The Annals of Statistics

Adaptive covariance estimation of locally stationary processes

Stéphane Mallat, George Papanicolaou, and Zhifeng Zhang

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It is shown that the covariance operator of a locally stationary process has approximate eigenvectors that are local cosine functions. We model locally stationary processes with pseudo-differential operators that are time-varying convolutions. An adaptive covariance estimation is calculated by searching first for a "best" local cosine basis which approximates the covariance by a band or a diagonal matrix. The estimation is obtained from regularized versions of the diagonal coefficients in the best basis.

Article information

Ann. Statist. Volume 26, Number 1 (1998), 1-47.

First available in Project Euclid: 28 August 2002

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Zentralblatt MATH identifier

Primary: 62M15: Spectral analysis
Secondary: 60G15: Gaussian processes

Locally stationary processes local cosine bases adaptive covariance estimation approximate Karhunen-Loeve basis


Mallat, Stéphane; Papanicolaou, George; Zhang, Zhifeng. Adaptive covariance estimation of locally stationary processes. Ann. Statist. 26 (1998), no. 1, 1--47. doi:10.1214/aos/1030563977.

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  • [1] Adak, S. (1995). Time-dependent spectral analysis of nonstationary time series. Technical report, Dept. Statistics, Stanford Univ.
  • [2] Asch, M., Kohler, W., Papanicolaou, G., Postel, M. and White, B. (1991). Frequency content of randomly scattered signals. SIAM Rev. 33 519-625.
  • [3] Benassi, A., Jaffard, S. and Roux, D. (1994). Elliptic Gaussian random processes. Preprint.
  • [4] Coifman, R. and Meyer, Y. (1991). Remarques sur l'analyse de Fourier a fen etre. C.R. Acad. Sci. Paris S´er. I 259-261.
  • [5] Coifman, R. and Wickerhauser, V. (1992). Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theory 38 713-718.
  • [6] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1-37.
  • [7] Donoho, D., Mallat, S. and von Sachs, R. (1996). Estimating covariances of locally stationary processes: consistency of best basis methods. In Proceedings of IEEE Time Frequence and Time-Scale Sy mposium, Paris, July 1996. IEEE, New York.
  • [8] Malvar, H. S. (1989). The LOT: transform coding without block effects. IEEE Trans. Acoust. Speech Signal Process. 37 553-559.
  • [9] Martin, W. and Flandrin, P. (1985). Wigner-Ville spectral analysis of non-stationary processes. IEEE Trans. Acoust. Speech Signal Process. 33 1461-1470.
  • [10] Meyer, Y. (1993). Wavelets and operators. Proceedings of Sy mposia in Applied Mathematics 47 35-57.
  • [11] Meyer, Y. (1993). Wavelets-algorithms and applications. SIAM.
  • [12] Neumann, M. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 38-76.
  • [13] Papanicolaou, G. and Weinry b, S. (1994). A functional limit theorem for waves reflected by a random medium. Appl. Math. Optim. 30 307-334.
  • [14] Priestley, M. B. (1965). Design relations for non-stationary processes. J. Roy. Statist. Soc. Ser. B 28 228-240.
  • [15] Priestley, M. B. (1965). Evolutionary spectra and non-stationary processes. J. Roy. Statist. Soc. Ser. B 27 204-237.
  • [16] Priestley, M. B. (1995). Wavelets and time-dependent spectral analysis. Technical Report 311, Dept. Statistics, Stanford Univ.
  • [17] Riedel, K. S. (1993). Optimal data-based kernel estimation of evolutionary spectra. IEEE Trans. Signal Proc. 41 2439-2447.
  • [18] Say eed, A. M. and Jones, D. L. (1995). Optimal kernels for nonstationary spectral estimation. IEEE Transactions on Signal Processing 43 478-491.
  • [19] von Sachs, R. and Schneider, K. (1993). Wavelet smoothing of evolutionary spectra by nonlinear thresholding. Journal of Appl. and Comput. Harmonic Analy sis 268-282.