Spectral Analysis for Signal Detection and Classification : Reducing Variance and Extracting Features

Spectral analysis encompasses several powerful signal processing methods. The papers in this thesis present methods for finding good spectral representations, and methods both for stationary and non-stationary signals are considered. Stationary methods can be used for real-time evaluation, analysing shorter segments of an incoming signal, while non-stationary methods can be used to analyse the instantaneous frequencies of fully recorded signals. All the presented methods aim to produce spectral representations that have high resolution and are easy to interpret. Such representations allow for detection of individual signal components in multi-component signals, as well as separation of close signal components. This makes feature extraction in the spectral representation possible, relevant features include the frequency or instantaneous frequency of components, the number of components in the signal, and the time duration of the components. Two methods that extract some of these features automatically for two types of signals are presented in this thesis. One adapted to signals with two longer duration frequency modulated components that detects the instantaneous frequencies and cross-terms in the Wigner-Ville distribution, the other for signals with an unknown number of short duration oscillations that detects the instantaneous frequencies in a reassigned spectrogram. This thesis also presents two multitaper methods that reduce the influence of noise on the spectral representations. One is designed for stationary signals and the other for non-stationary signals with multiple short duration oscillations. Applications for the methods presented in this thesis include several within medicine, e.g. diagnosis from analysis of heart rate variability, improved ultrasound resolution, and interpretation of brain activity from the electroencephalogram

Some topics in high-dimensional robust inference and graphical modeling

2021 Summer.Includes bibliographical references.In this dissertation, we focus on large-scale robust inference and high-dimensional graphical modeling. Especially, we study three problems: a large-scale inference method by a tail-robust regression, model specification tests for dependence structure of Gaussian Markov random fields, and a robust Gaussian graph estimation. First of all, we consider the problem of simultaneously testing a large number of general linear hypotheses, encompassing covariate-effect analysis, analysis of variance, and model comparisons. The new challenge that comes along with the overwhelmingly large number of tests is the ubiquitous presence of heavy-tailed and/or highly skewed measurement noise, which is the main reason for the failure of conventional least squares based methods. The new testing procedure is built on data-adaptive Huber regression, and a new covariance estimator of the regression estimate. Under mild conditions, we show that the proposed methods produce consistent estimates of the false discovery proportion. Extensive numerical experiments, along with an empirical study on quantitative linguistics, demonstrate the advantage of our proposal compared to many state-of-the-art methods when the data are generated from heavy-tailed and/or skewed distributions. In the next chapter, we focus on the Gaussian Markov random fields (GMRFs) and, by utilizing the connection between GMRFs and precision matrices, we propose an easily implemented procedure to assess the spatial structures modeled by GMRFs based on spatio-temporal observations. The new procedure is flexible to assess a variety of structures including the isotropic and directional dependence as well as the Matern class. A comprehensive simulation study has been conducted to demonstrate the finite sample performance of the procedure. Motivated from the efforts on modeling flu spread across the United States, we also apply our method to the Google Flu Trend data and report some very interesting epidemiological findings. Finally, we propose a high-dimensional precision matrix estimation method via nodewise distributionally robust regressions. The distributionally robust regression with an ambiguity set defined by Wasserstein-2 ball has a computationally tractable dual formulation, which is linked to square-root regressions. We propose an iterative algorithm that has a substantial advantage in terms of computation time. Extensive numerical experiments study the performance of the proposed method under various precision matrix structures and contamination models