CAR-Proper: The CAR-Proper Distribution

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

The CAR-Proper Distribution

Description

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

Usage

dcar_proper(
  x,
  mu,
  C = CAR_calcC(adj, num),
  adj,
  num,
  M = CAR_calcM(num),
  tau,
  gamma,
  evs = CAR_calcEVs3(C, adj, num),
  log = FALSE
)

rcar_proper(
  n = 1,
  mu,
  C = CAR_calcC(adj, num),
  adj,
  num,
  M = CAR_calcM(num),
  tau,
  gamma,
  evs = CAR_calcEVs3(C, adj, num)
)

Arguments

x	vector of values.
mu	vector of the same length as x, specifying the mean for each spatial location.
C	vector of the same length as adj, giving the weights associated with each pair of neighboring locations. See ‘Details’.
adj	vector of indices of the adjacent locations (neighbors) of each spatial location. This is a sparse representation of the full adjacency matrix.
num	vector giving the number of neighboring locations of each spatial location, with length equal to the number of locations.
M	vector giving the diagonal elements of the conditional variance matrix, with length equal to the number of locations. See ‘Details’.
tau	scalar precision of the Gaussian CAR prior.
gamma	scalar representing the overall degree of spatial dependence. See ‘Details’.
evs	vector of eigenvalues of the adjacency matrix implied by C, adj, and num. This parameter should not be provided; it will always be calculated using the adjacency information.
log	logical; if TRUE, probability density is returned on the log scale.
n	number of observations.

Details

If both C and M are omitted, then all weights are taken as one, and corresponding values of C and M are generated.

The C and M parameters must jointly satisfy a symmetry constraint: that M^(-1) %*% C is symmetric, where M is a diagonal matrix and C is the full weight matrix that is sparsely represented by the parameter vector C.

For a proper CAR model, the value of gamma must lie within the inverse minimum and maximum eigenvalues of M^(-0.5) %*% C %*% M^(0.5), where M is a diagonal matrix and C is the full weight matrix. These bounds can be calculated using the deterministic functions carMinBound(C, adj, num, M) and carMaxBound(C, adj, num, M), or simultaneously using carBounds(C, adj, num, M). In the case where C and M are omitted (all weights equal to one), the bounds on gamma are necessarily (-1, 1).

Value

dcar_proper gives the density, and rcar_proper generates random deviates.

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.

See Also

CAR-Normal, Distributions for other standard distributions

Examples

x <- c(1, 3, 3, 4)
mu <- rep(3, 4)
adj <- c(2, 1,3, 2,4, 3)
num <- c(1, 2, 2, 1)
 
## omitting C and M uses all weights = 1
dcar_proper(x, mu, adj = adj, num = num, tau = 1, gamma = 0.95)
 
## equivalent to above: specifying all weights = 1,
## then using as.carCM to generate C and M arguments
weights <- rep(1, 6)
CM <- as.carCM(adj, weights, num)
C <- CM$C
M <- CM$M
dcar_proper(x, mu, C, adj, num, M, tau = 1, gamma = 0.95)
 
## now using non-unit weights
weights <- c(2, 2, 3, 3, 4, 4)
CM2 <- as.carCM(adj, weights, num)
C2 <- CM2$C
M2 <- CM2$M
dcar_proper(x, mu, C2, adj, num, M2, tau = 1, gamma = 0.95)

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