# Probabilistic Fréchet means for time varying persistence diagrams

Journal Article (Journal Article)

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.

### Full Text

### Duke Authors

### Cited Authors

- Munch, E; Turner, K; Bendich, P; Mukherjee, S; Mattingly, J; Harer, J

### Published Date

- January 1, 2015

### Published In

### Volume / Issue

- 9 /

### Start / End Page

- 1173 - 1204

### International Standard Serial Number (ISSN)

- 1935-7524

### Digital Object Identifier (DOI)

- 10.1214/15-EJS1030

### Citation Source

- Scopus