on multivariate gaussian copulas - ut · on multivariate gaussian copulas ivan eºula choice of...

On multivariate Gaussian copulas

Ivan �eºula

Faculty of Science, P. J. �afárik University, Ko²ice

8th Tartu conference on multivariate statistics

OnmultivariateGaussiancopulas

Ivan �eºula

Notation

ϕ(x) standard normal density, �(x) standard normalcumulative distribution function, �−1(x)corresponding quantile function

general normal distribution has a density

f(1)(x) =1σ

ϕ(u) and cdf F(1)(x) = �(u), where

u =x− µ

σgeneral p-variate normal density can be expressed as

f(p)(x) =1

(2π)p2

p∏i=1

σi|R|12exp

{−12u′R−1u

}

where u = (u1, . . . , up)′, ui =

xi − µi

σi, and R is a

correlation matrix.


Ivan �eºula

Notation

ϕ(x) standard normal density, �(x) standard normalcumulative distribution function, �−1(x)corresponding quantile functiongeneral normal distribution has a density

f(1)(x) =1σ


u =x− µ

σ

general p-variate normal density can be expressed as

f(p)(x) =1

(2π)p2

p∏i=1

σi|R|12exp

{−12u′R−1u

}

where u = (u1, . . . , up)′, ui =

xi − µi

σi, and R is a

correlation matrix.


Ivan �eºula

Notation

ϕ(x) standard normal density, �(x) standard normalcumulative distribution function, �−1(x)corresponding quantile functiongeneral normal distribution has a density

f(1)(x) =1σ


u =x− µ

σgeneral p-variate normal density can be expressed as

f(p)(x) =1

(2π)p2

p∏i=1

σi|R|12exp

{−12u′R−1u

}

where u = (u1, . . . , up)′, ui =

xi − µi

σi, and R is a

correlation matrix.


Ivan �eºula

Introduction

for any multivariate absolutely continuousdistribution, with cdf F and marginal cdf's Fi,copula C is such distribution function on 〈0; 1〉p(with uniform one-dimensional marginals) that itholds

F (x1, . . . , xp) = C (F1(x1), . . . , Fp(xp))

copula density c is de�ned by

c =∂pC

∂F1 . . . ∂Fp;

then, joint density can be expressed as

f(x) = c (F1(x1), . . . , Fp(xp))p∏

i=1fi(xi)


Ivan �eºula

Introduction

Multivariate normal density can be written as

f(p)(x) =1

|R|12exp

{−12u′(R−1 − I

)u

} p∏i=1

1σi

ϕ (ui) ,

where ui = �−1 (Fi (xi)). Thus, if we allow Fi to be anarbitrary distribution function,

c(x) =1

|R|12exp

{−12u′(R−1 − I

)u

}is density of a Gaussian copula.

Gaussian copulas allow any marginal distribution andany p.d. correlation matrixGaussian copulas consider only pairwise dependencebetween individual components of a RV


Ivan �eºula

Introduction


f(p)(x) =1

|R|12exp

{−12u′(R−1 − I

)u

} p∏i=1

1σi

ϕ (ui) ,


c(x) =1

|R|12exp

{−12u′(R−1 − I

)u


Gaussian copulas allow any marginal distribution andany p.d. correlation matrix

Gaussian copulas consider only pairwise dependencebetween individual components of a RV


Ivan �eºula

Introduction


f(p)(x) =1

|R|12exp

{−12u′(R−1 − I

)u

} p∏i=1

1σi

ϕ (ui) ,


c(x) =1

|R|12exp

{−12u′(R−1 − I

)u


Gaussian copulas allow any marginal distribution andany p.d. correlation matrixGaussian copulas consider only pairwise dependencebetween individual components of a RV


Ivan �eºula

Special structures

Problems:

R can be di�cult to estimate, too many parameters

Gaussian densities are parameterized using Pearsoncorrelation coe�cients which are not invariant undermonotone transformations of original variables

Pearson ρ is not appropriate measure of dependencein many situations

Many times we meet simpler correlation structures, mostimportant of which are:

1 uniform correlation structure (ICC)2 serial (autoregressive) correlation structure

We can use them for construction of simple multivariateGaussian copulas with few parameters to estimate.


Ivan �eºula

Uniform correlation structure

R =

1 ρ . . . ρρ 1 . . . ρ...

.... . .

...ρ ρ . . . 1

= (1− ρ)Ip + ρ11′ ,

where ρ ∈⟨

−1p−1 ; 1

⟩.

It is easy to see that if ρ 6= 1 and ρ 6= −1p− 1

then R−1

exists and

R−1 =1

1− ρ

(Ip −

ρ

1 + (p− 1)ρ11′)

.


Ivan �eºula

Uniform correlation structure

Since |R| = [1 + (p− 1)ρ ](1− ρ)p−1, we get

c(x) =1

[1 + (p− 1)ρ ]12 (1− ρ)

p−12×

× exp{

−ρ

2(1− ρ)u′(

Ip −1

1 + (p− 1)ρ11′)

u

}.

Writing it componentwise:

c(x) =1√

[1 + (p− 1)ρ ](1− ρ)p−1exp

{−ρ

2(1− ρ)×

× 1[1 + (p− 1)ρ ]

(p− 1)ρp∑

i=1u2i − 2

∑∑i<j

uiuj

.

This is in accordance with Kelly and Krzysztofowicz [3]for p = 2.


Ivan �eºula

Serial correlation structure

R =

1 ρ . . . ρp−1

ρ 1 . . . ρp−2...

.... . .

...ρp−1 ρp−2 . . . 1

= Ip+p−1∑i=1

ρi(Ci1 + Ci

1′)

,

where C1 =( 0p−1 Ip−1

0 0′p−1

)and ρ ∈ 〈−1; 1〉.

If ρ 6= ±1 then R−1 exists and it holds

R−1 =1

1− ρ2[(1 + ρ2

)Ip − ρ2

(e1e′1 + epe

′p

)− ρ

(C1 + C ′

1)]

,

where ei contains 1 at i-th place and 0 at all other places.


Ivan �eºula

Serial correlation structure

Since |R| =(1− ρ2

)p−1, we get

c(x) =1

(1− ρ2)p−12×

×exp{

−ρ

2(1− ρ2)u′[2ρIp − ρ

(e1e′1 + epe

′p

)−(C1 + C ′

1)]

u

}.

Componentwise it is

c(x) =1√

(1− ρ2)p−1exp

{−ρ

2(1− ρ2)×

×

(2ρ

p∑i=1

u2i − ρ(u21 + u2p

)− 2

p−1∑i=1

uiui+1

)}.

This is again in accordance with Kelly and Krzysztofowicz[3] for p = 2.


Ivan �eºula

Choice of dependence measure

When working with non-elliptical distributions, it is betternot to use Pearson ρ. Usual alternatives are Kendall τand Spearman ρS :

they are both invariant on monotone transformations

they are both measures of concordance

Under normality, there is one-to-one relationship betweenthese coe�cients (Kruskal [4]):

τ = 2π arcsin ρ ⇔ ρ = sin πτ

2ρS = 6

π arcsin ρ2 ⇔ ρ = 2 sin πρS

6

Thus, we can estimate τ or ρS and convert it to ρ.Uniform correlation structure is not a�ected by this.What can we expect in the case of serial structure?


Ivan �eºula







2ρS = 6


6Thus, we can estimate τ or ρS and convert it to ρ.Uniform correlation structure is not a�ected by this.What can we expect in the case of serial structure?


Ivan �eºula







2

ρS = 6π arcsin ρ

2 ⇔ ρ = 2 sin πρS6



Ivan �eºula







2ρS = 6


6



Ivan �eºula







2ρS = 6


6Thus, we can estimate τ or ρS and convert it to ρ.Uniform correlation structure is not a�ected by this.What can we expect in the case of serial structure?


Ivan �eºula


Di�erences between powers of τ and ρ(p = 1, . . . , 5)

Maximum√

1− 4/π2 − 2/π · arcsin√

1− 4/π2 ≈ 0.211.


Ivan �eºula


Di�erences between powers of ρS and ρ(p = 1, . . . , 5)

Maximum 2√

1− 9/π2 − 6/π · arcsin√

1− 9/π2 ≈ 0.018.


Ivan �eºula


Di�erences ρkS

ρk−1S

− ρ Di�erences ρkS

ρS− ρk−1

(k = 2, . . . , 5) (k = 2, . . . , 5)


Ivan �eºula

Conclusions

Multivariate Gaussian copulas with uniform and serialcorrelation structures seem to be a simple tool formodeling dependence in complex situations.

Spearman ρS is a good dependence measure for use inthese situations, the di�erences in behaviour of powersbetween ρS and ρ are small.

Thank you for your attention!


Ivan �eºula

Literature

Aderman, V., Pihlak, M. (2005):Using copulas for modeling the dependence between

tree height and diameter at breast height. Acta etCommentationes Universitatis Tartuensis deMathematica, 9, 77�85.

Clemen, R.T., Reilly, T. (1999):Correlations and Copulas for Decision and Risk

Analysis. Management Science, 45(2), 208�224.

Kelly, K.S., Krzysztofowicz, R. (1997):A Bivariate Meta-Gaussian Density for Use in

Hydrology. Stochastic Hydrology and Hydraulics,11(1), 17�31.

Kruskal, W. (1958):Ordinal Measures of Association. JASA, 53, 814�861.

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