Electronic Journal of Statistics

Probabilistic Fréchet means for time varying persistence diagrams

Elizabeth Munch, Katharine Turner, Paul Bendich, Sayan Mukherjee, Jonathan Mattingly, and John Harer

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Abstract

In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in $(\mathcal{D}_{p},W_{p})$, the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards.

We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each $N$ a map $(\mathcal{D}_{p})^{N}\to\P(\mathcal{D}_{p})$. We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.

Article information

Source
Electron. J. Statist. Volume 9, Number 1 (2015), 1173-1204.

Dates
Received: November 2014
First available in Project Euclid: 1 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1433195858

Digital Object Identifier
doi:10.1214/15-EJS1030

Mathematical Reviews number (MathSciNet)
MR3354335

Subjects
Primary: 55 60

Keywords
Topological data analysis Fréchet mean time varying data

Citation

Munch, Elizabeth; Turner, Katharine; Bendich, Paul; Mukherjee, Sayan; Mattingly, Jonathan; Harer, John. Probabilistic Fréchet means for time varying persistence diagrams. Electron. J. Statist. 9 (2015), no. 1, 1173--1204. doi:10.1214/15-EJS1030. https://projecteuclid.org/euclid.ejs/1433195858.


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