Homology Groups of Embedded Fractional Brownian Motion

TL;DR

This study investigates the topological features of fractional Brownian motion data using persistent homology, revealing how embedding parameters affect the classification and robustness of fBm signals across scales.

Contribution

It introduces a method to analyze the homology groups of high-dimensional point clouds derived from fBm, highlighting the impact of embedding dimension and time-delay on topological measures.

Findings

Higher embedding dimensions increase Hurst exponent dependency in topological measures.

Small time-delays improve reliable classification of fBm.

Scale for path-connectedness and post-loopless regime are robust indicators.

Abstract

A well-known class of non-stationary self-similar time series is the fractional Brownian motion (fBm) considered to model ubiquitous stochastic processes in nature. In this paper, we study the homology groups of high-dimensional point cloud data (PCD) constructed from synthetic fBm data. We covert the simulated fBm series to a PCD, a subset of unit $D$-dimensional cube, employing the time delay embedding method for a given embedding dimension and a time-delay parameter. In the context of persistent homology (PH), we compute topological measures for embedded PCD as a function of associated Hurst exponent, $H$, for different embedding dimensions, time-delays and amount of irregularity existed in the dataset in various scales. Our results show that for a regular synthetic fBm, the higher value of the embedding dimension leads to increasing the $H$-dependency of topological measures based on zeroth and first homology groups. To achieve a reliable classification of fBm, we should consider the small value of time-delay irrespective of the irregularity presented in the data. More interestingly, the value of scale for which the PCD to be path-connected and the post-loopless regime scale are more robust concerning irregularity for distinguishing the fBm signal. Such robustness becomes less for the higher value of embedding dimension.