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Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. For analysing spatially indexed data, we address this difficulty by generalizing the joint quantile regression model of Yang and Tokdar (Journal of the American Statistical Association, 2017, 112(519), 1107–1120) and characterizing spatial dependence via a Gaussian or t-copula process on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out spatial quantile smoothing. An effective model comparison criteria is provided, particularly for selecting between different model specifications of tail heaviness and tail dependence. Extensive simulation studies and two real applications to particulate matter concentration and wildfire risk are presented to illustrate substantial gains in inference quality, prediction accuracy and uncertainty quantification over existing alternatives.

 

Quantile regression is widely seen as an ideal tool to understand complex predictor-response relations. Its biggest promise rests in its ability to quantify whether and how predictor effects vary across response quantile levels. But this promise has not been fully met due to a lack of statistical estimation methods that perform a rigorous, joint analysis of all quantile levels. This gap has been recently bridged by Yang and Tokdar [18]. Here we demonstrate how their joint quantile regression method, as encoded in the R package qrjoint, offers a comprehensive and model-based regression analysis framework. This chapter is an R vignette where we illustrate how to fit models, interpret coefficients, improve and compare models and obtain predictions under this framework. Our case study is an application to ecology where we analyse how the abundance of red maple trees depends on topographical and geographical features of the location. A complete absence of the species contributes excess zeros in the response data. We treat such excess zeros as left censoring in the spirit of a Tobit regression analysis. By utilising the generative nature of the joint quantile regression model, we not only adjust for censoring but also treat it as an object of independent scientific interest.