# inlabru implementation of the rational SPDE approach
Author: David Bolin and Alexandre B. Simas
Date: 2023-05-24

#### David Bolin and Alexandre B. Simas

#### 2023-05-24

## Introduction

In this vignette we will present the `inlabru` implementation of the covariance-based rational SPDE approach. For further technical details on the covariance-based approach, see the Rational approximation with the `rSPDE` package vignette and Xiong, Simas, and Bolin (2022).

We begin by providing a step-by-step illustration on how to use our implementation. To this end we will consider a real world data set that consists of precipitation measurements from the Paraná region in Brazil.

After the initial model fitting, we will show how to change some parameters of the model. In the end, we will also provide an example in which we have replicates.

The examples in this vignette are the same as those in the R-INLA implementation of the rational SPDE approach vignette. As in that case, it is important to mention that one can improve the performance by using the PARDISO solver. Please, go to https://www.pardiso-project.org/r-inla/#license to apply for a license. Also, use `inla.pardiso()` for instructions on how to enable the PARDISO sparse library.

## An example with real data

To illustrate our implementation of `rSPDE` in `inlabru` we will consider a dataset available in `R-INLA`. This data has also been used to illustrate the SPDE approach, see for instance the book Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA and also the vignette Spatial Statistics using R-INLA and Gaussian Markov random fields. See also Lindgren, Rue, and Lindström (2011) for theoretical details on the standard SPDE approach.

The data consist of precipitation measurements from the Paraná region in Brazil and were provided by the Brazilian National Water Agency. The data were collected at 616 gauge stations in Paraná state, south of Brazil, for each day in 2011.

### An rSPDE model for precipitation

We will follow the vignette Spatial Statistics using R-INLA and Gaussian Markov random fields. As precipitation data are always positive, we will assume it is Gamma distributed. `R-INLA` uses the following parameterization of the Gamma distribution, \[\Gamma(\mu, \phi): \pi (y) = \frac{1}{\Gamma(\phi)} \left(\frac{\phi}{\mu}\right)^{\phi} y^{\phi - 1} \exp\left(-\frac{\phi y}{\mu}\right) .\] In this parameterization, the distribution has expected value \(E(x) = \mu\) and variance \(V(x) = \mu^2/\phi\), where \(1/\phi\) is a dispersion parameter.

In this example \(\mu\) will be modelled using a stochastic model that includes both covariates and spatial structure, resulting in the latent Gaussian model for the precipitation measurements \[\begin{align} y_i\mid \mu(s_i), \theta &\sim \Gamma(\mu(s_i),c\phi)\\ \log (\mu(s)) &= \eta(s) = \sum_k f_k(c_k(s))+u(s)\\ \theta &\sim \pi(\theta) \end{align},\]

where \(y_i\) denotes the measurement taken at location \(s_i\), \(c_k(s)\) are covariates, \(u(s)\) is a mean-zero Gaussian Matérn field, and \(\theta\) is a vector containing all parameters of the model, including smoothness of the field. That is, by using the `rSPDE` model we will also be able to estimate the smoothness of the latent field.

### Examining the data

We will be using `inlabru`. The `inlabru` package is available on CRAN and also on GitHub.

We begin by loading some libraries we need to get the data and build the plots.

    library(ggplot2)
    library(INLA)
    library(inlabru)
    library(splancs)
    library(viridis)

Let us load the data and the border of the region

    data(PRprec)
    data(PRborder)

The data frame contains daily measurements at 616 stations for the year 2011, as well as coordinates and altitude information for the measurement stations. We will not analyze the full spatio-temporal data set, but instead look at the total precipitation in January, which we calculate as

    Y <- rowMeans(PRprec[, 3 + 1:31])

In the next snippet of code, we extract the coordinates and altitudes and remove the locations with missing values.

    ind <- !is.na(Y)
    Y <- Y[ind]
    coords <- as.matrix(PRprec[ind, 1:2])
    alt <- PRprec$Altitude[ind]

Let us build a plot for the precipitations:

    ggplot() +
      geom_point(aes(
        x = coords[, 1], y = coords[, 2],
        colour = Y
      ), size = 2, alpha = 1) +
      geom_path(aes(x = PRborder[, 1], y = PRborder[, 2])) +
      geom_path(aes(x = PRborder[1034:1078, 1], y = PRborder[
        1034:1078,
        2
      ]), colour = "red") +
      scale_color_viridis()

The red line in the figure shows the coast line, and we expect the distance to the coast to be a good covariate for precipitation.

This covariate is not available, so let us calculate it for each observation location:

    seaDist <- apply(spDists(coords, PRborder[1034:1078, ],
      longlat = TRUE
    ), 1, min)

Now, let us plot the precipitation as a function of the possible covariates:

    par(mfrow = c(2, 2))
    plot(coords[, 1], Y, cex = 0.5, xlab = "Longitude")
    plot(coords[, 2], Y, cex = 0.5, xlab = "Latitude")
    plot(seaDist, Y, cex = 0.5, xlab = "Distance to sea")
    plot(alt, Y, cex = 0.5, xlab = "Altitude")

    par(mfrow = c(1, 1))

### Creating the rSPDE model

To use the `inlabru` implementation of the `rSPDE` model we need to load the functions:

    library(rSPDE)

To create a `rSPDE` model, one would the `rspde.matern()` function in a similar fashion as one would use the `inla.spde2.matern()` function.

#### Mesh

We can use `R-INLA` for creating the mesh. Let us create a mesh which is based on a non-convex hull to avoid adding many small triangles outside the domain of interest:

    prdomain <- inla.nonconvex.hull(coords, -0.03, -0.05, resolution = c(100, 100))
    prmesh <- inla.mesh.2d(boundary = prdomain, max.edge = c(0.45, 1), cutoff = 0.2)
    plot(prmesh, asp = 1, main = "")
    lines(PRborder, col = 3)
    points(coords[, 1], coords[, 2], pch = 19, cex = 0.5, col = "red")

#### Setting up the data frame

In place of a `inla.stack`, we can set up a `data.frame()` to use `inlabru`. We refer the reader to vignettes in https://inlabru-org.github.io/inlabru/index.html for further details.

    prdata <- data.frame(long = coords[,1], lat = coords[,2],
                            seaDist = inla.group(seaDist), y = Y)
    coordinates(prdata) <- c("long","lat")

#### Setting up the rSPDE model

To set up an `rSPDE`model, all we need is the mesh. By default it will assume that we want to estimate the smoothness parameter \(\nu\) and to do a covariance-based rational approximation of order 2.

Later in this vignette we will also see other options for setting up `rSPDE` models such as keeping the smoothness parameter fixed and/or increasing the order of the covariance-based rational approximation.

Therefore, to set up a model all we have to do is use the `rspde.matern()` function:

    rspde_model <- rspde.matern(mesh = prmesh)

Notice that this function is very reminiscent of `R-INLA`’s `inla.spde2.matern()` function.

We will assume the following linkage between model components and observations \[\eta(s) \sim A x(s) + A \text{ Intercept} + \text{seaDist}.\] \(\eta(s)\) will then be used in the observation-likelihood, \[y_i\mid \eta(s_i),\theta \sim \Gamma(\exp(\eta (s_i)), c\phi).\]

### Model fitting

We will build a model using the distance to the sea \(x_i\) as a covariate through an improper CAR(1) model with \(\beta_{ij}=1(i\sim j)\), which `R-INLA` calls a random walk of order 1. We will fit it in `inlabru`’s style:

    cmp <- y ~ Intercept(1) + distSea(seaDist, model="rw1") +
    field(coordinates, model = rspde_model)

To fit the model we simply use the `bru()` function:

    rspde_fit <- bru(cmp, data = prdata,
      family = "Gamma",
      options = list(
        control.inla = list(int.strategy = "eb"),
        verbose = FALSE)
    )

### inlabru results

We can look at some summaries of the posterior distributions for the parameters, for example the fixed effects (i.e. the intercept) and the hyper-parameters (i.e. dispersion in the gamma likelihood, the precision of the RW1, and the parameters of the spatial field):

    summary(rspde_fit)

    ## inlabru version: 2.7.0
    ## INLA version: 23.05.22
    ## Components:
    ## Intercept: main = linear(1)
    ## distSea: main = rw1(seaDist)
    ## field: main = cgeneric(coordinates)
    ## Likelihoods:
    ##   Family: 'Gamma'
    ##     Data class: 'SpatialPointsDataFrame'
    ##     Predictor: y ~ .
    ## Time used:
    ##     Pre = 3.66, Running = 29.4, Post = 0.0985, Total = 33.2
    ## Fixed effects:
    ##           mean   sd 0.025quant 0.5quant 0.975quant mode kld
    ## Intercept 1.92 0.42      1.097     1.92      2.744 1.92   0
    ##
    ## Random effects:
    ##   Name     Model
    ##     distSea RW1 model
    ##    field CGeneric
    ##
    ## Model hyperparameters:
    ##                                                     mean       sd 0.025quant
    ## Precision parameter for the Gamma observations    13.483    0.374      12.76
    ## Precision for distSea                          16104.466 6273.652    6723.20
    ## Theta1 for field                                  -0.857    0.155      -1.14
    ## Theta2 for field                                   1.663    0.247       1.24
    ## Theta3 for field                                  -3.166    0.258      -3.61
    ##                                                 0.5quant 0.975quant    mode
    ## Precision parameter for the Gamma observations    13.480     14.299   13.30
    ## Precision for distSea                          14974.593  32640.913 2122.46
    ## Theta1 for field                                  -0.863     -0.534   -0.89
    ## Theta2 for field                                   1.700      2.050    1.66
    ## Theta3 for field                                  -3.127     -2.760   -3.17
    ##
    ## Deviance Information Criterion (DIC) ...............: 2497.69
    ## Deviance Information Criterion (DIC, saturated) ....: 701.14
    ## Effective number of parameters .....................: 89.78
    ##
    ## Watanabe-Akaike information criterion (WAIC) ...: 2499.16
    ## Effective number of parameters .................: 80.17
    ##
    ## Marginal log-Likelihood:  -1256.69
    ##  is computed
    ## Posterior summaries for the linear predictor and the fitted values are computed
    ## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')

Let \(\theta_1 = \textrm{Theta1}\), \(\theta_2=\textrm{Theta2}\) and \(\theta_3=\textrm{Theta3}\). In terms of the SPDE \[(\kappa^2 I - \Delta)^{\alpha/2}(\tau u) = \mathcal{W},\] where \(\alpha = \nu + d/2\), we have that \[\tau = \exp(\theta_1),\quad \kappa = \exp(\theta_2), \] and by default \[\nu = 4\Big(\frac{\exp(\theta_3)}{1+\exp(\theta_3)}\Big).\] The number 4 comes from the upper bound for \(\nu\), which is discussed in R-INLA implementation of the rational SPDE approach vignette.

In general, we have \[\nu = \nu_{UB}\Big(\frac{\exp(\theta_3)}{1+\exp(\theta_3)}\Big),\] where \(\nu_{UB}\) is the value of the upper bound for the smoothness parameter \(\nu\).

Another choice for prior for \(\nu\) is a truncated lognormal distribution and is also discussed in R-INLA implementation of the rational SPDE approach vignette.

### inlabru results in the original scale

We can obtain outputs with respect to parameters in the original scale by using the function `rspde.result()`:

    result_fit <- rspde.result(rspde_fit, "field", rspde_model)
    summary(result_fit)

    ##             mean        sd 0.025quant 0.5quant 0.975quant     mode
    ## std.dev 0.429425 0.0676978  0.3194420 0.421069   0.584116 0.401582
    ## range   5.369300 1.7212800  2.6364100 5.148280   9.375800 4.701170
    ## nu      0.164349 0.0528695  0.0796522 0.157823   0.286460 0.144392

We can also plot the posterior densities. To this end we will use the `gg_df()` function, which creates `ggplot2` user-friendly data frames:

    posterior_df_fit <- gg_df(result_fit)

    ggplot(posterior_df_fit) + geom_line(aes(x = x, y = y)) +
    facet_wrap(~parameter, scales = "free") + labs(y = "Density")

### Predictions

Let us now obtain predictions (i.e. do kriging) of the expected precipitation on a dense grid in the region.

We begin by creating the grid in which we want to do the predictions. To this end, we can use the `inla.mesh.projector()` function:

    nxy <- c(150, 100)
    projgrid <- inla.mesh.projector(prmesh,
      xlim = range(PRborder[, 1]),
      ylim = range(PRborder[, 2]), dims = nxy
    )

This lattice contains 150 × 100 locations. One can easily change the resolution of the kriging prediction by changing `nxy`. Let us find the cells that are outside the region of interest so that we do not plot the estimates there.

    xy.in <- inout(projgrid$lattice$loc, cbind(PRborder[, 1], PRborder[, 2]))

Let us plot the locations that we will do prediction:

    coord.prd <- projgrid$lattice$loc[xy.in, ]
    plot(coord.prd, type = "p", cex = 0.1)
    lines(PRborder)
    points(coords[, 1], coords[, 2], pch = 19, cex = 0.5, col = "red")

Let us now create a `data.frame()` of the coordinates:

    coord.prd.df <- data.frame(x1 = coord.prd[,1],
                                x2 = coord.prd[,2])
    coordinates(coord.prd.df) <- c("x1", "x2")

Since we are using distance to the sea as a covariate, we also have to calculate this covariate for the prediction locations. Finally, we add the prediction location to our prediction `data.frame()`, namely, `coord.prd.df`:

    seaDist.prd <- apply(spDists(coord.prd,
      PRborder[1034:1078, ],
      longlat = TRUE
    ), 1, min)
    coord.prd.df$seaDist <- seaDist.prd

    pred_obs <- predict(rspde_fit, coord.prd.df,
            ~exp(Intercept + field + distSea))

Let us now build the data frame with the results:

    pred_df <- pred_obs@data
    pred_df <- cbind(pred_df, pred_obs@coords)

Finally, we plot the results. First the predicted mean:

    ggplot(pred_df, aes(x = x1, y = x2, fill = mean)) +
      geom_raster() +
      scale_fill_viridis()

Then, the std. deviations:

    ggplot(pred_df, aes(x = x1, y = x2, fill = sd)) +
      geom_raster() + scale_fill_viridis()

## An example with replicates

For this example we will simulate a data with replicates. We will use the same example considered in the Rational approximation with the `rSPDE` package vignette (the only difference is the way the data is organized). We also refer the reader to this vignette for a description of the function `matern.operators()`, along with its methods (for instance, the `simulate()` method).

### Simulating the data

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

\[\begin{align} y_i &= x_1(\mathbf{s}_i)+\varepsilon_i,\\ \vdots &= \vdots\\ y_{i+29m} &= x_{30}(\mathbf{s}_i) + \varepsilon_{i+29m}, \end{align}\]

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

We use the basis function representation of \(x(\cdot)\) to define the \(A\) matrix linking the point locations to the mesh. We also need to account for the fact that we have 30 replicates at the same locations. To this end, the \(A\) matrix we need can be generated by `inla.spde.make.A()` function. The reason being that we are sampling \(x(\cdot)\) directly and not the latent vector described in the introduction of the Rational approximation with the `rSPDE` package vignette.

We begin by creating the mesh:

    m <- 200
    loc_2d_mesh <- matrix(runif(m * 2), m, 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 then compute the \(A\) matrix, which is needed for simulation, and connects the observation locations to the mesh:

    n.rep <- 30
    A <- inla.spde.make.A(
      mesh = mesh_2d,
      loc = loc_2d_mesh,
      index = rep(1:m, times = n.rep),
      repl = rep(1:n.rep, each = m)
    )

Notice that for the simulated data, we should use the \(A\) matrix from `inla.spde.make.A()` function.

We will 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 <- 0.5
    sigma <- 1
    range <- 0.1
    kappa <- sqrt(8 * nu) / range
    tau <- sqrt(gamma(nu) / (sigma^2 * kappa^(2 * nu) * (4 * pi) * gamma(nu + 1)))
    d <- 2
    operator_information <- matern.operators(
      mesh = mesh_2d,
      nu = nu,
      range = range,
      sigma = sigma,
      m = 2,
      parameterization = "matern"
    )

More details on this function can be found at the Rational approximation with the rSPDE package vignette.

To simulate the latent process all we need to do is to use the `simulate()` method on the `operator_information` object. We then obtain the simulated data \(y\) by connecting with the \(A\) matrix and adding the gaussian noise.

    set.seed(1)
    u <- simulate(operator_information, nsim = n.rep)
    y <- as.vector(A %*% as.vector(u)) +
      rnorm(m * n.rep) * 0.1

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

    proj <- inla.mesh.projector(mesh_2d, dims = c(100, 100))

    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 = y[1:m],
                          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()

### Fitting the inlabru rSPDE model

Let us then use the rational SPDE approach to fit the data.

We begin by creating the model object.

    rspde_model.rep <- rspde.matern(mesh = mesh_2d,
              parameterization = "spde")

Let us now create the `data.frame()` and the vector with the replicates indexes:

    rep.df <- data.frame(y = y, x1 = rep(loc_2d_mesh[,1], n.rep),
                          x2 = rep(loc_2d_mesh[,2], n.rep))
    coordinates(rep.df) <- c("x1", "x2")
    repl <- rep(1:n.rep, each=m)

Let us create the component and fit. It is extremely important not to forget the `replicate` when fitting model with the `bru()` function. It will not produce warning and might fit some meaningless model.

    cmp.rep <-
      y ~ -1 + field(coordinates,
        model = rspde_model.rep,
        replicate = repl
      )

    rspde_fit.rep <-
      bru(cmp.rep,
        data = rep.df,
        family = "gaussian"
      )

We can get the summary:

    summary(rspde_fit.rep)

    ## inlabru version: 2.7.0
    ## INLA version: 23.05.22
    ## Components:
    ## field: main = cgeneric(coordinates), replicate = iid(repl)
    ## Likelihoods:
    ##   Family: 'gaussian'
    ##     Data class: 'SpatialPointsDataFrame'
    ##     Predictor: y ~ .
    ## Time used:
    ##     Pre = 3.42, Running = 116, Post = 10.4, Total = 130
    ## Random effects:
    ##   Name     Model
    ##     field CGeneric
    ##
    ## Model hyperparameters:
    ##                                          mean    sd 0.025quant 0.5quant
    ## Precision for the Gaussian observations 95.89 3.796      89.53    95.62
    ## Theta1 for field                        -2.90 0.068      -3.01    -2.90
    ## Theta2 for field                         3.07 0.052       2.95     3.08
    ## Theta3 for field                        -1.69 0.030      -1.75    -1.69
    ##                                         0.975quant  mode
    ## Precision for the Gaussian observations     104.41 88.17
    ## Theta1 for field                             -2.75 -2.93
    ## Theta2 for field                              3.15  3.10
    ## Theta3 for field                             -1.63 -1.69
    ##
    ## Deviance Information Criterion (DIC) ...............: -5969.05
    ## Deviance Information Criterion (DIC, saturated) ....: 10497.43
    ## Effective number of parameters .....................: 4512.99
    ##
    ## Watanabe-Akaike information criterion (WAIC) ...: -6844.54
    ## Effective number of parameters .................: 2674.10
    ##
    ## Marginal log-Likelihood:  -4352.45
    ##  is computed
    ## Posterior summaries for the linear predictor and the fitted values are computed
    ## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')

and the summary in the user’s scale:

    result_fit_rep <- rspde.result(rspde_fit.rep, "field", rspde_model.rep)
    summary(result_fit_rep)

    ##             mean         sd 0.025quant   0.5quant 0.975quant       mode
    ## tau    0.0553396 0.00379791  0.0492655  0.0548614  0.0639945  0.0532736
    ## kappa 21.5619000 1.09142000 19.1784000 21.6828000 23.3518000 22.2276000
    ## nu     0.6234130 0.01544210  0.5928520  0.6235450  0.6535030  0.6242070

    result_df <- data.frame(
      parameter = c("tau", "kappa", "nu"),
      true = c(tau, kappa, nu),
      mean = c(
        result_fit_rep$summary.tau$mean,
        result_fit_rep$summary.kappa$mean,
        result_fit_rep$summary.nu$mean
      ),
      mode = c(
        result_fit_rep$summary.tau$mode,
        result_fit_rep$summary.kappa$mode,
        result_fit_rep$summary.nu$mode
      )
    )
    print(result_df)

    ##   parameter        true        mean        mode
    ## 1       tau  0.08920621  0.05533962  0.05327363
    ## 2     kappa 20.00000000 21.56190582 22.22762228
    ## 3        nu  0.50000000  0.62341287  0.62420727

## 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 to obtain the sample 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)

    # SPDE model
    op_cov_ns <- spde.matern.operators(mesh = mesh,
      theta = true_theta,
      nu = nu,
      B.sigma = B.sigma,
      B.range = B.range, m = 2,
      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

### Fitting the inlabru rSPDE model

Let us then use the rational SPDE approach to fit the data.

We begin by creating the model object. We are creating a new one so that we do not start the estimation at the true values.

    rspde_model_nonstat <- rspde.matern(mesh = mesh,
      B.sigma = B.sigma,
      B.range = B.range)

Let us now create the `data.frame()` and the vector with the replicates indexes:

    nonstat_df <- data.frame(y = y, x1 = loc_mesh[,1],
                          x2 = loc_mesh[,2])
    coordinates(nonstat_df) <- c("x1", "x2")

Let us create the component and fit. It is extremely important not to forget the `replicate` when fitting model with the `bru()` function. It will not produce warning and might fit some meaningless model.

    cmp_nonstat <-
      y ~ -1 + field(coordinates,
        model = rspde_model_nonstat
      )

    rspde_fit_nonstat <-
      bru(cmp_nonstat,
        data = nonstat_df,
        family = "gaussian",
        options = list(verbose = FALSE)
      )

We can get the summary:

    summary(rspde_fit_nonstat)

    ## inlabru version: 2.7.0
    ## INLA version: 23.05.22
    ## Components:
    ## field: main = cgeneric(coordinates)
    ## Likelihoods:
    ##   Family: 'gaussian'
    ##     Data class: 'SpatialPointsDataFrame'
    ##     Predictor: y ~ .
    ## Time used:
    ##     Pre = 3.18, Running = 35.4, Post = 0.586, Total = 39.2
    ## Random effects:
    ##   Name     Model
    ##     field CGeneric
    ##
    ## Model hyperparameters:
    ##                                           mean     sd 0.025quant 0.5quant
    ## Precision for the Gaussian observations 95.855 10.281     77.682   95.152
    ## Theta1 for field                        -0.067  0.140     -0.315   -0.074
    ## Theta2 for field                         0.942  0.170      0.588    0.947
    ## Theta3 for field                         0.757  0.328      0.163    0.748
    ## Theta4 for field                        -1.299  0.202     -1.662   -1.306
    ##                                         0.975quant   mode
    ## Precision for the Gaussian observations    118.114 55.997
    ## Theta1 for field                             0.233 -0.112
    ## Theta2 for field                             1.256  0.977
    ## Theta3 for field                             1.464  0.658
    ## Theta4 for field                            -0.865 -1.362
    ##
    ## Deviance Information Criterion (DIC) ...............: -684.00
    ## Deviance Information Criterion (DIC, saturated) ....: 939.65
    ## Effective number of parameters .....................: 338.53
    ##
    ## Watanabe-Akaike information criterion (WAIC) ...: -710.25
    ## Effective number of parameters .................: 236.16
    ##
    ## Marginal log-Likelihood:  -25.10
    ##  is computed
    ## Posterior summaries for the linear predictor and the fitted values are computed
    ## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')

We can obtain outputs with respect to parameters in the original scale by using the function `rspde.result()`:

    result_fit_nonstat <- rspde.result(rspde_fit_nonstat, "field", rspde_model_nonstat)
    summary(result_fit_nonstat)

    ##                    mean       sd 0.025quant   0.5quant 0.975quant      mode
    ## Theta1.matern -0.066862 0.139756  -0.315270 -0.0735089    0.23268 -0.112053
    ## Theta2.matern  0.941850 0.169625   0.587809  0.9471360    1.25573  0.977232
    ## Theta3.matern  0.756887 0.328460   0.163218  0.7479120    1.46447  0.658086
    ## nu             0.864626 0.138246   0.640878  0.8470200    1.17911  0.803650

Let us compare the mean to the true values of the parameters:

    summ_res_nonstat <- summary(result_fit_nonstat)
    result_df <- data.frame(
      parameter = result_fit_nonstat$params,
      true = c(true_theta, nu),
      mean = summ_res_nonstat[,1],
      mode = summ_res_nonstat[,6]
    )
    print(result_df)

    ##       parameter true      mean      mode
    ## 1 Theta1.matern  0.0 -0.066862 -0.112053
    ## 2 Theta2.matern  1.0  0.941850  0.977232
    ## 3 Theta3.matern  1.0  0.756887  0.658086
    ## 4            nu  0.8  0.864626  0.803650

We can also plot the posterior densities. To this end we will use the `gg_df()` function, which creates `ggplot2` user-friendly data frames:

    posterior_df_fit <- gg_df(result_fit_nonstat)

    ggplot(posterior_df_fit) + geom_line(aes(x = x, y = y)) +
    facet_wrap(~parameter, scales = "free") + labs(y = "Density")

## Further options of the `inlabru` implementation

There are several additional options that are available. For instance, it is possible to change the order of the rational approximation, the upper bound for the smoothness parameter (which may speed up the fit), change the priors, change the type of the rational approximation, among others. These options are described in the “Further options of the `rSPDE`-`INLA` implementation” section of the R-INLA implementation of the rational SPDE approach vignette. Observe that all these options are passed to the model through the `rspde.matern()` function, and therefore the resulting model object can directly be used in the `bru()` function, in an identical manner to the examples above.

## References

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.