ASSESSING PERIODICITY IN TIME SERIES DATA USING PERSISTENT HOMOLOGY

Abstract

Topological Data Analysis (TDA), a novel method rooted in algebraic topology, offers a powerful framework for quantifying the structure of high-dimensional datasets. This study focuses on Persistent Homology (PH), a key TDA method, to elucidate the underlying shape and structure of time series data, thereby facilitating the understanding of patterns and recurrence. By employing Takens’ embedding to represent the time series as a high-dimensional point cloud, this approach effectively encodes its topological characteristics, particularly periodicity. We utilize various PH summaries, in the form of persistence vectors (PV), to assess periodicity by quantifying connected components $(H_0)$ and one-dimensional holes $(H_1)$. The methodology is validated on ten synthetic datasets, each comprising 200 data points and exhibiting varying levels of periodicity, amplitude, and frequency. The optimal window size for the sliding window embedding is determined by maximizing the $PD_{\text{max}}$ score. Subsequently, the $L^1$-norm of the PVs is computed to quantify periodicity. Under optimal window selection $(D=14)$, the cosine signal achieves the highest periodicity, followed by damped cosine and linear trend. Formal quantitative stability analysis identifies one of the two Sliding Window 1-Persistence Scoring (SW1PerS) baselines, SW2, as the most stable representation under both Gaussian noise (Rank Instability Index, RII$=0.243$) and missing data (Mean Absolute Rank Deviation, MARD=0.117), with the Persistence Landscape Score (PLS) and SW1 ranking closely behind; differences under noise are not statistically significant (Friedman $p=0.42$), while those under missing data are significant $(p=0.001)$. A comparison against non-topological baselines (FFT-based spectral entropy, autocorrelation, and recurrence quantification analysis) demonstrates that PLS achieves Kendall $\tau=0.867$, substantially outperforming spectral entropy (SE: $\tau=0.733$) which is the only non-topological baseline that achieves statistically significant concordance $(p=0.002)$, while the autocorrelation peak ratio (APR: $\tau=-0.289$) and recurrence rate (RR: $\tau=-0.200$) yield non-significant negative concordance on these synthetic signals.