The CAR-Proper Distribution
Density function and random generation for the proper Gaussian conditional autoregressive (CAR) distribution.
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))
x |
vector of values. |
mu |
vector of the same length as |
C |
vector of the same length as |
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 |
log |
logical; if |
n |
number of observations. |
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).
dcar_proper gives the density, and rcar_proper generates random deviates.
Daniel Turek
Banerjee, S., Carlin, B.P., and Gelfand, A.E. (2015). Hierarchical Modeling and Analysis for Spatial Data, 2nd ed. Chapman and Hall/CRC.
CAR-Normal, Distributions for other standard distributions
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)
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