CAR-Proper | R Documentation |

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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