Wavelet-based methods for estimation and discrimination.

摘要

This manuscript consists of three papers which formulate novel wavelet-based methodologies.; In Chapter 1, we present a procedure for estimating the time-varying scaling function H(t) for a series that exhibits locally self-similar behavior. The procedure utilizes the discrete wavelet transformation (DWT) of the series, exploiting an approximate log-linear relationship between the scale and the energy content. Based on this relationship, we propose methods for constructing both pointwise confidence intervals and overall confidence bands. The effectiveness of the procedures for estimating H(t) and assigning error bounds is examined via simulations.; In Chapter 2, we propose and investigate a discriminant procedure for classifying a series of unknown origin that arises as a function corrupted by noise. No assumption is made regarding the form of the underlying function, but it is assumed that the noise process is a stationary Gaussian process with mean zero. The discriminant is developed using the distributional properties of the DWT coefficients of the series, and involves the coefficients as well as parameters representing the underlying populations. The denoising and decorrelating properties of the DWT make the procedure very robust, while the multiscale nature of the decomposition allows for a flexible formulation. We present simulation results that evaluate the performance of the discriminant in a variety of settings using split-sample validation. Our method appears to be effective even when the training set is very small and the noise levels are high.; In Chapter 3, we propose and investigate two additional discriminant procedures developed using Kullback information measures and the DWT. The first is designed for classifying stationary processes and is a natural competitor to more traditional spectral-based methods. The second is developed for categorizing nonstationary processes. Extensive simulations evaluate the performance of both discriminants in a variety of settings.