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Title: Dimension reduction and data visualization for Frechet regression Authors:  Qi Zhang - The Pennsylvania State University (United States) [presenting]
Bing Li - The Pennsylvania State University (United States)
Lingzhou Xue - The Pennsylvania State University (United States)
Abstract: With the rapid development of data collection techniques, complex data objects that are not in the Euclidean space are frequently encountered in new statistical applications. Frechet regression model provides a promising framework for regression analysis with metric space-valued responses, such as univariate probability distributions, covariance matrices and spherical data. We introduce a flexible sufficient dimension reduction method for Frechet regression to achieve two purposes: to mitigate the curse of dimensionality caused by the high-dimensional predictor, and to provide a tool for data visualization for Frechet regression. The approach is flexible enough to turn any existing SDR method for Euclidean $(X,Y)$ into one for Euclidean $X$ and metric space-valued $Y$. We derive the consistency and asymptotic convergence rate of the proposed methods. The finite-sample performance of the novel methods is illustrated through simulation studies of several commonly encountered metric spaces that include Wasserstein space, the space of symmetric positive definite matrices and the sphere. We illustrated the data visualization aspect of our method by the human mortality distribution data from the United Nation Databases and stroke data with CT hematoma densities.