CAR-Normal: The CAR-Normal Distribution

In nimble: MCMC, Particle Filtering, and Programmable Hierarchical Modeling

The CAR-Normal Distribution

Description

Density function and random generation for the improper (intrinsic) Gaussian conditional autoregressive (CAR) distribution.

Usage

dcar_normal(
  x,
  adj,
  weights = adj/adj,
  num,
  tau,
  c = CAR_calcNumIslands(adj, num),
  zero_mean = 0,
  log = FALSE
)

rcar_normal(
  n = 1,
  adj,
  weights = adj/adj,
  num,
  tau,
  c = CAR_calcNumIslands(adj, num),
  zero_mean = 0
)

Arguments

x	vector of values.
adj	vector of indices of the adjacent locations (neighbors) of each spatial location. This is a sparse representation of the full adjacency matrix.
weights	vector of symmetric unnormalized weights associated with each pair of adjacent locations, of the same length as adj. If omitted, all weights are taken to be one.
num	vector giving the number of neighboring locations of each spatial location, with length equal to the total number of locations.
tau	scalar precision of the Gaussian CAR prior.
c	integer number of constraints to impose on the improper density function. If omitted, c is calculated as the number of disjoint groups of spatial locations in the adjacency structure, which implicitly assumes a first-order CAR process for each group. Note that c should be equal to the number of eigenvalues of the precision matrix that are zero. For example, if the neighborhood structure is based on a second-order Markov random field in one dimension then the matrix has two zero eigenvalues and in two dimensions it has three zero eigenvalues. See Rue and Held (2005) and the NIMBLE User Manual for more information.
zero_mean	integer specifying whether to set the mean of all locations to zero during MCMC sampling of a node specified with this distribution in BUGS code (default 0). This argument is used only in BUGS model code when specifying models in NIMBLE. If 0, the overall process mean is included implicitly in the value of each location in a BUGS model; if 1, then during MCMC sampling, the mean of all locations is set to zero at each MCMC iteration, and a separate intercept term should be included in the BUGS model. Note that centering during MCMC as implemented in NIMBLE follows the ad hoc approach of WinBUGS and does not sample under the constraint that the mean is zero as discussed on p. 36 of Rue and Held (2005). See ‘Details’.
log	logical; if TRUE, probability density is returned on the log scale.
n	number of observations.

Details

When specifying a CAR distribution in BUGS model code, the zero_mean parameter should be specified as either 0 or 1 (rather than TRUE or FALSE).

Note that because the distribution is improper, rcar_normal does not generate a sample from the distribution. However, as discussed in Rue and Held (2005), it is possible to generate a sample from the distribution under constraints imposed based on the eigenvalues of the precision matrix that are zero.

Value

dcar_normal gives the density, while rcar_normal returns the current process values, since this distribution is improper.

Author(s)

Daniel Turek

References

Banerjee, S., Carlin, B.P., and Gelfand, A.E. (2015). Hierarchical Modeling and Analysis for Spatial Data, 2nd ed. Chapman and Hall/CRC.

Rue, H. and L. Held (2005). Gaussian Markov Random Fields, Chapman and Hall/CRC.

See Also

CAR-Proper, Distributions for other standard distributions

Examples

x <- c(1, 3, 3, 4)
num <- c(1, 2, 2, 1)
adj <- c(2, 1,3, 2,4, 3)
weights <- c(1, 1, 1, 1, 1, 1)
lp <- dcar_normal(x, adj, weights, num, tau = 1)

nimble documentation built on Sept. 11, 2024, 7:10 p.m.