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Copulas Univariate Functions
Gary G Venter
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.
Copulas Univariate Functions
Gary G Venter
Tails of Copulas
ASTIN 2001
Copulas Univariate Functions
Gary G Venter
Kendall correlation
is a constant of the copula = 4E[C(u,v)] – 1 = 2dE[C(u1, . . .,ud)] – 1
2d – 1 – 1
Copulas Univariate Functions
Gary G Venter
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
(a) = 1 – 4/a + 4/a2 0a t/(et-1) dt
For a<0 this will give negative values of . v = C1
-1(p|u) = -a-1ln{1+pg1/[1+gu(1–p)]}
Copulas Univariate Functions
Gary G Venter
0.01
0.1
1
10Frank's Copula Density on Log Scale =.5
Copulas Univariate Functions
Gary G Venter
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)]a-1 {1+(a-1)[(-ln u)a +(-ln v)a]-
1/a} (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
Copulas Univariate Functions
Gary G Venter
0.001
0.01
0.1
1
10
100
Gumbel Copula Log Scale =0.5
Copulas Univariate Functions
Gary G Venter
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
(a) = 1/(2a + 1) Can solve conditional distribution for v
Copulas Univariate Functions
Gary G Venter
0.1
1
10
100
Heavy Right Tail Copula Log Scale = .5
Copulas Univariate Functions
Gary G Venter
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]
Copulas Univariate Functions
Gary G Venter
Partial Perfect Correlation Copula Generator
Assume logical values 0 and 1 are arithmetic also
h : unit square unit interval H(x) = 0
xh(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)
Copulas Univariate Functions
Gary G Venter
h(u) = (u>a)
H(u) = (u – a)(u>a) (a) = (1 – a)4
PP Max Data Pairs t = .5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Copulas Univariate Functions
Gary G Venter
h(u) = ua
H(u) = ua+1/(a+1) (a) = 1/[3(a+1)4] +
8/[(a+1)(a+2)2(a+3)]
PP (uv)̂ a Data Pairs =.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Copulas Univariate Functions
Gary G Venter
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(1-a2)])} (a) = 2arcsin(a)/ a: 0.15643 0.38268 0.70711 0.92388 0.98769
0.10000 0.25000 0.50000 0.75000 0.90000
Copulas Univariate Functions
Gary G Venter
0.001
0.01
0.1
1
10
100
Normal Copula Log Scale =.5
Copulas Univariate Functions
Gary G Venter
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
Copulas Univariate Functions
Gary G Venter
HRT L and R Functions for = .1, .5, and .9
1
10
100
1000
Gumbel L and R functions for = .1, .5, and .9
1
10
100
1000Frank L and R Functions for = .1, .5, .9
1
10
100
1000
Normal L and R Functions for = .1, .5, .9
1
10
100
1000
PP Power L and R Functions for = .1, .5, .9
1
10
100
1000
PP Max L and R Functions for = .1, .5, and .9
1
10
100
1000
Copulas Univariate Functions
Gary G Venter
Cumulative Tau
–1+4010
1 C(u,v)c(u,v)dvdu
J(z) = –1+40z0
z C(u,v)c(u,v)dvdu/C(z,z)2
Generalizes to multi-variate case
Copulas Univariate Functions
Gary G Venter
Frank Cumulative Tau = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gumbel Cumulative Tau = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PP Power Cumulative Tau = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
HRT Cumulative Tau = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PP Max Cumulative Tau = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normal Cumulative Tau, t = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Copulas Univariate Functions
Gary G Venter
Cumulative Conditional Mean
M(z) = E(V|U<z) = z-10z0
1 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
Copulas Univariate Functions
Gary G Venter
Frank M(z) for = .1, .5, .9
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
HRT M(z), = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
Normal M(z) for = .1, .5, .9
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
PP Max M(z), = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
PP Power M(z), = .1, .5, .9
0
0.1
0.2
0.3
0.4
0.5
0.6
Gumbel M(z) for = .1, .5, .9
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Copulas Univariate Functions
Gary G Venter
MD & DE Joint Empirical Probabilities
DE vs. MD copula
-
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
- 0.200 0.400 0.600 0.800 1.000
HRT Gumbel Frank NormalParameter 0.968 1.67 4.92 0.624Ln Likelihood 124 157 183 176Tau 0.34 0.40 0.45 0.43
Copulas Univariate Functions
Gary G Venter
LR Function for DE/MD and Fits
1
10
100
Data
Frank
Normal
PP Pow er
J(z) Data and Fits
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Data
Normal
Frank
M(z) Data and Fits
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Data
Frank
Normal
Empirical K as Function of Frank K
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Data
Frank