Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis

Abstract

We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets, and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS which stands for Sliding Windows and 1-dimensional Persistence Scoring.

Synchronization among Groups of Spectators for Highlight Detection in Movies

“…In this subsection we detail the usage of the periodicity score to measure synchronization of signals [17], [18]. The overview is described in Figure 1.…”

“…Finally, we introduce the basics of persistent homology: filtrations and persistence diagrams [17], [18], [8] plotted in Figure 2. Once the sequence of S i,l…”

“…One important step was made by [21] to introduce generalized synchronization of coupled chaotic systems. In [17], [18] the authors proposed that a level of the periodicity score can measure the amount of pattern repetitions in signals.…”

Computational Topology to Monitor Human Occupancy

“…The research on TDA was recently boosted by the introduction of the Persistent Homology (PH) theory [12], along with fast algorithms for its computation [13] and efficient implementations [14]. The application scenarios include shape and texture analysis [15], biological and molecular data analysis [16], sensor networks [17], image and signal processing [18][19][20]. In what follows, we offer a brief intuition on PH; we refer the reader to [21] for a rigorous treatment of the subject.…”

“…Then, we cluster rows into sliding windows F i [19]: each F i corresponds to W time samples of all K sensors, with each window sharing W/2 samples with the previous one. Therefore, each F i can be seen as a point cloud, made of W points in the Euclidean space R K , where the coordinates correspond to the output of the different sensors.…”