basecor: A base Cholesky model of a correlation matrix

In graphpcor: Models for Correlation Matrices Based on Graphs

A base Cholesky model of a correlation matrix

Description

The basecor class contain a correlation matrix base, the parameter vector theta, that generates or is generated by base, the dimention p, the index iLtheta for theta in the (lower) Cholesky, and the Hessian around it I0, see details.

Usage

basecor(base, p, parametrization = "cpc", iparams, iLtheta)

## S3 method for class 'numeric'
basecor(base, p, parametrization = "cpc", iparams, iLtheta)

## S3 method for class 'matrix'
basecor(base, p, parametrization = "cpc", iparams, iLtheta)

## S3 method for class 'basecor'
print(x, ...)

Arguments

base	numeric vector/matrix used to define the base correlation matrix. If numeric vector with length 'm', 'm' should be 'p(p-1)/2' in the dense model case and 'length(iLtheta)' in the sparse model case.
p	integer with the dimension, the number of rows/columns of the correlation matrix.
parametrization	character to specify the parametrization used. The available ones are "cpc" (or "CPC") or "sap" (or "SAP"). See Details. The default is "cpc".
iparams	integer ordered vector with length equal the number of parameters used to specify common parameter values. If missing, assumed to be 1:length(theta). Example: By setting iparams = c(1,1,2,3) the first and second parameters are considered to be the same.
iLtheta	integer vector to specify the (vectorized) position where 'theta' will be placed in the (lower triangle) Cholesky factorization of the correlation matrix. Default (missing, or if NULL) and assumes iLtheta = which(lower.tri(...)).
x	a basecor object.
...	further arguments passed on.

Details

For 'parametrization' = "CPC" or 'parametrization' = "cpc": The Canonical Partial Correlation - CPC parametrization, Lewandowski, Kurowicka, and Joe (2009), compute r[k] = tanh(\theta[k]), for k=1,...,m, and the two p\times p matrices

A = \left[ \begin{array}{ccccc} 1 & & & & \\ r_1 & 1 & & & \\ r_2 & r_p & 1 & & \\ \vdots & \vdots & \ddots & \ddots & \\ r_{p-1} & r_{2p-3} & \ldots & r_m & 1 \end{array} \right] \textrm{ and } B = \left[ \begin{array}{ccccc} 1 & & & & \\ \sqrt{1-r_1^2} & 1 & & & \\ \sqrt{1-r_2^2} & \sqrt{1-r_p^2} & 1 & & \\ \vdots & \vdots & \ddots & \ddots & \\ \sqrt{1-r_{p-1}^2} & \sqrt{r_{2p-3}^2} & \ldots & \sqrt{1-r_m^2} & 1 \end{array} \right]

The matrices A and B are then used to build the Cholesky factor of the correlation matrix, given as

L = \left[ \begin{array}{ccccc} 1 & 0 & 0 & \ldots & 0\\ A_{2,1} & B_{2,1} & 0 & \ldots & 0\\ A_{3,1} & A_{3,2}B_{3,1} & B_{3,1}B_{3,2} & & \vdots \\ \vdots & \vdots & \ddots & \ddots & 0\\ A_{p,1} & A_{p,2}B_{p,1} & \ldots & A_{p,p-1}\prod_{k=1}^{p-1}B_{p,k} & \prod_{k=1}^{p-1}B_{p,k} \end{array} \right]

Note: The determinant of the correlation matriz is

\prod_{i=2}^p\prod_{j=1}^{i-1}B_{i,j} = \prod_{i=2}^pL_{i,i}

For 'parametrization' = "SAP" or 'parametrization' = "sap": The Standard Angles Parametrization - SAP, as described in Rapisarda, Brigo and Mercurio (2007), compute x[k] = \pi/(1+\exp(-\theta[k])), for k=1,...,m, and the two p\times p matrices

A = \left[ \begin{array}{ccccc} 1 & & & & \\ \cos(x_1) & 1 & & & \\ \cos(x_2) & \cos(x_p) & 1 & & \\ \vdots & \vdots & \ddots & \ddots & \\ \cos(x_{p-1}) & \cos(x_{2p-3}) & \ldots & \cos(x_m) & 1 \end{array} \right] \textrm{ and } B = \left[ \begin{array}{ccccc} 1 & & & & \\ \sin(x_1) & 1 & & & \\ \sin(x_2) & \sin(x_p) & 1 & & \\ \vdots & \vdots & \ddots & \ddots & \\ \sin(x_{p-1}) & \sin(x_{2p-3}) & \ldots & \sin(x_m) & 1 \end{array} \right]

The decomposition of the Hessian matrix around the base model, I0, formally \mathbf{I}(\theta_0), is numerically computed. This element has the following attributes: 'h.5' as \mathbf{I}^{1/2}(\theta_0), and 'hneg.5' as \mathbf{I}^{-1/2}(\theta_0).

Value

a basecor object

Methods (by class)

• basecor(numeric): Build a basecor from the parameter vector.

• basecor(matrix): Build a basecor from a correlation matrix.

Methods (by generic)

• print(basecor): Print method for 'basecor'

Functions

• basecor(): Build a basecor object.

References

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

Lewandowski, Kurowicka and Joe (2009) Generating Random Correlation Matrices Based on Vines and Extended Onion Method. Journal of Multivariate Analysis 100: 1989–2001. <doi: 10.1016/j.jmva.2009.04.008>

Examples

## A correlation matrix
c0 <- matrix(c( 1.0,  0.8, -0.5,
                0.8,  1.0, -0.4,
               -0.5, -0.4,  1.0), 3)

## build the 'basecor'
b3 <- basecor(c0) ## base as matrix
b3

## elements
b3$base
b3$theta

## from 'theta' 
th3 <- b3$theta
bth3 <- basecor(th3, p = 3) ## base as vector
bth3

## numerically the same
all.equal(c0, bth3$base, tol = 1e-4)

## from a numeric vector (theta)
th2 <- c(-1, -0.5)
b2 <- basecor(th2, p = 3, iLtheta = c(2,3))
b2

## from the correlation matrix
b2c <- basecor(b2$base, iLtheta = c(2,3))

all.equal(th2, b2c$theta, tol = 1e-4)

## Hessian around the base 
hessian(b3)
hessian(b3, ifixed = 3)
hessian(b2)

graphpcor documentation built on March 23, 2026, 9:07 a.m.