Clusterpath algorithm for Hüsler-Reiss models

Gradient descent based on the Clusterpath algorithm adapted to the likelihood of a graphical Hüsler-Reiss model. Usefull when the precision matrix $\Theta$ (or the variogram $\Gamma$) has a block matrix structure.

Usage

get_cluster(gamma, weights, eps_f, ...)

HR_Clusterpath(
  data,
  zeta,
  lambda,
  p = NULL,
  eps_g = 0.001,
  eps_f = 0.01,
  it_max = 1000
)

Arguments

gamma

For get_clusters(), a $d \times d$ matrix : the variogram matrix $\Gamma$.

weights

For get_clusters(), the $d \times d$ symmetric weightsmatrix with a zero diagonal.

eps_f

A positive number : tolerance threshold for merging clusters.

data

For HR_Clusterapth(), a $n \times d$ matrix : the matrix of data.

zeta

For HR_Clusterapth(), a positive number : tuned parameter for the exponential weights.

lambda

For get_clusters(), a positive number : the weight of the penalty.

For HR_Clusterpath(), a numerix vector of positive number : the grid line for $\lambda$.

p

For HR_Clusterapth(), a numeric between 0 and 1, or NULL. If NULL (default), it is assumed that the data are already on multivariate Pareto scale. Else, p is used as the probability in the function data2mpareto() to standardize the data (see graphicalExtremes documentation).

eps_g

A positive number : tolerance threshold for the convergence of the gradient descent.

it_max

An integer : maximal number of iteration of the gradient descent algorithm.

Value

HR_Clusterpath() is a gradient descent from the data with default values and method for the optimization : the variogram matrix $\Gamma$ is the empirical variogram and the weights are set as the exponential weights, which depends of only one tuning parameter $\zeta$ and where we have some theoretical results.

get_clusters() is a freer version of the previous function. You can use your own estimation for the variogram $\Gamma$ and customizable weights with the matrix $W$.

Both of them produce a list of results :

$R : the $R$ matrix of the clusters.
$clusters : a list of the variable indices, clustered.
$nllh : the value of the negative penalised negative loglikelihood.
$lambda : the value of lambda.
$message : a message about the optimization results.

In the case of HR_Clusterpath() it is a list of the previous list.

Details

The block matrix models is defined from two elements : the cluster's partition $\{C_1, \dots, C_K\}$, included in $V$ and the $R$ matrix which belongs to $\mathcal S_K(\mathbb R)$, the set of symmetric $K \times K$ matrix (See also build_theta() and extract_R_matrix()).

The Clusterpath aims to find optimum of some penalised negative loglikelihood defined in neg_likelihood_pen().

Some results for `HR_Clusterpath()`

When we get replications $X_1, \dots, X_n$ of a random vector $X$ which belongs to the attraction domain of a $K$-block Hüsler-Reiss graphical model, the minimum of the penalised negative loglikelihood conveges almost surely to the true precision matrix of the model $\Theta^*$, for all $\lambda$, provided that we choose a sequence $(\zeta_n)_{n\in \mathbb N^*}$ which grows slower than the $log(n)$ sequence. In general, the $\zeta \in [1,2]$ is a good choice.

Examples