predict.CBrSPDEobj {rSPDE} | R Documentation |
Prediction of a fractional SPDE using the covariance-based rational SPDE approximation
Description
The function is used for computing kriging predictions based
on data Y_i = u(s_i) + \epsilon_i
, where \epsilon
is mean-zero Gaussian measurement noise and u(s)
is defined by
a fractional SPDE (\kappa^2 I - \Delta)^{\alpha/2} (\tau u(s)) = W
,
where W
is Gaussian white noise and \alpha = \nu + d/2
,
where d
is the dimension of the domain.
Usage
## S3 method for class 'CBrSPDEobj'
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 |
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 |
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 |
n_samples |
Number of samples to be returned. Will only be used if |
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
set.seed(123)
# 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)
tau <- sqrt(gamma(nu) / (sigma^2 * kappa^(2 * nu) *
(4 * pi)^(1 / 2) * gamma(nu + 1 / 2)))
# Compute the covariance-based rational approximation
op_cov <- matern.operators(
loc_mesh = x, nu = nu,
range = range, sigma = sigma, d = 1, m = 2,
parameterization = "matern"
)
# Sample the model
u <- simulate(op_cov)
# 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 kriging predictions at the FEM grid
A.krig <- rSPDE.A1d(x, x)
u.krig <- predict(op_cov,
A = A, Aprd = A.krig, Y = Y, sigma.e = sigma.e,
compute.variances = TRUE
)
plot(obs.loc, Y,
ylab = "u(x)", xlab = "x", main = "Data and prediction",
ylim = c(
min(u.krig$mean - 2 * sqrt(u.krig$variance)),
max(u.krig$mean + 2 * sqrt(u.krig$variance))
)
)
lines(x, u.krig$mean)
lines(x, u.krig$mean + 2 * sqrt(u.krig$variance), col = 2)
lines(x, u.krig$mean - 2 * sqrt(u.krig$variance), col = 2)