# Prediction of a fractional SPDE using the covariance-based rational SPDE approximation

Source:`R/fractional.computations.R`

`predict.CBrSPDEobj.Rd`

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 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

`matern.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

```
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)
```