Quantile tomography: using quantiles with multivariate data

By Linglong Kong, Ivan Mizera. Statistica sinica, 2008

159 citations recorded by the source corpus.

The use of quantiles to obtain insights about multivariate data is ad- dressed. It is argued that incisive insights can be obtained by considering direc- tional 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 distri- butions (in particular multivariate normal) with density contours. Relevant ques- tions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approx- imation/asymptotic properties are studied. It is argued that analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic inter- pretation and thus conveys a concrete quantitative meaning; the directional defini- tion 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 this development.

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2c4f877555beb57e63d001be2c140f87666ffabc

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Citations in the corpus

Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation (2010, background): “Tyler, Critchley, Dümbgen, and Oja (TCDO) (2009) define ‘invariant coordinate system’ (ICS) functionals and study them in depth.”
Vector Quantile Regression (2014, background): “A closely related construction is provided by [23] who define the directional quantile associated to the index u ∈ Bd via: Q Y (u) = Qu>Y/‖u‖ (‖u‖)u/‖u‖, where Qu>Y/‖u‖ is the univariate quantile function of the random variable u >Y/‖u‖.”
Localized Conformal Prediction: A Generalized Inference Framework for Conformal Prediction (2021, background): “Various authors have worked on constructions of PIs for multi-dimensional responses to address this issue (Paindaveine & ˇSiman 2011, Kong & Mizera 2012) and Feldman et al. (2021) has incorporated such constructions into CP recently.”
The spatial distribution in infinite dimensional spaces and related quantiles and depths (2014, background): “These quantiles are defined for unit direction vectors in such spaces, and they extend the finite dimensional quantiles considered in [21].”