PERSISTENT HOMOLOGY

Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features as one varies a tuning parameter. We are bringing some statistical ideas to persistent homology. In particular, we derived confidence intervals that allow us to separate topological signal from topological noise. I am collaborating with Frederic Chazal,

__Brittany Fasy__, Bertrand Michel,

__Alessandro Rinaldo__, and

__Larry Wasserman__.

For more information on our work see the TopStat website.

We have written the R package TDA for Topological Data Analysis. See this page

DENSITY CLUSTERING

Density clustering allows us to identify and visualize the spatial organization of a point cloud, without specific knowledge about the data generating mechanism and in particular without any a priori information about the number of clusters. The main topological descriptor is the cluster tree, which provides a simple yet meaningful abstraction of the topological changes in the level sets of a density estimator.

I am working on density clustering with Brian Kent,

__Alessandro Rinaldo__and

__Larry Wasserman__.

METRIC GRAPH RECONSTRUCTION

A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. I am studying the statistical problem of reconstructing the topology of a metric graph from a random sample. See our

__paper__.

CHS DATA AND PREDICTION OF DEMENTIA

Alzheimer’s disease is the most common form of dementia in the elderly. There are complex relationships between age and other risk factors, producing highly variable natural histories from normal cognition through the prodromal stage of Mild Cognitive Impairment to clinical dementia. We are using a novel statistical approach, Mixed Membership Trajectory Models, (Manrique-Vallier and Fienberg, 2009) to capture the variety of such pathways or trajectories.

We are also working on a Fused Lasso model that predicts the onset of dementia using 4000 variables from the

__CHS__longitudinal study.

I am working on this project with Sam Adhikari, James Becker,

__Brian Junker__, Oscar Lopez and

__Ryan Tibshirani__