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

2025-01-06

AI Summary

This paper addresses the problem of quantifying the *conditional periodicity* of one time series given the influence of another. We propose the first theoretically grounded measure, denoted score(f₁|f₂), for this task. Methodologically, our approach integrates delay embedding, PCA-based dimensionality reduction, persistent homology, and topological feature extraction. We establish the first theoretical guarantee by proving that the score satisfies Lipschitz stability; further, we derive a novel stability bound under PCA projection and provide a lower bound on the embedding dimension ensuring ε-consistency. Experiments on synthetic signals demonstrate that our measure significantly outperforms conventional methods—such as cross-recurrence plots—in both accuracy and robustness for discriminating periodic similarity. Overall, this work delivers an interpretable, mathematically verifiable framework for causal periodic analysis of multivariate time series.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Web measurements
• Security and Privacy: Large-scale security measurements

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, 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.

Problem

• Measure conditional periodicity between time series pairs
• Develop stable scoring function using persistent homology
• Ensure stability under PCA dimension reduction

Innovation

• Uses persistent homology for periodicity scoring
• Introduces conditional periodicity score with stability guarantees
• Incorporates PCA for dimension reduction in scoring