The SLEX Model of a Non-Stationary Random Process

  • Hernando Ombao
  • Jonathan Raz
  • Rainer von Sachs
  • Wensheng Guo


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.

Bootstrap Fourier functions Haar wavelet representation locally stationary time series periodograms SLEX functions spectral estimation stationary time series 


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Copyright information

© The Institute of Statistical Mathematics 2002

Authors and Affiliations

  • Hernando Ombao
    • 1
  • Jonathan Raz
    • 2
  • Rainer von Sachs
    • 3
  • Wensheng Guo
    • 2
  1. 1.Departments of Statistics and PsychiatryUniversity of PittsburghPittsburghU.S.A
  2. 2.Division of BiostatisticsUniversity of PennsylvaniaPhiladelphiaU.S.A
  3. 3.Institut de StatistiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium

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