rSPDE (version 2.4.0)

matern.rational: Rational approximation of the Matern fields on intervals and metric graphs

Description

The function is used for computing an approximation, which can be used for inference and simulation, of the fractional SPDE $$(\kappa^2 - \Delta)^{\alpha/2} (\tau u(s)) = W$$ on intervals or metric graphs. Here \(W\) is Gaussian white noise, \(\kappa\) controls the range, \(\alpha = \nu + 1/2\) with \(\nu>0\) controls the smoothness and \(\tau\) is related to the marginal variances through $$\sigma^2 = \frac{\Gamma(\nu)}{\tau^2\Gamma(\alpha)2\sqrt{\pi}\kappa^{2\nu}}.$$

Usage

matern.rational(
  graph = NULL,
  loc = NULL,
  bc = c("free", "Neumann", "Dirichlet"),
  kappa = NULL,
  range = NULL,
  nu = NULL,
  sigma = NULL,
  tau = NULL,
  alpha = NULL,
  m = 2,
  parameterization = c("matern", "spde"),
  type_rational_approximation = "brasil",
  type_interp = "spline"
)

Value

A model object for the the approximation

Arguments

graph

Metric graph object. The default is NULL, which means that a stationary Matern model on the line is created.

loc

Locations where to evaluate the model.

bc

Specifies the boundary conditions. The default is "free" which gives stationary Matern models on intervals. Other options are "Neumann" or "Dirichlet".

kappa

Range parameter

range

practical correlation range

nu

Smoothness parameter

sigma

Standard deviation

tau

Precision parameter

alpha

Smoothness parameter

m

The order of the approximation

parameterization

Which parameterization to use? matern uses range, std. deviation and nu (smoothness). spde uses kappa, tau and alpha. The default is matern.

type_rational_approximation

Method used to compute the coefficients of the rational approximation.

type_interp

Interpolation method for the rational coefficients.

Examples

s <- seq(from = 0, to = 1, length.out = 101)
kappa <- 20
sigma <- 2
nu <- 0.8
r <- sqrt(8*nu)/kappa #range parameter
op_cov <- matern.rational(loc = s, nu = nu, range = r, sigma = sigma, m = 2, 
parameterization = "matern")
cov.true <- matern.covariance(abs(s-s[1]), kappa = kappa, sigma = sigma, nu = nu)
cov.approx <- op_cov$covariance(ind = 1)

plot(s, cov.true)
lines(s, cov.approx, col = 2)