Partial quantile regression

Abstract

Partial least squares regression (PLSR) is a method of finding a reliable predictor of the response variable when there are more regressors than observations. It does so by eliciting a small number of components from the regressors that are inherently informative about the response. Quantile regression (QR) estimates the quantiles of the response distribution by regression functions of the covariates, and so gives a fuller description of the response than does the usual regression for the mean value of the response. We extend QR to partial quantile regression (PQR) when there are more regressors than observations. For each percentile the method provides a low dimensional approximation to the joint distribution of the covariates and response with a given coverage probability and which, under further linearity assumptions, estimates the corresponding quantile of the conditional distribution. The methodology parallels the procedure for PLSR using a quantile covariance that is appropriate for predicting a quantile rather than the usual covariance which is appropriate for predicting a mean value. The analysis suggests a new measure of risk associated with the quantile regressions. Examples are given that illustrate the methodology and the benefits accrued, based on simulated data and the analysis of spectrometer data.