Spectral Density Function Estimation with Applications in Clustering and Classification

Abstract

Spectral density function (SDF) plays a critical role in spatio-temporal data analysis, where the data are analyzed in the frequency domain. Although many methods have been proposed for SDF estimation, real-world applications in many research fields, such as neuroscience and environmental science, call for better methodologies. In this thesis, we focus on the spectral density functions for time series and spatial data, develop new estimation algorithms, and use the estimators as features for clustering and classification purposes. The first topic is motivated by clustering electroencephalogram (EEG) data in the spectral domain. To identify synchronized brain regions that share similar oscillations and waveforms, we develop two robust clustering methods based on the functional data ranking of the estimated SDFs. The two proposed clustering methods use different dissimilarity measures and their performance is examined by simulation studies in which two types of contaminations are included to show the robustness. We apply the methods to two sets of resting-state EEG data collected from a male college student. Then, we propose an efficient collective estimation algorithm for a group of SDFs. We use two sets of basis functions to represent the SDFs for dimension reduction, and then, the scores (the coefficients of the basis) estimated by maximizing the penalized Whittle likelihood are used for clustering the SDFs in a much lower dimension. For spatial data, an additional penalty is applied to the likelihood to encourage the spatial homogeneity of the clusters. The proposed methods are applied to cluster the EEG data and the soil moisture data. Finally, we propose a parametric estimation method for the quantile spectrum. We approximate the quantile spectrum by the ordinary spectral density of an AR process at each quantile level. The AR coefficients are estimated by solving Yule- Walker equations using the Levinson algorithm. Numerical results from simulation studies show that the proposed method outperforms other conventional smoothing techniques. We build a convolutional neural network (CNN) to classify the estimated quantile spectra of the earthquake data in Oklahoma and achieve a 99.25% accuracy on testing sets, which is 1.25% higher than using ordinary periodograms.