Persistent homology of featured time series data and its applications

TL;DR

This paper introduces a method to incorporate domain-specific features into persistent homology analysis of time series data, enhancing its applicability and stability for real-world tasks like anomaly detection and music analysis.

Contribution

It proposes a novel framework for adjusting persistent homology calculations using domain knowledge, with a stability theorem ensuring reliable results.

Findings

Improved anomaly detection in stock data

Enhanced topological analysis of music data

Demonstrated stability of the method with influence vectors

Abstract

Recent studies have actively employed persistent homology (PH), a topological data analysis technique, to analyze the topological information in time series data. Many successful studies have utilized graph representations of time series data for PH calculation. Given the diverse nature of time series data, it is crucial to have mechanisms that can adjust the PH calculations by incorporating domain-specific knowledge. In this context, we introduce a methodology that allows the adjustment of PH calculations by reflecting relevant domain knowledge in specific fields. We introduce the concept of featured time series, which is the pair of a time series augmented with specific features such as domain knowledge, and an influence vector that assigns a value to each feature to fine-tune the results of the PH. We then prove the stability theorem of the proposed method, which states that adjusting the influence vectors grants stability to the PH calculations. The proposed approach enables the tailored analysis of a time series based on the graph representation methodology, which makes it applicable to real-world domains. We consider two examples to verify the proposed method's advantages: anomaly detection of stock data and topological analysis of music data.