Object-based log-likelihood function for latent Gaussian fractional SPDE model

This function evaluates the log-likelihood function for a fractional SPDE model \(L^\beta u(s) = W\) that is observed under Gaussian measurement noise: \(Y_i = u(s_i) + \epsilon_i\), where \(\epsilon_i\) are iid mean-zero Gaussian variables and \(x(s) = \mu(s) + u(s)\), where \(\mu(s)\) is the expectation vector of the latent field.

Usage

rSPDE.loglike(obj, Y, A, sigma.e, mu = 0)

Arguments

obj

The rational SPDE approximation, computed using fractional.operators(), matern.operators(), or spde.matern.operators().

Y

The observations, either a vector or a matrix where the columns correspond to independent replicates of observations.

A

An observation matrix that links the measurement location to the finite element basis.

sigma.e

The standard deviation of the measurement noise.

mu

Expectation vector of the latent field (default = 0).

Value

The log-likelihood value.

Note

This example below shows how the function can be used to evaluate the likelihood of a latent Matern model.

See also

spde.matern.loglike()

Examples

# Sample a Gaussian Matern process on R using a rational approximation
kappa <- 10
sigma <- 1
nu <- 0.8
sigma.e <- 0.3
range <- 0.2

# create mass and stiffness matrices for a FEM discretization
x <- seq(from = 0, to = 1, length.out = 101)
fem <- rSPDE.fem1d(x)

# compute rational approximation
op <- matern.operators(
  range = range, sigma = sigma, nu = nu,
  loc_mesh = x, d = 1,
  type = "operator", parameterization = "matern"
)

# Sample the model
u <- simulate(op)

# Create some data
obs.loc <- runif(n = 10, min = 0, max = 1)
A <- rSPDE.A1d(x, obs.loc)
Y <- as.vector(A %*% u + sigma.e * rnorm(10))

# compute log-likelihood of the data
lik1 <- rSPDE.loglike(op, Y, A, sigma.e)
cat(lik1)
#> -9.336895