2.1 Illustration data set nyspatial

To illustrate point reference spatio-temporal data modeling we use the nyspatial data set included in the package. This data set has 28 rows and 9 columns containing average ground level ozone air pollution data from 28 sites in the state of New York. The averages are taken over the 62 days in July and August 2006. The full spatio-temporal data set from 28 sites for 62 days is used to illustrate spatio-temporal modeling, see Section 3.1. Figure 2.1 represents a map of the state of New York together with the 28 monitoring locations where the three sites 1, 5 and 10 have been identified.

Figure 2.1: 25 fitted sites and 3 validation sites (numbered) in New York

For regression modeling purposes, the response variable is yo3 and the three important covariates are maximum temperature: xmaxtemp in degree Celsius, wind speed: xwdsp in nautical miles and percentage average relative humidity: xrh. This data set is included in the package and further information regarding this can be obtained from the help file ?nyspatial.

2.2 The Bspatial function for fitting spatial regression models

The bmstdr package includes the function Bspatial for fitting regression models to point referenced spatial data. The arguments to this function has been documented in the help file which can be viewed by issuing the R command ?Bspatial. The package manual also contains the full documentation. The discussion below highlights the main features of this model fitting function.

Besides the usual data and formula the argument scale.transform can take one of three possible values: NONE, SQRT and LOG. This argument defines the on the fly transformation for the response variable which appears on the left hand side of the formula.

Default values of the arguments prior.beta0, prior.M and prior.sigma2 defining the prior distributions for \(\mathbf{\beta}\) and \(1/\sigma^2_{\epsilon}\) are provided.

The options model="lm" and model="spat" are respectively used for fitting and analysis using the independent spatial regression model with exponential correlation function. If the latter regression model is to be fitted, the function requires three additional arguments, coordtype, coords and phi. The coords argument provides the coordinates of the data locations. The type of these coordinates, specified by the coordtype argument, taking one of three possible values: utm, lonlat and plain determines various aspects of distance calculation and hence model fitting. The default for this argument is utm when it is expected that the coordinates are supplied in units of meter. The coords argument provides the actual coordinate values and this argument can be supplied as a vector of size two identifying the two column numbers of the data frame to take as coordinates. Or this argument can be given as a matrix of number of sites by 2 providing the coordinates of all the data locations.

The parameter phi determines the rate of decay of the spatial correlation for the assumed exponential covariance function. The default value, if not provided, is taken to be 3 over the maximum distance between the data locations so that the effective range is the maximum distance.

The argument package chooses one package to fit the spatial model from among four possible choices. The default option none is used to fit the independent linear regression model and the also the spatial regression model without the nugget effect when the parameter phi is assumed to be known. The three other options are spBayes, stan and inla. Each of these options use the corresponding R packages for model fitting. The exact form of the models in each case is documented in Chapter 6 of the book Sahu (2022).

Calculation of model choice statistics is triggered by the option mchoice=T. In this case the DIC, WAIC and PMCC values are calculated.

An optional vector argument validrows providing the row numbers of the supplied data frame for model validation can also be given. The model choice statistics are calculated on the opted scale but model validations and their uncertainties are calculated on the original scale of the response for ease of interpretation. This strategy of a possible transformed modeling scale but predictions on the original scale is adopted throughout the package.

There are other arguments of Bspatial, e.g. verbose, which control various aspects of model fitting and return values. Some of these other arguments are only relevant for specifying prior distributions and performing specific tasks as we will see throughout the remainder of this section.

The return value of Bspatial is a list of class bmstdr providing parameter estimates, and if requested model choice statistics and validation predictions and statistics. The S3methods print, plot, summary, fitted, and residuals have been implemented for objects of the bmstdr class. Thus the user can give the commands such as summary(M1) and plot(M1) where M1 is the model fitted object .

2.3 Fitting independent error regression models

The bmstdr package allows us to fit the base linear regression model given by: \[\begin{equation} Y_i = \beta_1 x_{i1} + \ldots + \beta_p x_{ip} + \epsilon_i, i=1, \ldots, n \tag{2.1} \end{equation}\] where \(\beta_1, \ldots, \beta_p\) are unknown regression coefficients and \(\epsilon_i\) is the error term that we assume to follow the normal distribution with mean zero and variance \(\sigma^2_{\epsilon}\). The usual linear model assumes the errors \(\epsilon_i\) to be independent for \(i=1, \ldots, n\). With the suitable default assumptions regarding the prior distributions we can fit the above model (2.1) by using the following command:

M1 <- Bspatial(formula=yo3~xmaxtemp+xwdsp+xrh, data=nyspatial, mchoice=T)

2.4 Fitting linear models with spatial error distribution

The independent linear regression model (2.1) is now extended to have spatially colored covariance matrix \(\sigma^{2}_{\epsilon}H\) where \(H\) is a known correlation matrix of the error vector \(\mathbf{\epsilon}\), i.e. \(H_{ij}=\mbox{Cor}(\epsilon_i, \epsilon_j)\) for \(i, j=1, \ldots,n\), \[\begin{equation} {\bf Y} \sim N_n \left(X{\mathbf \beta}, \sigma^{2}_{\epsilon} H\right) \tag{2.2} \end{equation}\] Assuming the exponential correlation function, i.e., \(H_{ij} = \exp(-\phi d_{ij})\) where \(d_{ij}\) is the distance between locations \({\bf s}_i\) and \({\bf s}_j\) we can fit the model (2.2) by issuing the command:

M2 <- Bspatial(model="spat", formula=yo3~xmaxtemp+xwdsp+xrh, data=nyspatial, 
      coordtype="utm", coords=4:5, phi=0.4, mchoice=T)

We discuss the choice of the fixed value of the spatial decay parameter \(\phi=0.4\) in M2. We use cross-validation methods to find an optimal value for \(\phi\). We take a grid of values for \(\phi\) and calculate a cross-validation error statistics, e.g. root mean square-error (rmse), for each value of \(\phi\) in the grid. The optimal \(\phi\) is the one that minimizes the statistics.

To perform the grid search a simple R function, phichoice_sp is provided. The documentation of this function explains how to set the arguments. For example, the following commands work:

asave <- phichoice_sp(formula=yo3~xmaxtemp+xwdsp+xrh, data=nyspatial, coordtype="utm", coords=4:5, phis=seq(from=0.1, to=1, by=0.1), scale.transform="NONE", s=c(8,11,12,14,18,21,24,28), N=2000, burn.in=1000)
asave

For the nyspatial data example 0.4 turns out to be the optimal value for \(\phi\).

2.5 Fitting spatial models with nugget effect

A general spatial model with nugget effect is written as: \[\begin{equation} Y({\bf s}_i) = {\bf x}'({\bf s}_i) \mathbf{\beta} + w({\bf s}_i) + \epsilon( {\bf s}_i) \tag{2.3} \end{equation}\] for all \(i=1, \ldots, n\). In the above equation, the pure error term \(\epsilon({\bf s}_i)\) is assumed to follow the independent zero mean normal distribution with variance \(\sigma^2_{\epsilon}\), called the nugget effect, for all \(i=1. \ldots, n\). The stochastic process \(w({\bf s})\) is assumed to follow a zero mean Gaussian Process with the exponential covariance function, see Sahu (2022) for more details.

The un-observed random variables \(w({\bf s}_i)\), \(i=1, \ldots, n\), also known as the spatial random effects can be integrated out to arrive at the marginal model \[\begin{align} {\bf Y} & \sim N\left(X{\mathbf \beta}, \sigma^2_{\epsilon} \, I + \sigma^2_w S_w \right), \tag{2.4} \end{align}\] where the matrix \(S_w\) is determined using the exponential correlation function. This marginal model is fitted using any of the three packages mentioned above. The code for this model fitting is very similar to the one for fitting M2 above; the only important change is in the package= argument as noted below.

M3 <- Bspatial(package="spBayes", formula=yo3~xmaxtemp+xwdsp+xrh, data=nyspatial, 
               coordtype="utm", coords=4:5, prior.phi=c(0.005, 2), mchoice=T)
M4 <- Bspatial(package="stan", formula=yo3~xmaxtemp+xwdsp+xrh, data=nyspatial, 
               coordtype="utm", coords=4:5,phi=0.4, mchoice=T)
M5  <- Bspatial(package="inla",formula=yo3~xmaxtemp+xwdsp+xrh, data=nyspatial, 
               coordtype="utm", coords=4:5, mchoice=T)

Model fitting is very fast except for M4 with the stan package. The model run for M4 takes about 20 minutes on a fast personal computer. These run produces the values of various Bayesian model choice criteria shown in Table 2.2.

Mathematical expressions for all the quantities in the above table are provided in Sahu (2022). Here M0 is the intercept only model for which the results are obtained using the following bmstdr command,

Bmchoice(case="MC.sigma2.unknown", y=ydata).

The implementation using inla does not calculate the alternative values of the DIC and WAIC.

2.6 Illustrating the model validation statistics

The model fitting function Bspatial also calculates the values of four validation statistics: - root mean square-error (rmse), - mean absolute error (mae), - continuous ranked probability score (crps) and - coverage (cvg) if an additional argument validrows containing the row numbers of the supplied data frame to be validated is provided.

Data from eight validation sites 8, 11, 12, 14, 18, 21, 24 and 28 are set aside and model fitting is performed using the data from the remaining 20 sites.

The bmstdr command for performing validation needs an additional argument validrows which are the row numbers of the supplied data frame which should be used for validation.

s <- c(8,11,12,14,18,21,24,28)
f1 <- yo3~xmaxtemp+xwdsp+xrh
M1.v <-  Bspatial(package="none", model="lm", formula=f1, 
                  data=nyspatial, validrows=s)
M2.v <- Bspatial(package="none", model="spat", formula=f1, 
        data=nyspatial,   coordtype="utm", coords=4:5,phi=0.4,  validrows=s)
M3.v <-  Bspatial(package="spBayes", prior.phi=c(0.005, 2), formula=f1,
        data=nyspatial,   coordtype="utm", coords=4:5, validrows=s) 
M4.v  <- Bspatial(package="stan",formula=f1, 
    data=nyspatial,   coordtype="utm", coords=4:5,phi=0.4 , validrows=s) 
M5.v  <- Bspatial(package="inla", formula=f1, data=nyspatial,   
        coordtype="utm", coords=4:5, validrows=s) 

Table 2.4 presents the validation statistics for all five models. Coverage is 100% for all five models and the validation performances are comparable. Model M4 with \(\phi=0.4\) can be used as the best model if it is imperative that one must be chosen using the rmse criterion.

To illustrate \(K\)-fold cross-validation, the 28 observations in the nyspatial data set are randomly assigned to \(K=4\) groups of equal size.

set.seed(44)
x <- runif(n=28)
u <- order(x)
s1 <- u[1:7]
s2 <- u[8:14]
s3 <- u[15:21]
s4 <- u[22:28]

Now the M2.v command is called four times with the validrows argument taking values s1, ... s4. Table 2.6 presents the 4-fold cross-validation statistics for M2 only. It shows a wide variability in performance with a low coverage of 57.14% for Fold 3.

A validation plot is automatically drawn each time a validation is performed. Below, we include the validation plot for fold-3 only.

M2.v3 <- Bspatial(model="spat", formula=yo3~xmaxtemp+xwdsp+xrh, data=nyspatial, 
               coordtype="utm", coords=4:5, validrows= s3, phi=0.4, verbose = FALSE)

Figure 2.2: Prediction against observation plot with the prediction intervals included. The `in/out’ symbol in the plot indicates whether or not a prediction interval includes the 45 degree line.

In this particular instance four of the seven validation observations are over-predicted. The above figure shows low coverage and high rmse. However, these statistics are based on data from seven validation sites only and as a result these may have large variability explaining the differences in the \(K\)-fold validation results.

The above validation plot has been drawn using the bmstdr command obs_v_pred_plot. This validation plot may be drawn without the line segments, which is recommended when there are a large number of validation observations. The plot may also use the mean values of the predictions instead of the default median values. The documentation of the function explains how to do this. For example, having the fitted object M2.v3, we may issue the commands:

names(M2.v3)
psums <- get_validation_summaries(M2.v3$valpreds)
names(psums)
a <- obs_v_pred_plot(yobs=M2.v3$yobs_preds$yo3, predsums=psums, segments=FALSE, summarystat = "mean" )

3.1 Illustration data set nysptime

To illustrate point reference spatio-temporal data modeling we use the nysptime data set included in the package. This is a spatio-temporal version of the data set nyspatial introduced in Section 2.1. This data set, taken from Sahu and Bakar (2012), has 1736 rows and 12 columns containing ground level ozone air pollution data from 28 sites in the state of New York for the 62 days in July and August 2006.

For regression modeling purposes, the response variable is y8hrmax and the three important covariates are maximum temperature: xmaxtemp in degree Celsius, wind speed: xwdsp in nautical miles and percentage average relative humidity: xrh.

3.2 The Bsptime function for fitting spatio-temporal models

In this section we extend the spatial model (2.3) to the following spatio-temporal model. \[\begin{equation} Y({\bf s}_i, t) = {\bf x}'({\bf s}_i, t) \mathbf{\beta} + w({\bf s}_i, t) + \epsilon( {\bf s}_i, t) \tag{3.1} \end{equation}\] for \(i=1, \ldots, n\) and \(t=1, \ldots, T.\) Different distribution specifications for the spatio-temporal random effects \(w({\bf s}_i, t)\) and the observational errors \(\epsilon({\bf s}_i, t)\) give rise to different models. Variations of these models have been described in Sahu (2022). The bmstdr function Bsptime has been developed to fit these models.

Similar to the Bspatial function, the Bsptime function takes a formula and a data argument. It is important to note that the Bsptime function always assumes that the data frame is first sorted by space and then time within each site in space. Note that missing covariate values are not permitted.

The arguments defining the scale, scale.transform, and the hyper parameters of the prior distribution for the regression coefficients \({\mathbf \beta}\) and the variance parameters are also similar to the corresponding ones in the spatial model fitting case with Bspatial. Other important arguments are described below.

The arguments coordtype, coords, and validrows are also similarly defined as before. However, note that when the separable model is fitted the validrows argument must include all the rows of time points for each site to be validated.

The package argument can take one of six values: spBayes, stan, inla, spTimer, sptDyn and none with none being the default. Fittings using each of these package options are illustrated in the sections below. Only a limited number of models, specified by the model argument, can be fitted with each of these six choices. The model argument is described below.

In case the package is none, the model can either be lm or seperable. The lm option is for an independent error regression model while the other option fits a separable spatio-temporal model without any nugget effect. The separable model fitting method cannot handle missing data. All missing data points in the response variable will be replaced by the grand mean of the available observations. When the package option is one of the five named packages the model argument is passed to the chosen package.

For fitting a separable model Bsptime requires specification of two decay parameters \(\phi_s\) and \(\phi_t\). If these are not specified then values are chosen which correspond to the effective ranges as the maximum distance in space and length in time.

There are numerous other package specific arguments that define the prior distributions and many important behavioral aspects of the selected package. Those are not described here. Instead the user is directed to the documentation ?Bsptime and also the vignettes of the individual packages.

With the default value of package="none" the independent error regression model M1 and the separable model M2 are fitted using the commands:

f2 <- y8hrmax~xmaxtemp+xwdsp+xrh
M1 <- Bsptime(model="lm", formula=f2, data=nysptime, scale.transform = "SQRT")
M2 <- Bsptime(model="separable", formula=f2, data=nysptime, scale.transform = "SQRT",
              coordtype="utm", coords=4:5)

The fitted model objects M1 and M2 are of class bmstdr and these can be explored using the S3 methods print, plot, summary and residuals. To explore the model fitted object issue the command names(M2). Here we explore the residuals by issuing the command:

a <- residuals(M2)
#> 
#>  Note that the residuals are provided on the transformed scale. Please see the scale.transform argument.
#> 
#> Summary of the residuals
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.     NA's 
#> -3.13367 -0.98683 -0.49008 -0.49402  0.02336  2.11039       24

Figure 3.1: A multiple time series plot of residuals

This command renders a multiple time series plot of the residuals. However, the same command a <- residuals(M1) will not draw the residual plot since the independent error regression model is not aware of the temporal structure of the data. In this case it is possible to modify the command to

a <- residuals(M1, numbers=list(sn=28, tn=62))

to have the desired result, see ?residuals.bmstdr.

3.3 Exploring the separable model

The above command for fitting M2 does not specify the values of the spatial and temporal decay parameters \(\phi_s\) and \(\phi_t\). The adopted values of these parameters are printed by the command:

M2$phi.s; M2$phi.t

These values are approximately 0.005 and 0.048 which correspond to the spatial range (the value of distance by which spatial correlation dies down) of 591 kilometers and the temporal range of 62 days which are the maximum possible values for the spatial and temporal domains of the data.

Optimal values of \(\phi_s\) and \(\phi_t\) can be determined by performing a grid search as in the case spatial only model fitting with Bspatial. The bmstdr package contains a function called phichoice which does this grid search specifically for the nysptime data set using the eight validation sites noted earlier. The command is:

asave <- phichoicep(formula=y8hrmax ~ xmaxtemp+xwdsp+xrh, data=nysptime,
                coordtype="utm", coords=4:5, scale.transform = "SQRT",  phis=c(0.001,  0.005, 0.025, 0.125, 0.625), phit=c(0.05, 0.25, 1.25, 6.25), 
valids=c(8,11,12,14,18,21,24,28), N=2000, burn.in=1000)

The optimal values in this case turns out to be \(\phi_s=0.005\) and \(\phi_t=0.05\) approximately.

We now explore model validation for the separable model using the three sites chosen previously.

valids <-  c(1, 5, 10)
vrows <-  which(nysptime$s.index%in% valids)
M2.1 <- Bsptime(model="separable",  formula=f2, data=nysptime, 
             validrows=vrows, coordtype="utm", coords=4:5,
             phi.s=0.005, phi.t=0.05, scale.transform = "SQRT")

summary(M2.1)
#> Call:
#> Bsptime(formula = f2, data = nysptime, model = "separable", coordtype = "utm", 
#>     coords = 4:5, validrows = vrows, scale.transform = "SQRT", 
#>     phi.s = 0.005, phi.t = 0.05)
#> ##
#> # Total time taken:: 3.08  - Sec.
#> Model formula
#> y8hrmax ~ xmaxtemp + xwdsp + xrh
#> 
#> 
#> Parameter Estimates:
#>               mean    sd   2.5%  97.5%
#> (Intercept)  0.319 1.775 -3.161  3.799
#> xmaxtemp     0.256 0.038  0.182  0.330
#> xwdsp        0.010 0.036 -0.062  0.081
#> xrh         -0.024 0.113 -0.246  0.198
#> sigma2      12.442 0.447 11.597 13.348
#> 
#> Validation Statistics:
#>    rmse     mae    crps     cvg 
#>   6.895   5.583  11.868 100.000

The fitted model M2.1 can be passed to the plot function that may draw several plots depending on the contents of the model fitted object. For example, this always draws a residuals against fitted values plot. The plot command for objects of class bmstdr can take additional arguments such as segments = FALSE which will provide a plot of the predicted values against observations without the line segments.

3.4 Spatio-temporal model fitting with the spTimer package

The spTimer package Bakar and Sahu (2015) can be used to fit, predict and forecast using various spatio-temporal models. This package also offers a great deal of flexibility in data modeling since it can model segmented time series data and also mixture of discrete and continuous data such as precipitation. The bmstdr function Bsptime can implement these models by invoking the package="spTimer" option. For example:

M3 <- Bsptime(package="spTimer", formula=f2, data=nysptime, n.report=5, 
              coordtype="utm", coords=4:5, scale.transform = "SQRT")

As before, the fitted model object M3 can be explored using the print, summary, plot and residuals commands. The plot command will draw the MCMC trace and density plots for each model parameter. The output plots of these commands are omitted from this document for brevity. Instead, we show validation performance at three selected sites: 1, 5, and 10 as shown in the map.

We randomly select 31 time points for validation at the three selected sites and identify the validation rows by using the following commands.

set.seed(44)
tn <- 62
sn <- 28
valids <- c(1, 5, 10)
validt <- sort(sample(1:tn, size=31))
vrows <- getvalidrows(sn=sn, tn=tn, valids=valids, validt=validt)

The getvalidrows command is included in the databmstdr library. Now we use the spTimer package to fit and validate the model.

M31 <- Bsptime(package="spTimer",formula=f2, data=nysptime, 
               coordtype="utm", coords=4:5,
               validrows=vrows, model="GP", report=5, 
               mchoice=F, scale.transform = "NONE")

For ease of illustration we have chosen scale.transform = "NONE". More complicated additional coding will be required if a different scale is chosen.

We illustrate a plot of the observations, fitted values and the validation predictions and their uncertainty intervals as plotted in Figure 11.13 in the book Banerjee, Carlin, and Gelfand (2015). The databmstdr package includes the function fig11.13.plot to generate this plot. The arguments required for this function are prepared as follows. The spTimer package does not provide the 95% intervals for the fitted values nor does it provide the MCMC samples of the fitted values which can be used to construct the 95% intervals. Hence, in the code below we use a normal approximation to obtain the uncertainty intervals. Now we first organize the data for plotting.

modfit <- M31$fit
fitall <- data.frame(modfit$fitted)
fitall$s.index <- rep(1:sn, each=tn)
library(spTimer)
#> 
#> ## spTimer version: 3.3.1
vdat <- spT.subset(data=nysptime, var.name=c("s.index"), s=valids)
fitvalid <- spT.subset(data=fitall, var.name=c("s.index"), s=valids)
fitvalid$low <- fitvalid$Mean - 1.96 * fitvalid$SD
fitvalid$up <- fitvalid$Mean + 1.96 * fitvalid$SD
fitvalid$yobs <- vdat$y8hrmax
yobs <- matrix(fitvalid$yobs, byrow=T, ncol=tn)
y.valids.low <- matrix(fitvalid$low, byrow=T, ncol=tn)
y.valids.med <- matrix(fitvalid$Mean, byrow=T, ncol=tn)
y.valids.up <- matrix(fitvalid$up, byrow=T, ncol=tn)

Now we call the fig11.13.plot function to render the plot for each site.

p1 <- fig11.13.plot(yobs[1, ], y.valids.low[1, ], y.valids.med[1, ], 
                    y.valids.up[1, ], misst=validt)
p1 <- p1 + ggtitle("Validation for Site 1")
p2 <- fig11.13.plot(yobs[2, ], y.valids.low[2, ], y.valids.med[2, ], 
                    y.valids.up[2, ], misst=validt)
p2 <- p2 + ggtitle("Validation for Site 5")
p3 <- fig11.13.plot(yobs[3, ], y.valids.low[3, ], y.valids.med[3, ], 
                    y.valids.up[3, ], misst=validt)
p3 <- p3 + ggtitle("Validation for Site 10")
library(ggpubr)
#> 
#> Attaching package: 'ggpubr'
#> The following object is masked from 'package:huxtable':
#> 
#>     font
ggarrange(p1, p2, p3, common.legend = TRUE, legend = "top", nrow = 3, ncol = 1)

Figure 3.2: Time series of observed and predicted values at three sites.

This ability to fit and validate using user defined validation rows is an enhancement of the spTimer package since that only allows validation at all time points for any selected site. The original package does not allow selective predictions at a subset of time points.

3.4.1 Illustrating spatio-temporal predictions

The spTimer package includes a prediction method function that can be used to predict at a large number locations. It does these predictions at all time points of the modeling data hence the covariates used in the model must be available for all prediction locations and at all time points. We illustrate the predictions using the fitted model M3. We only show the average predicted pollution map over the 62 days. To do the site-wise averaging we use the sitemeans function:

sitemeans <- function(a, sn, tn=62) { 
   u <- matrix(a, nrow=sn, ncol=tn, byrow=T)
   b <- apply(u, 1, mean)
   as.vector(b)
}

The bmstdr package includes the data set gridnysptime which contains the prediction data for 100 locations within the state of New York. Here is the code-chunk to perform prediction at these 100 locations and then averaging:

post <- M3$fit
gpred <- predict(post, newdata=gridnysptime, newcoords=~Longitude+Latitude)
u <- gpred$pred.samples
v <- apply(u, 2, sitemeans, sn=100)
a <- get_parameter_estimates(t(v)) 
b <- data.frame(gridnyspatial[, 1:5], a) 

The data frame b contains the location information and the prediction summaries at the 100 prediction sites. To draw the prediction map we also include the fitted values from the 28 data modeling sites. We extract the fitted values as follows:

meanmat <- post$op
sig2eps <-  post$sig2ep
sige <- sqrt(sig2eps)
itmax <- ncol(meanmat)
nT <- nrow(nysptime)
sigemat <- matrix(rep(sige, each=nT), byrow=F, ncol=itmax)
a <- matrix(rnorm(nT*itmax), nrow=nT, ncol=itmax)
ypreds <- meanmat + a * sigemat
ypreds <-  (ypreds)^2
v <- apply(ypreds, 2, sitemeans, sn=28)
a <- get_parameter_estimates(t(v)) 
fits <- data.frame(nyspatial[, 1:5], a)

Finally we combine the predictions and the fitted values by the command:

b <- rbind(b, fits)

Now we can obtain the average pollution map by using the linear interpolation function interp in the akima library. Also we discard the interpolations outside the state of New York by using the function fnc.delete.map.XYZ included in the bmstdr package.

coord <- nyspatial[, c("Longitude","Latitude")]
library(akima)
xo <- seq(from=min(coord$Longitude)-0.5, to = max(coord$Longitude)+0.8, length=200)
yo <- seq(from=min(coord$Latitude)-0.25, to = max(coord$Latitude)+0.8, length=200)
surf <- interp(b$Longitude, b$Latitude, b$mean,  xo=xo, yo=yo)
v <- fnc.delete.map.XYZ(xyz=surf)

interp1 <- data.frame(long = v$x, v$z )
names(interp1)[1:length(v$y)+1] <- v$y
library(tidyr)
interp1 <- gather(interp1,key = lat,value =Predicted,-long,convert = TRUE)
library(ggplot2)
nymap <- map_data(database="state",regions="new york")
mappath <- cbind(nymap$long, nymap$lat)
zr <- range(interp1$Predicted, na.rm=T)
P <- ggplot() +  
geom_raster(data=interp1, aes(x = long, y = lat,fill = Predicted)) +
geom_polygon(data=nymap, aes(x=long, y=lat, group=group), color="black", size = 0.6, fill=NA) + 
geom_point(data=coord, aes(x=Longitude,y=Latitude))  +
stat_contour(data=na.omit(interp1), aes(x = long, y = lat,z = Predicted), colour = "black", binwidth =2) +
scale_fill_gradientn(colours=colpalette, na.value="gray95", limits=zr) +
theme(axis.text = element_blank(), axis.ticks = element_blank()) +
ggsn::scalebar(data =interp1, dist = 100, location = "bottomleft", transform=T, dist_unit = "km", st.dist = .05, st.size = 5, height = .06, st.bottom=T, model="WGS84") +
ggsn::north(data=interp1, location="topleft", symbol=12) +
labs(x="Longitude", y = "Latitude", size=2.5) 

Figure 3.3: Predicted map of average ozone air pollution in New York

Similar methods are used to obtain a map of the standard deviations of the predictions which has been omitted for brevity.

3.5 Marginal model fitting with the rstan package

The option package="stan" can be used to fit the only available model, which is a marginalized GP model, like (2.4) independently at each time point. Details for this model are provided in Chapter 7 of the book Sahu (2022). In this implementation the spatial decay parameter \(\phi\) is sampled and its estimate is made available in the parameter estimates table.

We illustrate model fitting by comparing the stan fitted model with the three previously fitted models. The commands for fitting all four models are:

M1.c <- Bsptime(model="lm", formula=f2, data=nysptime, 
          scale.transform = "SQRT", mchoice=T)
M2.c <- Bsptime(model="separable",  formula=f2, data=nysptime, 
          coordtype="utm", coords=4:5, phi.s=0.005, phi.t=0.05, 
          scale.transform = "SQRT", mchoice=T)
M3.c <- Bsptime(package="spTimer", model="GP", 
        formula=f2, data=nysptime, coordtype="utm", 
        coords=4:5, scale.transform = "SQRT", 
        mchoice=T, N=5000)
M4.c <- Bsptime(package="stan",formula=f2, data=nysptime, 
        coordtype="utm", coords=4:5, scale.transform = "SQRT", 
        N=1500, burn.in=500, mchoice=T, verbose = F)

The above commands have been executed to produce Table 3.2 of model choice criteria for the four models M1 to M4.

3.6 Autoregressive model fitting with spTimer and INLA

Temporal auto regressive (AR) models can be fitted using both the spTimer and INLA package. The command for fitting the AR model using the spTimer package requires the additional option model="AR":

M5 <- Bsptime(package="spTimer", model="AR", formula=f2, data=nysptime, 
                coordtype="utm", coords=4:5, scale.transform = "SQRT", 
                mchoice=T,  validrows = vrows)

To fit and validate with the AR model using the INLA package we issue the command:

M6 <- Bsptime(package="inla", model="AR", formula=f2, data=nysptime, 
        coordtype="utm", coords=4:5, scale.transform = "SQRT", 
        mchoice=T, validrows=vrows)

The AR models M5 and M6 produces the following model choice results in Table 3.4.

Parameter estimates from the two models are shown in Table 3.6. The INLA implemented model autoregressive model M6 has been fitted without an intercept as this has been seen to be better than spTimer model M5. Other parameter estimates are somewhat comparable but there are large differences also, see e.g., the estimates of the autoregressive parameter \(\rho\) and the spatial decay parameter \(\phi\). These differences are expected as the two packages use different model parameterisations and prior distributions. The current package bmstdr facilitates this sort of model comparison without much additional programming effort.

3.7 Spatio-temporal dynamic models using the spTDyn package

The spTDyn package can be used to fit dynamic models in both time and space. A spatial dynamic model allows the regression coefficient to vary in space. A temporal dynamic model allows the regression coefficient to have a dynamic equation in time. To invoke these dynamic models the regression variables are specified in the formula argument with the optional enclosures sp for spatially dynamic and tp for temporally dynamic. Here is an example where the model is specified to have spatially varying effects of maximum temperature and dynamic regression coefficients for wind speed.

library(spTDyn)
f3 <- y8hrmax~ xmaxtemp + sp(xmaxtemp)+ tp(xwdsp) + xrh
M7 <- Bsptime(package="sptDyn", model="GP", formula=f3, data=nysptime, 
      coordtype="utm", coords=4:5, scale.transform = "SQRT", n.report=2)

The model fitting results can be examined by the S3 methods functions as before. Here we explore the spatially varying regression coefficients as follows:

out <- M7$fit
dim(out$betasp)
#> [1]   28 1000
a <- out$betasp
u <- c(t(out$betasp))
sn <- nrow(a)
itmax <- ncol(a)
v <- rep(1:sn, each=itmax)
d <- data.frame(site=as.factor(v), sp = u)
p <- ggplot(data=d, aes(x=site, y=sp)) + 
   geom_boxplot(outlier.colour="black", outlier.shape=1,
                outlier.size=0.5) +
   geom_abline(intercept=0, slope=0, color="blue") + 
   labs(title= "Spatial effects of maximum temperature", x="Site", y = "Effects", size=2.5) 
p 

Figure 3.4: Spatial effects of maximum temperature from model M7

Similarly, we provide a plot of the temporal effects parameter.

b <- out$betatp
tn <- nrow(b)
itmax <- ncol(b)
tids <- 1:tn 
stat <- apply(b[tids,], 1, quantile, prob=c(0.025,0.5,0.975))
tstat <- data.frame(tids, t(stat))
dimnames(tstat)[[2]] <- c("Days", "low", "median", "up")
# head(tstat)
yr <- c(min(c(stat)),max(c(stat)))
p <- ggplot(data=tstat, aes(x=Days, y=median)) + 
   geom_point(size=3) + 
   ylim(yr) + 
   geom_segment(data=tstat, aes(x=Days, y=median, xend=Days, yend=low), linetype=1) +
   geom_segment(data=tstat, aes(x=Days, y=median, xend=Days, yend=up), linetype=1) +
   geom_abline(intercept=0, slope=0, col="blue") +
   labs(title="Temporal effects of wind speed", x="Days", y="Temporal effects") 
p 

Figure 3.5: Temporal effects of wind speed from model M7

3.8 Fitting dynamic models using spBayes

The package option package="spBayes" in the Bsptime function triggers model fitting by using a complex dynamic model due to Gelfand, Banerjee, and Gamerman (2005), see also Chapter 11 of the book Banerjee, Carlin, and Gelfand (2015). This model fitting is sensitive to the choice of the prior distributions and the options below leads to a reasonable model fit.

M8 <- Bsptime(package="spBayes",  formula=f2, data=nysptime, 
              prior.sigma2=c(2, 25),
              prior.tau2 =c(2, 25),
              prior.sigma.eta =c(2, 0.001),
              coordtype="utm", 
              coords=4:5, scale.transform = "SQRT", 
              mchoice=T,  N=5000,  n.report=200)

The dynamic model parameters are extracted using the code below.

modfit <- M8$fit
N <- 5000
burn.in <- 1000
tn <- 62
quant95 <- function(x){
   quantile(x, prob=c(0.025, 0.5, 0.975))
}
theta <- apply(modfit$p.theta.samples[burn.in:N,], 2, quant95)
sigma.sq <- theta[,grep("sigma.sq", colnames(theta))]
tau.sq <- theta[,grep("tau.sq", colnames(theta))]
phi <- theta[,grep("phi", colnames(theta))]

These extracted variance and the range (\(3/\phi\)) parameters are plotted using the ggplot function. The details are omitted from this document. We also do not provide parameter plots for the regression parameters.

Figure 3.6: Parameter estimates for the spBayes fitted spatio-temporal model M8

3.9 Autoregressive modeling using Gaussian predictive processes

This is a spatio-temporal model proposed by S. K. Sahu and Bakar (2012). This model is suitable for modeling temporal data from a large number of spatial locations. In addition to the fixed effects regression coefficients this model sets up random effects which are temporally autoregreesive but spatially based at a much smaller number of locations called the knots. These knots can be specified by a knots.coords argument in spTimer. In bmstdr the knots may also be specified by a g_size argument which may define a square or a rectangular equi-spaced grid covering the range of coordinates of the data locations.

The bmstdr command to fit a model with a square \(5 \times 5\) grid of knot locations is given by:

M9 <-  Bsptime(package="spTimer", model="GPP", g_size=5, 
               formula=f2, data=nysptime, n.report=5, 
               coordtype="utm", coords=4:5, scale.transform = "SQRT")

The grid size parameter g_size can be chosen by cross-validation methods as has been demonstrated in Sahu (2022). The model fitted object M9 can be examined using the suite of S3 functions as before.

3.10 Comparing all the spatio-temporal models

The bmstdr package function can be used compare all the model fits and the performance of the models as evaluated by the four validation statistics. We set aside all the data from the eight sites as noted previously. This gives us 496 (\(=8 \times 62\)) data points the validation set and the model is fitted with remaining 1240 space time observations from the 20 modeling sites. Table 3.8 reports the results.

In the above table we have omitted M8, the model based on spBayes because we were not able to produce comparable results. We do not report the WAIC and the DIC since those are not available in the bmstdr package for all the models. Sahu (2022) provides further discussion of the results.

We end this comparison with some words of caution. The comparison should not be generalized to make statements like package A performs better than package B. For example, the marginal GP model, M4, implemented using stan performed slightly worse than M9. But there may be another model, e.g. auto-regressive, implemented using stan, that may perform better than the spTimer models. The worth of this illustration lies in the comparison itself. Using the bmstdr package it is straightforward to compare different models implemented in different packages without having to learn and program the individual packages.

3.11 Modeling data from moving sensors in time

Sahu and Challenor (2008) model ocean temperature data from the roaming Argo floats in the North Atlantic Ocean. The locations of the roaming floats are seen in Figure 3.7.

atlmap <- map_data("world", xlim=c(-70, 10), ylim=c(15, 65))
atlmap <- atlmap[atlmap$long < 5, ]
atlmap <- atlmap[atlmap$long > -70, ]
atlmap <- atlmap[atlmap$lat < 65, ]
atlmap <- atlmap[atlmap$lat > 10, ]

argo <- argo_floats_atlantic_2003

deep <- argo[argo$depth==3, ]
deep$month <- factor(deep$month)

p <- ggplot() +
  geom_polygon(data=atlmap, aes(x=long, y=lat, group=group),
               color="black", size = 0.6, fill=NA) +
  geom_point(data =deep, aes(x=lon, y=lat, colour=month), size=1) +
  labs( title= "Argo float locations in deep ocean in 2003", x="Longitude", y = "Latitude") +
  ggsn::scalebar(data =atlmap, dist = 1000, location = "bottomleft", transform=T, dist_unit = "km",
                 st.dist = .05, st.size =5, height = .05, st.bottom=T, model="WGS84") +
  ggsn::north(data=atlmap, location="topright", symbol=12)
p

Figure 3.7: Locations of moving Argo floats in the deep ocean in 2003.

This data set is included in the package as the object argo_floats_atlantic_2003. In this section we model the temperature data only at the deep ocean using a marginalized Gaussian Process model \[\begin{equation} {\bf Y}_t \sim N\left({\bf X}_t {\bf \beta}, \ \ \sigma^2_{w} A_t S_w A_t' + \sigma^2_{\epsilon} I\right), t=1, \ldots, T \tag{3.2} \end{equation}\] where \({\bf Y}_t\) and \({\bf X}_t\) are the vector of observations and covariate values at the \(n_t\) locations at time \(t\). Here \(A_{t} = C_{t} S_{w}^{-1}\), where \(S_{w}\) is \(m \times m\) and has elements induced by the GP and \(C_t\) is \(n_t \times m\) having the \(j\)th row and \(k\)th column entry \(\exp(-\phi |{\bf s}_j - {\bf s}_k^*|)\) for \(j=1, \ldots, n_t\) and \(k=1, \ldots, m\). Thus, \(C_{t}\) captures the cross-correlation between the observation locations at time \(t\) and the \(m\) knot locations, \({\bf s}_k^*, k=1, \ldots, m\).

Assuming that the regression model formula is the object f2 and the subset data for the deep ocean is deep, we use the command

Natl <- 110
Nburn <- 10
options(warn = -1) 
M2atl <- Bmoving_sptime(formula=f2, data = deep, coordtype="lonlat", 
  coords = 1:2, N=Natl, burn.in=Nburn, validrows =NULL, mchoice =F)

to fit model (3.2) implemented by code written in rstan. Like the Bsptime function this model fitting function also renders a bmstdr object which can be further investigated.

The model output is not shown here. But we obtain an annual prediction map by averaging over the model equation (3.2) at each data location. The posterior samples contained in M2 are used to sample the annual predictions and then those samples are linearly interpolated using the akima library to obtain the prediction map in Figure 3.8.

Figure 3.8: A map of predicted temperatures in the deep ocean in 2003

4.1 Two illustration data sets on Covid-19 mortality from England

The engtotals data set presents the number of deaths due to Covid-19 during the peak from March 13 to July 31, 2020 in the 313 Local Authority Districts, Counties and Unitary Authorities in England. Sahu and Böhning (2021) provides further details of the data set and maps of the local areas.

The engdeaths data set contains 49,292 weekly recorded deaths during this period of 20 weeks. The boxplot of the weekly death rates shows the first peak during weeks 15 and 16 (April 10th to 23rd) and a very slow decline of the death numbers after the peak. The main purpose here is to model the spatio-temporal variation in the death rates.

engdeaths$covidrate <- 100000*engdeaths$covid/engdeaths$popn
ptime <- ggplot(data=engdeaths,  aes(x=factor(Weeknumber), y=covidrate)) +
  geom_boxplot() +
  labs(x = "Week", y = "Death rate per 100,000")  +
  stat_summary(fun=median, geom="line", aes(group=1, col="red")) +
  theme(legend.position = "none")
ptime

Figure 4.1: Weekly Covid-19 death rate per 100,000

4.2 The Bcartime function for fitting CAR models

The bmstdr package function Bcartime fits a variety of spatial and spatio-temporal models for areal data. These models are based on the generalized linear models with one of binomial, Poisson and Gaussian error distributions and with the canonical link in each case. Chapter 10 of the book by Sahu (2022) describe the models. The fitted output can be explored using the S3 methods functions as in the case of Bspatial and Bsptime for modeling point reference spatial data. More details are provided below.

To fit the Bayesian GLMs without any random effects Bcartime employs the S.glm function of the CARBayes package Lee (2021). Deploying the Bcartime function requires the following essential arguments:

• package can take one of three possible values: "CARBayes", "CARBayesST" or "inla". The default is "CARBayes".
• model defines the specific spatio temporal model to be fitted. If the package is "inla" then the model argument should be a vector with two elements giving the spatial model, e.g. "bym" as the first component and the temporal model which could be one of "iid", "ar1" or "none" as the second component. In case the second component is “none” then no temporal random effects will be fitted. No temporal random effects will be fitted in case model is supplied as a singleton.
• formula specifying the response and the covariates for forming the linear predictor \(\eta\) in a GLM.
• data containing the data set to be used; family being one of either "binomial", "poisson" "gaussian", "multinomial", or "zip". In this illustration we only consider the first three choices. If the binomial family is chosen, the trials argument must be provided. This should be a numeric vector containing the number of for each row of data.
• scol Either the name (character) or number of the column in the supplied data frame identifying the spatial units. The program will try to access data[, scol] to identify the spatial units. If this is omitted, no spatial modeling will be performed, instead an independent error GLM will be fitted using the "CARBayes" package.
• tcol Like the scol argument but for the time identifier. Either the name (character) or number of the column in the supplied data frame identifying the time indices. The program will try to access data[, tcol] to identify the time points. If this is omitted, no temporal modeling will be performed.
• W A non-negative K by K neighborhood matrix (where K is the number of spatial units). Typically a binary specification is used, where the \(jk\)th element equals one if areas (j, k) are spatially close (e.g. share a common border) and is zero otherwise. The matrix can be non-binary, but each row must contain at least one non-zero entry. This argument may not need to be specified if adj.graph is specified instead.
• adj.graph Adjacency graph which may be specified instead of the adjacency matrix matrix. This argument is used if W has not been supplied. The argument W is used in case both W and adj.graph are supplied.

There are numerous other arguments specifying more details of the models and the prior distributions. Those are documented in the help file ?Bcartime.

Like the Bsptime function, model validation is performed automatically by specifying the optional vector valued validrows argument containing the row numbers of the supplied data frame that should be used for model validation. As before, the user does not need to modify the data set for validation. This task is done by the Bcartime function.

The function Bcartime automatically chooses the default prior distributions which can be modified by the many optional arguments, see the documentation of this function and also the S.glm function from CARBayes. Three MCMC control parameters N, burn.in and thin determine the number of iterations, burn-in and thinning interval. The default values of these are 2000, 1000 and 10 respectively. In all of our analysis in this section, unless otherwise mentioned, we take these to be 50000, 10000 and 10 respectively.

Ncar <- 50000
burn.in.car <- 10000
thin <- 10

4.3 Modeling static areal unit data

In this section we model the static engtotals data set. Here we employ the conditionally auto regressive (CAR) models for the spatial random effects.

f1 <- noofhighweeks ~ jsa + log10(houseprice) + log(popdensity) + sqrt(no2)

M1 <- Bcartime(formula=f1,   data=engtotals, family="binomial",
trials=engtotals$nweek, N=Ncar, burn.in=burn.in.car, thin=thin) 

M1.leroux <- Bcartime(formula=f1, data=engtotals, scol="spaceid", 
model="leroux", W=Weng, family="binomial", trials=engtotals$nweek, 
N=Ncar, burn.in=burn.in.car, thin=thin)

M1.bym <- Bcartime(formula=f1, data=engtotals, 
scol="spaceid", model="bym", W=Weng, family="binomial", 
trials=engtotals$nweek, N=Ncar, burn.in=burn.in.car, thin=thin)

M1.inla.bym <- Bcartime(formula=f1, data=engtotals, scol ="spaceid", 
model=c("bym"),  W=Weng, family="binomial", trials=engtotals$nweek,
package="inla", N=Ncar, burn.in=burn.in.car, thin=thin) 

a <- rbind(M1$mchoice, M1.leroux$mchoice, M1.bym$mchoice)
a <- a[, -(5:6)]
a <- a[, c(2, 1, 4, 3)]
b <- M1.inla.bym$mchoice[1:4]
a <- rbind(a, b)
rownames(a) <- c("Independent", "Leroux", "BYM", "INLA-BYM")
colnames(a) <- c("pDIC", "DIC", "pWAIC",  "WAIC") 
table4.1 <- a
dput(table4.1, file=paste0(tablepath, "/table4.1.txt"))

f2 <-  covid ~ offset(logEdeaths) + jsa + log10(houseprice) + log(popdensity) + sqrt(no2) 

M2 <- Bcartime(formula=f2, data=engtotals, family="poisson",
              N=Ncar, burn.in=burn.in.car, thin=thin)

M2.leroux <- Bcartime(formula=f2, data=engtotals,
            scol="spaceid",  model="leroux",  family="poisson", W=Weng,
            N=Ncar, burn.in=burn.in.car, thin=thin)

M2.bym <- Bcartime(formula=f2, data=engtotals,
                   scol="spaceid",  model="bym",  family="poisson", W=Weng,
                   N=Ncar, burn.in=burn.in.car, thin=thin)

M2.inla.bym <- Bcartime(formula=f2, data=engtotals, scol ="spaceid",  
                        model=c("bym"), family="poisson", 
                        W=Weng, offsetcol="logEdeaths", link="log", 
                          package="inla", N=Ncar, burn.in = burn.in.car, thin=thin) 

f3 <-  sqrt(no2) ~  jsa + log10(houseprice) + log(popdensity) 

M3 <- Bcartime(formula=f3, data=engtotals, family="gaussian",
               N=Ncar, burn.in=burn.in.car, thin=thin)

M3.leroux <- Bcartime(formula=f3, data=engtotals,
                      scol="spaceid",  model="leroux",  family="gaussian", W=Weng,
                      N=Ncar, burn.in=burn.in.car, thin=thin)


M3.inla.bym <- Bcartime(formula=f3, data=engtotals, scol ="spaceid",  
                        model=c("bym"), family="gaussian", 
                        W=Weng,  package="inla", N=Ncar, burn.in =burn.in.car, thin=thin) 

a <- rbind(M3$mchoice, M3.leroux$mchoice)
a <- a[, -(5:6)]
a <- a[, c(2, 1, 4, 3)]
b <- M3.inla.bym$mchoice[1:4]
a <- rbind(a, b)
rownames(a) <- c("Independent", "Leroux",  "INLA-BYM")
colnames(a) <- c("pDIC", "DIC", "pWAIC",  "WAIC") 
table4.3 <- a
dput(table4.3, file=paste0(tablepath, "/table4.3.txt"))

4.4 Modeling temporal areal unit data

The data set used in this example is the engdeaths data set described earlier. In this section we will modify the Bcartime commands presented earlier to fit all the spatio-temporal models discussed in Chapter 10 of Sahu (2022). We will illustrate model fitting, choice and validation using the binomial, Poisson and normal distribution based models as before in the previous section. The user does not need to write any direct code for fitting the models using the CARBayesST package. The Bcartime function does this automatically and returns the fitted model object in its entirety and in addition, performs model validation for the named rows of the supplied data frame as passed on by the validrows argument.

The previously documented arguments of Bcartime for spatial model fitting remain the same for the corresponding spatio-temporal models. For example, the arguments formula, family, trials, scol and W are unchanged in spatial-temporal model fitting. The data argument is changed to the spatio-temporal data set data=engdeaths. We keep the MCMC control parameters N, burn.in and thin to be same as before.

The additional arguments are tcol, similar to scol, which identifies the temporal indices. Like the scol argument this may be specified as a column name or number in the supplied data frame.
The package argument must be specified as package="CARBayesST" to change the default CARBayes package. The model argument should be changed to one of four models, "linear", "anova", "sepspatial" and "ar". Other possibilities for this argument are "localised",“multilevel”and“dissimilarity”`, but those are not illustrated here. For the sake of brevity it is undesirable to report parameter estimates of all the models. Instead, below we report only selected results.

nweek <- rep(1, nrow(engdeaths))

f1 <- highdeathsmr ~  jsa + log10(houseprice) + log(popdensity) 

M1st <- Bcartime(formula=f1, data=engdeaths, scol=scol, tcol=tcol, trials=nweek, 
W=Weng, model="linear", family="binomial", package="CARBayesST",  N=Ncar,
burn.in=burn.in.car, thin=thin)

4.4.2 Spatio-temporal GLM fitting with Poisson distribution

For fitting the Poisson distribution based model we take the response variable as the column covid, which records the number of Covid-19 deaths, of the engdeaths data set. The column logEdeaths is used as an offset in the model with the default log link function.

The formula argument for the regression part of the linear predictor is chosen to be the same as the one used by Sahu and Böhning (2021) for a similar data set. The formula contains, in addition to the thee socio-economic variables, the log of the SMR for the number cases in the current week and three previous weeks denoted by n0, n1, n2 and n3. The formula is given below:

f2 <-  covid ~ offset(logEdeaths) + jsa + log10(houseprice) + log(popdensity) + n0 + n1 + n2 + n3

We now fit the Poisson model by keeping the other arguments same as before in the previous Section. The command for fitting the temporal auto-regressive model is:

The model argument can be changed to fit the other models. The resulting model fits are used to obtain the model choice statistics presented in Table 4.10.

Table 4.9: Table 4.10: Model choice criteria values for the spatio-temporal Poisson model.

Model	pDIC	DIC	pWAIC	WAIC
Linear	446.69	33586.19	1268.00	34622.01
Anova (without interaction)	243.24	34177.79	783.83	34801.49
Anova (with interaction)	3070.36	27771.87	2316.68	27725.65
Separable	2931.26	27722.78	2249.90	27710.16
AR (1)	2992.32	27711.43	2268.62	27670.95
AR (2)	2217.45	26921.31	1832.49	27044.86
INLA-BYM	208.72	25059.87	246.93	25112.86

To investigate the differences in model choice by DIC and WAIC we test both models for validation. We randomly select 10% data rows for validation by issuing the command

vs <- sample(nrow(engdeaths), 0.1*nrow(engdeaths))

This gives us 626 data points for validation purposes. We then refit the Anova model with interaction and both the AR (1) and AR (2) models and also the INLA based model by supplying the additional argument validrows=vs. The validation statistics are presented in Table 4.11. These statistics show that the INLA based model has less bias than the CARBayesST models but it fails to capture the full variability of the set aside data as its coverage percentage is very low. Figure 4.2 highlights this problem. The prediction intervals are too narrow for the INLA based model but AR (2) model gets this uncertainty exactly as expected. Again, we end this comparison with a word of caution that INLA is merely a computing platform and there can be other models which can achieve better coverage than the one reported here.

Table 4.11: Table 4.12: Model validation statistics for the Anova and AR model.

Model	rmse	mae	crps	cvg
Anova	5.73	3.34	2.52	98.24
AR (1)	5.94	3.36	3.09	97.92
AR (2)	5.75	3.03	2.11	95.37
INLA-BYM	3.13	1.98	0.29	22.20

f20 <-  covid ~ offset(logEdeaths) + jsa + log10(houseprice) + log(popdensity) + n0
model <- c("bym", "ar1")
f2inla <-  covid ~  jsa + log10(houseprice) + log(popdensity) + n0 
set.seed(5)
vs <- sample(nrow(engdeaths), 0.1*nrow(engdeaths))
M2st_ar2.0 <- Bcartime(formula=f20, data=engdeaths, scol="spaceid", tcol= "Weeknumber",  
                       W=Weng, model="ar", AR=2, family="poisson", package="CARBayesST", 
                       N=Ncar, burn.in=burn.in.car, thin=thin, 
                       validrows=vs, verbose=F)
M2stinla.0  <- Bcartime(data=engdeaths, formula=f2inla, W=Weng, scol ="spaceid", tcol="Weeknumber",  
                        offsetcol="logEdeaths",  model=model,  link="log", family="poisson", package="inla", validrow=vs, N=N, burn.in=0) 
yobspred <- M2st_ar2.0$yobs_preds
names(yobspred)
yobs <- yobspred$covid
predsums <- get_validation_summaries(t(M2st_ar2.0$valpreds))
dim(predsums)
b <- obs_v_pred_plot(yobs, predsums, segments=T) 
names(M2stinla.0)
inlapredsums <- get_validation_summaries(t(M2stinla.0$valpreds))
dim(inlapredsums)
a <- obs_v_pred_plot(yobs, inlapredsums, segments=T) 
inlavalid <- a$pwithseg
ar2valid <- b$pwithseg
library(ggpubr)
ggarrange(ar2valid, inlavalid, common.legend = TRUE, legend = "top", nrow = 2, ncol = 1)
ggsave(filename = paste0(figpath, "/inlavAR2.png"))

Figure 4.2: Predictions with 95% limits against observations for two models: AR (2) on the left panel and INLA on the right panel

M3st <- Bcartime(formula=f3, data=engdeaths, scol=scol, tcol=tcol, 
W=Weng, model="ar", family="gaussian", package="CARBayesST", 
N=Ncar, burn.in=burn.in.car, thin=thin)