Canonical quantile regression

In using multiple regression methods for prediction, one often considers the linear combination of explanatory variables as an index. Seeking a single such index when here are multiple responses is rather more complicated. One classical approach is to use the coefficients from the leading Canonical Correlation. However, methods based on variances are unable to disaggregate responses by quantile effects, lack robustness, and rely on normal assumptions for inference. An alternative canonical regression quantile (CanRQ) approach seeks to find the linear combination of explanatory variables that best predicts the τth quantile of the best linear combination of response variables. Applying this “regression” approach more generally, subsequent linear combinations are chosen to explain what earlier CanRQ components failed to explain. While numerous technical issues need to be addressed, the major methodological issue concerns directionality: a quantile analysis requires that the notion of a larger or smaller response be well-defined. To address this issue, the sign of at least one response coefficient will be assumed to be non-negative. CanRQ results can be quite different from those of classical canonical correlation, and can offer the kind of improvements offered by regression quantiles in linear models.