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.
翻译:众所周知的非静止自相类似时间序列的一类非静止自相类似时间序列是用于在性质上模拟无处不在的随机过程的分数的布朗运动(fBm) 。 在本文中,我们研究了从合成的fBm数据构建的高维点云数据(PCD)的同质组。 我们把模拟的fBm序列隐藏到一个 PCD, 单位为$D美元立方体的子集, 使用时间延迟嵌入特定嵌入维度和时间间隔参数的方法。 在持续同系(PH) 的背景下, 我们计算嵌入的PCD的表层测量度测量值是相关的 Hust Expent 函数的函数, $H$, 在不同比例的嵌入维度、 时间间隔和不规则的大小中存在。 我们的结果显示, 对于一个正常的 FBmmm, 嵌入维度的值越高, 导致基于零级和第一个同系组的表面测量度度度测量度度测量度度度测量值越高。 为了可靠地对 FBmelent 的准确度值进行分类, 我们考虑, rodeal- dealdealdealdealdealdeal dededeal deal deal dedealdealdealde rodudeal deal deal deal deal deal deal dealde rode rode rode 。