Regularized adaptive long autoregressive spectral analysis

This paper is devoted to adaptive long autoregressive spectral analysis when i) very few data are available and ii) information does exist beforehand concerning the spectral smoothness and time continuity of the analyzed signals. The contribution is founded on two papers by Kitagawa and Gersch (1985). The first one deals with spectral smoothness in the regularization framework, while the second one is devoted to time continuity in the Kalman formalism. The present paper proposes an original synthesis of the two contributions. A new regularized criterion is introduced that takes bath pieces of information into account. The criterion is efficiently optimized by a Kalman smoother. One of the major features of the method is that it is entirely unsupervised. The problem of automatically adjusting the hyperparameters that balance data-based versus prior-based information is solved by maximum likelihood (ML). The improvement is quantified in the field of meteorological radar.

[1]  T. Kailath,et al.  A state-space approach to adaptive RLS filtering , 1994, IEEE Signal Processing Magazine.

[2]  Ken D. Sauer,et al.  A generalized Gaussian image model for edge-preserving MAP estimation , 1993, IEEE Trans. Image Process..

[3]  Pascal Larzabal,et al.  A new parametric approach for wind profiling with Doppler Radar , 1997 .

[4]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[5]  Steven Kay.  Recursive maximum likelihood estimation of autoregressive processes , 1983 .

[6]  G. Kitagawa,et al.  A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series , 1985, IEEE Transactions on Automatic Control.

[7]  José M. N. Leitão,et al.  Nonparametric estimation of mean Doppler and spectral width , 2000, IEEE Trans. Geosci. Remote. Sens..

[8]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[9]  P. Green.  Bayesian reconstructions from emission tomography data using a modified EM algorithm. , 1990, IEEE transactions on medical imaging.

[10]  IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 34. NO. 4, JULY 1996 Universal Multifractal Scaling of Synthetic , 1996 .

[11]  S.M. Kay,et al.  Spectrum analysis—A modern perspective , 1981, Proceedings of the IEEE.

[12]  David Knox Barton,et al.  Radar Technology Encyclopedia , 1997 .

[13]  L. R. Rabiner,et al.  An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition , 1983, The Bell System Technical Journal.

[14]  W. M. Carey,et al.  Digital spectral analysis: with applications , 1986 .

[15]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[16]  David McLaughlin,et al.  Dual-polarized Doppler radar measurements of oceanic fronts , 1999, IEEE Trans. Geosci. Remote. Sens..

[17]  Pham Dinh Tuan,et al.  Maximum likelihood estimation of the autoregressive model by relaxation on the reflection coefficients , 1988, IEEE Trans. Acoust. Speech Signal Process..

[18]  E. Parzen.  Some recent advances in time series modeling , 1974 .

[19]  J.B. Allen,et al.  A unified approach to short-time Fourier analysis and synthesis , 1977, Proceedings of the IEEE.

[20]  Jont B. Allen,et al.  Short term spectral analysis, synthesis, and modification by discrete Fourier transform , 1977 .

[21]  Jérôme Idier,et al.  Convex half-quadratic criteria and interacting auxiliary variables for image restoration , 2001, IEEE Trans. Image Process..

[22]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[23]  G. Demoment,et al.  A Bayesian method for long AR spectral estimation: a comparative study , 1996, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[24]  D. M. Titterington,et al.  A Study of Methods of Choosing the Smoothing Parameter in Image Restoration by Regularization , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[25]  C. Swanson.  On spectral estimation , 1962 .

[26]  G. Kitagawa.  Non-Gaussian State—Space Modeling of Nonstationary Time Series , 1987 .

[27]  T. Ulrych,et al.  Time series modeling and maximum entropy , 1976 .

[28]  P. Hall,et al.  Common Structure of Techniques for Choosing Smoothing Parameters in Regression Problems , 1987 .

[29]  W. Gersch,et al.  A smoothness priors long AR model method for spectral estimation , 1985, IEEE Transactions on Automatic Control.

[30]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[31]  H. Akaike.  A new look at the statistical model identification , 1974 .

[32]  R. Shumway,et al.  AN APPROACH TO TIME SERIES SMOOTHING AND FORECASTING USING THE EM ALGORITHM , 1982 .

[33]  D. Titterington.  Common structure of smoothing techniques in statistics , 1985 .

[34]  Yves Goussard,et al.  GCV and ML Methods of Determining Parameters in Image Restoration by Regularization: Fast Computation in the Spatial Domain and Experimental Comparison , 1993, J. Vis. Commun. Image Represent..

[35]  H. Akaike.  Statistical predictor identification , 1970 .