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Modelling multivariate extremes through angular-radial decomposition of the density function

Ed Mackay, Philip Jonathan

Abstract

We present a new framework for modelling multivariate extremes, based on an angular-radial representation of the probability density function. Under this representation, the problem of modelling multivariate extremes is transformed to that of modelling an angular density and the tail of the radial variable, conditional on angle. Motivated by univariate theory, we assume that the tail of the conditional radial distribution converges to a generalised Pareto (GP) distribution. To simplify inference, we also assume that the angular density is continuous and finite and the GP parameter functions are continuous with angle. We refer to the resulting model as the semi-parametric angular-radial (SPAR) model for multivariate extremes. We consider the effect of the choice of polar coordinate system and introduce generalised concepts of angular-radial coordinate systems and generalised scalar angles in two dimensions. We show that under certain conditions, the choice of polar coordinate system does not affect the validity of the SPAR assumptions. However, some choices of coordinate system lead to simpler representations. In contrast, we show that the choice of margin does affect whether the model assumptions are satisfied. In particular, the use of Laplace margins results in a form of the density function for which the SPAR assumptions are satisfied for many common families of copula, with various dependence classes. We show that the SPAR model provides a more versatile framework for characterising multivariate extremes than provided by existing approaches, and that several commonly-used approaches are special cases of the SPAR model.

Modelling multivariate extremes through angular-radial decomposition of the density function

Abstract

We present a new framework for modelling multivariate extremes, based on an angular-radial representation of the probability density function. Under this representation, the problem of modelling multivariate extremes is transformed to that of modelling an angular density and the tail of the radial variable, conditional on angle. Motivated by univariate theory, we assume that the tail of the conditional radial distribution converges to a generalised Pareto (GP) distribution. To simplify inference, we also assume that the angular density is continuous and finite and the GP parameter functions are continuous with angle. We refer to the resulting model as the semi-parametric angular-radial (SPAR) model for multivariate extremes. We consider the effect of the choice of polar coordinate system and introduce generalised concepts of angular-radial coordinate systems and generalised scalar angles in two dimensions. We show that under certain conditions, the choice of polar coordinate system does not affect the validity of the SPAR assumptions. However, some choices of coordinate system lead to simpler representations. In contrast, we show that the choice of margin does affect whether the model assumptions are satisfied. In particular, the use of Laplace margins results in a form of the density function for which the SPAR assumptions are satisfied for many common families of copula, with various dependence classes. We show that the SPAR model provides a more versatile framework for characterising multivariate extremes than provided by existing approaches, and that several commonly-used approaches are special cases of the SPAR model.
Paper Structure (41 sections, 18 theorems, 234 equations, 17 figures)

This paper contains 41 sections, 18 theorems, 234 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Definition of the SPAR model
  4. Outline of the paper
  5. Generalised angular-radial coordinate systems
  6. Polar coordinate systems for star-shaped sets
  7. Generalised angles in R2
  8. Effects of the choice of coordinate system
  9. Bivariate density functions
  10. Multivariate density functions
  11. Angular-radial models for copulas
  12. Constant scaling order
  13. Different scaling orders
  14. Relations between models for constant and variable scaling orders
  15. Generalised angular-radial dependence functions
  16. ...and 26 more sections

Key Result

Proposition 3.1

Let $\cos_p$ and $\sin_p$ be pseudo-trigonometric functions for the $L^p$ norm. Then for $q\in\mathbb{R}$, When $p=1$, 2 or $\infty$, $J_p(q)$ is a constant and we have For $p\in(1,2)\cup(2,\infty)$, $J_p(q)$ is not constant.

Figures (17)

  • Figure 1: Example datasets and structural responses for motivating problems. Ellipses on plots (b) and (d) indicate regions of variable space of where offshore structures may experience large responses. Regions in the red ellipses are not the largest values in either variable, whereas regions in the blue ellipses are extreme in at least one variable. The aim of the methodology proposed in this work is to characterise all 'extreme regions' of a joint distribution using a single inference.
  • Figure 2: Examples of two possible transformations from Cartesian to polar coordinates, using data from the previous figure. After the polar coordinate transformation, the problem of modelling multivariate extremes is transformed to a problem of modelling univariate extremes conditional on angle.
  • Figure 3: Black dots: Random sample of $10^4$ points from bivariate normal distribution with $\rho=0.7$. Coloured lines in (a) are isodensity contours at equal logarithmic increments. Coloured lines in (b) are level sets of the joint survivor function at equal logarithmic increments. In (b) an angular-radial description is only useful in the upper-right quadrant of the plane.
  • Figure 4: Illustration of definition of gauge function $\mathcal{R}_*$ and pseudo-angle, $q$, with respect to boundary $\mathcal{S}_*$. The circumference of the boundary is $\mathcal{C}_*$. Pseudo-trigonometric functions $\sin_*(q)$ and $\cos_*(q)$ relate the pseudo-angle with the corresponding x- and y-coordinates on the unit circle.
  • Figure 5: Left: Unit circles for the $L^p$ norm for various values of $p$. When $0<p<1$, $\|\cdot\|_p$ is not a norm, but is a gauge function. Right: Corresponding pseudo-trigonometric functions.
  • ...and 12 more figures

Theorems & Definitions (53)

  • Definition 1: Star-shaped set
  • Definition 2: Gauge function
  • Definition 3: Norm
  • Definition 4: Unit sphere of a norm
  • Definition 5: $L^p$ norm
  • Definition 6: Polar coordinates in $\mathbb{R}^d$
  • Definition 7: Arc length functions
  • Definition 8: Pseudo-angles
  • Definition 9: Polar coordinates on $\mathbb{R}^2$
  • Definition 10: Pseudo-trigonometric functions
  • ...and 43 more