Log-Likelihood of an R-Vine Copula Model
This function calculates the log-likelihood of a d-dimensional R-vine copula model for a given copula data set.
RVineLogLik( data, RVM, par = RVM$par, par2 = RVM$par2, separate = FALSE, verbose = TRUE, check.pars = TRUE, calculate.V = TRUE )
data |
An N x d data matrix (with uniform margins). |
RVM |
An |
par |
A d x d matrix with the pair-copula parameters (optional;
default: |
par2 |
A d x d matrix with the second parameters of pair-copula
families with two parameters (optional; default: |
separate |
Logical; whether log-likelihoods are returned point wisely
(default: |
verbose |
In case something goes wrong, additional output will be plotted. |
check.pars |
logical; default is |
calculate.V |
logical; whether V matrices should be calculated. Default
is |
For observations u=(u_1',...,u_N')' the log-likelihood of a d-dimensional R-vine copula with d-1 trees and corresponding edge sets E_1,...,E_{d-1} is given by
loglik:=ll_{RVine}(θ|u)
=∑_{i=1}^N ∑_{k=1}^{d-1} ∑_{e\in E_k} \ln[c_{j(e),k(e)|D(e)}(F(u_{i,j(e)}|u_{i,D(e)}),F(u_{i,k(e)}|u_{i,D(e)})|θ_{j(e),k(e)|D(e)})],
where u_i=(u_{i,1},...,u_{i,d})'\in[0,1]^d, i=1,...,N. Further c_{j(e),k(e)|D(e)} denotes a bivariate copula density associated to an edge e and with parameter(s) θ_{j(e),k(e)|D(e)}. Conditional distribution functions such as F(u_{i,j(e)}|u_{i,D(e)}) are obtained recursively using the relationship
h(u|v,θ) := F(u|v) = d C_{uv_j|v_{-j}}(F(u|v_{-j}),F(v_j|v_{-j}))/d F(v_j|v_{-j}),
where C_{uv_j|v_{-j}} is a bivariate copula distribution function with parameter(s) θ and v_{-j} denotes a vector with the j-th component v_j removed. The notation of h-functions is introduced for convenience. For more details see Dissmann et al. (2013).
loglik |
The calculated log-likelihood value of the R-vine copula model. |
V |
The stored transformations (h-functions and
log-likelihoods of each pair-copula) which may be used for posterior updates
(three matrices: |
Ulf Schepsmeier, Jeffrey Dissmann, Jakob Stoeber
Dissmann, J. F., E. C. Brechmann, C. Czado, and D. Kurowicka (2013). Selecting and estimating regular vine copulae and application to financial returns. Computational Statistics & Data Analysis, 59 (1), 52-69.
# define 5-dimensional R-vine tree structure matrix
Matrix <- c(5, 2, 3, 1, 4,
0, 2, 3, 4, 1,
0, 0, 3, 4, 1,
0, 0, 0, 4, 1,
0, 0, 0, 0, 1)
Matrix <- matrix(Matrix, 5, 5)
# define R-vine pair-copula family matrix
family <- c(0, 1, 3, 4, 4,
0, 0, 3, 4, 1,
0, 0, 0, 4, 1,
0, 0, 0, 0, 3,
0, 0, 0, 0, 0)
family <- matrix(family, 5, 5)
# define R-vine pair-copula parameter matrix
par <- c(0, 0.2, 0.9, 1.5, 3.9,
0, 0, 1.1, 1.6, 0.9,
0, 0, 0, 1.9, 0.5,
0, 0, 0, 0, 4.8,
0, 0, 0, 0, 0)
par <- matrix(par, 5, 5)
# define second R-vine pair-copula parameter matrix
par2 <- matrix(0, 5, 5)
# define RVineMatrix object
RVM <- RVineMatrix(Matrix = Matrix, family = family,
par = par, par2 = par2,
names = c("V1", "V2", "V3", "V4", "V5"))
# simulate a sample of size 300 from the R-vine copula model
set.seed(123)
simdata <- RVineSim(300, RVM)
# compute the log-likelihood
ll <- RVineLogLik(simdata, RVM, separate = FALSE)
ll$loglik
# compute the pointwise log-likelihoods
ll <- RVineLogLik(simdata, RVM, separate = TRUE)
ll$loglikPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.