High-Dimensional Spatial Quantile Function-on-Scalar Regression

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.

Automated summary

This study introduces a novel spatial quantile function-on-scalar regression model to analyze the conditional spatial distribution of high-dimensional functional responses given scalar predictors. The method characterizes the conditional distribution of functional or image responses accurately and comprehensively, providing insights into the effect of scalar covariates across different quantile levels and offering a practical approach to generate new images. The study establishes minimax rates of convergence for estimating coefficient functions and develops an efficient primal-dual algorithm for handling high-dimensional image data, with simulations and real data analysis demonstrating the method's performance.