📝 SummarXiv

Abstract

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, 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 similarity of two univariate time series, denoted score(f1|f2), and derive theoretical stability results for the same. We prove stability of score(f1|f2) under orthogonal projection of the time series embeddings onto their 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. We derive a lower bound on the embedding dimension to use in our pipeline which guarantees that any two such embeddings produce scores that are linearly within epsilon of each other. We present a procedure for computing conditional periodicity scores and implement it on several types of synthetic signals. We experimentally compare our similarity measure to the most-similar statistical measure of percent determinism (%DET) and show greater stability of score(f1|f2). We also compare both measures on several pairs of real time series describing monthly proportions of incoming calls to a police agency and highlight the decreased stability of %DET on the same.