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Univariate Functions
Copulas
Conditioning with Copulas
Let C1(u,v) denote the first partial derivative of
C(u,v). F(x,y) = C(FX(x),FY(y)), distribution of Y|X=x
is given by:
FY|X(y) = C1(FX(x),FY(y))
C(u,v) = uv, the conditional distribution of V given
that U=u is C1(u,v) = v = Pr(V<v|U=u).
If C1 is simple enough to invert algebraically, then
the simulation of joint probabilities can be done using
the derived conditional distribution. That is, first
simulate a value of U, then simulate a value of V
from C1.
Gary G Venter
Univariate Functions
Copulas
Tails of Copulas
ASTIN 2001
Gary G Venter
Univariate Functions
Copulas
Kendall correlation
t is a constant of the copula
t = 4E[C(u,v)] – 1
t = 2dE[C(u1, . . .,ud)] – 1
2d – 1 – 1
Gary G Venter
Univariate Functions
Copulas
Frank’s Copula
Define gz = e-az – 1
Frank’s copula with parameter a 0 can be expressed
as:
C(u,v) = -a-1ln[1 + gugv/g1]
C1(u,v) = [gugv+gv]/[gugv+g1]
c(u,v) = -ag1(1+gu+v)/(gugv+g1)2
t(a) = 1 – 4/a + 4/a2 0a t/(et-1) dt
For a<0 this will give negative values of t.
v = C1-1(p|u) = -a-1ln{1+pg1/[1+gu(1–p)]}
Gary G Venter
Univariate Functions
Copulas
10
1
0.1
0.01
Gary G Venter
Frank's Copula Density on Log Scale t=.5
Univariate Functions
Copulas
Gumbel Copula
C(u,v) = exp{- [(- ln u)a + (- ln v)a]1/a}, a 1.
C1(u,v) = C(u,v)[(- ln u)a + (- ln v)a]-1+1/a(-ln u)a-1/u
c(u,v) = C(u,v)u-1v-1[(-ln u)a +(-ln v)a]-2+2/a[(ln u)(ln v)]a1
{1+(a-1)[(-ln u)a +(-ln v)a]-1/a}
t(a) = 1 – 1/a
Simulate two independent uniform deviates u and v
Solve numerically for s>0 with ues = 1 + as
The pair [exp(-sva), exp(-s(1-v)a)] will have the Gumbel
copula distribution
Gary G Venter
Univariate Functions
Copulas
Gumbel Copula Log Scale t =0.5
100
10
1
0.1
0.01
0.001
Gary G Venter
Univariate Functions
Copulas
Heavy Right Tail Copula
C(u,v) = u + v – 1 + [(1 – u)-1/a + (1 – v)-1/a – 1]-a
a>0
C1(u,v) = 1 – [(1 – u)-1/a + (1 – v)-1/a – 1] -a-1(1 – u)-1-1/a
c(u,v) = (1+1/a)[(1–u)-1/a +(1– v)-1/a –1] -a-2[(1–u)(1–
v)]-1-1/a
t(a) = 1/(2a + 1)
Can solve conditional distribution for v
Gary G Venter
Univariate Functions
Copulas
Heavy Right Tail Copula Log Scale t = .5
100
10
1
0.1
Gary G Venter
Univariate Functions
Copulas
Joint Burr
F(x) = 1 – (1 + (x/b)p)-a and G(y) = 1 – (1 + (y/d)q)-a
F(x,y) = 1 – (1 + (x/b)p)-a – (1 + (y/d)q)-a +
[1 + (x/b)p + (y/d)q]-a
The conditional distribution of y|X=x is also Burr:
FY|X(y|x) = 1 – [1 + (y/dx)q]-(a+1),
where dx = d[1 + (x/b)p/q]
Gary G Venter
Copulas
Univariate Functions
Partial Perfect Correlation
Copula Generator
Assume logical values 0 and 1 are arithmetic also
h : unit square unit interval
H(x) = 0xh(t)dt
C(u,v) = uv – H(u)H(v) + H(1)H(min(u,v))
C1(u,v) = v – h(u)H(v) + H(1)h(u)(v>u)
c(u,v) = 1 – h(u)h(v) + H(1)h(u)(u=v)
Gary G Venter
Univariate Functions
Copulas
h(u) = (u>a)
H(u) = (u –
a)(u>a)
t(a) = (1 – a)4
PP MaxData Pairs t = .5
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
Gary G Venter
0.2
0.4
0.6
0.8
1
Univariate Functions
Copulas
h(u) = ua
PP(uv)^aDataPairst =.5
ua+1/(a+1)
1
H(u) =
0.9
t(a) = 1/[3(a+1)4] + 0.8
8/[(a+1)(a+2)2(a+3)] 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
Gary G Venter
0.2
0.4
0.6
0.8
1
Copulas
Univariate Functions
The Normal Copula
N(x) = N(x;0,1)
B(x,y;a) = bivariate normal distribution function, = a
Let p(u) be the percentile function for the standard
normal:
N(p(u)) = u, dN(p(u))/du = N’(p(u))p’(u) = 1
C(u,v) = B(p(u),p(v);a)
C1(u,v) = N(p(v);ap(u),1-a2)
c(u,v) = 1/{(1-a2)0.5exp([a2p(u)2-2ap(u)p(v)+a2p(v)2]/[2(1a2)])}
t(a) = 2arcsin(a)/p
a: 0.15643 0.38268 0.70711 0.92388 0.98769
t: 0.10000 0.25000 0.50000 0.75000 0.90000
Gary G Venter
Univariate Functions
Copulas
Normal Copula Log Scale t =.5
100
10
1
0.1
0.01
0.001
Gary G Venter
Copulas
Univariate Functions
Tail Concentration Functions
L(z) = Pr(U<z,V<z)/z2
R(z) = Pr(U>z,V>z)/(1 – z)2
L(z) = C(z,z)/z2
1 - Pr(U>z,V>z) = Pr(U<z) + Pr(V<z) - Pr(U<z,V<z)
= z + z – C(z,z).
Then R(z) = [1 – 2z +C(z,z)]/(1 – z)2
Generalizes to multi-variate case
Gary G Venter
Univariate Functions
Copulas
Normal L and R Functions for t = .1, .5, .9
PP Power L and R Functions for t = .1, .5, .9
HRT L and R Functions for t = .1, .5, and .9
1000
1000
1000
100
100
100
10
10
10
1
1
1
PP Max L and R Functions for t = .1, .5, and .9
Gumbel L and R functions for t = .1, .5, and .9
Frank L and R Functions for t = .1, .5, .9
1000
1000
1000
100
100
100
10
10
10
1
1
1
Gary G Venter
Univariate Functions
Copulas
Cumulative Tau
t = –1+40101 C(u,v)c(u,v)dvdu
J(z) = –1+40z0z C(u,v)c(u,v)dvdu/C(z,z)2
Generalizes to multi-variate case
Gary G Venter
Univariate Functions
Copulas
Frank Cumulative Tau t = .1, .5, .9
HRT Cumulative Tau t = .1, .5, .9
1
1
0.9
0.9
0.8
0.8
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0.6
0.5
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0.2
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0.1
0.1
0
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Gumbel Cumulative Tau t = .1, .5, .9
PP Max Cumulative Tau t = .1, .5, .9
1
1
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0.5
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0.1
0.1
0
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PP Power Cumulative Tau t = .1, .5, .9
Normal Cumulative Tau, t = .1, .5, .9
1
1
0.9
0.9
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0.8
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0.6
0.5
0.5
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0
Gary G Venter
0
Copulas
Univariate Functions
Cumulative Conditional Mean
M(z) = E(V|U<z) = z-10z01 vc(u,v)dvdu
M(1) = ½
A pairwise concept
Copula Distribution Function
K(z) = Pr(C(u,v)<z)
Generalizes to multi-variate case
Gary G Venter
Univariate Functions
Copulas
Frank M(z) for t = .1, .5, .9
0.5
PP Max M(z), t = .1, .5, .9
0.6
0.45
0.5
0.4
0.35
0.4
0.3
0.25
0.3
0.2
0.15
0.2
0.1
0.1
0.05
0
0
PP Power M(z), t = .1, .5, .9
HRT M(z), t = .1, .5, .9
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
Gumbel M(z) for t = .1, .5, .9
Normal M(z) for t = .1, .5, .9
0.5
0.45
0.5
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
Gary G Venter
Univariate Functions
Copulas
MD & DE Joint Empirical Probabilities
DE vs. MD copula
1.000
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
-
Parameter
Ln Likelihood
Tau
Gary G Venter
0.200
0.400
0.600
0.800
HRT Gumbel Frank Normal
0.968 1.67 4.92 0.624
124 157 183 176
0.34 0.40 0.45 0.43
1.000
Univariate Functions
Copulas
0.6
100
J(z) Data and Fits
0.4
LR Function for DE/MD and Fits
Data
0.2
Frank
0
Normal
0
PP Pow er
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data
-0.2
Normal
-0.4
Frank
-0.6
-0.8
1
-1
1
0.5
Empirical K as Function of Frank K
0.9
M(z) Data and Fits
0.45
0.8
0.4
0.7
0.35
0.6
0.3
0.25
Data
0.2
Frank
0.15
Normal
0.5
Data
0.4
Frank
0.1
0.3
0.05
0.2
0
0.1
0
0
Gary G Venter
0.2
0.4
0.6
0.8
1