variogram.intrinsic.spde {rSPDE}

Variogram of intrinsic SPDE model

Description

Variogram \gamma(s_0,s) of intrinsic SPDE model

(-\Delta)^{\beta/2}(\kappa^2-\Delta)^{\alpha/2} (\tau u) = \mathcal{W}

with Neumann boundary conditions and a mean-zero constraint on a square [0,L]^d for d=1 or d=2.

Usage

variogram.intrinsic.spde(
  s0 = NULL,
  s = NULL,
  kappa = NULL,
  alpha = NULL,
  beta = NULL,
  tau = 1,
  L = NULL,
  N = 100,
  d = NULL
)

Arguments

s0

The location where the variogram should be evaluated, either a double for 1d or a vector for 2d

s

A vector (in 1d) or matrix (in 2d) with all locations where the variogram is computed

kappa

Range parameter.

alpha

Smoothness parameter.

beta

Smoothness parameter.

tau

Precision parameter.

L

The side length of the square domain.

N

The number of terms in the Karhunen-Loeve expansion.

d

The dimension (1 or 2).

Details

The variogram is computed based on a Karhunen-Loeve expansion of the covariance function.

See Also

intrinsic.matern.operators()

Examples

if (requireNamespace("RSpectra", quietly = TRUE)){
 x <- seq(from = 0, to = 10, length.out = 201)
 beta <- 1
 alpha <- 1
 kappa <- 1
 op <- intrinsic.matern.operators(kappa = kappa, tau = 1, alpha = alpha, 
                                 beta = beta, loc_mesh = x, d=1) 
 # Compute and plot the variogram of the model
 Sigma <- op$A %*% solve(op$Q,t(op$A))
 One <- rep(1, times = ncol(Sigma))
 D <- diag(Sigma)
 Gamma <- 0.5*(One %*% t(D) + D %*% t(One) - 2 * Sigma)
 k <- 100
 plot(x, Gamma[k, ], type = "l")
 lines(x, 
      variogram.intrinsic.spde(x[k], x, kappa, alpha, beta, L = 10, d = 1),
      col=2, lty = 2)
}