Contents

1 Introduction

Model-centric analysis of spatial and spatio-temporal data is essential in many applied areas of research such as atmospheric sciences, climatology, ecology, environmental health and oceanography. Such diversity in application areas is being serviced by the rich diversity of R contributed packages listed in the abstract and many others, see the CRAN Task Views: Handling and Analyzing Spatio-Temporal Data and Analysis of Spatial Data. Moreover, there are a number of packages and text books discussing handling of spatial and spatio-temporal data. For example, see the references E. J. Pebesma and Bivand (2005), Bivand, Pebesma, and Gomez-Rubio (2013), E. Pebesma (2012), Millo and Piras (2012), Banerjee, Carlin, and Gelfand (2015), Wikle, Zammit-Mangion, and Cressie (2019) and Sujit K. Sahu (2022). The diversity in packages, however, is also a source of challenge for an applied scientist who is also interested in exploring solutions offered by other models from rival packages. The challenge comes from the essential requirement to learn the package specific commands and setting up of the prior distributions that are to be used for the applied problem at hand.

The current package bmstdr sets out to help researchers in applied sciences model a large variety of spatial and spatio-temporal data using a multiplicity of packages but by using only three commands with different options. Point reference spatial data, where each observation comes with a single geo-coded location reference such as a latitude-longitude pair, can be analyzed by fitting several spatial and spatio-temporal models using R software packages such as spBayes (Finley, Banerjee, and Gelfand 2015), spTimer (Khandoker S. Bakar and Sahu 2015), INLA (Rue, Martino, and Chopin 2009), rstan (Stan Development Team 2020), and spTDyn (Khandoker Shuvo Bakar, Kokic, and Jin 2016).
A particular package is chosen with the package= option to the bmstdr model fitting routines Bspatial for point reference spatial only data and Bsptime for spatio-temporal data. In each of these cases a Bayesian linear model, which can be fitted with the option package="none" provides a base line for model comparison purposes. For areal unit data modeling the bmstdr function Bcartime provides opportunities for model fitting using three packages: CARBayes (D. Lee 2021), CARBayesST (Duncan Lee, Rushworth, and Napier 2018) and INLA (Blangiardo and Cameletti 2015). Here also a base line Bayesian generalized linear model for independent data, fitted using CARBayes, is included for model comparison purposes.

Models fitted using bmstdr can be validated using the optional argument validrows, which can be a vector of row numbers of the model fitting data frame, to any of the three model fitting functions. The package then automatically sets aside the nominated data rows as specified by the validrows argument and use the remaining data rows for model fitting. Inclusion of this argument also automatically triggers calculation of four popular model validation statistics: root mean square error, mean absolute error, continuous ranked probability score (Gneiting and Raftery 2007) and coverage percentage. While performing validation the package also produces a scatter plot of predictions against observations with further options controlling the behavior of this plot.

The generality of the platforms INLA and rstan allows tremendous flexibility in modeling and validation. However, bespoke code must be written to implement each different model. The current version of bmstdr provides a limited number of models which can be fitted using INLA and rstan. For point reference spatial only data a marginal model, after integrating the spatial random effects, with a known exponential correlation function is implemented using rstan and for spatio-temporal point reference data a marginal model has also been implemented using this package. A marginal model has also been implemented in a separate bmstdr model fitting function called Bmoving_sptime which facilitates modeling of point reference temporal data from moving sensors such as Argo floats in the oceans, see Section 3.11. INLA based models use a discretized Gaussian Markov random field with penalized complexity prior distribution (Fuglstad et al. 2018).

For areal unit data modeling the bmstdr function Bcartime provides opportunities for selected model fitting using CARBayes (D. Lee 2021), and CARBayesST (Duncan Lee, Rushworth, and Napier 2018) CARBayesST(Duncan Lee, Rushworth, and Napier 2018) and also INLA as illustrated by Blangiardo and Cameletti (2015). Here also a base line Bayesian generalized linear model for independent data, fitted using CARBayes, is included for model comparison purposes. The INLA based models can fit the celebrated Besag, York, and Mollié (1991) model and the Leroux model (Leroux, Lei, and Breslow 2000).

Models fitted using bmstdr can be validated using the optional argument validrows, which can be a vector of row numbers of the model fitting data frame, to any of the three model fitting functions. The package then automatically sets aside the nominated data rows as specified by the validrows argument and use the remaining data rows for model fitting. Inclusion of this argument also automatically triggers calculation of four popular model validation statistics: root mean square error, mean absolute error, continuous ranked probability score (Gneiting and Raftery 2007) and coverage percentage. While performing validation, the package also produces a scatter plot of predictions against observations with further options controlling the behavior of this plot.

The remainder of this vignette is organized as follows. Section 2 illustrates point reference spatial data modeling with Gaussian error distribution. Section 3 discusses Gaussian models for point reference spatio-temporal data. Area data are modeled in Section 4 where Section 4.3 illustrates models for static areal unit data and Section 4.4 considers areal temporal data. Some summary remarks are provided in Section 5.

2 Point reference spatial data modeling

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 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.

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

Figure 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 Sujit K. 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: \[\begin{equation} Y_i = \beta_1 x_{i1} + \ldots + \beta_p x_{ip} + \epsilon_i, i=1, \ldots, n \tag{1} \end{equation}\] 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 (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 (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 \(i, j=1, \ldots,n\), \[\begin{equation} {\bf Y} \sim N_n \left(X{\mathbf \beta}, \sigma^{2}_{\epsilon} H\right) \tag{2} \end{equation}\] 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) 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: \[\begin{equation} Y({\bf s}_i) = {\bf x}'({\bf s}_i) \mathbf{\beta} + w({\bf s}_i) + \epsilon( {\bf s}_i) \tag{3} \end{equation}\] 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 Sujit K. 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 \[\begin{align} {\bf Y} & \sim N\left(X{\mathbf \beta}, \sigma^2_{\epsilon} \, I + \sigma^2_w S_w \right), \tag{4} \end{align}\] 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 1.

Table 1: Model choice criteria for various models fitted to the nyspatial data set.
CriteriaM0M1M2M3M4M5
pdic2.074.994.985.175.314.17
pdicalt13.585.175.167.836.46   
dic169.20158.36158.06158.68158.75157.23
dicalt192.22158.72158.41163.99161.04   
pwaic11.825.204.934.884.934.73
pwaic22.526.325.916.775.96   
waic1168.95158.57157.51158.70157.92158.46
waic2170.35160.82159.47162.48159.99   
gof591.82327.98330.08323.56316.67334.03
penalty577.13351.52346.73396.63394.8639.17
pmcc1168.95679.50676.82720.18711.52373.19

Mathematical expressions for all the quantities in the above table are provided in Sujit K. 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 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.

Table 2: Model validation statistics for the five models fitted to the nyspatial data set.
CriteriaM1M2M3M4M5
rmse2.4472.4002.4282.4222.392
mae2.1352.0152.0432.0541.985
crps2.8912.8852.8912.0372.324

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 3 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.

Table 3: 4-fold cross-validation statistics for model M2 fitted to the nyspatial data set.
CriteriaFold1Fold2Fold3Fold4
rmse2.4415.0055.8652.508
mae1.7893.5455.4622.145
crps2.0852.0771.2282.072
cvg100.00085.71457.143100.000

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, plotit=FALSE)

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 Point reference spatio-temporal data modeling

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 S. K. 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 (3) to the following spatio-temporal model. \[\begin{equation} Y({\bf s}_i, t) = {\bf x}'({\bf s}_i, t) \mathbf{\beta} + w({\bf s}_i, t) + \epsilon( {\bf s}_i, t) \tag{5} \end{equation}\] 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 Sujit K. 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