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Visualization and Non-parametric Statistical Testing Methods for Multivariate Time Series


Many model-based methods have been developed over the last several decades for analysis of multivariate time series, such as electroencephalograms (EEG) in order to understand electrical neural data. In this dissertation, we propose to use the functional boxplot to analyze log periodograms of EEG time series data in the spectral domain. The functional boxplot approach produces a median curve -- which is not equivalent to connecting medians obtained from frequency-specific boxplots. In addition, this approach identifies a functional median, summarizes variability and detects potential outliers. By extending functional boxplots analysis from one-dimensional curves to surfaces, surface boxplots are also used to explore the variation of the spectral power for the alpha ($8-12$ Hertz) and beta ($16-32$ Hertz) frequency bands across the brain cortical surface. By using rank-based nonparametric tests, we also investigate the stationarity of EEG traces across an exam acquired during resting-state by comparing the spectrum during the early vs. late phases of a single resting-state EEG exam.

Moreover, we present an exploratory data analysis tool for visualizing and testing the symmetric positive definite matrices (e.g., covariance, spectral and coherence matrices) in a multi--subject experimental setting. %derived from electroencephalogram (EEG) recordings from several clinical subjects. Our work is motivated by the clinician's interest to determine associations between brain functional connectivity (as measured by coherence) and patients' response to treatment. For each study participant, the geometric surface boxplot is developed to characterize the distribution of coherence matrices through the median matrix and the 50\% most central region of the data. The surface boxplot will also be used to detect the outlier coherence matrices as in the classical boxplot. To investigate the treatment effect, we develop a rank--based non--parametric approach to test for significant differences in coherence matrices between treatment and control groups. As an application, we demonstrate our proposed methods on coherence matrices, derived from an electroencephalograms, to determine potential associations treatment effect on patients with infantile spasms and hypsarrhythmia both before and after treatment.

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