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.
翻译:在本文中,我们使用持久性同质学(PH)技术来检查分数高斯噪音(fGn)的地形特性。我们开发了加权自然可见度图表算法,而PH则通过过滤过程量化了相关的简易复合体。Betti数字所代表的同质组的演化表明对Hurst Exponent(H美元)的高度依赖性很强。在某一阈值上,以美元为单位的地表洞(k$-洞)的出生和死亡曲线的系数取决于美元,而美元几乎不受有限样本大小的影响。我们表明,美元洞的寿命寿命周期的分布功能会急剧衰减,相应的斜度是相对于$的日益增强的功能。更有趣的是,以Bettiphn数字表示的样本大小完全消失。随着一个几乎以美元为单位的系统可见度图形的大小增长而恒定。相反,当地统计特征无法确定FGncial-H$的直径值的直径值值值值值,同时显示FGn-Q-H值的正值的正值的直径值数据。