Prediction of a space-time SPDE

The function is used for computing kriging predictions based on data \(Y_i = u(s_i,t_i) + \epsilon_i\), where \(\epsilon\) is mean-zero Gaussian measurement noise and \(u(s,t)\) is defined by a spatio-temporal SPDE as described in spacetime.operators().

Usage

# S3 method for class 'spacetimeobj'
predict(
  object,
  A,
  Aprd,
  Y,
  sigma.e,
  mu = 0,
  compute.variances = FALSE,
  posterior_samples = FALSE,
  n_samples = 100,
  only_latent = FALSE,
  ...
)

Arguments

object

The covariance-based rational SPDE approximation, computed using spacetime.operators()

A

A matrix linking the measurement locations to the basis of the FEM approximation of the latent model.

Aprd

A matrix linking the prediction locations to the basis of the FEM approximation of the latent model.

Y

A vector with the observed data, can also be a matrix where the columns are observations of independent replicates of \(u\).

sigma.e

The standard deviation of the Gaussian measurement noise. Put to zero if the model does not have measurement noise.

mu

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

compute.variances

Set to also TRUE to compute the kriging variances.

posterior_samples

If TRUE, posterior samples will be returned.

n_samples

Number of samples to be returned. Will only be used if sampling is TRUE.

only_latent

Should the posterior samples be only given to the laten model?

...

further arguments passed to or from other methods.

Value

A list with elements

mean

The kriging predictor (the posterior mean of u|Y).

variance

The posterior variances (if computed).

Examples

s <- seq(from = 0, to = 20, length.out = 101)
t <- seq(from = 0, to = 20, length.out = 31)

op_cov <- spacetime.operators(space_loc = s, time_loc = t,
                             kappa = 5, sigma = 10, alpha = 1,
                             beta = 2, rho = 1, gamma = 0.05)
# generate data
sigma.e <- 0.01
n.obs <- 500
obs.loc <- data.frame(x = max(s)*runif(n.obs), 
                     t = max(t)*runif(n.obs))
A <- rSPDE.Ast(space_loc = s, time_loc = t, obs.s = obs.loc$x, obs.t = obs.loc$t)
Aprd <- Diagonal(dim(A)[2])
x <- simulate(op_cov, nsim = 1) 
Y <- A%*%x + sigma.e*rnorm(n.obs)
u.krig <- predict(op_cov, A, Aprd, Y, sigma.e)