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), as returned by build_plugin_field().

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.

Details

infer_kernel_params() maximises this quantity for you; call it directly when you want to score a candidate set of hyperparameters yourself (e.g. profiling a likelihood surface). The kernels should be correlation matrices (unit diagonal) as built by space_kernel() / time_kernel(), decomposed with eigen(., symmetric = TRUE).

Examples

n <- 5; nt <- 20
coords <- data.frame(id = 1:n, lon = runif(n), lat = runif(n))
g <- stats::rnorm(n * nt)
eig_s <- eigen(space_kernel(coords, length_scale = 1), symmetric = TRUE)
eig_t <- eigen(time_kernel(1:nt, periodic_scale = 1, long_term_scale = 50,
                           period = 52), symmetric = TRUE)
ll <- gp_marginal_loglik(g, n, nt, eig_s, eig_t, eta = 0.1)
ll
#> [1] -158.7387
#> attr(,"sigma2")
#> [1] 9.878955
attr(ll, "sigma2")  # the profiled global variance
#> [1] 9.878955