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Rational approximation with the rSPDE package

David Bolin, Alexandre B. Simas, Zhen Xiong

Created: 2021-12-04. Last modified: 2024-11-14.

Source: vignettes/rspde_cov.Rmd
rspde_cov.Rmd

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 idea behind the approach. In the SPDE approach, introduced in Lindgren, Rue, and Lindström (2011) we model uu as the solution of the following SPDE: Lα/2(τu)=W, L^{\alpha/2}(\tau u) = W, where L=−Δ+κ2IL = -\Delta +\kappa^2 I and WW is the standard Gaussian white noise. Here, α\alpha, κ\kappa and τ\tau are the parameters of the model. In the standard SPDE approach, α=ν+d/2\alpha = \nu + d/2 needs to be fixed to an integer value, where α=2\alpha = 2 is the usual default value. In the rational SPDE approach we can use any value of ν>0\nu>0 and also estimate it from data.

The main idea of the covariance-based rational SPDE approach is to perform the rational approximation of the covariance operator L−αL^{-\alpha}. To this end, we begin by obtaining an approximation of the random field uu, which is the solution of the SPDE above, by using the finite element method (FEM): uh(𝐬i)=∑j=1nhûjφj(𝐬i), u_h(\mathbf{s}_i)=\sum_{j=1}^{n_h} \hat{u}_j \varphi_j(\mathbf{s}_i), where {ûj}j=1nh\{\hat{u}_j\}_{j = 1}^{n_h} are stochastic weights and {φj(𝐬i)}j=1nh\{\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 LL, which is given by LhL_h, and the covariance operator of uhu_h is given by Lh−αL_h^{-\alpha}.

Now, by using the rational approximation on LhL_h, we can approximate covariance operator Lh−αL_h^{-\alpha} as Lh,m−α=Lh−⌊α⌋p(Lh−1)q(Lh−1)−1,L_{h,m}^{-\alpha} = L_h^{-\lfloor\alpha\rfloor} p(L_h^{-1})q(L_h^{-1})^{-1}, where ⌊α⌋\lfloor\alpha\rfloor denotes the integer part of α\alpha, mm is the order of rational approximation, p(Lh−1)=∑i=0maiLhm−ip(L_h^{-1}) = \sum_{i=0}^m a_i L_h^{m-i} and q(Lh−1)=∑j=0mbjLhm−iq(L_h^{-1}) = \sum_{j=0}^m b_j L_h^{m-i}, with {ai}i=0m\{a_i\}_{i = 0}^m and {bj}j=0m\{b_j\}_{j = 0}^m being known coefficients obtained from a rational approximation of the function xα−⌊α⌋x^{\alpha - \lfloor\alpha\rfloor}.

The next step is to perform a partial fraction decomposition of the rational function p(Lh−1)q(Lh−1)−1p(L_h^{-1})q(L_h^{-1})^{-1}, which yields the representation Lh,m−α=Lh−⌊α⌋(∑i=1mri(Lh−piI)−1+k).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 𝐮̂\hat{\textbf{u}}, where 𝐮̂=[û1,...,ûnh]⊤\hat{\textbf{u}}=[\hat{u}_1,...,\hat{u}_{n_h}]^\top, as 𝚺𝐮̂=(𝐋−1𝐂)⌊α⌋∑i=1mri(𝐋−pi𝐂)−1+𝐊,\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 𝐂={Cij}i,j=1nh\textbf{C} = \{C_{ij}\}_{i,j=1}^{n_h}, Cij=(φi,φj)L2(𝒟)C_{ij} = (\varphi_i,\varphi_j)_{L_2(\mathcal{D})}, is the mass matrix, 𝐋=κ2𝐂+𝐆\textbf{L} = \kappa^2\textbf{C}+\textbf{G}, 𝐆={Gij}i,j=1nh\textbf{G} = \{G_{ij}\}_{i,j=1}^{n_h}, Gij=(∇φi,∇φj)L2(𝒟)G_{ij}=(\nabla\varphi_i,\nabla\varphi_j)_{L_2(\mathcal{D})}, is the stiffness matrix, and 𝐊={k𝐂⌊α⌋=0k𝐋−1(𝐂𝐋−1)⌊α⌋−1⌊α⌋≥1. \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 𝐮̂\hat{\textbf{u}} as 𝐮̂=∑i=1m+1𝐱i, \hat{\textbf{u}}=\sum_{i=1}^{m+1}\textbf{x}_i, where 𝐱i=(xi,1,…,xi,nh)\textbf{x}_i = (x_{i,1}, \ldots, x_{i,n_h}), 𝐱i∼N(𝟎,𝐐i−1),\textbf{x}_i \sim N(\textbf{0},\textbf{Q}_i^{-1}), and 𝐐i\textbf{Q}_i is the precision matrix of 𝐱i\textbf{x}_i, which is given by 𝐐i={(𝐋−pi𝐂)(𝐂−1𝐋)⌊α⌋/ri,i=1,...,m𝐊−1,i=m+1. \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 𝐐=[𝐐1⋱𝐐m+1].\textbf{Q}=\begin{bmatrix}\textbf{Q}_1& &\\&\ddots&\\& &\textbf{Q}_{m+1}\end{bmatrix}. We thus have a Markov approximation which can be used for computationally efficient inference. For example, assume that we observe yj=uh(𝐬j)+εj,j=1,…,N,y_j = u_h(\mathbf{s}_j) + \varepsilon_j,\quad j=1,\ldots, N, where εj∼N(0,σε2)\varepsilon_j\sim N(0,\sigma_\varepsilon^2) are iid measurement noise. Then, we have that yj=uh(𝐬j)+εj=∑l=1nhûlφl(𝐬j)+εj=∑i=1m+1∑l=1nhxi,lφl(𝐬j)+εj. y_j = u_h(\mathbf{s}_j) + \varepsilon_j = \sum_{l=1}^{n_h} \hat{u}_l \varphi_l(\mathbf{s}_j) + \varepsilon_j = \sum_{i=1}^{m+1} \sum_{l=1}^{n_h} x_{i,l} \varphi_l(\mathbf{s}_j) + \varepsilon_j. This can be written in a matrix form as 𝐲=𝐀¯𝐗+𝛆,\textbf{y} = \overline{\textbf{A}} \textbf{X} + \boldsymbol{\varepsilon}, where 𝐲=[y1,…,yN]⊤,𝐗=[𝐱1⊤,…,𝐱m+1⊤]⊤\textbf{y} = [y_1,\ldots,y_N]^\top, \textbf{X} = [\textbf{x}_1^\top,\ldots,\textbf{x}_{m+1}^\top]^\top, 𝛆=[ε1,…,εN]⊤\boldsymbol{\varepsilon} = [\varepsilon_1,\ldots,\varepsilon_N]^\top, 𝐀¯=[𝐀⋯𝐀]n×nh(m+1),\overline{\textbf{A}}=\begin{bmatrix}\textbf{A}&\cdots&\textbf{A}\end{bmatrix}_{n\times n_h(m+1)}, and 𝐀=[φ1(s1)⋯φnh(s1)⋮⋮⋮φ1(sn)⋯φnh(sn)].\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: 𝐲∣𝐗∼N(0,σε𝐈)𝐗∼N(0,𝐐−1).\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. We will also illustrate how spatial models can be constructed if the FEM implementation of the fmesher package is used. 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:

Assume that we want to define a model on the interval [0,1][0,1]. We then start by defining a vector with mesh nodes sis_i where the basis functions φi\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=2m=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 u\boldsymbol{\mathrm{u}}. If we want to evaluate uh(s)u_h(s) at some locations s1,…,sns_1,\ldots, s_n, we need to multiply the weights with the basis functions φi(s)\varphi_i(s) evaluated at the locations. For this, we can construct the observation matrix A\boldsymbol{\mathrm{A}}, with elements Aij=φj(si)A_{ij} = \varphi_j(s_i), which links the FEM basis functions to the locations. This matrix can be constructed using the function fm_basis() from the fmesher package. However, as observed in the introduction of this vignette, we have decomposed the stochastic weights u\boldsymbol{\mathrm{u}} into a vector of latent variables. Thus, the AA matrix for the covariance-based rational approximation, which we will denote by A¯\overline{A}, is actually given by the m+1m+1-fold horizontal concatenation of these AA matrices, where mm is the order of the rational approximation.

To evaluate the accuracy of the approximation, let us compute the covariance function between the process at s=0.5s=0.5 and all other locations in s and compare with the true Matérn covariance function. The covariances can be calculated by using the cov_function_mesh() method. 

c_cov.approx <- op_cov$cov_function_mesh(0.5)

Let us now compute the true Matérn covariance function on the interval (0,1)(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 mm, and/or increase the number of basis functions used for the FEM approximation. Let us, for example, compute the approximation with m=4m=4 using the same mesh, as well as the approximation when we increase the number of basis functions and use m=2m=2 and m=4m=4. We will also load the fmesher package to use the fm_basis() and fm_mesh_1d() functions to map between the meshes.

library(fmesher)

op_cov2 <- matern.operators(
  range = r, sigma = sigma, nu = nu,
  loc_mesh = s, d = 1, m = 4,
  parameterization = "matern"
)

c_cov.approx2 <- op_cov2$cov_function_mesh(0.5)

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"
)

mesh_s2 <- fm_mesh_1d(s2)

# Map the mesh s2 to s
A2 <- fm_basis(mesh_s2, s)

c_cov.approx3 <- A2 %*% op_cov$cov_function_mesh(0.5)

op_cov <- matern.operators(
  range = r, sigma = sigma, nu = nu,
  loc_mesh = s2, d = 1, m = 4,
  parameterization = "matern"
)

c_cov.approx4 <- A2 %*% op_cov$cov_function_mesh(0.5)

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 mm, there is in general rarely a need for an approximation with a large value of mm. This is good because the size of Q\boldsymbol{\mathrm{Q}} increases with mm, which makes the approximation more computationally costly to use. To illustrate this, let us compute the norm of the approximation error for different mm.

# Mapping s2 to s
A2 <- fm_basis(mesh_s2, s)

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"
  )
  c_cov.approx <- A2 %*% op_cov$cov_function_mesh(0.5)
  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 mm from 11 to 44, 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)

Fitting a model

There is built-in support for computing log-likelihood functions and performing kriging prediction in the rSPDE package. To illustrate this, we use the simulation to create some noisy observations of the process. For this, we first construct the observation matrix linking the FEM basis functions to the locations where we want to simulate. We first randomly generate some observation locations and then construct the matrix.

set.seed(1)
s <- seq(from = 0, to = 1, length.out = 501)
n.obs <- 200
obs.loc <- runif(n.obs)
mesh_s <- fm_mesh_1d(s)
A <- fm_basis(x = mesh_s, loc = obs.loc)

We now generate the observations as Yi=2−x1+u(si)+εiY_i = 2 - x1 + u(s_i) + \varepsilon_i, where εi∼N(0,σe2)\varepsilon_i \sim N(0,\sigma_e^2) is Gaussian measurement noise, x1x1 is a covariate giving the observation location. We will assume that the latent process has a Matérn covariance with κ=20,σ=1.3\kappa=20, \sigma=1.3 and ν=0.8\nu=0.8:

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)

Let us create a new object to fit the model:

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

Let us now fit the model. To this end we will use the rspde_lme() function:

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

We can get a summary of the fit with the summary() method:

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)   1.2977    1.1935   1.087    0.277
## x1           -0.5586    2.0517  -0.272    0.785
## 
## Random effects:
##        Estimate Std.error z-value
## alpha  1.342557  0.053412  25.136
## tau    0.038168  0.009482   4.025
## kappa 16.477711  5.252625   3.137
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu     0.84256   0.05341  15.775
## sigma  1.47083   0.31851   4.618
## range  0.15756   0.04912   3.208
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev  0.33075   0.02231   14.83
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -145.9036 
## Number of function calls by 'optim' = 111
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  21.22376 secs

Let us compare the parameters of the latent model:

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     1.300000 0.1264911 0.8000000
## Estimates 1.470833 0.1575606 0.8425568
# Total time
print(fit$fitting_time)
## Time difference of 21.22377 secs

Let us take a glance at the fit:

glance(fit)
## # A tibble: 1 × 9
##    nobs sigma logLik   AIC   BIC deviance df.residual model                alpha
##   <int> <dbl>  <dbl> <dbl> <dbl>    <dbl>       <dbl> <chr>                <dbl>
## 1   200 0.331  -146.  304.  324.     292.         194 Covariance-Based Ma…  1.34

We can also speed up the optimization by setting parallel=TRUE (which uses implicitly the optimParallel function):

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

Here is the summary:

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)   1.2977    1.1935   1.087    0.277
## x1           -0.5586    2.0517  -0.272    0.785
## 
## Random effects:
##        Estimate Std.error z-value
## alpha  1.342557  0.053412  25.136
## tau    0.038168  0.009482   4.025
## kappa 16.477711  5.252625   3.137
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu     0.84256   0.05341  15.775
## sigma  1.47083   0.31851   4.618
## range  0.15756   0.04912   3.208
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev  0.33075   0.02231   14.83
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -145.9036 
## Number of function calls by 'optim' = 111
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  14.17469 secs 
##   set up the parallelization = 2.26773 secs

Let us compare with the true values and compare the time:

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     1.300000 0.1264911 0.8000000
## Estimates 1.470833 0.1575606 0.8425568
# 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 16.44243 secs

Kriging

Finally, we compute the kriging prediction of the process uu at the locations in s based on these observations.

Let us create the data.frame with locations in which we want to obtain the predictions. Observe that we also must provide the values of the covariates.

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

We can now perform kriging with the predict() method:

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

The simulated process, the observed data, and the kriging prediction 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(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)

We can also use the augment() function and pipe the results into a plot:

library(ggplot2)
library(dplyr)

augment(fit, newdata = df_pred, loc = "loc") %>% ggplot() + 
                aes(x = loc, y = .fitted) +
                geom_line(col="red") + 
                geom_point(data = df_data, aes(x = loc, y=Y))

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 YY:

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

Note that YY 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)
## Warning in rspde_lme(y_vec ~ -1, model = op_cov_est, repl = repl, data =
## df_data_repl, : The optimization failed to provide a numerically
## positive-definite Hessian. You can try to obtain a positive-definite Hessian by
## setting 'improve_hessian' to TRUE or by setting 'parallel' to FALSE, which
## allows other optimization methods to be used.
## Warning in sqrt(diag(inv_fisher)): NaNs produced

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  1.28321       NaN     NaN
## tau    0.04583       NaN     NaN
## kappa 20.01370       NaN     NaN
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.783209       NaN     NaN
## sigma 1.271305  0.052037   24.43
## range 0.125071  0.007832   15.97
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev 0.302680  0.004417   68.52
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -2758.186 
## Number of function calls by 'optim' = 34
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  13.56808 secs 
##   set up the parallelization = 2.25855 secs

and glance:

glance(fit_repl)
## # A tibble: 1 × 9
##    nobs sigma logLik   AIC   BIC deviance df.residual model                alpha
##   <int> <dbl>  <dbl> <dbl> <dbl>    <dbl>       <dbl> <chr>                <dbl>
## 1  4000 0.303 -2758. 5524. 5550.    5516.        3996 Covariance-Based Ma…  1.28

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 1.271305 0.1250709 0.7832089
# Total time
print(fit_repl$fitting_time)
## Time difference of 13.56808 secs

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  1.283209  0.013648   94.02
## tau    0.045834  0.003324   13.79
## kappa 20.013699  1.369621   14.61
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.783209  0.013648   57.39
## sigma 1.271305  0.052038   24.43
## range 0.125071  0.007832   15.97
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev 0.302680  0.004949   61.16
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -2758.186 
## Number of function calls by 'optim' = 34
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  11.19102 secs 
##   compute the Hessian = 5.95239 secs 
##   set up the parallelization = 2.26457 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 fmesher package. Let us start by defining a mesh over [0,1]×[0,1][0,1]\times [0, 1] and compute the mass and stiffness matrices for that mesh.

We consider a simple Gaussian linear model with 30 independent replicates of a latent spatial field u(𝐬)u(\mathbf{s}), observed at the same mm locations, {𝐬1,…,𝐬m}\{\mathbf{s}_1 , \ldots , \mathbf{s}_m \}, for each replicate. For each i=1,…,m,i = 1,\ldots,m, we have

yi=u1(𝐬i)+εi,⋮=⋮yi+29m=u30(𝐬i)+εi+29m,\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 ε1,…,ε30m\varepsilon_1,\ldots,\varepsilon_{30m} are iid normally distributed with mean 0 and standard deviation 0.1.

Let us create the FEM mesh:

n_loc <- 500
loc_2d_mesh <- matrix(runif(n_loc * 2), n_loc, 2)
mesh_2d <- fm_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=2m=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 σ=1\sigma=1 and range 0.10.1. We will use ν=0.5\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 function fm_basis() from the fmesher package. Recall that we will simulate the data with 30 replicates.

n.rep <- 30
u <- simulate(op_cov_2d, nsim = n.rep)
A <- fm_basis(
  x = 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 <- fm_evaluator(mesh_2d, dims = c(70, 70))

df_field <- data.frame(x = proj$lattice$loc[,1],
                        y = proj$lattice$loc[,2],
                        field = as.vector(fm_evaluate(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  1.52782   0.04909  31.123
## tau    0.07302   0.01346   5.423
## kappa 13.13775   0.80390  16.342
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.527816  0.049090   10.75
## sigma 1.365723  0.014512   94.11
## range 0.156410  0.005158   30.32
## 
## Measurement error:
##           Estimate Std.error z-value
## std. dev 0.1003218 0.0008777   114.3
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -5659.535 
## Number of function calls by 'optim' = 40
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  1.68804 mins 
##   set up the parallelization = 2.30177 secs

and glance:

glance(fit_2d)
## # A tibble: 1 × 9
##    nobs sigma logLik    AIC    BIC deviance df.residual model              alpha
##   <int> <dbl>  <dbl>  <dbl>  <dbl>    <dbl>       <dbl> <chr>              <dbl>
## 1 15000 0.100 -5660. 11327. 11358.   11319.       14996 Covariance-Based …  1.53

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 1.365723 0.1564103 0.5278161
# Total time
print(fit_2d$fitting_time)
## Time difference of 1.688046 mins

Let us now plot the prediction for replicate 3 by using the augment function. We begin by creating the data.frame we want to do prediction:

df_pred <- data.frame(x = proj$lattice$loc[,1],
                        y = proj$lattice$loc[,2])
augment(fit_2d, newdata = df_pred, loc = c("x","y"), which_repl = 3) %>% ggplot() +
              geom_raster(aes(x=x, y=y, fill = .fitted)) + xlim(0,1) + ylim(0,1) + 
              scale_fill_viridis()
## Warning: Removed 3744 rows containing missing values or values outside the scale range
## (`geom_raster()`).

An example with a non-stationary model

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

For this example we will consider simulated data.

Simulating the data

Let us consider a simple Gaussian linear model with a latent spatial field x(𝐬)x(\mathbf{s}), defined on the rectangle (0,10)×(0,5)(0,10) \times (0,5), where the std. deviation and range parameter satisfy the following log-linear regressions: log(σ(𝐬))=θ1+θ3b(𝐬),log(ρ(𝐬))=θ2+θ3b(𝐬),\begin{align} \log(\sigma(\mathbf{s})) &= \theta_1 + \theta_3 b(\mathbf{s}),\\ \log(\rho(\mathbf{s})) &= \theta_2 + \theta_3 b(\mathbf{s}), \end{align} where b(𝐬)=(s1−5)/10b(\mathbf{s}) = (s_1-5)/10. We assume the data is observed at mm locations, {𝐬1,…,𝐬m}\{\mathbf{s}_1 , \ldots , \mathbf{s}_m \}. For each i=1,…,m,i = 1,\ldots,m, we have

yi=x1(𝐬i)+εi,y_i = x_1(\mathbf{s}_i)+\varepsilon_i,

where ε1,…,εm\varepsilon_1,\ldots,\varepsilon_{m} are iid normally distributed with mean 0 and standard deviation 0.1.

We begin by defining the domain and creating the mesh:

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

mesh <- fm_mesh_2d(loc.domain = rec_domain, cutoff = 0.1, 
  max.edge = c(0.5, 1.5), offset = c(0.5, 1.5))

We follow the same structure as INLA. However, INLA only allows one to specify B.tau and B.kappa matrices, and, in INLA, if one wants to parameterize in terms of range and standard deviation one needs to do it manually. Here we provide the option to directly provide the matrices B.sigma and B.range.

The usage of the matrices B.tau and B.kappa are identical to the corresponding ones in inla.spde2.matern() function. The matrices B.sigma and B.range work in the same way, but they parameterize the stardard deviation and range, respectively.

The columns of the B matrices correspond to the same parameter. The first column does not have any parameter to be estimated, it is a constant column.

So, for instance, if one wants to share a parameter with both sigma and range (or with both tau and kappa), one simply let the corresponding column to be nonzero on both B.sigma and B.range (or on B.tau and B.kappa).

We will assume ν=0.8\nu = 0.8, θ1=0,θ2=1\theta_1 = 0, \theta_2 = 1 and θ3=1\theta_3=1. Let us now build the model with the spde.matern.operators() function:

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")

Let us now sample the data with the simulate() method:

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

Let us now obtain 600 random locations on the rectangle and compute the AA matrix:

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

A <- fm_basis(
  x = mesh,
  loc = loc_mesh
)

We can now generate the response vector y:

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

Let us now create the object to fit the data:

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

Let us also create the data.frame() that contains the data and the locations:

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

Fitting the non-stationary rSPDE model 

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

Let us get the summary:

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    1.89706   0.06386  29.705
## Theta 1 -0.29625   0.15621  -1.897
## Theta 2  0.90944   0.23431   3.881
## Theta 3  1.90932   0.40437   4.722
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev  0.10194   0.05288   1.928
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -89.47917 
## Number of function calls by 'optim' = 93
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  10.61764 secs 
##   set up the parallelization = 2.31235 secs

Let us now compare with the true values:

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      0.0000000 1.0000000 1.000000 1.800000
## Estimates -0.2962479 0.9094445 1.909317 1.897063

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  1.51696   0.04769  31.807
## tau    0.07206   0.01281   5.626
## kappa 13.98082   0.75501  18.517
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.516958  0.047693   10.84
## sigma 1.392506  0.014772   94.27
## range 0.145459  0.004507   32.27
## 
## Measurement error:
##           Estimate Std.error z-value
## std. dev 0.1003733 0.0008786   114.2
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -5658.37 
## Number of function calls by 'optim' = 30
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  32.21481 secs 
##   set up the parallelization = 2.20883 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     1.300000 0.1500000 0.8000000
## Estimates 1.392506 0.1454587 0.5169575

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)
## Warning in rspde_lme(y ~ -1, model = op_cov_2d_type, data = df_data_2d, : The
## optimization failed to provide a numerically positive-definite Hessian. You can
## try to obtain a positive-definite Hessian by setting 'improve_hessian' to TRUE
## or by setting 'parallel' to FALSE, which allows other optimization methods to
## be used.
## Warning in sqrt(diag(inv_fisher)): NaNs produced
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  1.49502       NaN     NaN
## tau    0.08003       NaN     NaN
## kappa 13.61506       NaN     NaN
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.495016       NaN     NaN
## sigma 1.375559  0.014745   93.29
## range 0.146162  0.004105   35.61
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev 0.100338  0.000878   114.3
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -5658.827 
## Number of function calls by 'optim' = 63
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  57.97109 secs 
##   set up the parallelization = 2.23421 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     1.300000 0.1500000 0.8000000
## Estimates 1.375559 0.1461621 0.4950159

References

Bolin, David, and Kristin Kirchner. 2020. “The Rational SPDE Approach for Gaussian Random Fields with General Smoothness.” Journal of Computational and Graphical Statistics 29 (2): 274–85.
Bolin, David, Alexandre B. Simas, and Zhen Xiong. 2023. “Covariance-Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference.” Journal of Computational and Graphical Statistics.
Hofreither, Clemens. 2021. “An Algorithm for Best Rational Approximation Based on Barycentric Rational Interpolation.” Numerical Algorithms 88 (1): 365–88.
Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society. Series B. Statistical Methodology 73 (4): 423–98.