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

In rSPDE: Rational Approximations of Fractional Stochastic Partial Differential Equations

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"
)

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.

Value

A model object for the the approximation

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)

rSPDE documentation built on Jan. 26, 2026, 9:06 a.m.