On adaptive covariance and spectrum estimation of locally stationary multivariate processes - Publication - Bridge of Knowledge


On adaptive covariance and spectrum estimation of locally stationary multivariate processes


When estimating the correlation/spectral structure of a locally stationary process, one has to make two important decisions. First, one should choose the so-called estimation bandwidth, inversely proportional to the effective width of the local analysis window, in the way that complies with the degree of signal nonstationarity. Too small bandwidth may result in an excessive estimation bias, while too large bandwidth may cause excessive estimation variance. Second, but equally important, one should choose the appropriate order of the spectral representation of the signal so as to correctly model its resonant structure – when the order is too small, the estimated spectrum may not reveal some important signal components (resonances), and when it is too high, it may indicate the presence of some nonexistent components. When the analyzed signal is not stationary, with a possibly time-varying degree of nonstationarity, both the bandwidth and order parameters should be adjusted in an adaptive fashion.Thepaperpresentsandcomparesthreeapproachesallowingforunifiedtreatmentoftheproblem of adaptive bandwidth and order selection for the purpose of identification of nonstationary vector autoregressive processes: the cross-validation approach, the full cross-validation approach, and the approach that incorporates the multivariate version of the generalized Akaike’s final prediction error criterion. It is shown that the latter solution yields the best results and, at the same time, is very attractive from the computational viewpoint.


  • 1 4


  • 0

    Web of Science

  • 1 4


Authors (3)

Cite as

Full text

download paper
downloaded 27 times
Publication version
Accepted or Published Version
Creative Commons: CC-BY-NC-ND open in new tab



artykuł w czasopiśmie wyróżnionym w JCR
Published in:
AUTOMATICA no. 82, pages 1 - 12,
ISSN: 0005-1098
Publication year:
Bibliographic description:
Niedźwiecki M., Ciołek M., Kajikawa Y.: On adaptive covariance and spectrum estimation of locally stationary multivariate processes// AUTOMATICA. -Vol. 82, (2017), s.1-12
Digital Object Identifier (open in new tab) 10.1016/j.automatica.2017.04.033
Bibliography: test
  1. Akaike, H. (1971). Autoregressive model fitting for control. Annals of the Institute of Statistical Mathematics, 23, 163-180. open in new tab
  2. Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716-723. open in new tab
  3. Bunke, O., Droge, B., & Polzehl, J. (1999). Model selection, transformations and variance estimation in nonlinear regression. Statistics, 33, 197-240. open in new tab
  4. Burg, J.P. (1967). Maximum entropy spectral analysis. In Proc. 37th meet. society of exploration geophysicists.
  5. Burg, J. P. (1975). Maximum entropy spectral analysis. (Ph.D. Dissertation), Stanford, CA: Dept. of Geophysics, Stanford University.
  6. Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. The Annals of Statistics, 30, 351-413. open in new tab
  7. Dahlhaus, R. (2009). Local inference for locally stationary time series based on the empirical spectral measure. Journal of Econometrics, 151, 101-112. open in new tab
  8. Dahlhaus, R. (2012). Locally stationary processes. Handbook of Statistics, 25, 1-37. open in new tab
  9. Dahlhaus, R., & Giraitis, L. (1998). On the optimal segment length for parameter estimates for locally stationary time series. Journal of Time Series Analysis, 19, 629-655. open in new tab
  10. Epanechnikov, V. A. (1969). Non-parametric estimation of a multivariate probabil- ity density. Theory of Probability and its Applications, 14, 153-158. open in new tab
  11. Ferrante, A., Masiero, C., & Pavon, M. (2012). Time and spectral domain relative entropy: A new approach to multivariate spectral estimation. IEEE Transactions on Automatic Control, 57, 2561-2575. open in new tab
  12. Friedl, H., & Stampfer, E. (2002). Cross-validation. In A. H. El-Shaarawi, & W. W. Piegorsch (Eds.), Encyclopedia of environmetrics. Vol. 1 (pp. 452-460). Wiley. open in new tab
  13. Fu, Z., Chan, S.-C., Di, X., Biswal, B., & Zhang, Z. (2014). Adaptive covariance estimation of non-stationary processes and its application to infer dynamic connectivity from fMRI. IEEE Transactions on Biomedical Circuits and Systems, 8, 228-239.
  14. Goldenshluger, A., & Nemirovski, A. (1997). On spatial adaptive estimation of nonparametric regression. Mathematical Methods of Statistics, 6, 135-170.
  15. Katkovnik, V. (1999). A new method for varying adaptive bandwidth selection. IEEE Transactions on Signal Processing, 47, 2567-2571. open in new tab
  16. Niedźwiecki, M. (1984). On the localized estimators and generalized Akaike's criteria. IEEE Transactions on Automatic Control, 29, 970-983. open in new tab
  17. Niedźwiecki, M. (1985). Bayesian-like autoregressive spectrum estimation in the case of unknown process order. IEEE Transactions on Automatic Control, 30, 950-961. open in new tab
  18. Niedźwiecki, M. (1993). Statistical reconstruction of multivariate time series. IEEE Transactions on Signal Processing, 41, 451-457. open in new tab
  19. Niedźwiecki, M. (2000). Identification of time-varying processes. Wiley, 2000. Niedźwiecki, M. (2010). Easy recipes for cooperative smoothing. Automatica, 46, 716-720. open in new tab
  20. Niedźwiecki, M. (2012). Locally adaptive cooperative Kalman smoothing and its application to identification of nonstationary stochastic systems. IEEE Transactions on Signal Processing, 60, 48-59. open in new tab
  21. Niedźwiecki, M., & Gackowski, S. (2011). On noncausal weighted least squares identification of nonstationary stochastic systems. Automatica, 47, 2239-2245. open in new tab
  22. Niedźwiecki, M., & Gackowski, S. (2013). New approach to noncausal identification of nonstationary stochastic FIR systems subject to both smooth and abrupt parameter changes. IEEE Transactions on Automatic Control, 58, 1847-1853. open in new tab
  23. Niedźwiecki, M., & Guo, L. (1991). Nonasymptotic results for finite-memory WLS filters. IEEE Transactions on Automatic Control, 36, 515-522. open in new tab
  24. Priestley, M. B. (1965). Evolutionary spectra and non-stationary processes. Journal of the Royal Statistical Society, Series B, 27, 204-237. open in new tab
  25. Rissanen, J. (1978). Modeling by shortest data descriptiona. Automatica, 14, 465-471. open in new tab
  26. Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461-464. open in new tab
  27. Söderström, T., & Stoica, P. (1988). System identification. Englewood Cliffs NJ: Prentice-Hall.
  28. Stanković, L. (2004). Performance analysis of the adaptive algorithm for bias-to- variance tradeoff. IEEE Transactions on Signal Processing, 52, 1228-1234. open in new tab
  29. Stoica, P., & Moses, R. L. (1997). Introduction to spectral analysis. Prentice Hall. open in new tab
  30. Taylor, R. L. (1978). Stochastic convergence of weighted sums of random elements in linear spaces. Lecture Notes in Mathematics, 672. open in new tab
Verified by:
Gdańsk University of Technology

seen 123 times

Recommended for you

Meta Tags