spde.matern.loglike: Parameter-based log-likelihood for a latent Gaussian Matern SPDE model using a rational SPDE approximation

Description

This function evaluates the log-likelihood function for observations of a Gaussian process defined as the solution to the SPDE $$(\kappa(s) - \Delta)^\beta (\tau(s)u(s)) = W.$$

Usage

spde.matern.loglike(
  object,
  Y,
  A,
  sigma.e,
  mu = 0,
  nu = NULL,
  kappa = NULL,
  tau = NULL,
  theta = NULL,
  m = NULL
)

Value

The log-likelihood value.

Arguments

object: The rational SPDE approximation, computed using 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: IF non-null, the standard deviation of the measurement noise will be kept fixed in the returned likelihood.
mu: Expectation vector of the latent field (default = 0).
nu: If non-null, the shape parameter will be kept fixed in the returned likelihood.
kappa: If non-null, updates the range parameter.
tau: If non-null, updates the parameter tau.
theta: If non-null, updates the parameter theta (that connects tau and kappa to the model matrices in object).
m: If non-null, update the order of the rational approximation, which needs to be a positive integer.

Details

The observations are assumed to be generated as $Y_i = u(s_i) + \epsilon_i$, where $\epsilon_i$ are iid mean-zero Gaussian variables. The latent model is approximated using a rational approximation of the fractional SPDE model.

Examples

Run this code

# this example illustrates how the function can be used for maximum
# likelihood estimation
# Sample a Gaussian Matern process on R using a rational approximation
sigma.e <- 0.1
n.rep <- 10
n.obs <- 100
n.x <- 51
# create mass and stiffness matrices for a FEM discretization
x <- seq(from = 0, to = 1, length.out = n.x)
fem <- rSPDE.fem1d(x)
tau <- rep(0.5, n.x)
nu <- 0.8
alpha <- nu + 1 / 2
kappa <- rep(1, n.x)
# compute rational approximation
op <- spde.matern.operators(
  kappa = kappa, tau = tau, alpha = alpha,
  parameterization = "spde", d = 1,
  loc_mesh = x
)
# Sample the model
u <- simulate(op, n.rep)
# Create some data
obs.loc <- runif(n = n.obs, min = 0, max = 1)
A <- rSPDE.A1d(x, obs.loc)
noise <- rnorm(n.obs * n.rep)
dim(noise) <- c(n.obs, n.rep)
Y <- as.matrix(A %*% u + sigma.e * noise)
# define negative likelihood function for optimization using matern.loglike
mlik <- function(theta) {
  return(-spde.matern.loglike(op, Y, A,
    sigma.e = exp(theta[4]),
    nu = exp(theta[3]),
    kappa = exp(theta[2]),
    tau = exp(theta[1])
  ))
}
#' #The parameters can now be estimated by minimizing mlik with optim
# \donttest{
# Choose some reasonable starting values depending on the size of the domain
theta0 <- log(c(1 / sqrt(var(c(Y))), sqrt(8), 0.9, 0.01))
# run estimation and display the results
theta <- optim(theta0, mlik)
print(data.frame(
  tau = c(tau[1], exp(theta$par[1])), kappa = c(kappa[1], exp(theta$par[2])),
  nu = c(nu, exp(theta$par[3])), sigma.e = c(sigma.e, exp(theta$par[4])),
  row.names = c("Truth", "Estimates")
))
# }