Statistical applications of wavelets on time series

Shelby Xiaobo Gao, Purdue University

Abstract

This thesis deals with the applications of wavelet theory to time series data. We first focus on stationary times series. For dependent observations forming a stationary time series, we study the autocovariance or autocorrelation structure of wavelet coefficients at each level of the discrete wavelet decomposition of the data. We show how the variance and autocovariance of wavelet coefficients differ across each level of the wavelet decompositon structure for correlated time series. Thus the thresholds for the wavelet shrinkage method proposed by Donoho and Johnstone should be level dependent. We show how the autocovariance of wavelet coefficients on a given level can be exactly derived from those on the previous level. The sum of the autocovariances of scaling coefficients and wavelet coefficients for a given level and a given lag is twice the autocovariance of scaling coefficients at the previous level at twice the lag. We provide a pair of two-level formulae, an algorithm, and a list of coefficients to perform the calculation. We show that the theory we develop in time domain has a nice counterpart in the frequency domain. In one application, we apply the wavelet filters to estimate the power spectrum of a time series from a sinusoidal model. In another application, we use the cumulative sum of squares (CUSUM) of the first level wavelet coefficients to detect multiple change of variances of a time series. Next, we study nonparametric regression with heteroscedastic time series. We use a two step method to estimate the nonconstant variances. In the first step, a weighted local average in quadratic form is used and the estimated variances are local but noisy. In the second step, wavelet nonparametric regression is used to smooth the estimated variance obtained in the first step. We assume that the observations obtained in the first step are $\phi$-mixing and stationary.

Degree

Ph.D.

Advisors

Bock, Purdue University.

Subject Area

Statistics

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