Type:	Package
Title:	Models for Correlation Matrices Based on Graphs
Version:	0.1.24
Maintainer:	Elias Teixeira Krainski <eliaskrainski@gmail.com>
Description:	Implement some models for correlation/covariance matrices including two approaches to model correlation matrices from a graphical structure. One use latent parent variables as proposed in Sterrantino et. al. (2024) <doi:10.1007/s10260-025-00788-y>. The other uses a graph to specify conditional relations between the variables. The graphical structure makes correlation matrices interpretable and avoids the quadratic increase of parameters as a function of the dimension. In the first approach a natural sequence of simpler models along with a complexity penalization is used. The second penalizes deviations from a base model. These can be used as prior for model parameters, considering C code through the 'cgeneric' interface for the 'INLA' package (https://www.r-inla.org). This allows one to use these models as building blocks combined and to other latent Gaussian models in order to build complex data models.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding:	UTF-8
RoxygenNote:	7.3.3
NeedsCompilation:	yes
Depends:	R (≥ 4.3), Matrix, igraph, INLAtools (≥ 0.0.8), numDeriv
Imports:	methods, stats, utils
Suggests:	knitr
BuildVignettes:	true
VignetteBuilder:	knitr
Packaged:	2026-03-20 21:09:51 UTC; eliask
Author:	Elias Teixeira Krainski [cre, aut, cph], Denis Rustand [aut, cph], Anna Freni-Sterrantino [aut, cph], Janet van Niekerk [aut, cph], Haavard Rue’ [aut]
Repository:	CRAN
Date/Publication:	2026-03-23 07:20:02 UTC

The Laplacian of a graph

Description

The (symmetric) Laplacian of a graph is a square matrix with dimention equal the number of nodes. It is defined as

L_{ij} = n_i \textrm{ if } i=j, -1 \textrm{ if } i\sim j, 0 \textrm{ otherwise}

where i~j means that there is an edge between nodes i and j and n_i is the number of edges including node i.

Usage

Laplacian(x)

## Default S3 method:
Laplacian(x)

## S3 method for class 'matrix'
Laplacian(x)

## S3 method for class 'Matrix'
Laplacian(x)

Arguments

Value

matrix as the Laplacian of a graph

Methods (by class)

• Laplacian(default): The Laplacian default method (none)

• Laplacian(matrix): The Laplacian of a matrix

• Laplacian(Matrix): The Laplacian of a Matrix

graphpcor: correlation from nodes and edges

Description

A graphpcor is a graph where a node represents a variable and an edge represent a conditional distribution. The correlation built from a graphpcor consider the parameters for the Cholesky of a precision matrix, whose non-zero pattern is given from the graph.

Usage

## S3 method for class 'graphpcor'
Laplacian(x)

graphpcor(...)

## S3 method for class 'list'
graphpcor(...)

## S3 method for class 'formula'
graphpcor(...)

## S3 method for class 'character'
graphpcor(...)

## S3 method for class 'matrix'
graphpcor(...)

## S3 method for class 'Matrix'
graphpcor(...)

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

## S3 method for class 'graphpcor'
summary(object, ...)

## S3 method for class 'graphpcor'
dim(x, ...)

## S3 method for class 'graphpcor'
plot(x, y, ...)

## S3 method for class 'graphpcor'
vcov(object, ...)

## S3 method for class 'graphpcor'
prec(model, ...)

## S3 method for class 'graphpcor'
cgeneric(model, ...)

## S3 method for class 'Matrix'
cgeneric(model, ...)

Arguments

Methods (by class)

• graphpcor(list): Each element may be a character or formula.

• graphpcor(formula): Each term represents a node, and each ~ an edge.

• graphpcor(character): Each term represents a node, and each ~ an edge.

• graphpcor(matrix): Build a graphpcor from a matrix object

• graphpcor(Matrix): Build a graphpcor for a Matrix object

Methods (by generic)

• Laplacian(graphpcor): The Laplacian method for a graphpcor

• print(graphpcor): The print method for graphpcor

• summary(graphpcor): The summary method for graphpcor

• dim(graphpcor): The dim method for graphpcor

• plot(graphpcor): The plot method for a graphpcor

• vcov(graphpcor): The vcov method for a graphpcor

• prec(graphpcor): The precision method for 'graphpcor'

• cgeneric(graphpcor): The cgeneric method for graphpcor uses cgeneric_graphpcor()

Functions

• cgeneric(Matrix): The cgeneric method for Matrix uses cgeneric_graphpcor()

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

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)

Functions used by/to basecor

Description

Functions used by/to basecor

Usage

cholcor(theta, p, iLtheta, parametrization = "cpc")

Arguments

Value

matrix with lower triangle as the Cholesky factor of a correlation matrix, and atttibutes: parametrization ("cpc" or "sap"), "theta" the parameter vector under the parametrization, "iLtheta" the index in the lower triangle matrix for theta, "logDeterminant" the log determinant for the correlation matrix.

Functions

• cholcor(): Build the (lower) Cholesky for a correlation matrix

A base precision's Cholesky model for a correlation matrix

Description

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

Usage

basepcor(base, p, iLtheta, d0, iparams)

## S3 method for class 'numeric'
basepcor(base, p, iLtheta, d0, iparams)

## S3 method for class 'matrix'
basepcor(base, p, iLtheta, d0, iparams)

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

Arguments

Details

The Inverse Transform Parametrization - ITP, is applied by starting with a

\mathbf{L}^{(0)} = \left[ \begin{array}{ccccc} \textrm{p} & 0 & 0 & \ldots & 0 \\ \theta_1 & \textrm{p-}1 & 0 & \ldots & 0 \\ \theta_2 & \theta_p & \textrm{p-}2 & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & 0 \\ \theta_{p-1} & \theta_{2p-3} & \ldots & \theta_m & 1 \end{array}\right] .

Then compute \mathbf{Q}^{(0)} = \mathbf{L}\mathbf{L}^{T}, \mathbf{V}^{(0)} = (\mathbf{Q}^{(0)})^{-1} and s_{i}^{(0)} = \sqrt{\mathbf{V}_{i,i}^{(0)}}, and define \mathbf{S}^{(0)} = diag(s_1^{(0)},\ldots,s_p^{(0)}) in order to have \mathbf{C}= \mathbf{S}^{-1}\mathbf{V}^{(0)}\mathbf{S}^{-1}.

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 basepcor object

a basepcor object

Methods (by class)

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

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

Methods (by generic)

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

Functions

• basepcor(): Build a basepcor object.

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)

## p = 3, m = 3
b1 <- basepcor(c0)
b1

## p = 3, m = 2
b2 <- basepcor(c0, iLtheta = c(2,3))
b2

all.equal(b1$base, b2$base)

## Hessian
hessian(b2)
hessian(b1, ifixed = 3)
hessian(b1)

## p = 4, m = 4
th4 <- c(0.5,-1,0.5,-0.3)
ith4 <- c(2,3,8,12)
b44 <- basepcor(th4, p = 4, iLtheta = ith4)
b44

Sparse(round(solve(b44$base), 4), zeros.rm = TRUE)

## p = 4, m = 3 (with some common theta)
th3 <- c(0.5, -1, -0.3)
ip3 <- c(1, 2, 1, 3) ## 1st == 3rd
b43 <- basepcor(th3, p = 4, iLtheta = ith4, iparams = ip3)

all.equal(b44$base, b43$base) ## TRUE

## parameter dimension is now reduced
hessian(b44)
hessian(b43)

## If a subset of the parameters are known (fixed), then the
## Hessian is only computed with respect to the unknown ones
hessian(basepcor(th4, p=4, iLtheta = ith4), ifixed = 2:3)
hessian(basepcor(th4, p=4, iLtheta = ith4), ifixed = 1)
hessian(basepcor(th4, p=4, iLtheta = ith4), ifixed = 3)

hessian(basepcor(th3, p=4, iLtheta = ith4,
                 iparams = ip3), ifixed = 3)
hessian(basepcor(th3, p=4, iLtheta = ith4,
                 iparams = ip3), ifixed = 2:3)
hessian(basepcor(th3, p=4, iLtheta = ith4,
                 iparams = ip3), ifixed = 1:2)

## check
hessian(basepcor(th3, p=4, iLtheta = ith4,
                 iparams = ip3), ifixed = NULL)

Functions used by/to basepcor

Description

Functions used by/to basepcor

Usage

fillLprec(L, lfi)

Lprec0(theta, p, iLtheta, d0)

Arguments

Details

The (lower triangle) Cholesky factor of the initial precision for a correlation matrix contains the parameters in the non-zero elements of the lower triangle side of the precision matrix. The filled-in elements are computed from them using fillLprec().

Value

lower triangular matrix with the filled-in elements thus Q0 can be computed.

lower triangular matrix

Functions

• fillLprec(): Function to fill-in a Cholesky matrix

• Lprec0(): Compute the (lower triangle) Cholesky of the initial precision Q0.

Build an cgeneric model for the LKG prior on correlation matrix.

Description

Build an cgeneric model for the LKG prior on correlation matrix.

Usage

cgeneric_LKJ(
  n,
  eta,
  sigma.prior.reference = rep(1, n),
  sigma.prior.probability = rep(NA, n),
  ...
)

Arguments

Value

a cgeneric object, see INLAtools::cgeneric() for details.

See Also

It uses the Cannonical Partial Correlation (CPC), parametrization, see basecor() for details.

Build an cgeneric to implement the Wishart prior for a precision matrix.

Description

Build an cgeneric to implement the Wishart prior for a precision matrix.

Usage

cgeneric_Wishart(n, dof, R, ...)

Arguments

Details

For a random p\times p precision matrix Q, given the parameters d and R, respectively scalar degree of freedom and the inverse scale p\times p matrix the Wishart density is

|Q|^{(d-p-1)/2}\textrm{e}^{-tr(RQ)/2}|R|^{p/2}2^{-dp/2}\Gamma_p(n/2)^{-1}

Value

a cgeneric, INLAtools::cgeneric() object.

Build an cgeneric for a (graph based) correlation matrix PC-prior.

Description

From either a graphpcor (see graphpcor()) or a square matrix (used as a graph), creates an cgeneric (see INLAtools::cgeneric()) to implement the Penalized Complexity prior using the Kullback-Leibler divergence - KLD from a base graphpcor.

Usage

cgeneric_graphpcor(
  model,
  lambda,
  base,
  sigma.prior.reference,
  sigma.prior.probability,
  iparams,
  cfixed,
  d0,
  ...
)

Arguments

Details

The parametrization is set as in basepcor() and the base is used to define an informative prior, as derived in the pcmultivariate vignette.

Value

cgeneric object.

See Also

graphpcor() and basepcor()

Build an cgeneric for a correlation matrix PC-prior.

Description

Build an cgeneric for a correlation matrix PC-prior.

Usage

cgeneric_pc_correl(n, base, iLtheta, iparams, ...)

## S3 method for class 'basecor'
cgeneric(
  model,
  lambda,
  sigma.prior.reference,
  sigma.prior.probability,
  iparams,
  cfixed,
  ...
)

Arguments

Details

The parametrization is set as in basecor() and the base is used to define an informative prior, as derived in the pcmultivariate vignette.

Value

a cgeneric object, see INLAtools::cgeneric() for details.

Functions

• cgeneric(basecor): Build a cgeneric for a basecor.

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>

Build an cgeneric for treepcor())

Description

This creates an cgeneric (see INLAtools::cgeneric()) containing the necessary data to implement the penalized complexity prior for a correlation matrix considering a three as proposed in Sterrantino et. al. 2025 doi:10.1007/s10260-025-00788-y.

Usage

cgeneric_treepcor(
  model,
  lambda,
  sigma.prior.reference,
  sigma.prior.probability,
  ...
)

Arguments

Details

The correlation prior as in the paper depends on the lambda value. The prior for each sigma_i is the Penalized-complexity prior which can be defined from the following probability statement P(sigma > U) = a. where "U" is a reference value and "a" is a probability. The values "U" and probabilities "a" for each sigma_i are passed in the sigma.prior.reference and sigma.prior.probability arguments. If a=0 then U is taken to be the fixed value of the corresponding sigma. E.g. if there are three sigmas in the model and one supply sigma.prior.reference = c(1, 2, 3) and sigma.prior.probability = c(0.05, 0.0, 0.01) then the sigma is fixed to 2 and not estimated.

Value

cgeneric/cgeneric object.

See Also

treepcor() and INLAtools::cgeneric()

The LKJ density for a correlation matrix

Description

The LKJ density for a correlation matrix

Usage

dLKJ(R, eta, log = FALSE)

Arguments

Value

numeric as the (log) density

Retrieve edges of an object

Description

Retrieve edges of an object

Usage

edges(object)

## Default S3 method:
edges(object)

## S3 method for class 'graphpcor'
edges(object)

## S3 method for class 'treepcor'
edges(object)

Arguments

Methods (by class)

• edges(default): Default method for edges

• edges(graphpcor): Extract the edges of a graphpcor.

• edges(treepcor): Extract the edges of a treepcor

Internal functions.

Description

Internal functions.

Usage

p_iLtheta_fncheck(p, iLtheta)

m_iparams_fncheck(m, iparams)

KLD10(C1, C0, L1, L0)

pcSigmasCheck(nsigmas, sigma.prior.reference, sigma.prior.probability)

Arguments

Details

By assuming equal mean vector we have

KLD = 0.5( tr(C0^{-1}C1) -p - log(|C1|) + log(|C0|) )

Functions

• p_iLtheta_fncheck(): Function to deal with p and iLtheta

• m_iparams_fncheck(): Function to deal with m and iparams

• KLD10(): Compute the KLD between two multivariate Gaussian distributions, assuming equal mean vector

• pcSigmasCheck(): Check the PC-prior arguments for sigma.

Functions for numerical algorithms

Description

Functions for numerical algorithms

Usage

dspd(M, decomposition = "svd")

## S3 method for class 'basecor'
hessian(func, x, method = "Richardson", method.args = list(), ...)

## S3 method for class 'basepcor'
hessian(func, x, method = "Richardson", method.args = list(), ...)

## S3 method for class 'graphpcor'
hessian(func, x, method = "Richardson", method.args = list(), ...)

Arguments

Value

dspd returns a list with the decomposition elements, "logDeterminant" (of the original matrix), "sqrt" (its 'square root') and "sqrtInv" (its inverse 'square root'). hessian returns the Hessian

matrix with the Hessian

Functions

• dspd(): hessian: Evaluate the Hessian of the KLD for a basecor. spd: decompose a positive definite matrix, and compute useful elements out of that.

• hessian(basecor): Evaluate the Hessian for a basecor.

• hessian(basepcor): Evaluate the hessian of the KLD for a basepcor.

• hessian(graphpcor): Evaluate the hessian of the KLD for a graphpcor correlation model around a base model.

Internal functions to map between Euclidean and spherical coordinates

Description

Internal functions to map between Euclidean and spherical coordinates

Usage

rphi2x(rphi)

x2rphi(x)

rtheta(n, lambda = 1, R, theta.base)

dtheta(theta, lambda, theta.base, H.elements)

Arguments

Details

For details, please see the wikipedia entry on 'N-sphere' at N-sphere

Functions

• rphi2x(): Map between spherical to Euclidean coordinates

• x2rphi(): Transform from Euclidean coordinates to spherical

• rtheta(): Drawn samples from the PC-prior for correlation

• dtheta(): PC-prior density for the correlation matrix

Build/add STAN code/data for the correlation's PC-prior.

Description

Build/add STAN code/data for the correlation's PC-prior.

Usage

stan_add(x, model, lambda, name)

stan_add_pc_correl(x, model, lambda, name)

stan_add_graphpcor(x, model, lambda, name)

stan_add_code(x, to_add)

Arguments

Details

The parametrization is set as in basecor() or basepcor(). If a basecor is provided, the prior would be considered for the Cholesky factor of a correlation matrix. If a basepcor is provided, the prior would be considered for a correlation matrix (parametrized from it's inverse). The base is used to define an informative prior, as derived in the pcmultivariate vignette.

Value

a list of two elements, one as a list of three additional code to be added into a STAN code and the other with the required additional data.

Functions

• stan_add_pc_correl(): method for basecor

• stan_add_graphpcor(): method for basepcor

• stan_add_code(): add code at the end of each section

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>

Examples

## STAN model code without the prior for L, as LCorr
Scode0 <- "
data {
  int<lower=1> p;
  int<lower=1> n;
  vector[p] y[n];
  vector[p] mu;
}
model {
  y ~ multi_normal_cholesky(mu, LCorr);
}
generated quantities {
  corr_matrix[p] Corr = tcrossprod(LCorr);
}
"

## add the pc_correl code
Scode <- stan_add(Scode0, 'pc_correl', lambda = 1, name = "LCorr")
Scode

Sdata0 <- list(
  p = as.integer(3), 
  n = as.integer(1), 
  y = matrix(0,1,3), 
  mu = rep(0.0, 3)
)

baseC <- basecor(rep(0,3), 3)

Sdata <- stan_add(Sdata0, baseC, lambda = 1, name = "LCorr")
str(Sdata)

treepcor: correlation from tree

Description

A tree with two kind of nodes, parents and children. The parents are nodes with children. The children are nodes with no children. This is used to model correlation matrices, where parents represent latent variables, and children represent the variables of interest. A tree is defined from a formula for each parent.

Usage

treepcor(...)

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

## S3 method for class 'treepcor'
summary(object, ...)

## S3 method for class 'treepcor'
dim(x, ...)

drop1(object)

## S3 method for class 'treepcor'
plot(x, y, ...)

## S3 method for class 'treepcor'
prec(model, ...)

etreepcor2precision(d.el)

## S3 method for class 'treepcor'
vcov(object, ...)

etreepcor2variance(d.el)

## S3 method for class 'treepcor'
cgeneric(model, ...)

Arguments

Details

In the formula, the left side are parent variables, and the right side include all the children and parents that are also children. The children variables are those with an ancestor (parent), and are identified as c1, ..., cn, where n is the total number of children variables. The parent variables are identified as p1, ..., pm, where the m is the number of parent variables. The main parent (first) should be identified as p1. Except p1 all the other parent variables have an ancestor, which is a parent variable.

Value

a treepcor object

Methods (by generic)

• print(treepcor): The print method for a treepcor

• summary(treepcor): The summary method for a treepcor

• dim(treepcor): The dim for a treepcor

• plot(treepcor): The plot method for a treepcor

• prec(treepcor): The prec for a treepcor

• vcov(treepcor): The vcov method for a treepcor

• cgeneric(treepcor): The cgeneric method for treepcor, uses cgeneric_treepcor()

Functions

• drop1(): The drop1 method for a treepcor

• etreepcor2precision(): Internal function to extract elements to build the precision from the treepcor edges.

• etreepcor2variance(): Internal function to extract elements to build the covariance matrix from a treepcor.

Examples

## for details see
## https://link.springer.com/article/10.1007/s10260-025-00788-y

if(FALSE) {
    
### examples of what is not allowed
    treepcor(p1 ~ p2)
    treepcor(p1 ~ c2)
    
    treepcor(
        p1 ~ c1 + c2,
        p2 ~ c3)
    
    treepcor(
        p1 ~ c1 + c2,
        p2 ~ p1 + c2 + c3)
    
    treepcor(
        p1 ~ c1 + c2,
        p2 ~ p3 + c2 + c3)
    
    treepcor(
        p1 ~ p2 + c1 + c2,
        p2 ~ c2 + c3)

}

### allowed cases

## 3 children and 1 parent
tree1 <- treepcor(p1 ~ c1 + c2 - c3)

tree1

dim(tree1)

summary(tree1)

plot(tree1)

prec(tree1)

(q1 <- prec(tree1, theta = c(0)))

v1 <- chol2inv(chol(q1))

v1

cov2cor(v1)

vcov(tree1, raw = TRUE)
cov2cor(vcov(tree1, raw = TRUE))
vcov(tree1)

vcov(tree1, theta = 0)
vcov(tree1, theta = -1)
vcov(tree1, theta = 1)

cov2cor(vcov(tree1))
cov2cor(vcov(tree1, theta = -1))
cov2cor(vcov(tree1, theta = 1))

## 4 children and 2 parent
tree2 <- treepcor(
    p1 ~ p2 + c1 + c2,
    p2 ~ -c3 + c4)
tree2
dim(tree2)
summary(tree2)

prec(tree2)
prec(tree2, theta = c(0, 0))
prec(tree2, theta = c(-1, 1))

vcov(tree2, raw = TRUE)
cov2cor(vcov(tree2, raw = TRUE))

cov2cor(solve(prec(tree2, theta = c(0,0))))
vcov(tree2, theta = c(0,0))
vcov(tree2, theta = c(log(4:1), 0,0))

vcov(tree2)

tree2

## 4 children and 2 parent (notice the signs)
tree2b <- treepcor(
    p1 ~ -p2 + c1 + c2,
    p2 ~ -c3 + c4)
tree2b
dim(tree2b)
summary(tree2b)

summary(tree2)
summary(tree2b)

par(mfrow = c(1, 2), mar = c(0,0,0,0))
plot(tree2)
plot(tree2b)

prec(tree2)
prec(tree2b)

## prec is not equal
all.equal(prec(tree2, theta = c(0, 0)),
          prec(tree2b, theta = c(0, 0)))

## vcov is equal
all.equal(vcov(tree2, theta = c(0, 0)),
          vcov(tree2b, theta = c(0, 0)))