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2015 Probabilistic Fréchet means for time varying persistence diagrams
Elizabeth Munch, Katharine Turner, Paul Bendich, Sayan Mukherjee, Jonathan Mattingly, John Harer
Electron. J. Statist. 9(1): 1173-1204 (2015). DOI: 10.1214/15-EJS1030

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.

Citation

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Elizabeth Munch. Katharine Turner. Paul Bendich. Sayan Mukherjee. Jonathan Mattingly. John Harer. "Probabilistic Fréchet means for time varying persistence diagrams." Electron. J. Statist. 9 (1) 1173 - 1204, 2015. https://doi.org/10.1214/15-EJS1030

Information

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

zbMATH: 1348.68285
MathSciNet: MR3354335
Digital Object Identifier: 10.1214/15-EJS1030

Subjects:
Primary: 55, 60

Rights: Copyright © 2015 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.9 • No. 1 • 2015
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