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# Probabilistic Fréchet means for time varying persistence diagrams

 dc.contributor.author Munch, E dc.contributor.author Turner, K dc.contributor.author Bendich, P dc.contributor.author Mukherjee, S dc.contributor.author Mattingly, J dc.contributor.author Harer, J dc.date.accessioned 2015-05-14T16:11:48Z dc.date.issued 2015-01-01 dc.identifier.issn 1935-7524 dc.identifier.uri https://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 (Dp, Wp), 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 (Dp)N→ℙ(Dp). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards. dc.publisher Institute of Mathematical Statistics 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 duke.contributor.id Bendich, P|0308528 duke.contributor.id Mattingly, J|0297691 duke.contributor.id Harer, J|0100474 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 duke.contributor.orcid Mattingly, J|0000-0002-1819-729X
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