Homology Groups of Embedded Fractional Brownian Motion

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.