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Central subspace data depth

Giacomo Francisci, Claudio Agostinelli

TL;DR

This work provides a general framework to construct statistical data depths which attain maximum value in a subspace, providing a center-outward ordering from that subspace, and introduces general notions of symmetry about a subspace for distributions.

Abstract

Statistical data depth plays an important role in the analysis of multivariate data sets. The main outcome is a center-outward ordering of the observations that can be used both to highlight features of the underlying distribution of the data and as input to further statistical analysis. An important property of data depth is related to symmetric distributions as the point with the highest depth value, the center, coincides with the point of symmetry. However, there are applications in which it is more natural to consider symmetry with respect to a subspace of a certain dimension rather than to a point, i.e. a subspace of dimension zero. We provide a general framework to construct statistical data depths which attain maximum value in a subspace, providing a center-outward ordering from that subspace. We refer to these data depths as central subspace data depths. Moreover, if the distribution is symmetric with respect to a subspace, then the depth is maximized at that subspace. We introduce general notions of symmetry about a subspace for distributions, study the properties of central subspace data depths and provide asymptotic convergence for the corresponding sample versions. Additionally, we discuss connections with projection pursuit and dimension reduction. An application based on custom data fraud detection shows the importance of the proposed approach and strengthens its potential.

Central subspace data depth

TL;DR

This work provides a general framework to construct statistical data depths which attain maximum value in a subspace, providing a center-outward ordering from that subspace, and introduces general notions of symmetry about a subspace for distributions.

Abstract

Statistical data depth plays an important role in the analysis of multivariate data sets. The main outcome is a center-outward ordering of the observations that can be used both to highlight features of the underlying distribution of the data and as input to further statistical analysis. An important property of data depth is related to symmetric distributions as the point with the highest depth value, the center, coincides with the point of symmetry. However, there are applications in which it is more natural to consider symmetry with respect to a subspace of a certain dimension rather than to a point, i.e. a subspace of dimension zero. We provide a general framework to construct statistical data depths which attain maximum value in a subspace, providing a center-outward ordering from that subspace. We refer to these data depths as central subspace data depths. Moreover, if the distribution is symmetric with respect to a subspace, then the depth is maximized at that subspace. We introduce general notions of symmetry about a subspace for distributions, study the properties of central subspace data depths and provide asymptotic convergence for the corresponding sample versions. Additionally, we discuss connections with projection pursuit and dimension reduction. An application based on custom data fraud detection shows the importance of the proposed approach and strengthens its potential.
Paper Structure (27 sections, 28 theorems, 112 equations, 11 figures, 1 table)

This paper contains 27 sections, 28 theorems, 112 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Symmetry
  3. Deeply immersion, central subspace and data depth
  4. Central subspace data depths
  5. Selection of the optimal subspace dimension
  6. Some properties of the dispersion measure and of the central subspace data depth
  7. Dimension reduction
  8. Data analysis
  9. Concluding remarks
  10. Introduction
  11. Symmetry
  12. Symmetry with respect to a point
  13. Symmetry with respect to a subspace
  14. Symmetry in a subspace
  15. Statistical data depths
  16. ...and 12 more sections

Key Result

Proposition 1

Let $\boldsymbol{X} \sim F$ be absolutely continuous and elliptically symmetric with mean $\boldsymbol{\mu}$ and covariance matrix $\Sigma$. Consider a statistical data depth $d(\cdot , \cdot)$ with respect to at least elliptical symmetry and the corresponding dispersion measure $\sigma(\cdot)$. If if and only if $\hat{S}_p$ and $\hat{S}_q$ are orthogonal subspaces in $\mathbb{R}^m$ generated by

Figures (11)

  • Figure 1: POD 33 data set. Weights and prices are in log scale. Usual data depth (left panel), central subspace data depth (right panel). Maximum depth is highlighted in yellow, value of the depth in gray scale for the central region with $0.5$ probability content. Points in green are in the outer region. In the right panel we further mark points with quantiles between $0.95$ and $0.975$ (blue) and those in a region with quantile order greater than $0.975$ (red). The analysis is performed using halfspace depth.
  • Figure 2: Sample points in simulation scenario (i). First and second coordinate (left panel), second and third coordinate (central panel), and first and third coordinate (right panel). The region of maximum depth is highlighted in yellow. The true maximal direction is plotted in red.
  • Figure 3: Sample points in simulation scenario (ii). First and second coordinate (left panel), second and third coordinate (central panel), and first and third coordinate (right panel). The region of maximum depth is highlighted in yellow. The true maximal direction is plotted in red.
  • Figure 4: Mixture of normal distributions. Dispersion measure of the projected distribution, as a function of $u \in [-1,1]$, of a mixture of normal distributions with variance $\eta^{2} I$ and means the vertices $(1,1)$, $(1,-1)$, $(-1,1)$, and $(-1,-1)$ of a square. The analysis is performed using halfspace depth.
  • Figure 5: Iris data set. Maximal dispersion measure for $p=1,2,3,4$ (on the left) and the depth values of the projected data (on the right). The analysis is performed using halfspace depth.
  • ...and 6 more figures

Theorems & Definitions (56)

  • Definition 1: Halfspace symmetry
  • Definition 2: Symmetry with respect to a subspace
  • Definition 3: romanazzi2009
  • Definition 4: Deeply immersion
  • Definition 5: Central subspace
  • Example 1: Spherical symmetry
  • Example 2: Elliptical symmetry
  • Definition 6: Central subspace data depths
  • Proposition 1
  • Definition 7: Spherical symmetry
  • ...and 46 more