Dimension Reduction and Data Visualization for Fréchet Regression

• Activity Number: 369 - Analysis of Random Objects
• Type: Invited
• Date/Time: Wednesday, August 10, 2022 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Qi Zhang and Lingzhou Xue* and Bing Li
• Companies: Penn State University and Penn State University and National Institute of Statistical Sciences and Penn State University
• Keywords: Ensembled sufficient dimension reduction; Forward regression; Inverse regression; Statistical objects; Universal kernel; Wasserstein space

Abstract

Fréchet regression model (Peterson & Müller 2019) provides a promising framework for regression analysis with metric space-valued responses. We introduce a flexible sufficient dimension reduction (SDR) method for Fréchet regression to achieve two purposes: to mitigate the curse of dimensionality caused by high-dimensional predictors, and to provide a tool for data visualization for Fréchet regression. Our approach is flexible enough to turn any existing SDR method for Euclidean (X, Y) into one for Euclidean X and metric space-valued Y. The basic idea is to first map the metric-space valued random object to a real-valued random variable using a class of functions, and then perform classical SDR to the transformed data. If the class of functions is sufficiently rich, then we are guaranteed to uncover the Fréchet SDR space. We showed that such a class, which we call an ensemble, can be generated by a universal kernel. We established the consistency and asymptotic convergence rate of the proposed methods. The finite-sample performance of the proposed methods is illustrated through simulation studies. We illustrated the data visualization aspect of our method in real applications.