Functional central limit theorems for persistent Betti numbers on cylindrical networks

关于持续 Betti 数的渐近正态性

On the asymptotic normality of persistent Betti numbers

本文证明持续 Betti 数具有强稳定化性质，并将其渐近正态性推广到多元情形及更广泛的 Poisson、二项过程，且结论不受渗流影响。

On the consistency and asymptotic normality of multiparameter persistent Betti numbers

The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or persistence diagram. While this approach to data science has been very successful, it suffers from sensitivity to outliers, and it does not allow for additional filtration parameters. Such parameters naturally appear when a cloud of data points comes together with additional measurements taken at the locations of the data. For these reasons, multiparameter persis…

持久Betti数的Bootstrap方法及其他稳定化统计量

Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics

本文研究了用于拓扑数据分析等稳定化统计量的多元Bootstrap方法，证明了平滑Bootstrap在标准非参数方法失效时能提供一致的置信区间估计，并放宽了无需连续底层密度等理论条件，最后通过模拟和SDSS宇宙网数据验证了其有效性。

Testing goodness of fit for point processes via topological data analysis

We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions…

用于在线变点检测的校准持久拉普拉斯累积和

Online Change-Point Detection with Persistent Laplacian Features

针对高维非线性时间序列在线变点检测，提出PL-CUSUM方法，将滑动窗口转为点云，用持久拉普拉斯谱构建监测分数，分析关键性能标准，给出阶段程序，实验表明其能稳定控制误报且检测性能有竞争力。

On limit theorems for persistent Betti numbers from dependent data

We study persistent Betti numbers and persistence diagrams obtained from time series and random fields. It is well known that the persistent Betti function is an efficient descriptor of the topology of a point cloud. So far, convergence results for the $(r,s)$-persistent Betti number of the $q$th homology group, $β^{r,s}_q$, were mainly considered for finite-dimensional point cloud data obtained from i.i.d. observations or stationary point processes such as a Poisson process. In this article, we extend these considerations. We derive limit theorems for the pointwise convergence of persistent B…