Initialize a coregionalization kernel from the data

Builds a coregionalization_kernel() whose \(B\) is the empirical covariance between the outputs, a starting point from which optimization can move \(W\).

Usage

initialize_coregionalization(output, y, x = NULL, rank = 1L, n_outputs = NULL)

Arguments

output

Output index of each observation, from 1 to n_outputs.

y

Numeric response of each observation.

x

Optional inputs of each observation, a vector or matrix without the output-index column. Covariances between two outputs are estimated from the inputs at which both are observed; without x, or with fewer than three shared inputs, they start at zero.

rank

Rank \(R\) of \(W\).

n_outputs

Number of outputs; by default the largest index in output.

Value

A gaussianprocesses_kernel.

Details

The empirical covariance \(\hat B\) of the centred responses is made positive semidefinite by setting negative eigenvalues to zero. \(W\) is taken from its leading \(R\) eigenvectors, scaled by the square roots of their eigenvalues, and \(\kappa\) from what remains on the diagonal, with at least 5% of each output's variance. The empirical variances include the observation noise, so \(B\) starts too large on the diagonal; optimization corrects that.

Stability

Experimental: this interface may change in a minor release, and every change is listed in NEWS. See gaussianprocesses-package for the policy.

See also

coregionalization_kernel(), gp_stack_outputs().

Examples

set.seed(1)
x <- seq(0, 5, length.out = 20)
y <- cbind(sin(x), 2 * sin(x) + 0.1) + rnorm(40, sd = 0.1)
stacked <- gp_stack_outputs(x, y)
kernel <- initialize_coregionalization(stacked$output, stacked$y,
                                       x = stacked$x[, 1])
coregionalization_matrix(kernel)
#> [[1]]
#>           [,1]     [,2]
#> [1,] 0.5452383 1.029026
#> [2,] 1.0290260 2.209153
#>