car_precision: CAR Precision Matrix

In holans/ST-COS: Space-Time Change of Support

CAR Precision Matrix

Description

A convenience function to compute the CAR precision matrix based on a given adjacency matrix.

Usage

car_precision(A, tau = 1, scale = FALSE)

Arguments

A	An adjacency matrix.
tau	The CAR dependency parameter \tau \in [0,1]. See "Value". Default: 1.
scale	Whether to scale matrix entries. See "Value". Default: FALSE.

Details

Suppose \bm{A} is an n \times n adjacency matrix and \bm{D} = \textrm{Diag}(\bm{A} \bm{1}) = \textrm{Diag}(a_{1+}, \ldots, a_{n+}). If scale is FALSE, return the CAR precision matrix

\bm{Q} = \bm{D} - \tau \bm{A}.

If scale is TRUE, return a scaled version

\tilde{\bm{Q}} = \bm{D}^{-1} \bm{Q}.

An error is thrown if scale = TRUE and any of \{ a_{1+}, \ldots, a_{n+} \} are equal to 0. Taking \tau = 1 corresponds to the Intrinsic CAR precision matrix.

Typically in a modeling context, the precision matrix will be multiplied by a scaling parameter; e.g., a CAR model for random effects \bm{\phi} could be

f(\bm{\phi} \mid \alpha) \propto \alpha^{-q} \exp\left\{ -\frac{1}{2 \alpha^2} \bm{\phi}^\top \bm{Q} \bm{\phi} \right\}.

where q = \textrm{rank}(Q).

Value

CAR precision matrix.

Examples

data("acs_sf")
dom = acs5_2013[1:4,]
A = adjacency_matrix(dom)
Q = car_precision(A)

holans/ST-COS documentation built on Aug. 28, 2023, 5:50 a.m.