R/models.R
precision_matrix_multivariate.RdCompute the precision matrix for multivariate model
precision_matrix_multivariate(
p,
operator_list,
rho,
theta = NULL,
Q = NULL,
scale = NULL
)dimension, should be integer and greater than 1
a list of ngme_operator object (length should be p)
vector with the p(p-1)/2 correlation parameters rho_11, rho_21, rho_22, ... rho_p1, rho_p2, ... rho_p(p-1)
parameter for Q matrix (length of 1 when p=2, length of 3 when p=3)
orthogonal matrix of dim p*p (provide when p > 3)
A vector of length p with constants to multiply each operator matrix with
the precision matrix of the multivariate model
The general model is defined as $D diag(L_1, ..., L_p) x = M$. D is the dependence matrix, it is paramterized by $D = Q(theta) * D_l(cor_mat)$, where $Q$ is the orthogonal matrix, and $D_l$ is matrix controls the cross-correlation. See the section 2.2 of Bolin and Wallin (2020) for exact parameterization of Dependence matrix.
Bolin, D. and Wallin, J. (2020), Multivariate type G Matérn stochastic partial differential equation random fields. J. R. Stat. Soc. B, 82: 215-239. https://doi.org/10.1111/rssb.12351
rho <- c(-0.5, 0.5,-0.25) #correlation parameters
operator_list <- list(ar1(1:5, rho=0.4), ar1(1:5, rho=0.5), ar1(1:5, rho=0.6))
precision_matrix_multivariate(3, operator_list, rho, theta=c(1,2,3))
#> 15 x 15 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1.3906250 -0.5562500 . . . 0.4513514
#> [2,] -0.5562500 1.6131250 -0.5562500 . . -0.2271007
#> [3,] . -0.5562500 1.6131250 -0.5562500 . .
#> [4,] . . -0.5562500 1.6131250 -0.5562500 .
#> [5,] . . . -0.5562500 1.3906250 .
#> [6,] 0.4513514 -0.2271007 . . . 1.3281250
#> [7,] -0.1816805 0.5450416 -0.2271007 . . -0.6640625
#> [8,] . -0.1816805 0.5450416 -0.2271007 . .
#> [9,] . . -0.1816805 0.5450416 -0.2271007 .
#> [10,] . . . -0.1816805 0.4542013 .
#> [11,] -0.4181080 0.2577699 . . . 0.3179182
#> [12,] 0.1718466 -0.5327244 0.2577699 . . -0.1601086
#> [13,] . 0.1718466 -0.5327244 0.2577699 . .
#> [14,] . . 0.1718466 -0.5327244 0.2577699 .
#> [15,] . . . 0.1718466 -0.4296165 .
#>
#> [1,] -0.1816805 . . . -0.4181080 0.1718466
#> [2,] 0.5450416 -0.1816805 . . 0.2577699 -0.5327244
#> [3,] -0.2271007 0.5450416 -0.1816805 . . 0.2577699
#> [4,] . -0.2271007 0.5450416 -0.1816805 . .
#> [5,] . . -0.2271007 0.4542013 . .
#> [6,] -0.6640625 . . . 0.3179182 -0.1601086
#> [7,] 1.6601563 -0.6640625 . . -0.1921303 0.4162824
#> [8,] -0.6640625 1.6601563 -0.6640625 . . -0.1921303
#> [9,] . -0.6640625 1.6601563 -0.6640625 . .
#> [10,] . . -0.6640625 1.3281250 . .
#> [11,] -0.1921303 . . . 1.3125000 -0.7875000
#> [12,] 0.4162824 -0.1921303 . . -0.7875000 1.7850000
#> [13,] -0.1601086 0.4162824 -0.1921303 . . -0.7875000
#> [14,] . -0.1601086 0.4162824 -0.1921303 . .
#> [15,] . . -0.1601086 0.3202172 . .
#>
#> [1,] . . .
#> [2,] 0.1718466 . .
#> [3,] -0.5327244 0.1718466 .
#> [4,] 0.2577699 -0.5327244 0.1718466
#> [5,] . 0.2577699 -0.4296165
#> [6,] . . .
#> [7,] -0.1601086 . .
#> [8,] 0.4162824 -0.1601086 .
#> [9,] -0.1921303 0.4162824 -0.1601086
#> [10,] . -0.1921303 0.3202172
#> [11,] . . .
#> [12,] -0.7875000 . .
#> [13,] 1.7850000 -0.7875000 .
#> [14,] -0.7875000 1.7850000 -0.7875000
#> [15,] . -0.7875000 1.3125000