## Probabilistic Fréchet means for time varying persistence diagrams

Publication ,  Journal Article
Munch, E; Turner, K; Bendich, P; Mukherjee, S; Mattingly, J; Harer, J
Published in: Electronic Journal of Statistics
January 1, 2015

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.

## Published In

Electronic Journal of Statistics

1935-7524

January 1, 2015

9

## Start / End Page

1173 / 1204

• 4905 Statistics
• 0104 Statistics

### Citation

APA
Chicago
ICMJE
MLA
NLM
Munch, E., Turner, K., Bendich, P., Mukherjee, S., Mattingly, J., & Harer, J. (2015). Probabilistic Fréchet means for time varying persistence diagrams. Electronic Journal of Statistics, 9, 1173–1204. https://doi.org/10.1214/15-EJS1030
Munch, E., K. Turner, P. Bendich, S. Mukherjee, J. Mattingly, and J. Harer. “Probabilistic Fréchet means for time varying persistence diagrams.” Electronic Journal of Statistics 9 (January 1, 2015): 1173–1204. https://doi.org/10.1214/15-EJS1030.
Munch E, Turner K, Bendich P, Mukherjee S, Mattingly J, Harer J. Probabilistic Fréchet means for time varying persistence diagrams. Electronic Journal of Statistics. 2015 Jan 1;9:1173–204.
Munch, E., et al. “Probabilistic Fréchet means for time varying persistence diagrams.” Electronic Journal of Statistics, vol. 9, Jan. 2015, pp. 1173–204. Scopus, doi:10.1214/15-EJS1030.
Munch E, Turner K, Bendich P, Mukherjee S, Mattingly J, Harer J. Probabilistic Fréchet means for time varying persistence diagrams. Electronic Journal of Statistics. 2015 Jan 1;9:1173–1204.

## Published In

Electronic Journal of Statistics

1935-7524

January 1, 2015

9

1173 / 1204