Rational approximation with the rSPDE package

Introduction

In this vignette we will introduce the covariance-based rational SPDE approach and illustrate how to perform statistical inference with it.

The covariance-based approach is an efficient alternative to the operator-based rational SPDE approach by Bolin and Kirchner (2020) which works when one has SPDE driven by Gaussian white noise. We refer the reader to Bolin, Simas, and Xiong (2023) for the theoretical details of the approach.

Details about the operator-based rational SPDE approach are given in the Operator-based rational approximation vignette. For the R-INLA and inlabru implementations of the covariance-based rational SPDE approach we refer the reader to the vignettes R-INLA implementation of the rational SPDE approach and inlabru implementation of the rational SPDE approach respectively.

Covariance-based rational SPDE approach

Let us first present the basic setup. We want to model the precipitation as a two-dimensional random field \(u\) within a bounded domain, where each location on the domain associates with a random variable which describe the local precipitation.

In the SPDE approach, introduced in Lindgren, Rue, and Lindström (2011) we model \(u\) as the solution of the following SPDE: \[L^{\alpha/2}(\tau u) = \mathcal{W},\] where \(L = -\Delta +\kappa^2 I\) and \(\mathcal{W}\) is the standard Gaussian white noise. Here, \(\alpha\), \(\kappa\) and \(\tau\) are three parameters we want to estimate. In the standard SPDE approach, we write, for a general dimension \(d\), \(\alpha = \nu + d/2\) and assume \(\nu\) to be fixed so that \(\alpha\) is an integer. In the rational SPDE approach we can use any value of \(\nu>0\) and also estimate it from data.

Now let us briefly describe how covariance based rational SPDE approach works in statistical inference. The main idea is to perform the rational approximation directly on the covariance operator, which is given by \(L^{-\alpha}\), instead of on the solution \(u\). To this end, we begin by obtaining an approximation of the random field \(u\), which is the solution of the SPDE above, by using the finite element method (FEM): \[u_h(\mathbf{s}_i)=\sum_{j=1}^{n_h} \hat{u}_j \varphi_j(\mathbf{s}_i),\] where \(\{\hat{u}_j\}_{j = 1}^{n_h}\) are stochastic weights and \(\{\varphi_j(\mathbf{s}_i)\}_{j = 1}^{n_h}\) are fixed piecewise linear and continuous basis functions obtained from a triangulation of the spatial domain. We then obtain a FEM approximation of the operator \(L\), which is given by \(L_h\), and the covariance operator of \(u_h\) is given by \(L_h^{-\alpha}\).

Now, by using the rational approximation on \(L_h\), we can approximate covariance operator \(L_h^{-\alpha}\) as \[L_{h,m}^{-\alpha} = L_h^{-\lfloor\alpha\rfloor} p(L_h^{-1})q(L_h^{-1})^{-1},\] where \(\lfloor\alpha\rfloor\) denotes the integer part of \(\alpha\), \(m\) is the order of rational approximation, \(p(L_h^{-1}) = \sum_{i=0}^m a_i L_h^{m-i}\) and \(q(L_h^{-1}) = \sum_{j=0}^m b_j L_h^{m-i}\), with \(\{a_i\}_{i = 0}^m\) and \(\{b_j\}_{j = 0}^m\) being known coefficients obtained from a rational approximation of the function \(x^{\alpha - \lfloor\alpha\rfloor}\).

The next step is to perform a partial fraction decomposition of the rational function \(p(L_h^{-1})q(L_h^{-1})^{-1}\), which yields the representation \[L_{h,m}^{-\alpha} =L_h^{-\lfloor\alpha\rfloor} \left(\sum_{i=1}^{m} r_i (L_h-p_i I)^{-1} +k\right).\] Based on the above operator equation, we can write the covariance matrix of the stochastic weights \(\hat{\textbf{u}}\), where \(\hat{\textbf{u}}=[\hat{u}_1,...,\hat{u}_{n_h}]^\top\), as \[\mathbf{\Sigma}_{\hat{\textbf{u}}} = (\textbf{L}^{-1}\textbf{C})^{\lfloor\alpha\rfloor} \sum_{i=1}^{m}r_i(\textbf{L}-p_i\textbf{C})^{-1}+\textbf{K}, \] where \(\textbf{C} = \{C_{ij}\}_{i,j=1}^{n_h}\), \(C_{ij} = (\varphi_i,\varphi_j)_{L_2(\mathcal{D})}\), is the mass matrix, \(\textbf{L} = \kappa^2\textbf{C}+\textbf{G}\), \(\textbf{G} = \{G_{ij}\}_{i,j=1}^{n_h}\), \(G_{ij}=(\nabla\varphi_i,\nabla\varphi_j)_{L_2(\mathcal{D})}\), is the stiffness matrix, and \[\textbf{K}=\left\{ \begin{array}{lcl} k\textbf{C} & & {\lfloor\alpha\rfloor=0}\\ k\textbf{L}^{-1}(\textbf{C}\textbf{L}^{-1})^{\lfloor\alpha\rfloor-1} & & {\lfloor\alpha\rfloor\geq 1}\\ \end{array} \right. .\]

The above representation shows that we can express \(\hat{\textbf{u}}\) as \[\hat{\textbf{u}}=\sum_{k=1}^{m+1}\textbf{x}_k,\] where \(\textbf{x}_k = (x_{k,1}, \ldots, x_{k,n_h})\), \[\textbf{x}_i \sim N(\textbf{0},\textbf{Q}_i^{-1}),\] and \(\textbf{Q}_i\) is the precision matrix of \(\textbf{x}_i\), which is given by \[\textbf{Q}_i=\left \{ \begin{array}{lcl} (\textbf{L}-p_i\textbf{C})(\textbf{C}^{-1}\textbf{L})^{\lfloor\alpha\rfloor}/r_i, & & {i = 1,...,m}\\ \textbf{K}^{-1}, & & {i = m+1}\\ \end{array}. \right.\]

We, then, replace the Matérn latent field by the latent vector given above, which has precision matrix given by \[\textbf{Q}=\begin{bmatrix}\textbf{Q}_1& &\\&\ddots&\\& &\textbf{Q}_{m+1}\end{bmatrix}.\] Now, assume we observe \[y_i = u_h(\mathbf{s}_i) + \varepsilon_i,\quad i=1,\ldots, N,\] where \(\varepsilon_i\sim N(0,\sigma_\varepsilon^2)\) are iid measurement noise. Then, we have that \[y_i = u_h(\mathbf{s}_i) + \varepsilon_i = \sum_{j=1}^{n_h} \hat{u}_j \varphi_j(\mathbf{s}_i) + \varepsilon_i = \sum_{k=1}^{m+1} \sum_{j=1}^{n_h} x_{k,j} \varphi(\mathbf{s}_i) + \varepsilon_i.\] This can be written in a matrix form as \[\textbf{y} = \overline{\textbf{A}} \textbf{X} + \boldsymbol{\varepsilon},\] where \(\textbf{y} = [y_1,\ldots,y_N]^\top, \textbf{X} = [\textbf{x}_1^\top,\ldots,\textbf{x}_{m+1}^\top]^\top\), \(\boldsymbol{\varepsilon} = [\varepsilon_1,\ldots,\varepsilon_N]^\top\), \[\overline{\textbf{A}}=\begin{bmatrix}\textbf{A}&\cdots&\textbf{A}\end{bmatrix}_{n\times n_h(m+1)},\] and \[\textbf{A}=\begin{bmatrix}\varphi_1(s_1)&\cdots&\varphi_{n_h}(s_1)\\\vdots&\vdots&\vdots\\\varphi_1(s_n)&\cdots&\varphi_{n_h}(s_n)\end{bmatrix}.\] We then arrive at the following hierarchical model: \[\begin{align} \textbf{y}\mid \textbf{X} &\sim N(0,\sigma_\varepsilon\textbf{I})\\ \textbf{X}&\sim N(0,\textbf{Q}^{-1}) \end{align}.\]

With these elements, we can, for example, use R-INLA to compute the posterior distribution of the three parameters we want to estimate.

Constructing the approximation

In this section, we explain how to to use the function matern.operators() with the default argument type, that is, type="covariance", which is constructs the covariance-based rational approximation. We will also illustrate the usage of several methods and functions related to the covariance-based rational approximation. We will use functions to sample from Gaussian fields with stationary Matérn covariance function, compute the log-likelihood function, and do spatial prediction.

The first step for performing the covariance-based 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 also illustrate how spatial models can be constructed if the FEM implementation of the R-INLA package is used instead. When using the R-INLA package, we also recommend the usage of our R-INLA implementation of the rational SPDE approach. For more details, see the R-INLA implementation of the rational SPDE approach vignette.

We begin by loading the rSPDE package:

library(rSPDE)

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)

We can now use matern.operators() to construct a rational SPDE approximation of order \(m=2\) for a Gaussian random field with a Matérn covariance function on the interval. We also refer the reader to the Operator-based rational approximation for a similar comparison made for the operator-based rational approximation.

kappa <- 20
sigma <- 2
nu <- 0.8
r <- sqrt(8*nu)/kappa #range parameter
op_cov <- matern.operators(loc_mesh = s, nu = nu,
  range = r, sigma = sigma, d = 1, m = 2, parameterization = "matern"
)

The object op_cov 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(). However, as observed in the introduction of this vignette, we have decomposed the stochastic weights \(\boldsymbol{\mathrm{u}}\) into a vector of latent variables. Thus, the \(A\) matrix for the covariance-based rational approximation, which we will denote by \(\overline{A}\), is actually given by the \(m+1\)-fold horizontal concatenation of these \(A\) matrices, where \(m\) is the order of the rational approximation.

To compute the precision matrix from the covariance-based rational approximation one can use the precision() method on the CBrSPDEobj object (the object returned by the matern.operators() function with the default type, which is type="covariance"):

Q <- precision(op_cov)

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. The covariances can be calculated as \[ \overline{\boldsymbol{\mathrm{A}}} \boldsymbol{\mathrm{Q}}^{-1}\overline{\boldsymbol{\mathrm{v}}}. \] Here, \(\boldsymbol{\mathrm{Q}}\) is the precision matrix obtained from the covariance-based rational approximation, \(\boldsymbol{\mathrm{A}}\) is an identity matrix since we are evaluating the approximation in the nodes of the FEM mesh, \(\overline{\boldsymbol{\mathrm{v}}}\) is the \((m+1)\)-fold vertical concatenation of the vector \(\boldsymbol{\mathrm{v}}\), where \(\boldsymbol{\mathrm{v}}\) is a vector with all basis functions evaluated in \(s=0.5\).

v <- t(rSPDE.A1d(s, 0.5))
v_bar <- kronecker(matrix(1, nrow = 3), v)
A <- Diagonal(101)
A_bar <- kronecker(matrix(1, ncol = 3), A)
c_cov.approx <- (A_bar) %*% solve(Q, v_bar)

Let us now compute the true Matérn covariance function on the interval \((0,1)\), 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.

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_cov.approx, col = 2)
legend("topright",
  bty = "n",
  legend = c("Matérn", "Rational"),
  col = c("black", "red"),
  lty = rep(1, 2), ncol = 1,
  cex = 0.8
)

plot(s, c.true - c_cov.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, for example, compute the approximation with \(m=4\) using the same mesh, as well as the approximation when we increase the number of basis functions and use \(m=2\) and \(m=4\).

op_cov2 <- matern.operators(
  range = r, sigma = sigma, nu = nu,
  loc_mesh = s, d = 1, m = 4,
  parameterization = "matern"
)
Q2 <- precision(op_cov2)
v_bar2 <- kronecker(matrix(1, nrow = 5), v)
A_bar2 <- kronecker(matrix(1, ncol = 5), A)
c_cov.approx2 <- (A_bar2) %*% solve(Q2, v_bar2)

s2 <- seq(from = 0, to = 1, length.out = 501)
op_cov <- matern.operators(
  range = r, sigma = sigma, nu = nu,
  loc_mesh = s2, d = 1, m = 2,
  parameterization = "matern"
)
Q3 <- precision(op_cov)
A2 <- rSPDE.A1d(s2, s)
v2 <- t(rSPDE.A1d(s2, 0.5))
v2_bar <- kronecker(matrix(1, nrow = 3), v2)
A2_bar <- kronecker(matrix(1, ncol = 3), A2)
c_cov.approx3 <- (A2_bar) %*% solve(Q3, v2_bar)

op_cov <- matern.operators(
  range = r, sigma = sigma, nu = nu,
  loc_mesh = s2, d = 1, m = 4,
  parameterization = "matern"
)
Q4 <- precision(op_cov)
v2_bar2 <- kronecker(matrix(1, nrow = 5), v2)
A2_bar2 <- kronecker(matrix(1, ncol = 5), A2)
c_cov.approx4 <- (A2_bar2) %*% solve(Q4, v2_bar2)

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_cov.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_cov.approx2, col = 2)
lines(s, c.true - c_cov.approx3, col = 3)
lines(s, c.true - c_cov.approx4, col = 4)
legend("bottomright",
  bty = "n",
  legend = c("m=2 coarse mesh", "m=4 coarse mesh", "m=2 fine mesh", "m=4 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 in general rarely a need for an approximation with a large value of \(m\). This is good because the size of \(\boldsymbol{\mathrm{Q}}\) increases with \(m\), which makes the approximation more computationally costly to use. 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_cov <- matern.operators(
    range = r, sigma = sigma, nu = nu,
    loc_mesh = s2, d = 1, m = i,
    parameterization = "matern"
  )
  Q <- precision(op_cov)
  v_bar <- kronecker(matrix(1, nrow = i + 1), v2)
  A_bar <- kronecker(matrix(1, ncol = i + 1), A2)
  c_cov.approx <- (A_bar) %*% solve(Q, v_bar)
  errors[i] <- norm(c.true - c_cov.approx)
}
print(errors)

## [1] 0.977500618 0.086659189 0.017335545 0.008432137

We see that the error decreases very fast when we increase \(m\) from \(1\) to \(4\), without any numerical instability. This is an advantage of the covariance-based rational approximation when compared to the operator-based rational approximation. See Operator-based rational approximation for details on the numerical instability of the operator-based rational approximation.

Using the approximation

When we use the function matern.operators(), we can simulate from the model using the simulate() method. To such an end we simply apply the simulate() method to the object returned by the matern.operators() function:

u <- simulate(op_cov)

If we want replicates, we simply set the argument nsim to the desired number of replicates. For instance, to generate two replicates of the model, we simply do:

u.rep <- simulate(op_cov, nsim = 2)

set.seed(1)
s <- seq(from = 0, to = 1, length.out = 501)
n.obs <- 200
obs.loc <- runif(n.obs)
A <- rSPDE.A1d(s, obs.loc)

kappa <- 20
sigma <- 1.3
nu <- 0.8
r <- sqrt(8*nu)/kappa
op_cov <- matern.operators(
  loc_mesh = s, nu = nu,
  range = r, sigma = sigma, d = 1, m = 2,
  parameterization = "matern"
)

u <- simulate(op_cov)

sigma.e <- 0.3

x1 <- obs.loc

Y <- 2 - x1 + as.vector(A %*% u + sigma.e * rnorm(n.obs))

df_data <- data.frame(y = Y, loc = obs.loc, x1 = x1)

op_cov_est <- matern.operators(
  loc_mesh = s, d = 1, m = 2
)

fit <- rspde_lme(y~x1, model = op_cov_est,
                    data = df_data, loc = "loc")

## Warning in sqrt(diag(inv_fisher)): NaNs produced

summary(fit)

## 
## Latent model - Whittle-Matern
## 
## Call:
## rspde_lme(formula = y ~ x1, loc = "loc", data = df_data, model = op_cov_est)
## 
## Fixed effects:
##             Estimate Std.error z-value Pr(>|z|)
## (Intercept)   1.2753    1.1979   1.065    0.287
## x1           -0.5042    2.0595  -0.245    0.807
## 
## Random effects:
##       Estimate Std.error z-value
## alpha  1.34266       NaN     NaN
## tau    0.03817       NaN     NaN
## kappa 16.42896   4.76648   3.447
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu      0.8427       NaN     NaN
## sigma   1.4741    0.3208   4.595
## range   0.1580    0.0495   3.193
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev  0.33071   0.02211   14.96
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -145.9037 
## Number of function calls by 'optim' = 99
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  43.10063 secs

print(data.frame(
  sigma = c(sigma, fit$matern_coeff$random_effects[2]), 
  range = c(r, fit$matern_coeff$random_effects[3]),
  nu = c(nu, fit$matern_coeff$random_effects[1]),
  row.names = c("Truth", "Estimates")
))

##             sigma     range        nu
## Truth     1.30000 0.1264911 0.8000000
## Estimates 1.47409 0.1580380 0.8426612

# Total time
print(fit$fitting_time)

## Time difference of 43.10063 secs

fit_par <- rspde_lme(y~x1, model = op_cov_est,
                    data = df_data, loc = "loc", parallel = TRUE)

summary(fit_par)

## 
## Latent model - Whittle-Matern
## 
## Call:
## rspde_lme(formula = y ~ x1, loc = "loc", data = df_data, model = op_cov_est, 
##     parallel = TRUE)
## 
## Fixed effects:
##             Estimate Std.error z-value Pr(>|z|)
## (Intercept)   1.2753    1.1979   1.065    0.287
## x1           -0.5042    2.0595  -0.245    0.807
## 
## Random effects:
##       Estimate Std.error z-value
## alpha  1.34266       NaN     NaN
## tau    0.03817       NaN     NaN
## kappa 16.42896   4.76648   3.447
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu      0.8427       NaN     NaN
## sigma   1.4741    0.3208   4.595
## range   0.1580    0.0495   3.193
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev  0.33071   0.02211   14.96
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -145.9037 
## Number of function calls by 'optim' = 99
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  23.80119 secs 
##   set up the parallelization = 1.73388 secs

print(data.frame(
  sigma = c(sigma, fit_par$matern_coeff$random_effects[2]), range = c(r, fit_par$matern_coeff$random_effects[3]),
  nu = c(nu, fit_par$matern_coeff$random_effects[1]),
  row.names = c("Truth", "Estimates")
))

##             sigma     range        nu
## Truth     1.30000 0.1264911 0.8000000
## Estimates 1.47409 0.1580380 0.8426612

# Total time (time to fit plus time to set up the parallelization)
total_time <- fit_par$fitting_time + fit_par$time_par
print(total_time)

## Time difference of 25.53508 secs

df_pred <- data.frame(loc = s, x1 = s)

u.krig <- predict(fit, data = df_pred, loc = "loc")

opar <- par(mgp = c(1.3, 0.5, 0), mar = c(2, 2, 0.5, 0.5) + 0.1)
plot(obs.loc, Y,
  ylab = "u(s)", xlab = "s",
  ylim = c(min(c(min(u), min(Y))), max(c(max(u), max(Y)))),
  cex.main = 0.8, cex.axis = 0.8, cex.lab = 0.8
)
lines(s, u.krig$mean, col = 2)
par(opar)

Fitting a model with replicates

Let us illustrate how to simulate a dataset with replicates and then fit a model to such data. Recall that to simulate a latent model with replicates, all we do is set the nsim argument to the number of replicates.

We will use the CBrSPDEobj object (returned from the matern.operators() function) from the previous example, namely op_cov.

set.seed(123)
n.rep <- 20
u.rep <- simulate(op_cov, nsim = n.rep)

Now, let us generate the observed values \(Y\):

sigma.e <- 0.3
Y.rep <- A %*% u.rep + sigma.e * matrix(rnorm(n.obs * n.rep), ncol = n.rep)

Note that \(Y\) is a matrix with 20 columns, each column containing one replicate. We need to turn y into a vector and create an auxiliary vector repl indexing the replicates of y:

y_vec <- as.vector(Y.rep)
repl <- rep(1:n.rep, each = n.obs)

df_data_repl  <- data.frame(y = y_vec, loc = rep(obs.loc, n.rep))

We can now fit the model in the same way as before by using the rspde_lme() function:

fit_repl <- rspde_lme(y_vec ~ -1, model = op_cov_est, repl = repl, 
      data = df_data_repl, loc = "loc", parallel = TRUE)

Let us see a summary of the fit:

summary(fit_repl)

## 
## Latent model - Whittle-Matern
## 
## Call:
## rspde_lme(formula = y_vec ~ -1, loc = "loc", data = df_data_repl, 
##     model = op_cov_est, repl = repl, parallel = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##        Estimate Std.error z-value
## alpha  1.222484  0.008837  138.33
## tau    0.061557  0.002829   21.76
## kappa 17.582448  1.227953   14.32
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.722484  0.008837   81.75
## sigma 1.279659  0.053954   23.72
## range 0.136735  0.009331   14.65
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev  0.29909   0.00491   60.91
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -2757.298 
## Number of function calls by 'optim' = 71
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  1.55411 mins 
##   set up the parallelization = 1.7297 secs

Let us compare with the true values:

print(data.frame(
  sigma = c(sigma, fit_repl$matern_coeff$random_effects[2]), 
  range = c(r, fit_repl$matern_coeff$random_effects[3]),
  nu = c(nu, fit_repl$matern_coeff$random_effects[1]),
  row.names = c("Truth", "Estimates")
))

##              sigma     range        nu
## Truth     1.300000 0.1264911 0.8000000
## Estimates 1.279659 0.1367350 0.7224839

# Total time
print(fit_repl$fitting_time)

## Time difference of 1.554118 mins

We can obtain better estimates of the Hessian by setting improve_hessian to TRUE, however this might make the process take longer:

fit_repl2 <- rspde_lme(y_vec ~ -1, model = op_cov_est, repl = repl, 
      data = df_data_repl, loc = "loc", parallel = TRUE, 
      improve_hessian = TRUE)

Let us get a summary:

summary(fit_repl2)

## 
## Latent model - Whittle-Matern
## 
## Call:
## rspde_lme(formula = y_vec ~ -1, loc = "loc", data = df_data_repl, 
##     model = op_cov_est, repl = repl, parallel = TRUE, improve_hessian = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##        Estimate Std.error z-value
## alpha  1.222484  0.034505  35.429
## tau    0.061557  0.009594   6.417
## kappa 17.582448  1.588939  11.066
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.722484  0.034505   20.94
## sigma 1.279659  0.053954   23.72
## range 0.136735  0.009331   14.65
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev 0.299094  0.005145   58.13
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -2757.298 
## Number of function calls by 'optim' = 71
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  1.40495 mins 
##   compute the Hessian = 19.14015 secs 
##   set up the parallelization = 1.84044 secs

Spatial data and parameter estimation

The functions used in the previous examples also work for spatial models. We then need to construct a mesh over the domain of interest and then compute the matrices needed to define the operator. These tasks can be performed, for example, using the R-INLA package. Let us start by defining a mesh over \([0,1]\times [0, 1]\) and compute the mass and stiffness matrices for that mesh.

It is important to mention that when using the R-INLA package we recommend the usage of the R-INLA implementation of the rational SPDE approach. The purpose of this section is to show that we can estimate the model without R-INLA and using a maximum likelihood approach instead.

We consider a simple Gaussian linear model with 30 independent replicates of a latent spatial field \(u(\mathbf{s})\), observed at the same \(m\) locations, \(\{\mathbf{s}_1 , \ldots , \mathbf{s}_m \}\), for each replicate. For each \(i = 1,\ldots,m,\) we have

\[\begin{align} y_i &= u_1(\mathbf{s}_i)+\varepsilon_i,\\ \vdots &= \vdots\\ y_{i+29m} &= u_{30}(\mathbf{s}_i) + \varepsilon_{i+29m}, \end{align}\]

where \(\varepsilon_1,\ldots,\varepsilon_{30m}\) are iid normally distributed with mean 0 and standard deviation 0.1.

Let us create the FEM mesh:

library(INLA)
n_loc <- 500
loc_2d_mesh <- matrix(runif(n_loc * 2), n_loc, 2)
mesh_2d <- inla.mesh.2d(
  loc = loc_2d_mesh,
  cutoff = 0.05,
  offset = c(0.1, 0.4),
  max.edge = c(0.05, 0.5)
)
plot(mesh_2d, main = "")
points(loc_2d_mesh[, 1], loc_2d_mesh[, 2])

We can now use this mesh to define a rational SPDE approximation of order \(m=2\) for a Matérn model in the same fashion as we did above in the one-dimensional case. We now simulate a latent process with standard deviation \(\sigma=1\) and range \(0.1\). We will use \(\nu=0.5\) so that the model has an exponential covariance function. To this end we create a model object with the matern.operators() function:

nu <- 0.7
sigma <- 1.3
range <- 0.15
d <- 2
op_cov_2d <- matern.operators(
  mesh = mesh_2d,
  nu = nu,
  range = range,
  sigma = sigma,
  m = 2,
  parameterization = "matern"
)
tau <- op_cov_2d$tau

Now let us simulate some noisy data that we will use to estimate the parameters of the model. To construct the observation matrix, we use the R-INLA function inla.spde.make.A(). Recall that we will simulate the data with 30 replicates.

n.rep <- 30
u <- simulate(op_cov_2d, nsim = n.rep)
A <- inla.spde.make.A(
  mesh = mesh_2d,
  loc = loc_2d_mesh
)
sigma.e <- 0.1
Y <- A %*% u + matrix(rnorm(n_loc * n.rep), ncol = n.rep) * sigma.e

The first replicate of the simulated random field as well as the observation locations are shown in the following figure.

library(viridis)
library(ggplot2)
proj <- inla.mesh.projector(mesh_2d, dims = c(70, 70))

df_field <- data.frame(x = proj$lattice$loc[,1],
                        y = proj$lattice$loc[,2],
                        field = as.vector(inla.mesh.project(proj, 
                        field = as.vector(u[, 1]))),
                        type = "field")

df_loc <- data.frame(x = loc_2d_mesh[, 1],
                      y = loc_2d_mesh[, 2],
                      field = as.vector(Y[,1]),
                      type = "locations")
df_plot <- rbind(df_field, df_loc)

ggplot(df_plot) + aes(x = x, y = y, fill = field) +
        facet_wrap(~type) + xlim(0,1) + ylim(0,1) + 
        geom_raster(data = df_field) +
        geom_point(data = df_loc, aes(colour = field),
        show.legend = FALSE) + 
        scale_fill_viridis() + scale_colour_viridis()

Let us now create a new object to fit the model:

op_cov_2d_est <- matern.operators(
  mesh = mesh_2d,
  m = 2
)

We can now proceed as in the previous cases. We set up a vector with the response variables and create an auxiliary replicates vector, repl, that contains the indexes of the replicates of each observation, and then we fit the model:

y_vec <- as.vector(Y)
repl <- rep(1:n.rep, each = n_loc)
df_data_2d <- data.frame(y = y_vec, x_coord = loc_2d_mesh[,1],
                      y_coord = loc_2d_mesh[,2])

fit_2d <- rspde_lme(y ~ -1, model = op_cov_2d_est, 
          data = df_data_2d, repl = repl,
          loc = c("x_coord", "y_coord"), parallel = TRUE)

Let us get a summary:

summary(fit_2d)

## 
## Latent model - Whittle-Matern
## 
## Call:
## rspde_lme(formula = y ~ -1, loc = c("x_coord", "y_coord"), data = df_data_2d, 
##     model = op_cov_2d_est, repl = repl, parallel = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##        Estimate Std.error z-value
## alpha  1.594423  0.005331  299.08
## tau    0.056567  0.001354   41.79
## kappa 14.077540  0.437694   32.16
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.594423  0.005331  111.50
## sigma 1.342997  0.014343   93.63
## range 0.154905  0.004760   32.55
## 
## Measurement error:
##           Estimate Std.error z-value
## std. dev 0.1003547 0.0008783   114.3
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -5669.867 
## Number of function calls by 'optim' = 30
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  3.60441 mins 
##   set up the parallelization = 1.72729 secs

Let us compare the estimated results with the true values:

print(data.frame(
  sigma = c(sigma, fit_2d$matern_coeff$random_effects[2]), 
  range = c(range, fit_2d$matern_coeff$random_effects[3]),
  nu = c(nu, fit_2d$matern_coeff$random_effects[1]),
  row.names = c("Truth", "Estimates")
))

##              sigma     range        nu
## Truth     1.300000 0.1500000 0.7000000
## Estimates 1.342997 0.1549051 0.5944225

# Total time
print(fit_2d$fitting_time)

## Time difference of 3.604419 mins

An example with a non-stationary model

Our goal now is to show how one can fit model with non-stationary \(\sigma\) (std. deviation) and non-stationary \(\rho\) (a range parameter). One can also use the parameterization in terms of non-stationary SPDE parameters \(\kappa\) and \(\tau\).

For this example we will consider simulated data.

rec_domain <- cbind(c(0, 1, 1, 0, 0) * 10, c(0, 0, 1, 1, 0) * 5)

mesh <- inla.mesh.2d(loc.domain = rec_domain, cutoff = 0.1, 
  max.edge = c(0.5, 1.5), offset = c(0.5, 1.5))

nu <- 0.8
true_theta <- c(0,1, 1)
B.sigma = cbind(0, 1, 0, (mesh$loc[,1] - 5) / 10)
B.range = cbind(0, 0, 1, (mesh$loc[,1] - 5) / 10)
alpha <- nu + 1 # nu + d/2 ; d = 2

# SPDE model
op_cov_ns <- spde.matern.operators(mesh = mesh, 
  theta = true_theta,
  nu = nu,
  B.sigma = B.sigma, 
  B.range = B.range,
  parameterization = "matern")

u <- as.vector(simulate(op_cov_ns, seed = 123))

m <-600
loc_mesh <- cbind(runif(m) * 10, runif(m) * 5)

A <- inla.spde.make.A(
  mesh = mesh,
  loc = loc_mesh
)

y <- as.vector(A %*% as.vector(u)) + rnorm(m) * 0.1

op_cov_ns_est <- op_cov_ns <- spde.matern.operators(mesh = mesh, 
  B.sigma = B.sigma, 
  B.range = B.range,
  parameterization = "matern")

df_data_ns <- data.frame(y= y, x_coord = loc_mesh[,1], y_coord = loc_mesh[,2])

fit_ns <- rspde_lme(y ~ -1, model = op_cov_ns_est, 
          data = df_data_ns, loc = c("x_coord", "y_coord"), 
          parallel = TRUE)

summary(fit_ns)

## 
## Latent model - Generalized Whittle-Matern
## 
## Call:
## rspde_lme(formula = y ~ -1, loc = c("x_coord", "y_coord"), data = df_data_ns, 
##     model = op_cov_ns_est, parallel = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##         Estimate Std.error z-value
## alpha   1.913838  0.065086  29.405
## Theta 1 0.007294  0.171096   0.043
## Theta 2 1.127656  0.266190   4.236
## Theta 3 1.572143  0.454988   3.455
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev -2.29871   0.05383  -42.71
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -103.7192 
## Number of function calls by 'optim' = 66
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  21.80644 secs 
##   set up the parallelization = 1.64565 secs

print(data.frame(
  theta1 = c(true_theta[1], fit_ns$coeff$random_effects[2]), 
  theta2 = c(true_theta[2], fit_ns$coeff$random_effects[3]),
  theta3 = c(true_theta[3], fit_ns$coeff$random_effects[4]), 
  alpha = c(alpha, fit_ns$coeff$random_effects[1])),
  row.names = c("Truth", "Estimates")
)

##                theta1   theta2   theta3    alpha
## Truth     0.000000000 1.000000 1.000000 1.800000
## Estimates 0.007293681 1.127656 1.572143 1.913838

Changing the type and the order of the rational approximation

We have three rational approximations available. The BRASIL algorithm Hofreither (2021), and two “versions” of the Clenshaw-Lord Chebyshev-Pade algorithm, one with lower bound zero and another with the lower bound given in Bolin, Simas, and Xiong (2023).

The type of rational approximation can be chosen by setting the type_rational_approximation argument in the matern.operators function. The BRASIL algorithm corresponds to the choice brasil, the Clenshaw-Lord Chebyshev pade with zero lower bound and non-zero lower bounds are given, respectively, by the choices chebfun and chebfunLB.

For instance, we can create an rSPDE object with a chebfunLB rational approximation by

op_cov_2d_type <- matern.operators(
  mesh = mesh_2d,
  m = 2,
  type_rational_approximation = "chebfunLB"
)
tau <- op_cov_2d_type$tau

We can check the order of the rational approximation with the rational.order() function and assign a new order with the rational.order<-() function:

rational.order(op_cov_2d_type)

## [1] 2

rational.order(op_cov_2d_type) <- 1

Let us fit a model using the data from the previous example:

fit_order1 <- rspde_lme(y ~ -1, model = op_cov_2d_type, 
          data = df_data_2d,repl = repl,
          loc = c("x_coord", "y_coord"), parallel = TRUE)

summary(fit_order1)

## 
## Latent model - Whittle-Matern
## 
## Call:
## rspde_lme(formula = y ~ -1, loc = c("x_coord", "y_coord"), data = df_data_2d, 
##     model = op_cov_2d_type, repl = repl, parallel = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##       Estimate Std.error z-value
## alpha  1.50787   0.04760  31.680
## tau    0.07456   0.01328   5.614
## kappa 13.77272   0.76605  17.979
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.507873  0.047596   10.67
## sigma 1.401220  0.014924   93.89
## range 0.146353  0.004591   31.88
## 
## Measurement error:
##           Estimate Std.error z-value
## std. dev 0.1003261 0.0008778   114.3
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -5669.386 
## Number of function calls by 'optim' = 43
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  2.31786 mins 
##   set up the parallelization = 1.63354 secs

Let us compare with the true values:

print(data.frame(
  sigma = c(sigma, fit_order1$matern_coeff$random_effects[2]), 
  range = c(range, fit_order1$matern_coeff$random_effects[3]),
  nu = c(nu, fit_order1$matern_coeff$random_effects[1]),
  row.names = c("Truth", "Estimates")
))

##             sigma     range        nu
## Truth     1.30000 0.1500000 0.8000000
## Estimates 1.40122 0.1463534 0.5078728

Finally, we can check the type of rational approximation with the rational.type() function and assign a new type by using the rational.type<-() function:

rational.type(op_cov_2d_type)

## [1] "chebfunLB"

rational.type(op_cov_2d_type) <- "brasil"

Let us now fit this model, with the data from the previous example, with brasil rational approximation:

fit_brasil <- rspde_lme(y ~ -1, model = op_cov_2d_type, 
          data = df_data_2d,repl = repl,
          loc = c("x_coord", "y_coord"), parallel = TRUE)

summary(fit_brasil)

## 
## Latent model - Whittle-Matern
## 
## Call:
## rspde_lme(formula = y ~ -1, loc = c("x_coord", "y_coord"), data = df_data_2d, 
##     model = op_cov_2d_type, repl = repl, parallel = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##        Estimate Std.error z-value
## alpha  1.493997  0.003203  466.47
## tau    0.080390  0.001073   74.94
## kappa 13.500477  0.380314   35.50
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.493997  0.003203  154.24
## sigma 1.380200  0.014911   92.56
## range 0.147251  0.004147   35.51
## 
## Measurement error:
##           Estimate Std.error z-value
## std. dev 0.1003686 0.0008785   114.2
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -5669.887 
## Number of function calls by 'optim' = 38
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  2.06148 mins 
##   set up the parallelization = 1.65444 secs

Let us compare with the true values:

print(data.frame(
  sigma = c(sigma, fit_brasil$matern_coeff$random_effects[2]), 
  range = c(range, fit_brasil$matern_coeff$random_effects[3]),
  nu = c(nu, fit_brasil$matern_coeff$random_effects[1]),
  row.names = c("Truth", "Estimates")
))

##            sigma     range        nu
## Truth     1.3000 0.1500000 0.8000000
## Estimates 1.3802 0.1472509 0.4939965