Exact separable-GP marginal log-likelihood of a field, with a nugget

Evaluates the exact log-density of `g ~ N(0, sigma^2 ((R_s (x) R_t) + eta I))` given the eigendecompositions of the spatial and temporal correlation kernels, via the Kronecker log-determinant and quadratic-form identities. The global variance `sigma^2` is profiled out (concentrated log-likelihood) and the profiled value is attached as `attr(., "sigma2")`.

Usage

gp_marginal_loglik(g, n, nt, eig_s, eig_t, eta)

Arguments

g

Plug-in field, length `n * nt`, ordered sites x times (time fastest).

n

Number of sites.

nt

Number of time points.

eig_s

Eigendecomposition (`eigen` object) of the spatial correlation kernel.

eig_t

Eigendecomposition (`eigen` object) of the temporal correlation kernel.

eta

Noise-to-signal ratio (nugget), a non-negative scalar.

Value

The concentrated log-likelihood (numeric scalar), with the profiled `sigma2` attached as an attribute.