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Probabilistic Fréchet means for time varying persistence diagrams

dc.contributor.author Bendich, Paul L
dc.contributor.author Harer, John
dc.contributor.author Mattingly, Jonathan Christopher
dc.contributor.author Mukherjee, Sayan
dc.contributor.author Munch, E
dc.contributor.author Turner, K
dc.date.accessioned 2015-05-14T16:11:48Z
dc.date.issued 2015-01-01
dc.identifier.issn 1935-7524
dc.identifier.uri http://hdl.handle.net/10161/10051
dc.description.abstract © 2015, Institute of Mathematical Statistics. All rights reserved.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 (D<inf>p</inf>, W<inf>p</inf>), 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 (D<inf>p</inf>)<sup>N</sup>→ℙ(D<inf>p</inf>). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.
dc.relation.ispartof Electronic Journal of Statistics
dc.relation.isversionof 10.1214/15-EJS1030
dc.title Probabilistic Fréchet means for time varying persistence diagrams
dc.type Journal article
pubs.begin-page 1173
pubs.end-page 1204
pubs.organisational-group Basic Science Departments
pubs.organisational-group Biostatistics & Bioinformatics
pubs.organisational-group Computer Science
pubs.organisational-group Duke
pubs.organisational-group Mathematics
pubs.organisational-group School of Medicine
pubs.organisational-group Statistical Science
pubs.organisational-group Trinity College of Arts & Sciences
pubs.publication-status Published
pubs.volume 9


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