A Multivariate Generalization of Quantile-Quantile Plots

Abstract

In this article we present a multivariate generalization of quantile-quantile (Q-Q) plots. Like univariate Q-Q plots, these plots are useful for examining the distributional shape of multivariate point clouds. These plots are based on finding a matching between the points of the data set whose shape is being examined and a reference sample. Graphical displays of how well the point clouds match are then developed. The reference sample used as the basis for comparison is typically derived from a random sample from a known multivariate distribution. The approach presented in this article is both a direct extension of the usual univariate Q-Q plot and truly multivariate in nature. It is truly multivariate in that the displays we develop show different aspects of one multivariate comparison between the data and the reference sample. This is unlike most generalizations of Q-Q plots to the multivariate case, which are based on making standard univariate Q-Q plots after some function of the multivariate observations has been used to reduce the dimension of the problem. Our method is also not tied to any specific reference distribution such as the multivariate normal. Furthermore, because it is truly multivariate, it is capable of uncovering certain kinds of features in the data that can be very difficult to detect using standard approaches.