# Rational approximation with the rSPDE package
Author: David Bolin, Alexandre B. Simas, Zhen Xiong
Date: 2023-05-24

#### David Bolin, Alexandre B. Simas, Zhen Xiong

#### 2023-05-24

## 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 Xiong, Simas, and Bolin (2022) 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)

### 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)
    A <- rSPDE.A1d(s, obs.loc)

We now generate the observations as \(Y_i = 2 + x1 - x2 + u(s_i) + \varepsilon_i\), where \(\varepsilon_i \sim N(0,\sigma_e^2)\) is Gaussian measurement noise, \(x1\) and \(x2\) are covariates generated by uniform and exponential distributions, respectively. We will assume that the latent process has a Matérn covariance with \(\kappa=20, \sigma=2\) and \(\nu=0.8\):

    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)   1.3029    1.1854   1.099    0.272
    ## x1           -0.5822    2.0382  -0.286    0.775
    ##
    ## Random effects:
    ##        Estimate Std.error z-value
    ## alpha  1.350481  0.054381  24.834
    ## tau    0.036745  0.009266   3.965
    ## kappa 16.764259  5.281681   3.174
    ##
    ## Random effects (Matern parameterization):
    ##       Estimate Std.error z-value
    ## nu     0.85048   0.05438  15.639
    ## sigma  1.46819   0.31566   4.651
    ## range  0.15559   0.04788   3.249
    ##
    ## Measurement error:
    ##          Estimate Std.error z-value
    ## std. dev  0.33090   0.02229   14.84
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## Log-Likelihood:  -145.9047
    ## Number of function calls by 'optim' = 88
    ## Optimization method used in 'optim' = L-BFGS-B
    ##
    ## Time used to:     fit the model =  34.25525 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 1.468192 0.1555940 0.8504809

    # Total time
    print(fit$fitting_time)

    ## Time difference of 34.25526 secs

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)   1.3029    1.1854   1.099    0.272
    ## x1           -0.5822    2.0382  -0.286    0.775
    ##
    ## Random effects:
    ##        Estimate Std.error z-value
    ## alpha  1.350481  0.054381  24.834
    ## tau    0.036745  0.009266   3.965
    ## kappa 16.764259  5.281681   3.174
    ##
    ## Random effects (Matern parameterization):
    ##       Estimate Std.error z-value
    ## nu     0.85048   0.05438  15.639
    ## sigma  1.46819   0.31566   4.651
    ## range  0.15559   0.04788   3.249
    ##
    ## Measurement error:
    ##          Estimate Std.error z-value
    ## std. dev  0.33090   0.02229   14.84
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## Log-Likelihood:  -145.9047
    ## Number of function calls by 'optim' = 88
    ## Optimization method used in 'optim' = L-BFGS-B
    ##
    ## Time used to:     fit the model =  19.44282 secs
    ##   set up the parallelization = 1.42516 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 1.468192 0.1555940 0.8504809

    # 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 20.868 secs

### Kriging

Finally, we compute the kriging prediction of the process \(u\) 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, data = 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)

## 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  1.216669  0.008626  141.04
    ## tau    0.063212  0.002858   22.12
    ## kappa 17.514161  1.217836   14.38
    ##
    ## Random effects (Matern parameterization):
    ##       Estimate Std.error z-value
    ## nu    0.716669  0.008626   83.08
    ## sigma 1.273971  0.053210   23.94
    ## range 0.136715  0.009298   14.70
    ##
    ## Measurement error:
    ##          Estimate Std.error z-value
    ## std. dev  0.29928   0.00492   60.83
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## Log-Likelihood:  -2757.29
    ## Number of function calls by 'optim' = 56
    ## Optimization method used in 'optim' = L-BFGS-B
    ##
    ## Time used to:     fit the model =  1.04659 mins
    ##   set up the parallelization = 1.40993 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 0.800000
    ## Estimates 1.273971 0.1367146 0.716669

    # Total time
    print(fit_repl$fitting_time)

    ## Time difference of 1.046596 mins

We can obtain better estimates of the Hessian by setting `improve_hessian` to `FALSE`, 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  1.21667   0.04494  27.073
    ## tau    0.06321   0.01378   4.587
    ## kappa 17.51416   2.02835   8.635
    ##
    ## Random effects (Matern parameterization):
    ##       Estimate Std.error z-value
    ## nu    0.716669  0.044941   15.95
    ## sigma 1.273971  0.053210   23.94
    ## range 0.136715  0.009298   14.70
    ##
    ## Measurement error:
    ##          Estimate Std.error z-value
    ## std. dev 0.299282  0.005528   54.13
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## Log-Likelihood:  -2757.29
    ## Number of function calls by 'optim' = 56
    ## Optimization method used in 'optim' = L-BFGS-B
    ##
    ## Time used to:     fit the model =  54.91622 secs
    ##   compute the Hessian = 15.89359 secs
    ##   set up the parallelization = 1.44927 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 =  3.15143 mins
    ##   set up the parallelization = 1.48666 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 3.151431 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.

### Simulating the data

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

\[y_i = x_1(\mathbf{s}_i)+\varepsilon_i,\]

where \(\varepsilon_1,\ldots,\varepsilon_{m}\) 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 <- inla.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 \(\nu = 0.8\), \(\theta_1 = 0, \theta_2 = 1\) and \(\theta_3=1\). 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 \(A\) matrix:

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

    A <- inla.spde.make.A(
      mesh = 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)

    ##
    ## 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 =  32.88776 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     0.000000000 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 Xiong, Simas, and Bolin (2022).

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 =  2.04484 mins
    ##   set up the parallelization = 1.44035 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 =  1.80698 mins
    ##   set up the parallelization = 1.66654 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

## 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.
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.
Xiong, Zhen, Alexandre B. Simas, and David Bolin. 2022. “Covariance-Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference.” arXiv:2209.04670.