# MGPs for univariate spatial gridded data

#### M Peruzzi

#### 2022-09-19

This is the simplest case for Meshed GPs. We start by generating some data

    library(magrittr)
    library(dplyr)
    library(ggplot2)
    library(meshed)
    set.seed(2021)

    SS <- 50 # coord values for jth dimension
    dd <- 2 # spatial dimension
    n <- SS^2 # number of locations
    p <- 3 # number of covariates

    xlocs <- seq(0, 1, length.out=SS)
    coords <- expand.grid(list(xlocs, xlocs))

    sigmasq <- 1.5
    nu <- 0.5
    phi <- 10
    tausq <- .1

    # covariance at coordinates
    CC <- sigmasq * exp(-phi * as.matrix(dist(coords)))
    # cholesky of covariance matrix
    LC <- t(chol(CC))
    # spatial latent effects are a GP
    w <- LC %*% rnorm(n)
    # measurement errors
    eps <- tausq^.5 * rnorm(n)

    # covariates and coefficients
    X <- matrix(rnorm(n*p), ncol=p)
    Beta <- matrix(rnorm(p), ncol=1)

    # univariate outcome, fully observed
    y_full <- X %*% Beta + w + eps

    # now introduce some NA values in the outcomes
    y <- y_full %>% matrix(ncol=1)
    y[sample(1:n, n/5, replace=FALSE), ] <- NA

    simdata <- coords %>%
      cbind(data.frame(Outcome_full= y_full,
                       Outcome_obs = y,
                       w = w))

    simdata %>% ggplot(aes(Var1, Var2, fill=w)) +
        geom_raster() +
        scale_fill_viridis_c() +
      theme_minimal() + ggtitle("w: Latent GP")

    simdata %>% ggplot(aes(Var1, Var2, fill=y)) +
        geom_raster() +
        scale_fill_viridis_c() +
      theme_minimal() + ggtitle("y: Observed outcomes")

We now fit the following spatial regression model \[ y = X \beta + w + \epsilon, \] where \(w \sim MGP\) are spatial random effects, and \(\epsilon \sim N(0, \tau^2)\). For spatial data, an exponential covariance function is used by default: \(C(h) = \sigma^2 \exp( -\phi h )\) where \(h\) is the spatial distance.

The prior for the spatial decay \(\phi\) is \(U[l,u]\) and the values of \(l\) and \(u\) must be specified. We choose \(l=1\), \(u=30\) for this dataset.1

Setting `verbose` tells `spmeshed` how many intermediate messages to output while running MCMC. We set up MCMC to run for 3000 iterations, of which we discard the first third. We use the argument `block_size=25` to specify the coarseness of domain partitioning. In this case, the same result can be achieved by setting `axis_partition=c(10, 10)`.

Finally, we note that `NA` values for the outcome are OK since the full dataset is on a grid. `spmeshed` will figure this out and use settings optimized for partly observed lattices, and automatically predict the outcomes at missing locations. On the other hand, `X` values are assumed to not be missing.

    mcmc_keep <- 200 # too small! this is just a vignette.
    mcmc_burn <- 400
    mcmc_thin <- 2

    mesh_total_time <- system.time({
      meshout <- spmeshed(y, X, coords,
                          #axis_partition=c(10,10), #same as block_size=25
                          block_size = 25,
                          n_samples = mcmc_keep, n_burn = mcmc_burn, n_thin = mcmc_thin,
                          n_threads = 2,
                          verbose = 0,
                          prior=list(phi=c(1,30))
      )})

We can now do some postprocessing of the results. We extract posterior marginal summaries for \(\sigma^2\), \(\phi\), \(\tau^2\), and \(\beta_2\). The model that `spmeshed` targets is a slight reparametrization of the above:2 \[ y = X \beta + \lambda w + \epsilon, \] where \(w\sim MGP\) has unitary variance. This model is equivalent to the previous one and in fact we find \(\sigma^2=\lambda^2\).

    summary(meshout$lambda_mcmc[1,1,]^2)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   1.310   1.833   1.976   2.140   2.319   4.758
    summary(meshout$theta_mcmc[1,1,])
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   2.839   5.933   7.305   7.137   7.949  12.564
    summary(meshout$tausq_mcmc[1,])
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #> 0.08837 0.11358 0.12587 0.12465 0.13383 0.17332
    summary(meshout$beta_mcmc[1,1,])
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #> -0.8278 -0.8049 -0.7970 -0.7958 -0.7860 -0.7515

We proceed to plot predictions across the domain along with the recovered latent effects.

    # process means
    wmesh <- data.frame(w_mgp = meshout$w_mcmc %>% summary_list_mean())
    # predictions
    ymesh <- data.frame(y_mgp = meshout$yhat_mcmc %>% summary_list_mean())

    outdf <- meshout$coordsdata %>%
      cbind(wmesh, ymesh) %>%
      left_join(simdata)

    outdf %>%
        ggplot(aes(Var1, Var2, fill=w_mgp)) +
        geom_raster() +
        scale_fill_viridis_c() +
        theme_minimal() + ggtitle("w: recovered")

    outdf %>%
        ggplot(aes(Var1, Var2, fill=y_mgp)) +
        geom_raster() +
        scale_fill_viridis_c() +
        theme_minimal() + ggtitle("y: predictions")

1. `spmeshed` implements a model which can be interpreted as assigning \(\sigma^2\) a folded-t via parameter expansion if `settings$ps=TRUE`, or an inverse Gamma with parameters \(a=2\) and \(b=1\) if `settings$ps=FALSE`, which cannot be changed at this time. \(\tau^2\) is assigned an exponential prior.↩︎

2. At its core, `spmeshed` implements the spatial factor model \(Y(s) = X(s)\beta + \Lambda v(s) + \epsilon(s)\) where \(w(s) = \Lambda v(s)\) is modeled via linear coregionalization.↩︎