cgeneric_pc_prec_correl: Build an `inla.cgeneric` to implement the PC-prior of a precision matrix as inverse of a correlation matrix.

Description

Build an inla.cgeneric to implement the PC-prior of a precision matrix as inverse of a correlation matrix.

Usage

cgeneric_pc_prec_correl(
  n,
  lambda,
  theta.base,
  debug = FALSE,
  useINLAprecomp = TRUE,
  libpath = NULL
)

Value

a inla.cgeneric, cgeneric() object.

Arguments

n: integer to define the size of the matrix
lambda: numeric (positive), the penalization rate parameter
theta.base: numeric vector with the model parameters at the base model
debug: integer, default is zero, indicating the verbose level. Will be used as logical by INLA.
useINLAprecomp: logical, default is TRUE, indicating if it is to be used the shared object pre-compiled by INLA. This is not considered if 'libpath' is provided.
libpath: string, default is NULL, with the path to the shared object.

Details

The precision matrix parametrization step 1: $$Q0 = \left[ \begin{array}{ccccc} 1 & & & & \\ \theta_1 & 1 & & & \\ \theta_2 & \theta_n & & & \\ \vdots & & \ldots & \ddots & \\ \theta_{n-1} & \theta_{2n-3} \ldots & \theta_m & 1 \end{array} \right] $$

step 2: $V = Q0^{-1}$

step 3: $S = diag(V)^{1/2}$

step 4: $C = SVS$

step 5: $Q = C^{-1}$

$$p(Q|\lambda) = p(\theta[1:m] | lambda) =$$ $$ p_C(C(Q)) | Jacobian C(Q) |$$ where p_C is the PC-prior for correlation, see section 6.2 of Simpson et. al. (2017), which is based on the hypersphere decomposition.

The hypershere decomposition, as proposed in Rapisarda, Brigo and Mercurio (2007) consider $\theta[k] \in [0, \infty], k=1,...,m=n(n-1)/2$ compute $x[k] = pi/(1+exp(-theta[k]))$ organize it as a lower triangle of a $n \times n$ matrix $$B[i,j] = \left\{\begin{array}{cc} cos(x[i,j]) & j=1 \\ cos(x[i,j])prod_{k=1}^{j-1}sin(x[i,k]) & 2 <= j <= i-1 \\ prod_{k=1}^{j-1}sin(x[i,k]) & j=i \\ 0 & j+1 <= j <= n \end{array}\right.$$ Result $$\gamma[i,j] = -log(sin(x[i,j]))$$ $$KLD(R) = \sqrt(2\sum_{i=2}^n\sum_{j=1}^{i-1} \gamma[i,j]$$

References

Daniel Simpson, H\aa vard Rue, Andrea Riebler, Thiago G. Martins and Sigrunn H. S\o rbye (2017). Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors Statistical Science 2017, Vol. 32, No. 1, 1–28. <doi 10.1214/16-STS576>

Rapisarda, Brigo and Mercurio (2007). Parameterizing correlations: a geometric interpretation. IMA Journal of Management Mathematics (2007) 18, 55-73. <doi 10.1093/imaman/dpl010>