High-Dimensional Spatial Quantile Function-on-Scalar Regression

Abstract

This article develops a novel spatial quantile function-on-scalar regression model, which studies the conditional spatial distribution of a high-dimensional functional response given scalar predictors. With the strength of both quantile regression and copula modeling, we are able to explicitly characterize the conditional distribution of the functional or image response on the whole spatial domain. Our method provides a comprehensive understanding of the effect of scalar covariates on functional responses across different quantile levels and also gives a practical way to generate new images for given covariate values. Theoretically, we establish the minimax rates of convergence for estimating coefficient functions under both fixed and random designs. We further develop an efficient primal-dual algorithm to handle high-dimensional image data. Simulations and real data analysis are conducted to examine the finite-sample performance.

KEYWORDS:

• Copula
• Function-on-scalar regression
• Image analysis
• Minimax rate of convergence
• Quantile regression
• Reproducing kernel Hilbert space

Supplementary Materials

The supplementary materials contain proofs of main theorems and lemmas, additional numerical experiments, and the details of the ADMM algorithm.

Additional information

Funding

Dr. Zhang’s research was partially supported by NIH grants MH118927 and AG066970. Dr. Wang’s research was supported by the NSF grant DMS-1613060. Dr. Kong’s research was partially supported by the Canadian Statistical Sciences Institute Collaborative Research Team Projects (CANSSI-CRT), the Natural Sciences and Engineering Research Council of Canada (NSERC) and Canada Research Chair (CRC) in Statistical Learning. Dr. Zhu’s research was partially supported by NIH grants MH116527 and MH086633.