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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.
OnmultivariateGaussiancopulas
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
ϕ(u) and cdf F(1)(x) = �(u), where
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.
OnmultivariateGaussiancopulas
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
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.
OnmultivariateGaussiancopulas
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)
OnmultivariateGaussiancopulas
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)
OnmultivariateGaussiancopulas
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
OnmultivariateGaussiancopulas
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 matrix
Gaussian copulas consider only pairwise dependencebetween individual components of a RV
OnmultivariateGaussiancopulas
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
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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′)
.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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.
OnmultivariateGaussiancopulas
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?
OnmultivariateGaussiancopulas
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?
OnmultivariateGaussiancopulas
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?
OnmultivariateGaussiancopulas
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
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?
OnmultivariateGaussiancopulas
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
2
ρS = 6π arcsin ρ
2 ⇔ ρ = 2 sin πρS6
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?
OnmultivariateGaussiancopulas
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?
OnmultivariateGaussiancopulas
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
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?
OnmultivariateGaussiancopulas
Ivan �eºula
Choice of dependence measure
Di�erences between powers of τ and ρ(p = 1, . . . , 5)
Maximum√
1− 4/π2 − 2/π · arcsin√
1− 4/π2 ≈ 0.211.
OnmultivariateGaussiancopulas
Ivan �eºula
Choice of dependence measure
Di�erences between powers of ρS and ρ(p = 1, . . . , 5)
Maximum 2√
1− 9/π2 − 6/π · arcsin√
1− 9/π2 ≈ 0.018.
OnmultivariateGaussiancopulas
Ivan �eºula
Choice of dependence measure
Di�erences ρkS
ρk−1S
− ρ Di�erences ρkS
ρS− ρk−1
(k = 2, . . . , 5) (k = 2, . . . , 5)
OnmultivariateGaussiancopulas
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!
OnmultivariateGaussiancopulas
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!
OnmultivariateGaussiancopulas
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!
OnmultivariateGaussiancopulas
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.