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   -> Volume 7, Issue 10


Thesis: Assessing Nonstationary Time Series Using Wavelets
 
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"Whitcher, B." (whitcher@eurandom.tue.nl)
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PostPosted: Mon Oct 12, 1998 2:10 pm    
Subject: Thesis: Assessing Nonstationary Time Series Using Wavelets
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#11 Thesis: Assessing Nonstationary Time Series Using Wavelets

In July 1998 I finished my thesis concerning wavelet analysis of
nonstationary time series with application to the physical sciences.

With respect to univariate time series, I investigated the ability to
detect and locate changes of variance (single and multiple) in time
series with long-range dependence; i.e., slowly decaying
autocorrelations. This is done on a scale by scale basis using output
from the discrete wavelet transform (both orthogonal and
translation-invariant).

I extend the notion of analysis of variance in time series by
considering the analysis of covariance between bivariate time series.
The concepts of wavelet cross-covariance and cross-correlation are
intuitively defined. The basic property that the wavelet covariance
decomposes the process covariance on a scale by scale basis is proved,
and confidence intervals for estimators of these quantities are
established.

Examples taken from the physical sciences include:

Nile River minimum water levels (Toussoun 1925),
vertical ocean shear measurements (Percival and Guttorp 1994),
the Southern Oscillation Index (Walker 1928), and
the Madden-Julian oscillation (Madden and Julian 1971).

The entire thesis may be downloaded from

http://www.eurandom.tue.nl/whitcher/papers/

and the abstract is as follows:

The discrete wavelet transform has be used extensively in the field of
statistics, mostly in the area of "denoising signals" or nonparametric
regression. This thesis provides a new application for the discrete
wavelet transform, assessing nonstationary events in time series --
especially long memory processes. Long memory processes are those
which exhibit substantial correlations between events separated by a
long period of time.

Departures from stationarity in these heavily autocorrelated time
series, such as an abrupt change in the variance at an unknown
location or "bursts" of increased variability, can be detected and
accurately located using discrete wavelet transforms -- both
orthogonal and overcomplete. A cumulative sum of squares method,
utilizing a Kolomogorov--Smirnov-type test statistic, and an
information criterion method are investigated. By analyzing a time
series on a scale by scale basis, each scale corresponding to a range
of frequencies, the ability to detect and locate a sudden change in
the variance in the time series is introduced. Using this same
procedure to detect a change in the long memory parameter is also
investigated. Applications involve the Nile River minimum water
levels and vertical ocean shear measurements.20

In the atmospheric sciences, broadband features in the spectrum of
recorded time series have been hypothesized to be nonstationary
events; e.g., the Madden--Julian oscillation. The Madden--Julian
oscillation is a result of large-scale circulation cells oriented in
the equatorial plane from the Indean Ocean to the central Pacific. The
oscillation has been noted to have higher frequencies during warm
events in El NiF1o--Southern Oscillation (ENSO) years. The concepts of
wavelet covariance and wavelet correlation are introduced and applied
to this problem as an alternative to cross-spectrum analysis. The
wavelet covariance is shown to decompose the covariance between two
stationary processes on a scale by scale basis. Asymptotic normality
of estimators of the wavelet covariance and correlation is shown in
order to construct approximate confidence intervals. Both quantities
are generalized into the wavelet cross-covariance and
cross-correlation in order to investigate possible lead/lag relations
in bivariate time series.20

Atmospheric measurements (such as station pressure and zonal wind
speeds) from a single station at Canton Island (2.8B0S, 171.7B0W) are
analyzed and nicely replicate the results found in Madden and Julian
(1971). To highlight that the wavelet methods can provide insight over
and above traditional spectral methods (including multitaper
techniques) a daily "Southern Oscillation Index" and station pressure
series from Truk Island (7.4B0N, 151.8B0W) are analyzed. The wavelet
cross-covariance nicely decomposes the usual cross-covariance into
scales which are more easily associated with physical phenomena. The
time-varying wavelet covariance is used to show the increase in
positive correlation between the SOI and Truk Island station pressure
in the first half of each year versus latter half.20

Brandon Whitcher
EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
+31 40 247 8104 (voice) +31 40 247 8190 (fax)
http://www.eurandom.tue.nl/whitcher
All times are GMT + 1 Hour
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