Topological feature selection for time series data

Abstract

We use tools from applied topology for feature selection on time series data. We develop a method for scoring the variables in a multivariate time series that reflects their contributions to the topological features of the corresponding point cloud. Our approach produces a piecewise-linear Lipschitz gradient path in the standard geometric simplex that starts at the barycenter, which weights the variables equally, and ends at the score. Taking the mean of stochastic gradient descent results in a mean gradient path that satisfies a strong law of large numbers and central limit theorem. Our theory is motivated by the analysis of the neuronal activities of the nematode C. elegans, and our method selects an informative subset of the neurons that optimizes the coordinated dynamics.

31 pages. Numerous improvements based on referee comments. We now use stochastic subgradient descent which has guarantees of convergence