Multivariate Non-Normally Distributed Random Variables

Transcription

1 Multivariate Non-Normally Distributed Random Variables An Introduction to the Copula Approach Workgroup seminar on climate dynamics Meteorological Institute at the University of Bonn 18 January 2008, Bonn Christian Schölzel 1(2) and Petra Friederichs 1 1) Meteorological Institute at the University of Bonn Bonn, Germany 2) Laboratoire des Sciences du Climat et l Environnement (LSCE) Gif-sur-Yvette, France

2 Motivation Multivariate non-normally distributed random variables Interest Estimating multivariate probability density functions Problem At least one component might be precipitation wind speed and direction relative humidity cloud cover... extremes Question How to describe random vectors with non-normal marginals?

3 Background General problem with multivariate random variables Standard textbooks Storch, H. v. and Zwiers, F. W. (1999): Statistical Analysis in Climate Research Wilks, D. S. (2005): Statistical Methods in the Atmospheric Sciences multivariate normal distribution Marginal distributions numerous parametric descriptions Dependence allows for more complexity degrees-of-freedom no canonical extension

4 Related Approaches How has this problem been addressed so far? Multivariate normal distribution* *) Not an option True/native multivariate distribution Best option, but often restricted in modeling marginals and dependence Mixture models Flexible w.r.t. covariance structure, complex in higher dimensions, no account for the original marginal distribution Marginal and conditional pdf Complexity of the estimation process grows exponentially Multivariate KDE* *) Not a parametric description Johnson distribution Simple concept, limited dependence strucure (precursor & special case)

5 The copula approach Basics The copula approach A brief (and incomplete) historical overview : johnson distribution and other precursors 1959: the term copula has been introdced (Sklar) 2000: applications in econometry 2007: applications in hydrology

6 Copula Definition Sklar s therorm (1959) Assume a m-dimensional random vector X with marginal cumulative distribution functions F X1,..., F Xm Definition copula A copula C X is a multivariate distribution with standard uniform marginal distributions Sklar s theorem (1959) Every joint distribution F X can be written as a function of its marginal distributions F X ( x) = C X (F X1 (x 1 ),..., F Xm (x m)) Note: C X is a copula, since U j = F Xj (X j ) = U j U(0, 1)

7 Definition Copula Copula density Copula density Furthermore, C X is unique and can be expressed by with u j = F Xj (x j ) u 1 u m Z Z C X (u 1,..., u m) = c X (u 1,..., u m) du 1... u m {z } 0 0 copula density Consequence of Sklar s theorem Every joint probability density can be written as f X ( x) = f X1 (x 1 ) f Xm (x m) c X (u 1,..., u m)

8 Bivariate Examples Normal-gamma and beta-beta marginals

9 Copula Families Elliptical and Archimedian copulas family Copula Families Two major classes are Elliptical copulas and Archimedian copulas Elliptical copulas Derived from elliptical distributions Normal (Gaussian) copula * Student s t-copula Archimedian copulas Base on so called generator functions C X (u 1,..., u m) = φ 1 (φ(u 1 ) + + φ(u m)) Clayton copula: φ C (u) = u θ C 1 Frank copula: φ F (u) = log( eθ F u 1 e θ F 1 ) Gumbel copula: φ G (u) = ( log u) θ G

10 Normal Copulas Derivation based on Sklar s theorem Additional transformation U j = F X (X j ) U(0, 1) Z j = F 1 N (0,1) (U j) N (0, 1) Z = (Z 1,..., Z m) T N (0, Σ) Normal copula Sklar s theorem leads to the copula C X (u 1,..., u m) = F N (0,Σ)`F 1 N (0,1) (u 1),..., F 1 N (0,1) (um) and the copula density c X (u 1,..., u m) = u 1 = f N (0,Σ) C X (u 1,..., u m) u m 1 `F N (0,1) (u 1),..., F (um) 1 N (0,1) Q (um) m 1 j=1 f N (0,1)`F N (0,1)

11 Title Subtitile Z j = F 1 N (0,1) (U j) U j = F Xj (X j ) X j f 1 N (Σ,1) c X f X

12 Student t-copula A more flexible case of elliptical copulas Student t-copula Based on an extension of the multivariate t-distribution C X (u 1,..., u m) = F t(ν,σ)`f 1 t(ν) (u 1),..., F 1 t(ν) (um), Tail dependence Definition of upper tail dependence (lower analogously) λ up = lim P X 1 > F 1 X u 1 1 (u) X 2 > F 1 X 2 (u)

13 Different Copulas Dependence structure and upper/lower tail dependence

14 Estimation The aim is to fit the whole multivariate distribution Full maximum likelihood (ML) Maximize the log-likelihood functions for the full pdf f X consistent estimates for all parameters often only numerically feasible Inference functions for margins (IFM) Two-stage estimation process (1. margins, 2. copula) computationally more efficient than ML predefined estimators for margins and copula no independent parameter estimates Canonical maximum likelihood (CML) Like IFM but use the empirical CDF of each margin instead consistent estimates of the copula parameters practical use questionable (see discussion)

15 Goodness-of-Fit The probability integral transform (PIT) Dimension reduction Transform X = (X 1,..., X m) T to a set of independent, uniform variables The probability integral transform (PIT) The PIT of X is defined as T 1 (X 1 ) = F X1 (X 1 ) T 2 (X 2 ) = F X2 X 1 (X 2 X 1 ). T m(x m) = F Xm X 1,...,X m 1 (X m X 1,..., X m 1 ) Hypothesis H 0 X comes from the specified multivariate model (F X ) = Z j = T j (X j ) iid U(0, 1)

16 Goodness-of-Fit The probability integral transform (PIT) Dimension reduction With the multivariate random variable (above) Z j = T j (X j ) iid U(0, 1) it follows that Y = mx j=1 F 1 N (0,1) (Z j ) has a χ 2 -distribution with m degrees of freedom, so that W = F χ 2 m (Y ) should be a univariate random variable from U(0, 1) univariate GoF tests

17 Applications Precipitation and minimum temperature at stations Berlin and Potsdam Example A: different random variables Example B: different locations

18 Multivariate Extremes How are copulas related to multivariate extremes? Copulas are related to multivariate extremes (later)

19 Capabilities What are copulas good for? The copula approach Simple and straight forward method to find parametric descriptions of multivariate non-normally distributed random variables Characteristics Catch different dependence structures Keep proper parametric descriptions of the margins Low number of parameters Mixture models (alternative and enhancement) More degrees-of-freedom Allow only approximative descriptions of the marginal distributions Limited in modelling tail dependence Based on the assumption of different populations mixture models and copulas can be combined

20 Limitations What are typical pitfalls? Critics: Mikosch, T. (2006), Copulas: Tales and facts, Extremes, 9, 3 20, Limitations Limited to the existing copula families No general procedure for selecting the copula class Pitfalls Questionable practices in finance, risk management, or insurance Theoretical value of the copula must not be exaggerated There is no dependence separately from the marginal distributions Copulas do not solve the problem of dimensionality

21 Outlook Where to go? Outlook After numerous successful applications in risk management, financial research and more recently in hydrology, it is very likely that copulas will have growing impact in meteorology and climate research

22 Navigation An Introduction to the Copula Approach 1 Introduction Motivation Background and Related Work 2 The Copula Approach Basics Copula Families Estimation Goodness-of-Fit 3 Applications 4 Multivariate Extremes 5 Discussion Capabilities Limitations Outlook Appendix