Self-organizing map visualizing conditional quantile functions with multidimensional covariates
Two existing methods, namely, local linear quantile regression and self-organizing map (SOM) are combined. The combination provides a fully operational method for the visualization of the t:th quantile q_t(x) in the conditional distribution of a dependent variable Y given the value X=x of a vector of many covariates. Quantile regression is used to provide a picture of the effect of x on the distribution of Y covering not only the center of the distribution, but also the upper and lower tails. Since the local linear quantile regression model is nonparametric, the shape of the estimate for q_t(x) may vary both by values of t and by values of x. The novelty of the proposed methodology ensues from the capability to track these changes in the regression surface via a two-dimensional SOM component plane representation. The methodology eases the interpretation of the dependence between the t:th quantile and covariates that is captured by the conditional quantile function. Moreover, the methodology reveals the sensitivity of this relationship to changes in x that is captured by the gradient of the conditional quantile function. Examples using both simulated and real data are provided to illustrate the methodology.