Persistent Homology of Fractional Gaussian Noise

TL;DR

This study applies persistent homology to fractional Gaussian noise, revealing how topological features depend on the Hurst exponent and providing insights into EEG data analysis for health diagnostics.

Contribution

It introduces a weighted natural visibility graph algorithm combined with persistent homology to analyze the topological properties of fractional Gaussian noise, highlighting Hurst exponent dependencies.

Findings

Betti numbers depend on Hurst exponent $H$

Lifetime distribution of $k$-holes decays exponentially with $H$

Persistence entropy grows logarithmically with visibility graph size

Abstract

In this paper, we employ the persistent homology (PH) technique to examine the topological properties of fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm, and the associated simplicial complexes through the filtration process are quantified by PH. The evolution of the homology group dimension represented by Betti numbers demonstrates a strong dependency on the Hurst exponent ($H$). The coefficients of the birth and death curve of the $k$-dimensional topological holes ($k$-holes) at a given threshold depend on $H$ which is almost not affected by finite sample size. We show that the distribution function of a lifetime for $k$-holes decays exponentially and the corresponding slope is an increasing function versus $H$, and more interestingly, the sample size effect completely disappears in this quantity. The persistence entropy logarithmically grows with the size of the visibility graph of a system with almost $H$-dependent prefactors. On the contrary, the local statistical features are not able to determine the corresponding Hurst exponent of fGn data, while the moments of eigenvalue distribution ($M_{n}$) for $n\ge1$ reveal a dependency on $H$, containing the sample size effect. Finally, the PH shows the correlated behavior of electroencephalography for both healthy and schizophrenic samples.