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   -> Volume 7, Issue 1

Preprint: On wavelets and non-stationary time series (von Sachs)
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Rainer von Sachs (

PostPosted: Wed Dec 17, 1997 5:47 pm    
Subject: Preprint: On wavelets and non-stationary time series (von Sachs)
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#1 Preprint: On wavelets and non-stationary time series (von Sachs)

With some delay, I finally managed to put up this information about a
series of statistical papers of mine, together with various
co-authors, on using wavelet methods (time-frequency/time-scale
methods) for modeling and estimation of the time-varying structure of
non-stationary time series. Here is a complete list of those, with
reference to my

homepage in Germany and my

homepage in the USA,

for fast internet access.

I added the abstracts of the more recent technical reports (5)-(8).

(1) von Sachs, R., Schneider, K. Wavelet Smoothing of Evolutionary
Spectra by Non-linear Thresholding. Appl. Comp. Harmonic Anal. 3
(1996), 268--282.

(2) Neumann, M.H., von Sachs, R. Wavelet Thresholding in Anisotropic
Function Classes and Application to Adaptive Estimation of
Evolutionary Spectra. Annals of Statistics 25 (1997), 38--76.

(3) Dahlhaus, R., Neumann, M.H., von Sachs, R. Non-linear Wavelet Estimation
of Time--Varying Autoregressive Processes. Bernoulli 4 (1998), to

(4) von Sachs, R. Modelling and Estimation of the Time-varying Structure of
Nonstationary Time Series.
Technical Report 503, Stanford Statistics Department, and REBRAPE
(Brazilian Journal of Probability and Statistics), 10, 1996, 181--204.

(5) von Sachs, R., MacGibbon, B. Non-parametric Curve Estimation by Wavelet
Thresholding with Locally Stationary Errors. Technical Report AGTM 179,
University of Kaiserslautern, 1997, submitted for publication.

Abstract: We use non-linear wavelet thresholding to estimate a
regression or trend function from noisy time-varying data which is an
important aspect, e.g, in the modeling of biological phenomena. Here,
the covariance of the noise does not need to be stationary, but is
allowed to change slowly over time. We develop a procedure to adapt
existing threshold rules to such situations. Moreover, in the model of
curve estimation in Besov class with locally stationary errors, we
derive a near-optimal rate for the risk between the unknown function
and our estimator. In the special case of Gaussian errors, a lower
bound on the asymptotic minimax rate in the wavelet coefficient domain
is also obtained. We finally indicate how to derive a stronger
adaptivity result by modifying existing SURE estimators. Our method is
illustrated on both an interesting simulated example and a
biostatistical data-set, which exhibits a clear nonstationarity in its

(6) von Sachs, R., Neumann, M.H. A Wavelet-based Test for Stationarity.
Technical Report AGTM 182, University of Kaiserslautern, 1997,
submitted for publication.

Abstract: We develop a test for stationarity of a time series against
the alternative of a time-changing covariance structure. Using
localized versions of the periodogram, we obtain empirical versions of
a reasonable notion of a time-varying spectral density. Coefficients
with respect to a Haar wavelet series expansion of such a time-varying
periodogram are a possible indicator whether there is some deviation
from covariance stationarity. We propose a test based on the limit
distribution of these empirical coefficients. Some simulations
illustrate the performance of our procedure both on the
null-hypothesis of stationarity and on the alternative.

(7) von Sachs, R., Nason, G.P., Kroisandt, G. Wavelet Processes and
Adaptive Estimation of the Evolutionary Wavelet Spectrum.
Technical Report 516, Stanford Statistics Department, 1997, submitted
for publication.

Abstract: We define a new class of non-stationary random processes
which are characterized by a time-scale (spectral) representation with
respect to a family of non--decimated or stationary wavelets. Using
our new model of ``locally stationary wavelet" processes, we develop
a theory of how to define and estimate an ``evolutionary wavelet
spectrum". Our asymptotics are based on renormalizing in
time-location which permits rigorous estimation starting from a single
stretch of observations of the process. The wavelet spectrum measures
the local power in the variance-covariance decomposition of the
process at a certain scale and (renormalized) time location. To
estimate the wavelet spectrum we use (corrected and appropriately
locally smoothed) ``wavelet periodograms". Further, we suggest an
inverse transformation of the smoothed wavelet periodogram that
estimates the local autocovariances of the original stochastic
process. Some simulations and an application to a medical time series
indicate the usefulness of our new approach.

(8) Donoho, D.L., Mallat, S., von Sachs, R. Estimating Covariances of Locally
Stationary Processes: Rates of Convergence of Best Basis Methods.
Technical Report 517, Stanford Statistics Department, 1997, to be submitted
for publication.

Abstract: Mallat, Papanicolaou and Zhang recently proposed a method
for approximating the covariance of a locally stationary process by a
covariance which is diagonal in a specially constructed Coifman--Meyer
basis of cosine packets. In this paper we extend this approach to
estimating the covariance from sampled data. Our method combines both
wavelet shrinkage and cosine-packet best-basis selection in a simple
and natural way. The resulting algorithm is fast and automatic. The
method has an interpretation as a nonlinear, adaptive form of
anisotropic time-frequency smoothing. We introduce a new class of
locally stationary processes which exhibits a form of inhomogeneous
nonstationarity; our processes have covariances which typically change
little from row to row, but might occasionally change abruptly. We
study performance in an asymptotic setting involving triangular arrays
of processes which are becoming increasingly stationary, and are able
to prove rates of convergence results for our estimator. For this
class of processes, the algorithm has advantages over traditional
approaches like fixed-window-length segmentation followed by
autocovariance estimation.

Yours sincerely,

Rainer von Sachs

Department of Mathematics, University of Kaiserslautern, Germany.
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