Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics

The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. The present work relates to other general results in the area of stabilizing statistics, including central limit theorems for functionals of Poisson and Binomial processes in the critical regime. Specific statistics considered include the persistent Betti numbers of \v{C}ech and Vietoris-Rips complexes over point sets in $\mathbb R^d$, along with Euler characteristics, and the total edge length of the $k$-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS). Source code is available at github.com/btroycraft/stabilizing_statistics_bootstrap.Comment: 59 pages, 3 figures. Restructured paper with alternate problem settings moved to appendix. Rewrote data analysis and simulations study sections to be more comprehensive, moved each to the end of the pape

Asymptotic Inference and Network Analysis with Topological Statistics

In this work we establish a sequence of asymptotic statistical results for important quantities in Topological Data Analysis (TDA). Furthermore, we investigate the use of topological methods to analyze protein coexpression data from a structural perspective. We give an initial introduction to the techniques of topological data analysis, and outline existing statistical results in the large-sample setting.We first investigate multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. Specific statistics considered include the persistent Betti numbers of Cech and Vietoris-Rips complexes over point sets in Euclidean space, along with Euler characteristics, and the total edge length of the k-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS).Next we study approximation theorems for the Euler characteristic of the Vietoris-Rips and Cech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.Finally, we examine the problem of network analysis from the topological perspective, using the techniques of persistence homology to extract meaningful structural features from protein coexpression data. We propose novel topological separation metrics in an optimization framework, and make an application to the protein coexpression for three-spine stickleback (gasterosteus aculeatus)

Asymptotic Inference and Network Analysis with Topological Statistics

In this work we establish a sequence of asymptotic statistical results for important quantities in Topological Data Analysis (TDA). Furthermore, we investigate the use of topological methods to analyze protein coexpression data from a structural perspective. We give an initial introduction to the techniques of topological data analysis, and outline existing statistical results in the large-sample setting.We first investigate multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. Specific statistics considered include the persistent Betti numbers of Cech and Vietoris-Rips complexes over point sets in Euclidean space, along with Euler characteristics, and the total edge length of the k-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS).Next we study approximation theorems for the Euler characteristic of the Vietoris-Rips and Cech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.Finally, we examine the problem of network analysis from the topological perspective, using the techniques of persistence homology to extract meaningful structural features from protein coexpression data. We propose novel topological separation metrics in an optimization framework, and make an application to the protein coexpression for three-spine stickleback (gasterosteus aculeatus)

Bootstrapping persistent Betti numbers and other stabilizing statistics

We investigate multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. The work relates to other general results in the area of stabilizing statistics, including central limit theorems for geometric and topological functionals of Poisson and binomial processes in the critical regime, where limit theorems prove difficult to use in practice, motivating the use of a bootstrap approach. A smoothed bootstrap procedure is shown to give consistent estimation in these settings. Specific statistics considered include the persistent Betti numbers of Čech and Vietoris–Rips complexes over point sets in Rd, along with Euler characteristics, and the total edge length of the k-nearest neighbor graph. Special emphasis is given to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. Numerical studies illustrate the performance of the proposed method