A Subsequence Approach to Topological Data Analysis for Irregularly-Spaced Time Series

TL;DR

This work introduces a novel {\em subsequence} embedding method for irregularly-spaced time-series data and shows that this method preserves the original state space topology while reducing spurious homological features.

Abstract

A time-delay embedding (TDE), grounded in the framework of Takens's Theorem, provides a mechanism to represent and analyze the inherent dynamics of time-series data. Recently, topological data analysis (TDA) methods have been applied to study this time series representation mainly through the lens of persistent homology. Current literature on the fusion of TDE and TDA are adept at analyzing uniformly-spaced time series observations. This work introduces a novel {\em subsequence} embedding method for irregularly-spaced time-series data. We show that this method preserves the original state space topology while reducing spurious homological features. Theoretical stability results and convergence properties of the proposed method in the presence of noise and varying levels of irregularity in the spacing of the time series are established. Numerical studies and an application to real data illustrates the performance of the proposed method.

Figures (18)

• Figure 1: Illustration of VR complexes and persistence diagram. The zero-simplices (black points) were sampled around three circles. Balls of diameter $0.8$ (a) and $1.5$ (b) are drawn around the points, resulting in one-simplices (black lines) and two-simplices (yellow triangles). The associated persistence diagram (c) has $H_0$ features (red points) and $H_1$ features (blue triangles) that represent the three circles.
• Figure 2: Illustration of the embedding process. Top-left: the state space, typically not observed. Middle-bottom: the time series obtained via the measurement function $h(\cdot)$. Top-right: the reconstructed space from the TDE matrix $F$, which preserves the topology of the original state space.
• Figure 3: SSE method illustration. (a) One thousand time series measurements (blue and orange points). About $20\%$ were designated as missing (hollow blue diamonds) to obtain irregularly-spaced observations (orange points). The TDE of the full time series ((b)-top) and the SSE of the irregularly-spaced time series ((b)-bottom); both time series were embedded in $\mathbb{R}^4$ and their first three principal components are plotted. The persistence diagram of the TDE (c) and SSE (d).
• Figure 4: The Hénon map used in assessing reconstruction accuracy. (a) The Hénon map with 500 points (blue and orange) where the blue diamonds are designated as missing. (b) The $h$-dimension of the Hénon map; only 200 points are displayed for visual clarity.
• Figure 5: Reconstructed state spaces of the Hénon map for: (a) proposed SSE method, (b) KS imputation, and (c) LOCF imputation.

Theorems & Definitions (11)

• Remark 1
• Proposition 4.1
• Remark 2
• Lemma 4.2: Topology-preserving transform
• Proposition 4.3
• Corollary 4.4
• Theorem 4.5
• Corollary 4.6
• proof
• proof