Novel Wavelet-Based Statistical Methods with Applications in Classification, Shrinkage, and Nano-Scale Image Analysis

Given the recent popularity and clear evidence of wide applicability of wavelets, this thesis is devoted to several statistical applications of Wavelet transforms. Statistical multiscale modeling has, in the most recent decade, become a well-established area in both theoretical and applied statistics, with impact on developments in statistical methodology. Wavelet-based methods are important in statistics in areas such as regression, density and function estimation, factor analysis, modeling and forecasting in time series analysis, assessing self-similarity and fractality in data, and spatial statistics. In this thesis we show applicability of the wavelets by considering three problems: First, we consider a binary wavelet-based linear classifier. Both consistency results and implemental issues are addressed. We show that under mild assumptions wavelet-based classification rule is both weakly and strongly universally consistent. The proposed method is illustrated on synthetic data sets in which the truth is known and on applied classification problems from the industrial and bioengineering fields. Second, we develop wavelet shrinkage methodology based on testing multiple hypotheses in the wavelet domain. The shrinkage/thresholding approach by implicit or explicit simultaneous testing of many hypotheses had been considered by many researchers and goes back to the early 1990's. We propose two new approaches to wavelet shrinkage/thresholding based on local False Discovery Rate (FDR), Bayes factors and ordering of posterior probabilities. Finally, we propose a novel method for the analysis of straight-line alignment of features in the images based on Hough and Wavelet transforms. The new method is designed to work specifically with Transmission Electron Microscope (TEM) images taken at nanoscale to detect linear structure formed by the atomic lattice.Ph.D.Committee Chair: Vidakovic, Brani; Committee Member: Hayter, Anthony; Committee Member: Heil, Chris; Committee Member: Huo, Xiaoming; Committee Member: Wang, Yan

Linear Feature Identification and Inference in Nano-Scale Images

In this paper a novel method for the analysis of straight line alignment of features in the images based on Hough and Wavelet transforms is proposed. The new method is designed to work specifically with nanoscale images, to detect linear structure formed by the atomic lattice

Bayesian False Discovery Rate Wavelet Shrinkage: Theory and Applications

Statistical inference in the wavelet domain remains vibrant area of contemporary statistical research because desirable properties of wavelet representations and the need of scientific community to process, explore, and summarize massive data sets. Prime examples are biomedical, geophysical, and internet related data. In this paper we develop wavelet shrinkage methodology based on testing multiple hypotheses in the wavelet domain. The shrinkage/thresholding approach by implicit or explicit simultaneous testing of many hypotheses had been considered by many researchers and goes back to the early 1990’s. Even the early proposal, the universal thresholding, could be interpreted as a test of multiple hypotheses in the wavelet domain. We propose two new approaches to wavelet shrinkage/thresholding. (i) In the spirit of Efron and Tibshirani’s recent work on local false discovery rate, we propose the theoretical counterpart Bayesian Local False Discovery Rate, BLFDR, where the underlying model assumes unknown variances. This approach to wavelet shrinkage can also be connected with shrinkage based on Bayes factors. (ii) The second proposal to wavelet shrinkage explored in this paper is Bayesian False Discovery Rate, BaFDR. This proposal is based on ordering of posterior probabilities of hypotheses in Bayesian testing of multiple hypotheses. We demonstrate that both approaches result in competitive shrinkage methods by contrasting them to some popular shrinkage technique