A Stable Measure for Conditional Periodicity of Time Series using Persistent Homology

Given a pair of time series, we study how the periodicity of one influences the periodicity of the other. There are several known methods to measure the similarity between a pair of time series, such as cross-correlation, coherence, cross-recurrence, and dynamic time warping. But we have yet to find any measures with theoretical stability results. Persistence homology has been utilized to construct a scoring function with theoretical guarantees of stability that quantifies the periodicity of a single univariate time series f1, denoted score(f1). Building on this concept, we propose a conditional periodicity score that quantifies the periodicity of one univariate time series f1 given another f2, denoted score(f1|f2), and derive theoretical stability results for the same. With the use of dimension reduction in mind, we prove a new stability result for score(f1|f2) under principal component analysis (PCA) when we use the projections of the time series embeddings onto their respective first K principal components. We show that the change in our score is bounded by a function of the eigenvalues corresponding to the remaining (unused) N-K principal components and hence is small when the first K principal components capture most of the variation in the time series embeddings. Finally we derive a lower bound on the minimum embedding dimension to use in our pipeline which guarantees that any two such embeddings give scores that are within a given epsilon of each other. We present a procedure for computing conditional periodicity scores and implement it on several pairs of synthetic signals. We experimentally compare our similarity measure to the most-similar statistical measure of cross-recurrence, and show the increased accuracy and stability of our score when predicting and measuring whether or not the periodicities of two time series are similar.