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# Homology, Homotopy and Applications

## Volume 9 (2007)

### Number 2

### A statistical approach to persistent homology

Pages: 337 – 362

DOI: http://dx.doi.org/10.4310/HHA.2007.v9.n2.a12

#### Authors

#### Abstract

Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach to this problem. We assume that the data is randomly sampled from an unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using statistical estimators for samples from certain families of distributions, we show that we can recover the persistent homology of the underlying distribution.

#### Keywords

persistent homology; point cloud data; directional statistics; parametric statistics; expected persistent homology

#### 2010 Mathematics Subject Classification

55Nxx, 62H11