# Prediction of a fractional SPDE using the covariance-based rational SPDE approximation 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. ```r ## 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 `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. ## Returns A list with elements - **mean**: The kriging predictor (the posterior mean of u|Y). - **variance**: The posterior variances (if computed). ## Examples ```r 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) ```

predict.CBrSPDEobj function