# Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces

@article{Blumberg2014RobustSH, title={Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces}, author={Andrew J. Blumberg and Itamar Gal and Michael A. Mandell and Matthew Pancia}, journal={Foundations of Computational Mathematics}, year={2014}, volume={14}, pages={745-789} }

We study distributions of persistent homology barcodes associated to taking subsamples of a fixed size from metric measure spaces. We show that such distributions provide robust invariants of metric measure spaces and illustrate their use in hypothesis testing and providing confidence intervals for topological data analysis.

## 63 Citations

Tropical Sufficient Statistics for Persistent Homology

- Computer ScienceSIAM J. Appl. Algebra Geom.
- 2019

The sufficiency result presented in this work allows for classical probability distributions to be assumed on the tropical geometric representation of barcodes and makes a variety of parametric statistical inference methods amenable to barcodes, all while maintaining their initial interpretations.

Confidence sets for persistence diagrams

- MathematicsThe Annals of Statistics
- 2014

This paper derives confidence sets that allow us to separate topological signal from topological noise, and brings some statistical ideas to persistent homology.

Persistent homology and applied homotopy theory

- Mathematics
- 2020

This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence…

Parametric Inference using Persistence Diagrams: A Case Study in Population Genetics

- Mathematics, Computer Science
- 2014

This work shows that, in certain models, parametric inference can be performed using statistics defined on the computed invariants of persistent homology, and develops this idea with a model from population genetics, the coalescent with recombination.

Topological Consistency via Kernel Estimation

- Mathematics, Computer Science
- 2014

We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based…

Statistical topological data analysis using persistence landscapes

- MathematicsJ. Mach. Learn. Res.
- 2015

A new topological summary for data that is easy to combine with tools from statistics and machine learning and obeys a strong law of large numbers and a central limit theorem is defined.

A topological study of functional data and Fréchet functions of metric measure spaces

- MathematicsJ. Appl. Comput. Topol.
- 2019

This work studies the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces and investigates the stability of these invariants using metrics that downplay regions where signals are weak.

Topological spaces of persistence modules and their properties

- MathematicsJ. Appl. Comput. Topol.
- 2018

This work considers various classes of persistence modules, including many of those that have been previously studied, and describes the relationships between them, and undertake a systematic study of the resulting topological spaces and their basic topological properties.

Topological and metric properties of spaces of generalized persistence diagrams

- Mathematics
- 2022

. Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a…

Multiple testing with persistent homology

- Computer ScienceArXiv
- 2018

This paper proposes a null model based approach to testing for acyclicity, coupled with a Family-Wise Error Rate (FWER) control method that does not suffer from computational costs, and adapt an False Discovery Rate (FDR) control approach to the topological setting.

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