Abstract
We propose a new model for non-stationary random processes to represent time series with a time-varying spectral structure. Our SLEX model can be considered as a discrete time-dependent Cramér spectral representation. It is based on the so-called Smooth Localized complex EXponential basis functions which are orthogonal and localized in both time and frequency domains. Our model delivers a finite sample size representation of a SLEX process having a SLEX spectrum which is piecewise constant over time segments. In addition, we embed it into a sequence of models with a limit spectrum, a smoothly in time varying “evolutionary” spectrum. Hence, we develop the SLEX model parallel to the Dahlhaus (1997, Ann. Statist., 25, 1–37) model of local stationarity, and we show that the two models are asymptotically mean square equivalent. Moreover, to define both the growing complexity of our model sequence and the regularity of the SLEX spectrum we use a wavelet expansion of the spectrum over time. Finally, we develop theory on how to estimate the spectral quantities, and we briefly discuss how to form inference based on resampling (bootstrapping) made possible by the special structure of the SLEX model which allows for simple synthesis of non-stationary processes.
Similar content being viewed by others
References
Brillinger, D. R. (1981). Time Series: Data Analysis and Theory, McGraw-Hill, New York.
Coifman, R. and Wickerhauser, M. (1992). Entropy based algorithms for best basis selection, IEEE Trans. Inform. Theory, 32, 712–718
Dahlhaus, R. (1997). Fitting time series models to nonstationary processes, Ann. Statist., 25, 1–37.
Dahlhaus, R. and Janas, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis, Ann. Statist., 24, 1934–1963.
Daubechies, I. (1992). Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.
Donoho, D., Mallat, S. and von Sachs, R. (1998). Estimating covariances of locally stationary processes: Rates of convergence of best basis methods, Tech. Report, No. 517, Statistics Department, Stanford University, California.
Franke, J. and Härdle, W. (1992). On bootstrapping kernel spectral estimates, Ann. Statist., 20, 121–145.
Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications, Lecture Notes in Statist., No. 129, Springer, New York.
Nason, G., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum, J. Roy. Statist. Soc. Ser. B, 62, 271–292.
Neumann, M. H. (2000). Multivariate wavelet thresholding in anisotropic function spaces, Statist. Sinica, 10, 399–432.
Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra, Ann. Statist., 25, 38–76.
Ombao, H. (1999). Statistical analysis of non-stationary time series, Ph.D. dissertation, Department of Biostatistics, University of Michigan.
Ombao, H., von Sachs, R. and Guo, W. (2000). Estimation and inference for time-varying spectra of locally stationary SLEX processes, Discussion Paper, No. 00/27, Institut de Statistique, UCL, Louvain-la-Neuve and Proceedings of the 2nd International Symposium on Frontiers of Time Series Modeling, Nara, Japan, December 14–17, 2000.
Ombao, H., Raz, J., von Sachs, R. and Malow, B. (2001a). Automatic statistical analysis of bivariate nonstationary time series, J. Amer. Statist. Assoc., 96, 543–560.
Ombao, H., Raz, J., Strawderman, R. and von Sachs, R. (2001b). A simple generalised cross validation method of span selection for periodogram smoothing, Biometrika, 88(4) (to appear).
Priestley, M. (1965). Evolutionary spectra and non-stationary processes, J. Roy. Statist. Soc. Ser. B, 28, 228–240.
Priestley, M. (1981). Spectral Analysis and Time Series, Academic Press, London.
Wickerhauser, M. (1994). Adapted Wavelet Analysis from Theory to Software, IEEE Press, Wellesley, Massachusetts.
Author information
Authors and Affiliations
About this article
Cite this article
Ombao, H., Raz, J., von Sachs, R. et al. The SLEX Model of a Non-Stationary Random Process. Annals of the Institute of Statistical Mathematics 54, 171–200 (2002). https://doi.org/10.1023/A:1016130108440
Issue Date:
DOI: https://doi.org/10.1023/A:1016130108440