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

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