Prepare data for the CAR model
Cliff A, Ord J (1981). Spatial Processes: Models and Applications. Pion.
Cressie N (2015 [1993]). Statistics for Spatial Data. Revised edition. John Wiley & Sons.
Cressie N, Perrin O, Thomas-Agnan C (2005). “Likelihood-based estimation for Gaussian MRFs.” Statistical Methodology, 2(1), 1–16.
Cressie N, Wikle CK (2011). Statistics for Spatio-Temporal Data. John Wiley & Sons.
Donegan, Connor (2021). Spatial conditional autoregressive models in Stan. OSF Preprints. doi:10.31219/osf.io/3ey65 .
Haining RP, Li G (2020). Modelling Spatial and Spatio-Temporal Data: A Bayesian Approach. CRC Press.
Binary adjacency matrix; for style = DCAR, provide a symmetric matrix of distances instead. The distance matrix should be sparse, meaning that most distances should be zero (usually obtained by setting some threshold distance beyond which all are zero).
Specification for the connectivity matrix (C) and conditional variances (M); one of "WCAR", "ACAR", or "DCAR".
For style = DCAR, distances will be raised to the -k power (d^-k).
For style = DCAR, distances will be offset by gamma before raising to the -kth power.
If TRUE, return eigenvalues required for calculating the log determinant of the precision matrix and for determining the range of permissible values of rho. These will also be printed with a message if lambda = TRUE.
If cmat = TRUE, return the full matrix C (in sparse matrix format).
Two computational methods are available for CAR models using stan_car: car\_normal\_lpdf and wcar\_normal\_lpdf. For WCAR models, either method will work but wcar\_normal\_lpdf is faster. To force use car\_normal\_lpdf when style = 'WCAR', provide stan_fn = "car_normal_lpdf".
A list containing all of the data elements required by the CAR model in stan_car.
The CAR model is:
Normal(Mu, Sigma), Sigma = (I - rho * C)^-1 * M * tau^2,where I is the identity matrix, rho is a spatial autocorrelation parameter, C is a connectivity matrix, and M * tau^2 is a diagonal matrix with conditional variances on the diagonal. tau^2 is a (scalar) scale parameter.
In the WCAR specification, C is the row-standardized version of A. This means that the non-zero elements of A will be converted to 1/N_i where N_i is the number of neighbors for the ith site (obtained using Matrix::rowSums(A). The conditional variances (on the diagonal of M * tau^2), are also proportional to 1/N_i.
The ACAR specification is from Cressie, Perrin and Thomas-Agnon (2005); also see Cressie and Wikle (2011, p. 188) and Donegan (2021).
The DCAR specification is inverse distance-based, and requires the user provide a (sparse) distance matrix instead of a binary adjacency matrix. (For A, provide a symmetric matrix of distances, not inverse distances!) Internally, non-zero elements of A will be converted to: d_{ij} = (a_{ij} + gamma)^(-k) (Cliff and Ord 1981, p. 144; Donegan 2021). Default values are k=1 and gamma=0. Following Cressie (2015), these values will be scaled (divided) by their maximum value. For further details, see the DCAR_A specification in Donegan (2021).
For inverse-distance weighting schemes, see Cliff and Ord (1981); for distance-based CAR specifications, see Cressie (2015 [1993]), Haining and Li (2020), and Donegan (2021).
When using stan_car, always use cmat = TRUE (the default).
Details on CAR model specifications can be found in Table 1 of Donegan (2021).
data(georgia)
## use a binary adjacency matrix
A <- shape2mat(georgia, style = "B")
## WCAR specification
cp <- prep_car_data(A, "WCAR")
1 / range(cp$lambda)
## ACAR specification
cp <- prep_car_data(A, "ACAR")
## DCAR specification (inverse-distance based)
A <- shape2mat(georgia, "B")
D <- sf::st_distance(sf::st_centroid(georgia))
A <- D * A
cp <- prep_car_data(A, "DCAR", k = 1)