Persistent homology: an approach for high dimensional data analysis

Topological data analysis (TDA) has been popularized since its development in early 2000. TDA has shown its effectiveness in discerning true features from noise in high-dimensional data. In this talk, we will introduce persistent homology, a particular branch of computational topology and discuss how it can be incorporated to classical statistics and techniques in machine learning. We will demonstrate its usefulness in classifying ADHD subjects. This is a joint project with Rui Hu, Zhichun Zhai, Linglong Kong and Bei Jiang.Non UBCUnreviewedAuthor affiliation: University of AlbertaFacult

Exploring uses of persistent homology for statistical analysis of landmark-based shape data

A method for the use of persistent homology in the statistical analysis of landmark-based shape data is given. Three-dimensional landmark configurations are used as input for separate filtrations, persistent homology is performed, and persistence diagrams are obtained. Groups of configurations are compared using distances between persistence diagrams combined with dimensionality reduction methods. A three-dimensional landmark-based data set is used from a longitudinal orthodontic study, and the persistent homology method is able to distinguish clinically relevant treatment effects. Comparisons are made with the traditional landmark-based statistical shape analysis methods of Dryden and Mardia, and Euclidean Distance Matrix Analysis.Persistent homology Wasserstein distance Landmark-based data Shape analysis Persistence diagrams Multidimensional scaling

Topological Hidden Markov Models

The hidden Markov model (HMM) is a classic modeling tool with a wide swath of applications. Its inception considered observations restricted to a finite alphabet, but it was quickly extended to multivariate continuous distributions. In this article, we further extend the HMM from mixtures of normal distributions in $d$-dimensional Euclidean space to general Gaussian measure mixtures in locally convex topological spaces. The main innovation is the use of the Onsager-Machlup functional as a proxy for the probability density function in infinite dimensional spaces. This allows for choice of a Cameron-Martin space suitable for a given application. We demonstrate the versatility of this methodology by applying it to simulated diffusion processes such as Brownian and fractional Brownian sample paths as well as the Ornstein-Uhlenbeck process. Our methodology is applied to the identification of sleep states from overnight polysomnography time series data with the aim of diagnosing Obstructive Sleep Apnea in pediatric patients. It is also applied to a series of annual cumulative snowfall curves from 1940 to 1990 in the city of Edmonton, Alberta.Comment: 32 pages, 8 figures, 6 table

Restricted Minimax Robust Designs for Misspecified Regression Models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of e#ciency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures su#ciently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression. R ESUM E Les auteurs proposent et explorent de nouveaux plans experimentaux pour la regression. Ces plans sont minimax par rapport a une classe parametrique restreinte et s'averent a la fois robustes au biais dua un mauvais choix de modele et raisonnablement e#caces. L'introduction de cette classe restreinte de plans est motivee par le desir d'evit..