# 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 (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.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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