Applications of Topological Data Analysis and Sliding Window Embeddings for Learning on Novel Features of Time-Varying Dynamical Systems

This work introduces geometric and topological data analysis (TDA) tools that can be used in conjunction with sliding window transformations, also known as delay-embeddings, for discovering structure in time series and dynamical systems in an unsupervised or supervised learning framework. For signals of unknown period, we introduce an intuitive topological method to discover the period, and we demonstrate its use in synthetic examples and real temperature data. Alternatively, for almost-periodic signals of known period, we introduce a metric called Geometric Complexity of an Almost Periodic signal (GCAP), based on a topological construction, which allows us to continuously measure the evolving variation of its periods. We apply this method to temperature data collected from over 200 weather stations in the United States and describe the novel patterns that we observe. Next, we show how geometric and TDA tools can be used in a supervised learning framework. Seizure-detection using electroencephalogram (EEG) data is formulated as a binary classification problem. We define new collections of geometric and topological features of multi-channel data, which utilizes temporal and spatial context of EEG, and show how it results in better overall performance of seizure detection than using the usual time-domain and frequency domain features. Finally, we introduce a novel method to sonify persistence diagrams, and more generally any planar point cloud, using a modified version of the harmonic table. This auditory display can be useful for finding patterns that visual analysis alone may miss.