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MATLAB Statistics Toolbox™ 估計 Copula 與 Copula 亂數求取方法
來源：Copulafit 函數說明
在 MATLAB 的 Statistics Toolbox 中，可運用下列提供方法估計 Copula 的參數與模擬。其提供的 Copula
套裝函數共有
1.
Gaussian Copula
2.
t Copula
3.
Archimedean copula 包含'Clayton', 'Frank', or 'Gumbel'。
運用指令為

copulafit: Fit copula to data

copularnd: Copula random numbers
Gaussian Copula
所謂的 Gaussian Copula 即為多元常態分配下的 copula 函數，其定義為：b 若 X  ( X 1 , X 2 ,..., X n ) 是
多元常態分配，則其邊際函數 F1 ,..., Fn 皆為常態分配，且存在唯一的 Copula 函數，使得
C RN (u1 ,..., u n )   R ( 1 (u1 ),...,  1 (u n ))
其中  R ：標準的多元常態分配，存在著對稱且正定的相關矩陣 R。 1 則是單維標準常態分配的反函數。
當 n=2 時，我們可以得到此時的 Copula 函數為：
 1 ( u )  1 ( v )
C (u, v) 
N
R
 


s 2  2 R12 st  t 2
1
exp{ 
}dsdt
2 (1  R122 )1 / 2
2(1  R122 )
t Copula
t-copula 是指多元 Student’s t 分配下的 copula 函數，假設 X  ( X 1 , X 2 ,..., X n ) 服從多元標準常態分配，
其相關矩陣為 R ， Y 是  2 分配的隨機變數，自由度為 v ，則 t-copula 函數為：
Cvt , R (u1 ,..., un )  t v, R (t v1 (u1 ),..., t v1 (un ))
其中 u i 
C
t
v,R
v
Y
X i , i  1,..., n 。當 n=2 時，我們可以得到 t-copula 函數為：
(u, v) 
t v1 ( u ) tv1 ( v )
 


s 2  2 R12 st  t 2 ( v  2) / 2
1
{
1

}
dsdt
2 (1  R122 )1 / 2
v(1  R122 )
Archimedean copula
多元 Archimedean Copulas
Clayton-n-Copula 函數：當 α>0
( x)  x   1
=>
 n 

C (u1 ,..., un )   ui  n  1
 i 1

1 
Gumbel-n-Copula 函數 ：當 α>1
( x)  ( ln x)
=>
1
  n

 
C (u1 ,..., un )  exp     ( ln ui )  
 
  i 1
Frank-n-Copula 函數 ：當 α>0,n>3
x
 1 e 
( x)   ln 
 
 1 e 
=>
一、首先，若已知邊際分配情況
n

e ui  1

1 

C (u1...un )   ln 1  i 1
n 1
 
ei  1




MATLAB 連續分配套件有下列指令
Continuous Distributions (Data)

pdf — Probability density functions

cdf — Cumulative distribution functions

inv — Inverse cumulative distribution functions

stat — Distribution statistics functions

fit — Distribution fitting functions

like — Negative log-likelihood functions

rnd — Random number generators
MATLAB
Statistics Toolbox for Copula 運用
2

 




Continuous Distributions (Data)
Name
Beta
Exponential
Extreme value
pdf
cdf
inv
betapdf,
betacdf,
betainv,
pdf
cdf
icdf
exppdf,
expcdf,
expinv,
pdf
cdf
icdf
stat
betastat
fit
betafit, mle
like
rnd
betalike
betarnd, random,
randtool
expstat
expfit, mle,
explike
dfittool
evpdf, pdf evcdf, cdf evinv, icdf evstat
evfit, mle,
randtool
evlike
dfittool
Gamma
gampdf,
gamcdf,
gaminv,
pdf
cdf
icdf
Generalized
gevpdf,
gevcdf,
gevinv,
extreme value
pdf
cdf
icdf
Generalized
gppdf, pdf gpcdf, cdf gpinv,
Pareto
gamstat
gevstat
lognpdf,
logncdf,
logninv,
pdf
cdf
icdf
Normal
normpdf,
normcdf,
norminv,
(Gaussian)
pdf
cdf
icdf
pdf
cdf
icdf
Rayleigh
raylpdf,
raylcdf,
raylinv,
pdf
cdf
icdf
gpstat
gpfit,
lognstat lognfit, mle,
gevlike
raylstat
gevrnd, random,
randtool
gplike
gprnd, random,
randtool
lognlike lognrnd, random,
dfittool
normstat normfit, mle,
gamrnd, randg,
random, randtool
mle,dfittool
Rician
randtool
normlike normrnd, randn,
dfittool
random, randtool
pearsrnd
pearsrnd
paretotails
random
raylfit, mle,
raylrnd, random,
dfittool
randtool
dfittool
Uniform
unifpdf,
unifcdf,
unifinv,
(continuous)
pdf
cdf
icdf
Weibull
wblpdf,
wblcdf,
wblinv,
pdf
cdf
icdf
MATLAB
gamlike
mle,dfittool
Pearson system
Piecewise
gevfit,
evrnd, random,
randtool
dfittool
icdf
Lognormal
gamfit, mle,
exprnd, random,
Statistics Toolbox for Copula 運用
unifstat
unifit, mle
unifrnd, rand,
random
wblstat
wblfit, mle,
dfittool
3
wbllike
wblrnd, random
Name
pdf
cdf
inv
stat
(反函數)
(統計
rnd
量)
Chi-square
chi2pdf, pdf chi2cdf, cdf chi2inv, icdf chi2stat
chi2rnd, random,
randtool
F
fpdf, pdf
fcdf, cdf
finv, icdf
fstat
frnd, random, randtool
Noncentral
ncx2pdf,
ncx2cdf,
ncx2inv,
ncx2stat ncx2rnd, random,
chi-square
pdf
cdf
icdf
Noncentral F
ncfpdf, pdf
ncfcdf, cdf
ncfinv, icdf
randtool
ncfstat
ncfrnd, random,
randtool
Noncentral t
nctpdf, pdf
nctcdf, cdf
nctinv, icdf
nctstat
nctrnd, random, randtool
Student's t
tpdf, pdf
tcdf, cdf
tinv, icdf
tstat
trnd, random, randtool
例如，若虛設一組資料是 N(3,16)，另一組是 Gamma(0.2,0.3)的資料如下：
data = 3+4*randn(2000,1);
data1= gamrnd(0.2,0.3,1,2000);
函數含義 ：R = gamrnd(A,B,m,n) generates gamma random numbers with parameters A and B, where scalars
m and n are the row and column dimensions of R.
he gamma pdf is
則可用 mle 指令求得
phat =
3.0746
4.0807
pci =
2.8956
3.9590
3.2536
4.2123
0.0587
0.1174%
=> returns MLEs and 95% confidence intervals for the parameters.
MATLAB
Statistics Toolbox for Copula 運用
4
[nlogL,AVAR] = normlike(phat ,data)
nlogL =
5.6504e+003
AVAR =
0.0083
-0.0000
-0.0000
0.0042
函數含義 ：nlogL = normlike(params,data) returns the negative of the normal log-likelihood function for the
parameters params(1) = mu and params(2) = sigma, given the vector data.
[nlogL,AVAR] = normlike(params,data) also returns the inverse of Fisher's information matrix, AVAR.
If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of
AVAR are their asymptotic variances. AVAR is based on the observed Fisher's information, not the
expected information.
t_value=phat./sqrt(diag(AVAR)')
t_value =
33.6955
63.2456
[phat1,pci] = mle(data1,'distribution','gamma')
phat1 =
0.1932
0.2882
pci =
0.1843
0.2581
0.2026
0.3219
[nlogL,AVAR1] = gamlike(phat1 ,data1)
nlogL =
-7.8126e+003
AVAR1 =
1.0e-003 *
0.0218
-0.0326
-0.0326
0.2636
t_value=phat1./sqrt(diag(AVAR1)')
MATLAB
Statistics Toolbox for Copula 運用
5
t_value =
41.3397
17.7522
%以下要給定 Copula 的 u and v
u = normcdf(data,phat(1),phat(2));
v=gamcdf(data1,phat1(1),phat1(2));
scatterhist(u,v);xlabel('u');ylabel('v');
Rho = copulafit('Gaussian',[u v'])
Rho =
1.0000
-0.0111
-0.0111
1.0000
[Rho,nu] = copulafit('t',[u v']) %[u v'] 是 2000*1 矩陣
Rho =
1.0000
-0.0112
-0.0112
1.0000
nu =
108.7465
MATLAB
Statistics Toolbox for Copula 運用
6
函數含義 ：HOHAT = copulafit('Gaussian',U) returns an estimate RHOHAT of the matrix of linear correlation
parameters for a Gaussian copula, given data in U. U is an n-by-p matrix of values in the open interval (0,1)
representing n points in the p-dimensional unit hypercube.
函數含義 ：[RHOHAT,nuhat] = copulafit('t',U) returns an estimate RHOHAT of the matrix of linear
correlation parameters for a t copula and an estimate nuhat of the degrees of freedom parameter, given data in U.
U is an n-by-p matrix of values in the open interval (0,1) representing n points in the p-dimensional unit
hypercube.
Generate a random sample from the t copula:
r = copularnd('t',Rho,nu,1000);
u1 = r(:,1);
v1 = r(:,2);
函數含義 ：U = copularnd('t',rho,NU,N) returns N random vectors generated from a t copula with linear
correlation parameters rho and degrees of freedom NU. If rho is a p-by-p correlation matrix, U is an n-by-p
matrix. If rho is a scalar correlation coefficient, copularnd generates U from a bivariate t copula. Each column
of U is a sample from a Uniform(0,1) marginal distribution.
scatterhist(u1,v1)
xlabel('u')
ylabel('v')
MATLAB
Statistics Toolbox for Copula 運用
7
Transform the random sample back to the original scale of the data:
x=norminv(u1,phat(1),phat(2));
y=gaminv(v1,phat1(1),phat1(2));
運用上述步驟就可求一組資料是 N(3,16)，另一組是 Gamma(0.2,0.3)的隨機樣本 1000 組。
二、未知邊際分配 (來源: Statistics Toolbox™ copulafit)
Load and plot simulated stock return data:
load stockreturns
x = stocks(:,1);
y = stocks(:,2);
scatterhist(x,y)
Transform the data to the copula scale (unit square) using a kernel estimator of the cumulative
distribution function: (無母數用法)
u = ksdensity(x,x,'function','cdf');%=>自動將 x 的資料轉成 CDF 資料
v = ksdensity(y,y,'function','cdf'); %=>自動將 y 的資料轉成 CDF 資料
scatterhist(u,v);xlabel('u');ylabel('v')
MATLAB
Statistics Toolbox for Copula 運用
8
函數含義 ：[f,xi] = ksdensity(x) computes a probability density estimate of the sample in the vector x. f is the
vector of density values evaluated at the points in xi. (Kernel smoothing density estimate)
The function type to estimate, chosen from among 'pdf', 'cdf', 'icdf', 'survivor', or 'cumhazard' for the
density, cumulative probability, inverse cumulative probability, survivor, or cumulative hazard functions, respectively.
Fit a Gaussian copula:
Rho = copulafit('Gaussian',[u v])
Rho =
1.0000
0.7221
0.7221
1.0000
補充：paramhat = copulafit(family,U) returns an estimate paramhat of the copula parameter for an
Archimedean copula specified by family, given data in U. U is an n-by-2 matrix of values in the open interval (0,1)
representing n points in the unit square. Family
is one of 'Clayton', 'Frank', or 'Gumbel'.
Generate a random sample from the Gaussian copula
r = copularnd('t',Rho,nu,1000);
u1 = r(:,1);
v1 = r(:,2);
scatterhist(u1,v1);xlabel('u');ylabel('v')
MATLAB
Statistics Toolbox for Copula 運用
9
Transform the random sample back to the original scale of the data:
x1 = ksdensity(u,u1,'function','icdf');
y1 = ksdensity(v,v1,'function','icdf');
scatterhist(x1,y1)
MATLAB
Statistics Toolbox for Copula 運用
10