分数阶高斯噪声的持久同调性
Persistent Homology of Fractional Gaussian Noise
摘要：
在本文中，我们采用持久同源（PH）技术研究分数高斯噪声（fGn）的拓扑性质。我们开发的加权自然可见性图算法，并通过过滤过程中的相关单纯形复合物量化的PH值。Betti数表示的同源群维数的演变表现出很强的依赖于赫斯特指数（$H$）。给定阈值下$k$维拓扑洞（$k$ -洞）的生死曲线系数取决于$H$，而$H$几乎不受有限样本量的影响。我们发现，$k$ -孔的寿命的分布函数呈指数衰减和相应的斜率是一个增加的函数与$H$，更有趣的是，在这个数量的样本尺寸效应完全消失。持久性熵的增长与系统的可见性图的大小几乎$H$依赖的前因子。相反，局部统计特征不能确定fGn数据的相应Hurst指数，而$n\ge1$的特征值分布矩（$M_{n}$）显示对$H$的依赖性，包含样本量效应。最后，PH显示了健康和精神分裂症样本的脑电图的相关行为。
英文摘要：
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.
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