Persistent homology of fractional Gaussian noise

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 curves 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 $n\ge 1$ 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.

Figure 1

The schematic representation of making a network for a typical data set using VG algorithm. (a) The HVG and (b) NVG of a synthetic fGn with $H=0.5$ (white noise). To make more sense, we took the sample size equal to $N=32$.

Figure 2

Probability distribution function of local features of WNVGs constructed from fGns for various Hurst exponents. Panel (a) is the $p\left(c^E\right)$ while panel (b) shows the probability distribution of betweenness centrality for $N=2^{10}$ in log-log scale. Panels (c) and (d) respectively illustrate the $p\left(c^C\right)$ and corresponding moments, $M_n$, for $n=2$ (solid line) and $n=3$ (dashed line) versus $H$. Different symbols are taken for various sizes.

Figure 3

(a) The probability distribution function of eigenvalue for $N=2^{10}$ when the vertical axis is plotted in log scale. (b) The corresponding moments, $M_n$, for $n=2$ (solid line) and $n=3$ (dashed line) versus $H$. Different symbols are taken for various sizes.

Figure 4

(a) The normalized ${\beta }_0$ curve in log scale for clique complexes of WNVGs associated with fGns of various Hurst exponents versus threshold. (b) The $w_0$ as a function of $H$. (c) The normalized ${\beta }_0^{\left(\mathrm{death}\right)}$ in log scale as a function of $w^2$ (see the text) for various Hurst exponents as a function of threshold. (d) The value of ${\alpha }_0^{\left(\mathrm{death}\right)}$ as a function of Hurst exponent.

Figure 5

(a) The normalized ${\beta }_1$ curve for clique complexes of WNVGs associated with fGns of various Hurst exponents versus threshold. (b) The $w_1$ as a function of $H$ for different lengths of data set. (c) The distribution of the persistence diagram in log scale versus the birth axis as a function of threshold. (d) The corresponding coefficient of ${\beta }_1^{\left(\mathrm{birth}\right)}$ in terms of threshold known as ${\alpha }_1^{\left(\mathrm{birth}\right)}$ as a function of Hurst exponent. Panels (e) and (f) are the same as panels (c) and (d), respectively, just for dying 1-hole statistics. In this plot for computing ${\beta }_1$, we took $N=2^{10}$.

Figure 6

(a) The probability distribution of lifetime for 1-dimensional holes. Different symbols correspond to various $H$. This diagram has been obtained by doing an ensemble average. We set the vertical axis in log scale. (b) The ${\alpha }_1^{\left(\mathrm{lifetime}\right)}$ coefficient for different sizes represented by different symbols.

Figure 7

The persistence diagram and persistence barcode (inset) of weighted clique complex of WNVG corresponding to fGn with (a) $H=0.2$ (anticorrelated noise), (b) $H=0.5$ (uncorrelated noise), and (c) $H=0.8$ (correlated noise). The blue circle and orange triangle symbols are considered for the 0-dimensional and 1-dimensional homology generators, respectively.

Figure 8

(a) The persistence entropy for the 0-homology group, ${\mathrm{PE}}_0$. (b) The ${\mathrm{PE}}_1$ for 1 hole as a function of sample size in the log scale. Different symbols correspond to various values of the Hurst exponent. (c) The prefactor of persistence entropy as a function of $H$ computed by the ensemble average.

Figure 9

(a) The distribution of persistence diagram versus birth axis (${\beta }_1^{\left(\mathrm{birth}\right)}$) in log scale for clique complexes of WNVGs associated with EEG signals for healthy (blue circle symbols) and schizophrenic cases (orange triangle symbols) as a function of threshold. (b) The ${\beta }_1^{\left(\mathrm{death}\right)}$ in log scale as a function of threshold. (c) The lifetime distribution in log scale for 1-homology generator statistics.

Figure 10

Low-dimensional simplices: (a) 0-simplex (point), (b) 1-simplex (line segment), (c) 2-simplex (filled triangle), and (d) 3-simplex (filled tetrahedron).

Figure 11

Betti numbers of some topological spaces: point, line, circle, sphere, torus, 2-torus.

Figure 12

(a) Clique complex of (b) a typical network.