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sigma_L Structure

tmpl_sigma_L.Rd

Create templates for the left square root \(L\) of the noise covariance matrix \(\Sigma = LL'\). This means that \(L\) is parametrized as $$\mbox{vec}(L) = h + H \theta$$ with a (\(k\)-dimensional) parameter vector \(\theta\).

Usage

tmpl_sigma_L(
  sigma_L,
  structure = c("as_given", "chol", "symm", "identity", "full_normalized")
)

Arguments

sigma_L

numeric (n x n) matrix, where the free entries are coded with NAs

structure

character string, determines the "structure" of sigma_L, see the examples.

Value

List with slots

  • h (\(n^2\)-dimensional vector),

  • H (\((n^2, k)\)-dimensional matrix, where \(k\) denotes the number of free/deep parameters) and

  • n.par (integer) is the number of free/deep parameters (\(=k\)).

Details

The parameter structure has the following meaning

as_given

Use the given parameter sigma_L to construct the template: NA entries are considered as free and all other entries as fixed.

chol

Set all entries of sigma_L above the diagonal to zero and then proceed as above.

symm

First make sigma_L symmetric (sigma_L = (sigma_L + t(sigma_L))/2) and then use this sigma_L as template. However, h, H are constructed such that \(h + H\theta\) gives a symmetric matrix! Note that NAs overwrite fixed values, see examples.

identity

Use the identity matrix as template. In this case there are no free parameters, i.e. \(\theta\) is an empty vector (vector with zero length).

full_normalized

Ones on the diagonal, otherwise all parameters are free.

Examples

sigma_L = matrix(c(0, NA, 1, 0, 2, 3, NA, 1, 1), nrow = 3, ncol = 3)
sigma_L
#>      [,1] [,2] [,3]
#> [1,]    0    0   NA
#> [2,]   NA    2    1
#> [3,]    1    3    1

tmpl = tmpl_sigma_L(sigma_L, structure = 'as_given')
th = -(1:tmpl$n.par)
matrix(tmpl$h + tmpl$H %*% th, nrow = 3, ncol = 3)
#>      [,1] [,2] [,3]
#> [1,]    0    0   -2
#> [2,]   -1    2    1
#> [3,]    1    3    1

tmpl = tmpl_sigma_L(sigma_L, structure = 'chol')
th = -(1:tmpl$n.par)
matrix(tmpl$h + tmpl$H %*% th, nrow = 3, ncol = 3)
#>      [,1] [,2] [,3]
#> [1,]    0    0    0
#> [2,]   -1    2    0
#> [3,]    1    3    1

tmpl = tmpl_sigma_L(sigma_L, structure = 'symm')
th = -(1:tmpl$n.par)
matrix(tmpl$h + tmpl$H %*% th, nrow = 3, ncol = 3)
#>      [,1] [,2] [,3]
#> [1,]    0   -1   -2
#> [2,]   -1    2    2
#> [3,]   -2    2    1

tmpl = tmpl_sigma_L(sigma_L, structure = 'identity')
tmpl$n.par # = 0
#> [1] 0
matrix(tmpl$h, nrow = 3, ncol = 3)
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1

tmpl = tmpl_sigma_L(sigma_L, structure = 'full_normalized')
th = -(1:tmpl$n.par)
matrix(tmpl$h + tmpl$H %*% th, nrow = 3, ncol = 3)
#>      [,1] [,2] [,3]
#> [1,]    1   -3   -5
#> [2,]   -1    1   -6
#> [3,]   -2   -4    1