Mathematical and Statistical Analysis of Non-stationary Time Series Data

Non-stationary time series, with intrinsic properties constantly changing over time, present significant challenges for analysis in various scientific fields, particularly in biomedical signal analysis. This dissertation presents novel methodologies for analyzing and classifying highly noisy and non-stationary signals with applications to electroencephalograms (EEGs) and electrocardiograms (ECGs).

The first part of the dissertation focuses on a framework integrating pseudo-differential operators with convolutional neural networks (CNNs). We present their synergistic potential for signal classification from an innovative perspective.

Building on the fundamental concept of pseudo-differential operators, the dissertation further proposes a novel methodology that addresses the challenges of applying time-variant filters or transforms to non-stationary signals. This approach enables the neural network to learn a convolution kernel that changes over time or location, providing a refined strategy to effectively handle these dynamic signals.

This dissertation also introduces a hybrid convolutional neural network that integrates both complex-valued and real-valued components with the discrete Fourier transform (DFT) for EEG signal classification. This fusion of techniques significantly enhances the neural network's ability to utilize the phase information contained in the DFT, resulting in substantial accuracy improvements for EEG signal classification.

In the final part of this dissertation, we apply a conventional machine learning approach for the detection and localization of myocardial infarctions (MIs) in electrocardiograms (ECGs) and vectorcardiograms (VCGs), using the innovative features extracted from the geometrical and kinematic properties within VCGs. This boosts the accuracy and sensitivity of traditional MI detection.