Using the package to perform operator-based rational approximations
The main purpose of the rSPDE package is to provide
functions for creating the rational approximation. In this vignette we
focus on the operator-based rational approximation, which means
assembling the matrices \(\boldsymbol{\mathrm{Q}}\) and \(\boldsymbol{\mathrm{P}}_r\). There are
three functions for computing the rational approximation. The most
general function is fractional.operators(), which works for
a wide class of models with a general differential operator \(L\). For the stationary Matérn case, where
\(L = \kappa^2 - \Delta\), the function
matern.operators() provides a simplified model
specification. For the generalized non-stationary Matérn model, defined
through the SPDE \[
(\kappa(s)^2 - \Delta)^\beta (\tau(s)u(s)) = \mathcal{W},
\] the function spde.matern.operators() can be
used.
For the alternative covariance-based rational approximation, we refer the reader to the Rational approximation with the rSPDE package vignette. It is worth noting that the covariance-based rational approximation only applies to fractional SPDE models with Gaussian noise, whereas the operator-based rational approximation can be used for more general models such as the models driven by type-G Lévy noise considered in Wallin and Bolin (2015), Bolin (2013), and Asar et al. (2020).
Once the approximation has been constructed, it can be included
manually in statistical models just as for the non-fractional case. The
package has some built-in functions for basic use of the approximation,
such as simulate() which can be applied for simulation of
the field. There are also functions for likelihood evaluation and
kriging prediction for geostatistical models with Gaussian measurement
noise. In the following sections, we illustrate the usage of these
functions.
Constructing the approximation
In this section, we explain how the different main functions can be
used for constructing the rational approximation. The first step for
constructing the rational SPDE approximation is to define the FEM mesh.
In this section, we use the simple FEM implementation in the
rSPDE package for models defined on an interval.
Assume that we want to define a model on the interval \([0,1]\). We then start by defining a vector with mesh nodes \(s_i\) where the basis functions \(\varphi_i\) are centered.
s <- seq(from = 0, to = 1, length.out = 101)Based on these nodes, we use (implicitly) the built-in function
rSPDE.fem1d() to assemble two matrices needed for creating
the approximation of a basic Matérn model. These matrices are the mass
matrix \(\boldsymbol{\mathrm{C}}\),
with elements \(C_{ij} = \int \varphi_j(s)
\varphi_i(s) ds\), and the stiffness matrix \(\boldsymbol{\mathrm{G}}\), with elements
\(G_{ij} = \int \nabla\varphi_j(s) \cdot
\nabla\varphi_i(s) ds\).
We can now use matern.operators() to construct a
rational SPDE approximation of degree \(m=1\) for a Gaussian random field with a
Matérn covariance function on the interval. Since we are using the
operator-based approximation, we must set type to
"operator".
kappa <- 20
sigma <- 2
nu <- 0.8
r <- sqrt(8*nu)/kappa
op <- matern.operators( sigma = sigma,
range = r,
nu = nu,
loc_mesh = s, d = 1, m = 1,
type = "operator",
parameterization = "matern"
)The object op contains the matrices needed for
evaluating the distribution of the stochastic weights \(\boldsymbol{\mathrm{u}}\). If we want to
evaluate \(u_h(s)\) at some locations
\(s_1,\ldots, s_n\), we need to
multiply the weights with the basis functions \(\varphi_i(s)\) evaluated at the locations.
For this, we can construct the observation matrix \(\boldsymbol{\mathrm{A}}\) with elements
\(A_{ij} = \varphi_j(s_i)\), which
links the FEM basis functions to the locations. This matrix can be
constructed using the function rSPDE.A1d().
To evaluate the accuracy of the approximation, let us compute the
covariance function between the process at \(s=0.5\) and all other locations in
s and compare with the true covariance function, which is
the folded Matérn covariance, see Theorem 1 in An explicit link between
Gaussian fields and Gaussian Markov random fields: the stochastic
partial differential equation approach. The covariances can be
calculated as \[
\boldsymbol{\mathrm{A}} \boldsymbol{\mathrm{P}}_r
\boldsymbol{\mathrm{Q}}^{-1}\boldsymbol{\mathrm{P}}_r^T\boldsymbol{\mathrm{v}}.
\] Here \(\boldsymbol{\mathrm{A}}\) is an identity
matrix since we are evaluating the approximation in the nodes of the FEM
mesh and \(\boldsymbol{\mathrm{v}}\) is
a vector with all basis functions evaluated in \(s=0.5\).
v <- t(rSPDE.A1d(s, 0.5))
c.approx <- op$Pr %*% solve(op$Q, op$Pr %*% v)
c.true <- folded.matern.covariance.1d(rep(0.5, length(s)),
abs(s), kappa, nu, sigma)The covariance function and the error compared with the Matérn covariance are shown in the following figure.
opar <- par(
mfrow = c(1, 2), mgp = c(1.3, 0.5, 0),
mar = c(2, 2, 0.5, 0.5) + 0.1
)
plot(s, c.true,
type = "l", ylab = "C(|s-0.5|)", xlab = "s", ylim = c(0, 5),
cex.main = 0.8, cex.axis = 0.8, cex.lab = 0.8
)
lines(s, c.approx, col = 2)
legend("topright",
bty = "n",
legend = c("Matérn", "m=1 rSPDE"),
col = c("black", "red"),
lty = rep(1, 2), ncol = 1,
cex = 0.8
)
plot(s, c.true - c.approx,
type = "l", ylab = "Error", xlab = "s",
cex.main = 0.8, cex.axis = 0.8, cex.lab = 0.8
)
par(opar)
To improve the approximation we can increase the degree of the polynomials, by increasing \(m\), and/or increase the number of basis functions used for the FEM approximation. Let us, as an example, compute the approximation with \(m=2\) using the same mesh, as well as the approximation when we increase the number of basis functions and use \(m=1\) and \(m=2\).
op2 <- matern.operators(
range = r, sigma = sigma, nu = nu,
loc_mesh = s, d = 1, m = 2,
type = "operator",
parameterization = "matern"
)
c.approx2 <- op2$Pr %*% solve(op2$Q, op2$Pr %*% v)
s2 <- seq(from = 0, to = 1, length.out = 501)
fem2 <- rSPDE.fem1d(s2)
op <- matern.operators(
range = r, sigma = sigma, nu = nu,
loc_mesh = s2, d = 1, m = 1,
type = "operator",
parameterization = "matern"
)
A <- rSPDE.A1d(s2, s)
v <- t(rSPDE.A1d(s2, 0.5))
c.approx3 <- A %*% op$Pr %*% solve(op$Q, op$Pr %*% v)
op <- matern.operators(
range = r, sigma = sigma, nu = nu,
loc_mesh = s2, d = 1, m = 2,
type = "operator",
parameterization = "matern"
)
c.approx4 <- A %*% op$Pr %*% solve(op$Q, op$Pr %*% v)The resulting errors are shown in the following figure.
opar <- par(mgp = c(1.3, 0.5, 0), mar = c(2, 2, 0.5, 0.5) + 0.1)
plot(s, c.true - c.approx,
type = "l", ylab = "Error", xlab = "s", col = 1,
cex.main = 0.8, cex.axis = 0.8, cex.lab = 0.8
)
lines(s, c.true - c.approx2, col = 2)
lines(s, c.true - c.approx3, col = 3)
lines(s, c.true - c.approx4, col = 4)
legend("bottomright",
bty = "n",
legend = c("m=1 coarse mesh", "m=2 coarse mesh",
"m=1 fine mesh", "m=2 fine mesh"),
col = c(1, 2, 3, 4),
lty = rep(1, 2), ncol = 1,
cex = 0.8
)
par(opar)
Since the error induced by the rational approximation decreases exponentially in \(m\), there is rarely a need for an approximation with a large value of \(m\). This is good because the number of non-zero elements in \(\boldsymbol{\mathrm{P}}_r\) and \(\boldsymbol{\mathrm{Q}}\) increases with \(m\), which makes the approximation more computationally costly to use. Further, the condition numbers of \(\boldsymbol{\mathrm{P}}_r\) and \(\boldsymbol{\mathrm{Q}}\) increase with \(m\), which can cause numerical problems when working with these matrices. To illustrate this, let us compute the norm of the approximation error for different \(m\).
errors <- rep(0, 4)
for (i in 1:4) {
op <- matern.operators(
range = r, sigma = sigma, nu = nu,
loc_mesh = s2, d = 1, m = i,
type = "operator",
parameterization = "matern"
)
c.app <- A %*% op$Pr %*% solve(op$Q, op$Pr %*% v)
errors[i] <- norm(c.true - c.app)
}
print(errors)
#> [1] 1.012017 0.114015 51.947037 54.648301We see that the error decreases when increasing \(m\) from \(1\) to \(2\), but is very large for \(m=3\) and \(m=4\). The reason for this is not that the approximation is bad, but that the numerical accuracy of the product \(\boldsymbol{\mathrm{P}}_r\boldsymbol{\mathrm{Q}}^{-1}\boldsymbol{\mathrm{P}}_rv\) is low due to the large condition numbers of the matrices.
It is important to note that the alternative covariance-based rational approximation is more numerically stable. The main reason for this is that it relies on a decomposition of the field \(u\) into a sum of random fields, which removes the need of computing higher order finite element matrices for large values of \(m\). See the Rational approximation with the rSPDE package vignette for further details.
To handle this issue for the operator-based rational approximation,
the package contains functions for performing operations such as \(\boldsymbol{\mathrm{P}}_rv\) or \(\boldsymbol{\mathrm{P}}_r^{-1}v\) that
takes advantage of the structure of \(\boldsymbol{\mathrm{P}}_r\) to avoid
numerical instabilities. A complete list of these function can be seen
by typing ?operator.operations. One of these functions is
Sigma.mult(), which performs the multiplication \(\boldsymbol{\mathrm{P}}_r\boldsymbol{\mathrm{Q}}^{-1}\boldsymbol{\mathrm{P}}_rv\)
in a more numerically stable way. Let us use this function to compute
the errors of the approximations again to see that we indeed get better
approximations as \(m\) increases.
errors2 <- rep(0, 4)
for (i in 1:4) {
op <- matern.operators(
range = r, sigma = sigma, nu = nu,
loc_mesh = s2, d = 1, m = i,
type = "operator",
parameterization = "matern"
)
c.app <- A %*% Sigma.mult(op, v)
errors2[i] <- norm(c.true - c.app)
}
print(errors2)
#> [1] 1.01201540 0.10479953 0.02474815 0.01788324
We can now use these matrices to define a rational SPDE approximation of
degree 