTeX; PostScript; PDF)]
  • outline of discussion on multitapering [7:2]
  • basics of multitapering [7:3 and 7:4]
  • DPSS multitapering [7:5 and 7:6]
• lecture 23 (Friday 2/15):
  • sine multitapering [7:7 and 7:8]
  • information recovery via multitapering [7:9]
  • multitapering of white noise [7:10 to 7:13]
• lecture 24 (Wednesday 2/20, first hour):
  • definition of quadratic spectral estimators [7:14]
  • examples of quadratic spectral estimators [7:15]
  • multitaper representation of quadratic spectral estimators [7:16 and 7:17]
  • first moment of quadratic estimators
    • time/frequency domain formulations [7:18]
    • white noise and unbiasedness [7:19]
  • first moment of quadratic estimators
    • incorportion of resolution bandwidth [7:20]
  • broad-band and local bias
    • bounding local bias [7:21]
    • bounding broad-band bias [7:22]
  • minimization of broad-band bias [7:23]
  • managing bias and variance [7:24]
• lecture 25 (Wednesday 2/20, second hour):
  • adaptive multitaper estimation [7:25 and 7:26]
  • example: ocean wave data [7:27]
  • basic ideas behind parametric spectral estimation [9:1 (LaTeX; PostScript; PDF)]
  • rationale for using autoregressive models [9:2]
  • Yule-Walker equations [9:3]
  • Yule-Walker estimator of AR parameters [9:4 and 9:5]
• lecture 26 (Friday 2/22):
  • best linear predictors [9:6 to 9:8]
  • time-reversed processes and forward/backward prediction errors [9:9]
  • best predictors [9:10] [SKIPPED]
  • Levinson-Durbin recursions [9:11 to 9:13]
• lecture 27 (Monday 2/25):
  • reflection coefficients [9:14]
  • reversed Levinson-Durbin recursions [9:15]
  • relationship to Cholesky decompostion [9:16 and 9:17]
  • simulation of AR processes [9:18 and 9:19]
  • Yule-Walker estimator and examples [9:20]
• lecture 28 (Wednesday 2/27, first hour):
  • Burg's algorithm and example [9:21 to 9:23]
  • maximum entropy (ME) argument [9:24]
  • maximum entropy spectral analysis (MESA) [9:25]
  • critique of ME principle [9:26]
  • forward least squares estimator [9:27]
  • backward and forward/backward least squares estimators [9:28]
  • least squares estimators and the Yule-Walker estimator [9:29]
• lecture 29 (Wednesday 2/27, second hour):
  • maximum likelihood (ML) estimators [9:30 to 9:33]
  • large sample distribution for ML and other estimators [9:34]
  • AR order selection [9:35]
  • pacs criterion [9:36]
  • Akaike's FPE criterion [9:37]
  • Akaike's AIC criterion [9:38]
  • fitted AR models and prewhitening [9:39 and 9:40]
• lecture 30 (Friday 2/29):
  • implicit ARMA(p,q) modeling [9:41]
  • example: ocean wave data [9:42 and 9:43]
  • confidence intervals based on AR spectral estimators [9:44]
  • sdf estimation via moving average models [9:45 to 9:47]
  • harmonic analysis: basic concepts [10:1 (LaTeX; PostScript; PDF)]
  • purely discrete (line), discrete and mixed spectra [10:2]
  • discrete spectra with known frequency [10:3]
  • least square estimation of parameters for harmonic model [10:4]
  • least squares estimates at Fourier frequencies [10:5]
• lecture 31 (Monday 3/3):
  • connection between least squares estimators and the periodogram [10:6]
  • determination of unknown frequencies in discrete spectra [10:7 and 10:8]
  • role of data tapering in frequency estimation [10:9 and 10:10]
• lecture 32 (Monday 3/10):
  • tests for periodicity (introduction) [10:11]
  • Schuster's test [10:12]
  • Fisher's test [10:13]
  • Siegel's test [10:14 and 10:15]
  • example on ocean noise time series [10:16 and 10:17]
  • Thomson's F-test for periodicity [10:18]

Upcoming Lectures

• lecture 33 (Wednesday 3/12, first hour):
  • multiple time series: general comments [v2-1:1 (LaTeX; PostScript; PDF)]
  • example: bivariate ocean wave series [v2-1:2]
  • bivariate stationarity and the cross covariance sequence [v2-1:3]
  • the cross correlation sequence [v2-1:4]
  • examples of bivariate stationary processes
    • uncorrelated processes [v2-1:4]
    • processes correlated at simultaneous times [v2-1:5]
    • linear regression with delay [v2-1:6]
    • linear filtering with noise [v2-1:7]
  • definition of cross spectrum [v2-1:8]
  • interpretation of cross spectrum via spectral representation theorem [v2-1:9]
  • co-spectrum and quadrature spectrum [v2-1:10 and v2-1:11]
  • cross amplitude and phase spectra; group delay [v2-1:11]
  • complex coherency; magnitude squared coherence; spectral matrix [v2-1:12]
  • examples of cross spectra and related spectra
    • two uncorrelated processes [v2-1:13]
    • processes correlated at simultaneous times [v2-1:14]
    • linear regression with delay [v2-1:15]
    • linear filtering with noise [v2-1:16]
    • theoretical model for bivariate ocean wave data [v2-1:17 to v2-1:19]
• lecture 34 (Wednesday 3/12, second hour):
  • estimation of ccvs & ccs [v2-1:20]
  • example: bivariate ocean wave series [v2-1:20]
  • cross periodogram [v2-1:21]
  • alignment bias [v2-1:22]
  • aleviating alignment bias [v2-1:23]
  • estimation of msc [v2-1:24 and v2-1:25]
  • direct cross spectrum estimator [v2-1:26]
  • multitaper cross spectrum estimator [v2-1:27]
  • multitaper-based estimator of msc [v2-1:28]
  • example: bivariate ocean wave series [v2-1:29 and v2-1:30]

Material to be Covered if Time Permits

• from Chapter 10
  • complex-valued frequency domain regression [10:19]
  • F-test for periodicity [10:20]
  • reshaped sdf estimator [10:21]
  • completing a harmonic analysis: overview [10:22]
  • approximate conditional least squares [10:23]
  • exact conditional least squares [10:24]
  • exact unconditional least squares [10:25]
  • frequency domain regression [10:26]
  • completing a harmonic analysis: residuals [10:27]
  • completing a harmonic analysis: handling the background continuum [10:28]
  • harmonic analysis of river flow data [10:29 to 10:34]
  • parametric harmonic analysis & pseudo-AR processes [10:35 and 10:36]
  • frequency estimation via AR spectral estimators [10:37]
  • problems with AR harmonic analysis [10:38]
  • parametric harmonic analysis of river flow data [10:39 and 10:40]
  • use of SVD in harmonic analysis [10:41 to 10:46]
  • SVD approach used with river flow data [10:47 to 10:48]
• from Chapter 1, volume 2
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