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 |
|