CAR-models

Conditional autoregressive (CAR) models

Let \(A\) be an \(n \times n\) symmetric adjacency/weight matrix for a graph. In trafficCAR, \(A\) may be binary (0/1) or contain nonnegative edge weights; the diagonal is ignored (set to 0). If symmetrize = TRUE, the input is replaced by \((A + A^\top)/2\) before construction.

Define the diagonal “degree” matrix \[ D = \mathrm{diag}(A \mathbf{1}), \] i.e., \(D_{ii} = \sum_j A_{ij}\).

ICAR model

The intrinsic CAR (ICAR) precision matrix is \[ Q = \tau (D - A). \]

Key properties:

\(Q\) is singular (improper prior).
Rank deficiency equals the number of connected components.
Disconnected graphs and isolated nodes increase the rank deficiency.
Constraints are typically imposed for identifiability (e.g. sum-to-zero).

In trafficCAR, ICAR precision matrices are constructed with trafficCAR::car_precision(type = "icar") (internally using \(\rho = 1\)). If the graph contains isolated nodes (degree 0), car_precision() issues a warning because those components necessarily contribute to singularity.

library(Matrix)
library(trafficCAR)

A <- matrix(0, 4, 4)
A[1,2] <- A[2,1] <- 1
A[2,3] <- A[3,2] <- 1
A[3,4] <- A[4,3] <- 1

Q_icar <- car_precision(A, type = "icar", tau = 1)
Q_icar
#> 4 x 4 sparse Matrix of class "dsCMatrix"
#>                 
#> [1,]  1 -1  .  .
#> [2,] -1  2 -1  .
#> [3,]  . -1  2 -1
#> [4,]  .  . -1  1

Proper CAR model

The proper CAR precision matrix is \[ Q = \tau (D - \rho A). \]

In general, admissible values of \(\rho\) depend on the spectrum of a normalized adjacency/weight matrix. trafficCAR does not compute tight spectral bounds. Instead, trafficCAR::car_precision(type = "proper", check = TRUE) enforces the simple condition \(|\rho| < 1\) as a conservative, widely used safeguard.

Values with \(|\rho| \ge 1\) are rejected because they can yield singular or indefinite precision matrices under common graph structures and weight choices.

trafficCAR additionally rejects proper CAR models on graphs with isolated nodes (degree 0) when check = TRUE, since the construction \(D - \rho A\) cannot be positive definite in that case.

Q_prop <- car_precision(A, type = "proper", rho = 0.9, tau = 2)
Q_prop
#> 4 x 4 sparse Matrix of class "dsCMatrix"
#>                         
#> [1,]  2.0 -1.8  .    .  
#> [2,] -1.8  4.0 -1.8  .  
#> [3,]  .   -1.8  4.0 -1.8
#> [4,]  .    .   -1.8  2.0

As \(\rho \to 1\), the proper CAR model approaches the ICAR model.

Disconnected graphs and isolated nodes

Disconnected graphs imply multiple connected components. For ICAR, each connected component contributes to rank deficiency, so \(Q\) is singular even when all nodes have positive degree.

Isolated nodes (degree 0) are a special case of disconnectedness:

For ICAR: car_precision(type = "icar") warns when isolates are present, and the resulting precision is singular for those components.
For proper CAR: car_precision(type = "proper", check = TRUE) errors if any isolates are present.

The following example adds an isolated node (node 5 has no neighbors):

A_iso <- matrix(0, 5, 5)
A_iso[1,2] <- A_iso[2,1] <- 1
A_iso[2,3] <- A_iso[3,2] <- 1
A_iso[3,4] <- A_iso[4,3] <- 1
# node 5 is isolated

# ICAR: warning about isolated node(s)
Q_icar_iso <- car_precision(A_iso, type = "icar", tau = 1)
#> Warning in car_precision(A_iso, type = "icar", tau = 1): adjacency has isolated
#> node(s) with degree 0; Q will be singular for those components
Q_icar_iso
#> 5 x 5 sparse Matrix of class "dsCMatrix"
#>                   
#> [1,]  1 -1  .  . .
#> [2,] -1  2 -1  . .
#> [3,]  . -1  2 -1 .
#> [4,]  .  . -1  1 .
#> [5,]  .  .  .  . .

# Proper CAR: error when isolates are present (with check = TRUE)
try(car_precision(A_iso, type = "proper", rho = 0.9, tau = 1))
#> Error in car_precision(A_iso, type = "proper", rho = 0.9, tau = 1) : 
#>   proper CAR not defined for graphs with isolated nodes

Scaling (Sørbye–Rue)

ICAR precision matrices are often scaled so that the geometric mean of the marginal variances equals 1, making the precision parameter \(\tau\) more comparable across graphs and resolutions.

Implementation note. In trafficCAR, scaling of ICAR precision matrices is implemented as a best-effort calibration step intended to make the precision parameter \(\tau\) comparable across graphs and resolutions. The scaling constant is computed using a Cholesky-based approximation to the diagonal of the generalized inverse. On some graph structures or platforms, this computation may fail due to numerical or definiteness issues; in that case, a warning is issued and the unscaled ICAR precision matrix is returned.

This behavior preserves correctness of the ICAR model and ensures robust construction of precision matrices across platforms. When scaling succeeds, the resulting precision matrix is a positive scalar multiple of the unscaled ICAR precision, leaving the dependence structure unchanged while rescaling the marginal variances.

Future versions may adopt component-wise constrained scaling to guarantee the geometric-mean marginal variance normalization described by Sørbye and Rue (2014) in all cases. In trafficCAR, scaling is available via trafficCAR::intrinsic_car_precision(scale = TRUE). If the internal Cholesky factorization needed for scaling fails, the function warns and returns the unscaled ICAR precision matrix.

Q_icar_scaled <- intrinsic_car_precision(A, tau = 1, scale = TRUE)
#> Warning in .local(A, ...): CHOLMOD warning 'matrix not positive definite' at
#> file 'Supernodal/t_cholmod_super_numeric_worker.c', line 1114
#> Warning in intrinsic_car_precision(A, tau = 1, scale = TRUE): scaling failed:
#> Cholesky factorization unsuccessful
Q_icar_scaled
#> 4 x 4 sparse Matrix of class "dsCMatrix"
#>                 
#> [1,]  1 -1  .  .
#> [2,] -1  2 -1  .
#> [3,]  . -1  2 -1
#> [4,]  .  . -1  1