Negative log-likelihood for Hüsler-Reiss graphical models

The penalised negative loglikelihood of the Hüsler-Reiss graphical models can be written as follow : $$ L_{\mathcal P}^{(n)}(\Theta, \lambda) = \underbrace{-\log(|\Theta|_+) - \frac 12 \text{tr}(\hat \Gamma^{(n)}\Theta)}_{L^{(n)}(\Theta)} + \lambda \underbrace{\sum_{i<j} w_{ij} d^2_{ij}(\Theta)}_{\mathcal P(\Theta)} $$ where $|\cdot|_+$ is the generalised determinant, $n$ is the sample size, $\hat \Gamma^{(n)}$ an estimation of the variogram matrix $\Gamma$ and $w_{ij}>0$ the weights.

Usage

neg_likelihood(gamma)

penalty(weights)

neg_likelihood_pen(gamma, weights)

Arguments

gamma: A $d \times d$ matrix: the variogram matrix $\Gamma$.
weights: The $d \times d$ symmetric matrix $W$ with a zero diagonal.

Value

For neg_likelihood(), it produces a function with two parameters :

R the $K \times K$ clusters matrix
clusters a list of indices associated to a partition of $V$ and compute the value of the non penalised negative log-likelihood $L^{(n)}(\Theta)$.

For penalty(), it produces a function with two parameters :

R the $K \times K$ clusters matrix
clusters a list of indices associated to a partition of $V$ and compute the value of the penalty $\mathcal P(\Theta)$.

For neg_likelihood_pen(), it produces a function with three parameters :

R : the $K \times K$ clusters matrix
clusters : a list of indices associated to a partition of $V$
lambda : a positive number, the weight of the penalty. and compute the value of the penalised negative log-likelihood $L_{\mathcal P}^{(n)}(\Theta, \lambda)$.

Details

We use the block matrix structure to compute the likelihood here (see build_theta() and extract_R_matrix()), that is why we use two parameters for the computation instead of only one matrix $\Theta$.

Examples

############################################################
#                      INITIALIZATION
############################################################
# Block matrix structure
R <- matrix(c(0.5, -1,
              -1, -1), nr = 2)
clusters <- list(c(1,3), c(2,4))

# Weight matrix
W <- matrix(c(0, 1, 1, 1,
              1, 0, 1, 1,
              1, 1, 0, 1,
              1, 1, 1, 0), nc = 4)

# Random variogram
gamma <- matrix(c(0, 2, 1, 0,
                  2, 0, 4, 1,
                  1, 4, 0, 7,
                  0, 1, 7, 0), nc = 4)

# Regularization parameter
lambda <- 2.5

############################################################
#                   NEGATIVE LOG-LIKEHOOD
############################################################
nllh <- neg_likelihood(gamma)
nllh(R, clusters)
#> [1] 10.72741

############################################################
#                          PENALTY
############################################################
pen <- penalty(W)
pen(R, clusters)
#> [1] 18

############################################################
#             PENALISED NEGATIVE LOG-LIKELIHOOD
############################################################
nllh_pen <- neg_likelihood_pen(gamma, W)
nllh_pen(R, clusters, lambda)
#> [1] 55.72741