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 covariance-based rational approximation, we refer the reader to the Rational approximation with the rSPDE package vignette.
It is noteworthy 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 noises such as the type-G Lévy noises. We refer the reader to (Wallin and Bolin 2015)(https://onlinelibrary.wiley.com/doi/full/10.1111/sjos.12141), (Bolin 2013)(https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12046) and (Asar et al. 2020)(https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/rssc.12405) for examples of models driven by type-G Lévy noises. We also refer the reader to the ngme package where one can fit such models.
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 where \(u\) is included. In the following sections, we will 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. For illustration purposes, the rSPDE package contains a simple FEM implementation for models on \(\mathbb{R}\) which we will use first. We will then illustrate how spatial models can be constructed, if the FEM implementation of the R-INLA package is used instead.
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.
Based on these nodes, we use 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
op <- matern.operators(kappa = kappa, sigma = sigma, nu = nu,
G = fem$G, C = fem$C, d = 1, m = 1,
type="operator")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 Matérn covariance function, which is the folded Matérn, see Theorem 1 in An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach for further details. 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(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. 
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, for 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(kappa = kappa, sigma = sigma, nu = nu,
G = fem$G, C = fem$C, d = 1, m = 2,
type="operator")
c.approx2 <- op2$Pr %*% solve(op2$Q, op2$Pr %*% v)
s2 <- seq(from = 0, to = 1, length.out = 501)
fem2 <- rSPDE.fem1d(s2)
op <- matern.operators(kappa = kappa, sigma = sigma, nu = nu,
G = fem2$G, C = fem2$C, d = 1, m=1,
type="operator")
A <- rSPDE.A1d(s2,s)
v <- t(rSPDE.A1d(s2,0.5))
c.approx3 <- A%*%op$Pr %*% solve(op$Q, op$Pr %*% v)
op <- matern.operators(kappa = kappa, sigma = sigma, nu = nu,
G = fem2$G, C = fem2$C, d = 1, m=2,
type="operator")
c.approx4 <- A%*%op$Pr %*% solve(op$Q, op$Pr %*% v)The resulting errors are shown in the following figure.
Since the error induced by the rational approximation decreases exponentially in \(m\), there is in general 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 number of \(\boldsymbol{\mathrm{P}}_r\) and \(\boldsymbol{\mathrm{Q}}\) increases 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(kappa = kappa, sigma = sigma, nu = nu,
G = fem2$G, C = fem2$C, d = 1, m = i,
type = "operator")
c.app <- A%*%op$Pr %*% solve(op$Q, op$Pr %*% v)
errors[i] <- norm(c.true-c.app)
}
print(errors)
#> [1] 1.0120165 0.1100542 52.4841240 54.6499071We 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 high condition numbers of the matrices.
It is important to notice that the covariance-based rational approximation is more numerically stable, the main reason being that its clever decomposition of the latent field into a vector of random fields remove the need of higher order finite element matrices to have higher order rational approximations. We refer the reader to the Rational approximation with the rSPDE package vignette for further details.
To solve 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 issues. 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.
