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Functions to navigate between \(R\) and \(\Theta\) for given clusters

theta-r.Rd

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. The expression of the precision \(\Theta\) which statisfies the block matrix model is $$ \Theta = U R U^t + A $$ where \(U\) is the cluster matrix and \(A\) a diagonal matrix such that the rows of theta sum to zero : \( a_{ii} = - \sum_l p_l r_{kl} \) for \(i\) in cluster \(C_k\).

Usage

build_theta(R, clusters)

extract_R_matrix(Theta, clusters)

Arguments

R

For build_theta(), the \(K \times K\) matrix of the clusters.

clusters

A list of indices associated to a partition of \(V\).

Theta

For extract_R_matrix(), a \(d \times d\) block matrix which can be factorizable.

Value

For extract_R_matrix(), a matrix on size the number of clusters (i.e. \(K\)) corresponding to the reduced matrix \(R\) of the original matrix \(\Theta\).

For build_theta(), a matrix on size the number of variables (i.e. \(d\)) corresponding to the precision matrix \(\Theta\) induced by \(R\).

Examples

##############################################################
#                       FROM THETA TO R
##############################################################
Theta <- matrix(
 c(4.5, .5, .5, .5, -2, -2, -2,
   .5, 4.5, .5, .5, -2, -2, -2,
   .5, .5, 4.5, .5, -2, -2, -2,
   .5, .5, .5, 4.5, -2, -2, -2,
   -2, -2, -2, -2, 6, 1, 1,
   -2, -2, -2, -2, 1, 6, 1,
   -2, -2, -2, -2, 1, 1, 6),
 nc = 7
)

clusters <- list(c(1,2,3,4), c(5,6,7))

extract_R_matrix(Theta, clusters)
#>      [,1] [,2]
#> [1,]  0.5   -2
#> [2,] -2.0    1

##############################################################
#                       FROM R TO THETA
##############################################################

R <- matrix(c(1, -3, 0,
              -3, 2, -2,
              0, -2, 1), nc = 3)

clusters <- list(1:4, 5:8, 9:12)

build_theta(R, clusters)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
#>  [1,]    9    1    1    1   -3   -3   -3   -3    0     0     0     0
#>  [2,]    1    9    1    1   -3   -3   -3   -3    0     0     0     0
#>  [3,]    1    1    9    1   -3   -3   -3   -3    0     0     0     0
#>  [4,]    1    1    1    9   -3   -3   -3   -3    0     0     0     0
#>  [5,]   -3   -3   -3   -3   14    2    2    2   -2    -2    -2    -2
#>  [6,]   -3   -3   -3   -3    2   14    2    2   -2    -2    -2    -2
#>  [7,]   -3   -3   -3   -3    2    2   14    2   -2    -2    -2    -2
#>  [8,]   -3   -3   -3   -3    2    2    2   14   -2    -2    -2    -2
#>  [9,]    0    0    0    0   -2   -2   -2   -2    5     1     1     1
#> [10,]    0    0    0    0   -2   -2   -2   -2    1     5     1     1
#> [11,]    0    0    0    0   -2   -2   -2   -2    1     1     5     1
#> [12,]    0    0    0    0   -2   -2   -2   -2    1     1     1     5