Chapter 9 Spatial models

NIMBLE supports two variations of conditional autoregressive (CAR) model structures: the improper intrinsic Gaussian CAR (ICAR) model, and a proper Gaussian CAR model. This includes distributions to represent these spatially-dependent model structures in a BUGS model, as well as specialized MCMC samplers for these distributions.

9.1 Intrinsic Gaussian CAR model: dcar_normal

The intrinsic Gaussian conditional autoregressive (ICAR) model used to model dependence of block-level values (e.g., spatial areas or temporal blocks) is implemented in NIMBLE as the dcar_normal distribution. Additional details for using this distribution are available using help('CAR-Normal').

ICAR models are improper priors for random fields (e.g., temporal or spatial processes). The prior is a joint prior across a collection of latent process values. For more technical details on CAR models, including higher-order CAR models, please see Rue and Held (2005), Banerjee, Carlin, and Gelfand (2015), and Paciorek (2009). Since the distribution is improper it should not be used as the distribution for data values, but rather to specify a prior for an unknown process. As discussed in the references above, the distribution can be seen to be a proper density in a reduced dimension subspace; thus the impropriety only holds on one or more linear combinations of the latent process values.

In addition to our focus here on CAR modeling for spatial data, the ICAR model can also be used in other contexts, such as for temporal data in a discrete time context.

9.1.1 Specification and density

NIMBLE uses the same parameterization as WinBUGS / GeoBUGS for the dcar_normal distribution, providing compatibility with existing WinBUGS code. NIMBLE also provides the WinBUGS name car.normal as an alias.

9.1.1.1 Specification

The dcar_normal distribution is specified for a set of N spatially dependent regions as:

x[1:N] ~ dcar_normal(adj, weights, num, tau, c, zero_mean)

The adj, weights and num parameters define the adjacency structure and associated weights of the spatially-dependent field. See help('CAR-Normal') for details of these parameters. When specifying a CAR distribution, these parameters must have constant values. They do not necessarily have to be specified as constants when creating a model object using nimbleModel, but they should be defined in a static way: as right-hand-side only variables with initial values provided as constants, data or inits, or using fixed numerical deterministic declarations. Each of these two approaches for specifying values are shown in the example.

The adjacency structure defined by adj and the associated weights must be symmetric. That is, if region \(i\) is neighbor of region \(j\), then region \(j\) must also be a neighbor of region \(i\). Further, the weights associated with these reciprocating relationships must be equal. NIMBLE performs a check of these symmetries and will issue an error message if asymmetry is detected.

The scalar precision tau may be treated as an unknown model parameter and itself assigned a prior distribution. Care should be taken in selecting a prior distribution for tau, and WinBUGS suggests that users be prepared to carry out a sensitivity analysis for this choice.

When specifying a higher-order CAR process, the number of constraints c can be explicitly provided in the model specification. This would be the case, for example, when specifying a thin-plate spline (second-order) CAR model, for which c should be 2 for a one-dimensional process and 3 for a two-dimensional (e.g., spatial) process, as discussed in Rue and Held (2005) and Paciorek (2009). If c is omitted, NIMBLE will calculate c as the number of disjoint groups of regions in the adjacency structure, which implicitly assumes a first-order CAR process for each group.

By default there is no zero-mean constraint imposed on the CAR process, and thus the mean is implicit within the CAR process values, with an implicit improper flat prior on the mean. To avoid non-identifiability, one should not include an additional parameter for the mean (e.g., do not include an intercept term in a simple CAR model with first-order neighborhood structure). When there are disjoint groups of regions and the constraint is not imposed, there is an implicit distinct improper flat prior on the mean for each group, and it would not make sense to impose the constraint since the constraint holds across all regions. Similarly, if one sets up a neighborhood structure for higher-order CAR models, it would not make sense to impose the zero-mean constraint as that would account for only one of the eigenvalues that are zero. Imposing this constraint (by specifying the parameter zero_mean = 1) allows users to model the process mean separately, and hence a separate intercept term should be included in the model.

NIMBLE provides a convenience function as.carAdjacency for converting other representations of the adjacency information into the required adj, weights, num format. This function can convert:

  • A symmetric adjacency matrix of weights (with diagonal elements equal to zero), using as.carAdjacency(weightMatrix)
  • Two length-N lists with numeric vector elements giving the neighboring indices and associated weights for each region, using as.carAdjacency(neighborList, weightList)

These conversions should be done in R, and the resulting adj, weights, num vectors can be passed as constants into nimbleModel.

9.1.1.2 Density

For process values \(x = (x_1, \ldots, x_N)\) and precision \(\tau\), the improper CAR density is given as:

\[p(x | \tau) \propto \tau^{(N-c)/2} \; e^{ -\tfrac{\tau}{2} \sum_{i\ne j} w_{ij} \, (x_i-x_j)^2 }\]

where the summation over all \((i,j)\) pairs, with the weight between regions \(i\) and \(j\) given by \(w_{ij}\), is equivalent to summing over all pairs for which region \(i\) is a neighbor of region \(j\). Note that the value of \(c\) modifies the power to which the precision is raised, accounting for the impropriety of the density based on the number of zero eigenvalues in the implicit precision matrix for \(x\).

For the purposes of MCMC sampling the individual CAR process values, the resulting conditional prior of region \(i\) is:

\[p(x_i | x_{-i}, \tau) \sim \text{N} \left( \tfrac{1}{w_{i+}} \textstyle\sum_{j \in \mathcal{N}_i } w_{ij} \, x_j, \; w_{i+} \tau \right)\]

where \(x_{-i}\) represents all elements of \(x\) except \(x_{i}\), the neighborhood \(\mathcal{N}_i\) of region \(i\) is the set of all \(j\) for which region \(j\) is a neighbor of region \(i\), \(w_{i+} = \sum_{j \in \mathcal{N}_i} w_{ij}\), and the Normal distribution is parameterized in terms of precision.

9.1.2 Example

Here we provide an example model using the intrinsic Gaussian dcar_normal distribution. The CAR process values are used in a spatially-dependent Poisson regression.

To mimic the behavior of WinBUGS, we specify zero_mean = 1 to enforce a zero-mean constraint on the CAR process, and therefore include a separate intercept term alpha in the model. Note that we do not necessarily recommend imposing this constraint, per the discussion earlier in this chapter.

code <- nimbleCode({
    alpha ~ dflat()
    beta ~ dnorm(0, 0.0001)
    tau ~ dgamma(0.001, 0.001)
    for(k in 1:L)
        weights[k] <- 1
    s[1:N] ~ dcar_normal(adj[1:L], weights[1:L], num[1:N], tau, zero_mean = 1)
    for(i in 1:N) {
        log(lambda[i]) <- alpha + beta*x[i] + s[i]
        y[i] ~ dpois(lambda[i])
    }
})

constants <- list(N = 4, L = 8, num = c(3, 2, 2, 1),
                  adj = c(2,3,4,1,3,1,2,1), x = c(0, 2, 2, 8))
data <- list(y = c(6, 9, 7, 12))
inits <- list(alpha = 0, beta = 0, tau = 1, s = c(0, 0, 0, 0))
Rmodel <- nimbleModel(code, constants, data, inits)

The resulting model may be carried through to MCMC sampling. NIMBLE will assign a specialized sampler to the update the elements of the CAR process. See Chapter 7 for information about NIMBLE’s MCMC engine, and Section 9.3 for details on MCMC sampling of the CAR processes.

9.2 Proper Gaussian CAR model: dcar_proper

The proper Gaussian conditional autoregressive model used to model dependence of block-level values (e.g., spatial areas or temporal blocks) is implemented in NIMBLE as the dcar_proper distribution. Additional details of using this distribution are available using help('CAR-Proper').

Proper CAR models are proper priors for random fields (e.g., temporal or spatial processes). The prior is a joint prior across a collection of latent process values. For more technical details on proper CAR models please see Banerjee, Carlin, and Gelfand (2015), including considerations of why the improper CAR model may be preferred.

In addition to our focus here on CAR modeling for spatial data, the proper CAR model can also be used in other contexts, such as for temporal data in a discrete time context.

9.2.1 Specification and density

NIMBLE uses the same parameterization as WinBUGS / GeoBUGS for the dcar_proper distribution, providing compatibility with existing WinBUGS code. NIMBLE also provides the WinBUGS name car.proper as an alias.

9.2.1.1 Specification

The dcar_proper distribution is specified for a set of N spatially dependent regions as:

x[1:N] ~ dcar_proper(mu, C, adj, num, M, tau, gamma)

There is no option of a zero-mean constraint for proper CAR process, and instead the mean for each region is specified by the mu parameter. The elements of mu can be assigned fixed values or may be specified using one common, or multiple, prior distributions.

The C, adj, num and M parameters define the adjacency structure, normalized weights, and conditional variances of the spatially-dependent field. See help('CAR-Proper') for details of these parameters. When specifying a CAR distribution, these parameters must have constant values. They do not necessarily have to be specified as constants when creating a model object using nimbleModel, but they should be defined in a static way: as right-hand-side only variables with initial values provided as constants, data or inits, or using fixed numerical deterministic declarations.

The adjacency structure defined by adj must be symmetric. That is, if region \(i\) is neighbor of region \(j\), then region \(j\) must also be a neighbor of region \(i\). In addition, the normalized weights specified in C must satisfy a symmetry constraint jointly with the conditional variances given in M. This constraint requires that \(M^{-1}C\) is symmetric, where \(M\) is a diagonal matrix of conditional variances and \(C\) is the normalized (each row sums to one) weight matrix. Equivalently, this implies that \(C_{ij}M_{jj} = C_{ji}M_{ii}\) for all pairs of neighboring regions \(i\) and \(j\). NIMBLE performs a check of these symmetries and will issue an error message if asymmetry is detected.

Two options are available to simplify the process of constructing the C and M arguments; both options are demonstrated in the example. First, these arguments may be omitted from the dcar_proper specification. In this case, values of C and M will be generated that correspond to all weights being equal to one, or equivalently, a symmetric weight matrix containing only zeros and ones. Note that C and M should either both be provided, or both be omitted from the specification.

Second, a convenience function as.carCM is provided to generate the C and M arguments corresponding to a specified set of symmetric unnormalized weights. If weights contains the non-zero weights corresponding to an unnormalized weight matrix (weights is precisely the argument that can be used in the dcar_normal specification), then a list containing C and M can be generated using as.carCM(adj, weights, num). In this case, the resulting C contains the row-normalized weights, and the resulting M is a vector of the inverse row-sums of the unnormalized weight matrix.

The scalar precision tau may be treated as an unknown model parameter and itself assigned a prior distribution. Care should be taken in selecting a prior distribution for tau, and WinBUGS suggests that users be prepared to carry out a sensitivity analysis for this choice.

An appropriate value of the gamma parameter ensures the propriety of the dcar_proper distribution. The value of gamma must lie between fixed bounds, which are given by the reciprocals of the largest and smallest eigenvalues of \(M^{-1/2}CM^{1/2}\). These bounds can be calculated using the function carBounds or separately using the functions carMinBound and carMaxBound. For compatibility with WinBUGS, NIMBLE provides min.bound and max.bound as aliases for carMinBound and carMaxBound. Rather than selecting a fixed value of gamma within these bounds, it is recommended that gamma be assigned a uniform prior distribution over the region of permissible values.

Note that when C and M are omitted from the dcar_proper specification (and hence all weights are taken as one), or C and M are calculated from a symmetric weight matrix using the utility function as.carCM, then the bounds on gamma are necessarily \((-1, 1)\). In this case, gamma can simply be assigned a prior over that region. This approach is shown in both examples.

9.2.1.2 Density

The proper CAR density is given as:

\[p(x | \mu, C, M, \tau, \gamma) \sim \text{MVN} \left( \mu, \; \tfrac{1}{\tau} (I-\gamma C)^{-1} M \right)\]

where the multivariate normal distribution is parameterized in terms of covariance.

For the purposes of MCMC sampling the individual CAR process values, the resulting conditional prior of region \(i\) is:

\[p(x_i | x_{-i}, \mu, C, M, \tau, \gamma) \sim \text{N} \left( \mu_i + \textstyle\sum_{j \in \mathcal{N}_i } \gamma \, C_{ij} \, (x_j - \mu_i), \; \tfrac{M_{ii}}{\tau} \right)\]

where \(x_{-i}\) represents all elements of \(x\) except \(x_{i}\), the neighborhood \(\mathcal{N}_i\) of region \(i\) is the set of all \(j\) for which region \(j\) is a neighbor of region \(i\), and the Normal distribution is parameterized in terms of variance.

9.2.2 Example

We provide two example models using the proper Gaussian dcar_proper distribution. In both, the CAR process values are used in a spatially-dependent logistic regression to model binary presence/absence data. In the first example, the C and M parameters are omitted, which uses weights equal to one for all neighbor relationships. In the second example, symmetric unnormalized weights are specified, and as.carCM is used to construct the C and M parameters to the dcar_proper distribution.

# omitting C and M sets all non-zero weights to one
code <- nimbleCode({
    mu0 ~ dnorm(0, 0.0001)
    tau ~ dgamma(0.001, 0.001)
    gamma ~ dunif(-1, 1)
    s[1:N] ~ dcar_proper(mu[1:N], adj=adj[1:L], num=num[1:N], tau=tau, 
                         gamma=gamma)
    for(i in 1:N) {
        mu[i] <- mu0
        logit(p[i]) <- s[i]
        y[i] ~ dbern(p[i])
    }
})

adj <- c(2, 1, 3, 2, 4, 3)
num <- c(1, 2, 2, 1)
constants <- list(adj = adj, num = num, N = 4, L = 6)
data <- list(y = c(1, 0, 1, 1))
inits <- list(mu0 = 0, tau = 1, gamma = 0, s = rep(0, 4))
Rmodel <- nimbleModel(code, constants, data, inits)

# specify symmetric unnormalized weights, use as.carCM to generate C and M
code <- nimbleCode({
    mu0 ~ dnorm(0, 0.0001)
    tau ~ dgamma(0.001, 0.001)
    gamma ~ dunif(-1, 1)
    s[1:N] ~ dcar_proper(mu[1:N], C[1:L], adj[1:L], num[1:N], M[1:N], tau, 
                         gamma)
    for(i in 1:N) {
        mu[i] <- mu0
        logit(p[i]) <- s[i]
        y[i] ~ dbern(p[i])
    }
})

weights <- c(2, 2, 3, 3, 4, 4)
CM <- as.carCM(adj, weights, num)
constants <- list(C = CM$C, adj = adj, num = num, M = CM$M, N = 4, L = 6)
Rmodel <- nimbleModel(code, constants, data, inits)

Each of the resulting models may be carried through to MCMC sampling. NIMBLE will assign a specialized sampler to update the elements of the CAR process. See Chapter 7 for information about NIMBLE’s MCMC engine, and Section 9.3 for details on MCMC sampling of the CAR processes.

9.3 MCMC Sampling of CAR models

NIMBLE’s MCMC engine provides specialized samplers for the dcar_normal and dcar_proper distributions. These samplers perform sequential univariate updates on the components of the CAR process. Internally, each sampler assigns one of three specialized univariate samplers to each component, based on inspection of the model structure:

  1. A conjugate sampler in the case of conjugate Normal dependencies.
  2. A random walk Metropolis-Hastings sampler in the case of non-conjugate dependencies.
  3. A posterior predictive sampler in the case of no dependencies.

Note that these univariate CAR samplers are not the same as NIMBLE’s standard conjugate, RW, and posterior_predictive samplers, but rather specialized versions for operating on a CAR distribution. Details of these assignments are strictly internal to the CAR samplers.

In future versions of NIMBLE we expect to provide block samplers that update the entire CAR process as a single sample. This may provide improved MCMC performance by accounting for dependence between elements, particularly when conjugacy is available.

9.3.1 Initial values

Valid initial values should be provided for all elements of the process specified by a CAR structure before running an MCMC. This ensures that the conditional prior distribution is well-defined for each region. A simple and safe choice of initial values is setting all components of the process equal to zero, as is done in the preceding CAR examples.

For compatibility with WinBUGS, NIMBLE also allows an initial value of NA to be provided for regions with zero neighbors. This particular initialization is required in WinBUGS, so this allows users to make use of existing WinBUGS code.

9.3.2 Zero-neighbor regions

Regions with zero neighbors (defined by a 0 appearing in the num parameter) are a special case for both the dcar_normal and dcar_proper distribution. The corresponding neighborhood \(\mathcal{N}\) of such a region contains no elements, and hence the conditional prior is improper and uninformative, tantamount to a dflat prior distribution. Thus, the conditional posterior distribution of those regions is entirely determined by the dependent nodes, if any. Sampling of these zero-neighbor regions proceeds as:

  1. In the conjugate case, sampling proceeds according to the conjugate posterior.
  2. In the non-conjugate case, sampling proceeds using random walk Metropolis-Hastings, where the posterior is determined entirely by the dependencies.
  3. In the case of no dependents, the posterior is entirely undefined. Here, no changes will be made to the process value, and it will remain equal to its initial value throughout. By virtue of having no neighbors, this region does not contribute to the density evaluation of the subsuming dcar_normal node nor to the conditional prior of any other regions, hence its value (even NA) is of no consequence.

This behavior is different from that of WinBUGS, where the value of zero-neighbor regions of car.normal nodes is set to and fixed at zero.

9.3.3 Zero-mean constraint

A zero-mean constraint is available for the intrinsic Gaussian dcar_normal distribution. This constraint on the ICAR process values is imposed during MCMC sampling, if the argument zero_mean = 1, mimicking the behavior of WinBUGS. Following the univariate updates on each component, the mean is subtracted away from all process values, resulting in a zero-mean process.

Note that this is not equivalent to sampling under the constraint that the mean is zero (see p. 36 of Rue and Held (2005)) so should be treated as an ad hoc approach and employed with caution.

References

Rue, H., and L. Held. 2005. Gaussian Markov Random Fields: Theory and Applications. Boca Raton: Chapman & Hall.

Banerjee, S., B.P. Carlin, and A.E. Gelfand. 2015. Hierarchical Modeling and Analysis for Spatial Data. 2nd ed. Boca Raton: Chapman & Hall.

Paciorek, C.J. 2009. “Understanding Intrinsic Gaussian Markov Random Field Spatial Models, Including Intrinsic Conditional Autoregressive Models.” University of California, Berkeley. http://www.stat.berkeley.edu/~paciorek/research/techVignettes/techVignette5.pdf.