Quantile tomography: using quantiles with multivariate data

TL;DR

This paper introduces directional quantiles and quantile envelopes for multivariate data, providing a new way to interpret and analyze complex data structures with probabilistic meaning and applications to growth charts.

Contribution

It proposes directional quantile envelopes as a novel tool for multivariate analysis, linking them to halfspace depth and enabling new insights and applications.

Findings

Directional quantile envelopes coincide with density contours for elliptic distributions.

The approach offers a straightforward probabilistic interpretation.

Applications include multivariate growth charts and regression of depth contours.

Abstract

The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development.