Time Series Classification via Topological Data Analysis

In this study, we use persistent homology to perform classification tasks on two publicly available multivariate time series datasets [19, 11] that include physiological data collected during stressful and non stressful tasks. Instead of directly computing signal-specific features from sliding windows and subwindows on modalities such as electrocardiogram and wrist temperature (Figure 7), we extracted features using persistence diagrams and their statistical properties. Subwindowing method allowed us to reduce noise without incurring an extra computational cost. We then developed machine learning models and assess the performance of our models by varying window sizes and using different flavors of persistence diagrams. Topological Data Analysis (TDA) techniques usually work with points embedded in an affine space of large enough dimension. However, TDA techniques can still be applied to time series data sets whether they are univariate or multivariate. One can convert a univariate time series into a finite collection of points in a -dimensional affine space using delay embedding methods, of which one can compute persistent homology. Since Taken's Theorem implies that the delay embeddings produces topologically invariant subsets on a non-chaotical dynamical system [21], one can reasonably expect that persistent homology produces features that would distinguish different time series. There is a handful of research on the persistent homology of delay embeddings for time series classification [23, 20, 1].