Package 'HMMcopula'

CopulaFamiliesCDF

Description

COPULACDF Cumulative probability function for a copula with linear correlation parameters RHO

Usage

CopulaFamiliesCDF(family, u, ...)

Arguments

family	copula familly= "gaussian" , "t" , "clayton" , "frank" , "gumbel"
u	is an N-by-P matrix of values in [0,1], representing N points in the P-dimensional unit hypercube
...	additionnal parameter like RHO a P-by-P correlation matrix.

Value

Y = COPULACDF('Gaussian',U,RHO) returns the cumulative probability of the Gaussian copula with linear correlation parameters RHO, evaluated at the points in U. U is an N-by-P matrix of values in [0,1], representing N points in the P-dimensional unit hypercube. RHO is a P-by-P correlation matrix. If U is an N-by-2 matrix, RHO may also be a scalar correlation coefficient.

Y = COPULACDF('t',U,RHO,NU) returns the cumulative probability of the t copula with linear correlation parameters RHO and degrees of freedom parameter NU, evaluated at the points in U. U is an N-by-P matrix of values in [0,1]. RHO is a P-by-P correlation matrix. If U is an N-by-2 matrix, RHO may also be a scalar correlation coefficient.

Y = COPULACDF(FAMILY,U,ALPHA) returns the cumulative probability of the bivariate Archimedean copula determined by FAMILY, with scalar parameter ALPHA, evaluated at the points in U. FAMILY is 'Clayton', 'Frank', ort 'Gumbel'. U is an N-by-2 matrix of values in [0,1].

Examples

u = seq(0,1,0.1);
    U1=matrix(rep(u,length(u)),nrow=length(u),byrow = TRUE); U2=t(U1)
   F = CopulaFamiliesCDF('clayton',cbind(c(U1), c(U2)),1)

---

Dilogarithm function

Description

Computation of the dilogarithm function by nemerical integration.

Usage

dilog(x)

Arguments

x	a real number

Value

out	dilogarithm

---

Estimation of bivariate Markov regime switching bivariate copula model

Description

Estimation of parameters from a bivariate Markov regime switching bivariate copula model

Usage

EstHMMCop(y, reg, family, max_iter, eps)

Arguments

y	(nx2) data matrix (observations or residuals) that will be transformed to pseudo-observations
reg	number of regimes
family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
max_iter	maximum number of iterations of the EM algorithm
eps	precision (stopping criteria); suggestion 0.0001.

Value

theta	(1 x reg) estimated parameter of the copula according to CRAN copula package (except for Frank copula, where theta = log(theta_R_Package)) for each regime (except for degrees of freedom)
dof	estimated degree of freedom, only for the Student copula
Q	(reg x reg) estimated transition matrix
eta	(n x reg) conditional probabilities of being in regime k at time t given observations up to time t
tau	estimated Kendall tau for each regime
U	(n x 2) matrix of Rosenblatt transforms
cvm	Cramer-von-Mises statistic for goodness-of-fit
W	regime probabilities for the conditional distribution given the past Kendall's tau

Author(s)

Mamadou Yamar Thioub and Bruno Remillard, April 12, 2018

References

<doi::10.1002/cjs.11534>

Examples

Q <- matrix(c(0.8, 0.3, 0.2, 0.7),2,2) ; kendallTau <- c(0.3 ,0.7) ;
data <- SimHMMCop(Q, 'clayton', kendallTau, 10)$SimData;
estimations <- EstHMMCop(data,2,'clayton',10000,0.0001)

---

Sample Kendall's tau Estimation

Description

This function estimates the sample Kendall's tau of a bivariate data matrix

Usage

EstKendallTau(X)

Arguments

X	(n x 2) matrix

Value

KendallTau

estimated sample Kendall's tau of the data

---

Estimation of bivariate mixture bivariate copula model

Description

Estimation of parameters from a mixture of bivariate copula models

Usage

EstMixtureCop(y, reg, family, max_iter, eps)

Arguments

y	(nx2) data matrix (observations or residuals) that will be transformed to pseudo-observations
reg	number of regimes
family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
max_iter	maximum number of iterations of the EM algorithm
eps	precision (stopping criteria); suggestion 0.0001.

Value

theta	(1 x reg) estimated parameter of the copula according to CRAN copula package (except for Frank copula, where theta = log(theta_R_Package)) for each component (except for degrees of freedom)
dof	estimated degree of freedom, only for the Student copula
Q	(1 x reg) estimated weights vector
eta	(n x reg) conditional probabilities of being in regime k at time t given observations up to time t
tau	estimated Kendall tau for each regime
U	(n x 2) matrix of Rosenblatt transforms
cvm	Cramer-von-Mises statistic for goodness-of-fit

Author(s)

Mamadou Yamar Thioub and Bruno Remillard, April 12, 2018

References

<doi::10.1002/cjs.11534>

---

Goodness-of-fit of Markov regime switching bivariate copula model

Description

Goodness-of-fit test of a Markov regime switching bivariate copula model

Usage

GofHMMCop(R, reg, family, max_iter, eps, n_sample, n_cores)

Arguments

R	(n x 2) data matrix that will be transformed to pseudo-observations
reg	number of regimes
family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
max_iter	maxmimum number of iterations of the EM algorithm
eps	precision (stopping criteria); suggestion 0.0001
n_sample	number of bootstrap; suggestion 1000
n_cores	number of cores to use in the parallel computing

Value

pvalue	pvalue (significant when the result is greater than 5)
theta	(1 x reg) estimated parameter of the copula according to CRAN copula package (except for Frank copula, where theta = log(theta_R_Package)) for each regime (except for degrees of freedom)
dof	estimated degree of freedom, only for the Student copula
Q	(reg x reg) estimated transition matrix
eta	(n x reg) conditional probabilities of being in regime k at time t given observations up to time t
tau	estimated Kendall tau for each regime
U	(n x 2) matrix of Rosenblatt transforms
cvm	Cramer-von-Mises statistic for goodness-of-fit
W	regime probabilities for the conditional distribution given the past Kendall's tau

References

<doi::10.1002/cjs.11534>

---

Goodness-of-fit of mixture bivariate copula model

Description

Goodness-of-fit test of a mixture bivariate copula model

Usage

GofMixtureCop(R, reg, family, max_iter, eps, n_sample, n_cores)

Arguments

R	(nx2) data matrix (observations or residuals) that will be transformed to pseudo-observations
reg	number of regimes
family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
max_iter	maxmimum number of iterations of the EM algorithm
eps	precision (stopping criteria); suggestion 0.0001
n_sample	number of bootstrap; suggestion 1000
n_cores	number of cores to use in the parallel computing

Value

pvalue	pvalue (significant when the result is greater than 5)
theta	(1 x reg) estimated parameter of the copula according to CRAN copula package (except for Frank copula, where theta = log(theta_R_Package)) for each component (except for degrees of freedom)
dof	estimated degree of freedom, only for the Student copula
Q	(1 x reg) estimated weights vector
eta	(n x reg) conditional probabilities of being in regime k at time t given observations up to time t
tau	estimated Kendall tau for each regime
U	(n x 2) matrix of Rosenblatt transforms
cvm	Cramer-von-Mises statistic for goodness-of-fit

Author(s)

By Bruno Remillard, Nov 28, 2010

References

<doi::10.1002/cjs.11534>

---

Kendall's tau of a copula

Description

Computation of Kendall's tau of a copula family with an unconstrainted parameter alpha.

Usage

KendallTau(family, alpha)

Arguments

family	"gaussian" , "t" , "clayton" , "frank" , "gumbel"
alpha	unconstrainted parameters of the copula family

Value

tau	estimated Kendall's tau

---

Theta estimation

Description

Parameters of a copula according to CRAN copula package (except for Frank copula, where theta = log(theta_R_Package)), corresponding to the unconstrainted parameters alpha.

Usage

ParamCop(family, alpha)

Arguments

family	"gaussian" , "t" , "clayton" , "frank" , "gumbel"
alpha	unconstrainted parameters of the copula family

Value

theta

matlab parameters

---

Alpha estimation

Description

Unconstrainted parameter for a given Kendall's tau.

Usage

ParamTau(family, tau)

Arguments

family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
tau	Kendall's tau of the copula family

Value

alpha

estimated unconstrainted parameter

---

Rosenblatt transform for Clayton copula

Description

Computation of the Rosenblatt transform fot Clayton's copula

Usage

RosenblattClayton(u, theta)

Arguments

u	(n x d) matrix of pseudos-observations (normalized ranks)
theta	parameter of the Clayton copula

Value

R	Rosenblatt transform

---

Rosenblatt transform for Frank copula

Description

Computation of the Rosenblatt transform for Frank's copula

Usage

RosenblattFrank(U, theta)

Arguments

U	(n x d) matrix of pseudos-observations (normalized ranks)
theta	parameter of the Frank copula

Value

R	Rosenblatt transform

---

Rosenblatt transform for Gaussian copula

Description

Computation of the Rosenblatt transform for the Gaussian copula

Usage

RosenblattGaussian(u, rho)

Arguments

u	(n x d) matrix of pseudos-observations (normalized ranks)
rho	(d x d) correlation matrix, or the correlation coefficient (if, d = 2)

Value

R	Rosenblatt transform

---

Rosenblatt transform for Gumbel copula

Description

Computation of the Rosenblatt transform for Gumbel's copula

Usage

RosenblattGumbel(U, theta)

Arguments

U	(n x d) matrix of pseudos-observations (normalized ranks)
theta	parameter of the Gumbel copula

Value

R	Rosenblatt transform

---

Rosenblatt transform for Student copula

Description

Computation of the Rosenblatt transform for the Student copula

Usage

RosenblattStudent(u, rho, nu)

Arguments

u	(n x d) matrix of pseudos-observations (normalized ranks)
rho	(d x d) correlation matrix
nu	degrees of freedom

Value

R	Rosenblatt transform

---

Simulation of bivariate Markov regime switching copula model

Description

Simulaion of values from a bivariate Markov regime switching copula model

Usage

SimHMMCop(Q, family, KendallTau, n, DoF)

Arguments

Q	Transition probality matrix (d x d);
family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
KendallTau	Kendall's rank correlation
n	number of simulated vectors
DoF	degree of freedom only for the Student copula

Value

SimData	Simulated Data
MC	Markov chain regimes
alpha	parameters alpha

Examples

Q <- matrix(c(0.8, 0.3, 0.2, 0.7),2,2) ; kendallTau <- c(0.3 ,0.7) ;
simulations <- SimHMMCop(Q, 'gumbel', kendallTau, 300)

---

Markov chain simulation

Description

Simulation of n consecutive values of a Markov chain with transition matrix Q, starting from a state eta0 or the uniform distribution on the set 1,..., r.

Usage

SimMarkovChain(Q, n, eta0)

Arguments

Q	Transition probability matrix (d x d)
n	number of simulated vectors
eta0	variable eta

Value

x	Simulated Markov chain sequence

---

Simulation of bivariate mixture copula model

Description

Simulation of observations from a bivariate mixture copula model

Usage

SimMixtureCop(Q, family, KendallTau, n, DoF)

Arguments

Q	Weights vector (1 x component);
family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
KendallTau	Kendall's rank correlation
n	number of simulated vectors
DoF	vector of degree of freedom only for the Student copula

Value

SimData	Simulated Data
MC	Markov chain regimes
alpha	parameters alpha

Examples

Q <- matrix(c(0.8, 0.2),1,2) ; kendallTau <- c(0.3 ,0.7) ;
simulations <- SimMixtureCop(Q, 'gaussian', kendallTau, 300)

---

Cramer-von Mises statistic SnB for GOF based on the Rosenblatt transform

Description

Computation of the Cramer-von Mises statistic SnB for GOF based on the Rosenblatt transform

Usage

SnB(E)

Arguments

E	(n x d) matrix of pseudos-observations (normalized ranks)

Value

Sn	Cramer-von Mises statistic

---

Spearman's rho

Description

Value of Spearman's rho corresponding to a constrainted (matlab) parameter theta for a copula family.

Usage

Tau2Rho(family, theta)

Arguments

family	'gaussian' , 't' , 'clayton' , 'frank' , 'gumbel'
theta	parameter of the copula according to CRAN copula package (except for Frank copula, where theta = log(theta_R_Package))

Value

rho	estimated Spearman's rho

---