Nothing
Tree based latent variables correlation model
In graphpcor: Models for Correlation Matrices Based on Graphs
knitr::opts_chunk$set(
echo = TRUE,
message = FALSE,
warning = FALSE,
fig.width = 9,
fig.height = 5,
out.width = '0.49\\textwidth',
fig.align = 'center')
library(graphpcor)
chinclude = TRUE
This material illustrate and detail the model properties of the
graphical approach to model correlations proposed in @sterrantino2025graphical.
We first formalize the specific kind of graph used
to define the model and provides an intuitive visualization.
We use some examples for illustration and
show how to implement the tree based correlation
model proposed.
We add details of the model definition at the end.
The directed tree model definition
A graph is defined from a set of nodes and edges,
where the nodes are linked by the edges.
A directed acyclic graph consider
directed edges linking the nodes.
In the later case there is no closed loop.
Yet, a tree is a graph where there is only one
possible path between a pair of nodes.
Furthermore, the edges of a tree are directed edges.
This also impose a hierarchy on the nodes.
The parents and children tree
We consider a particular tree definition
where we have two different class of nodes to
represent two kind of variables.
We classify the nodes as parent or children.
The nodes classified as children are leafs of the tree,
which are nodes with an ancestor (parent)
but with no children.
The nodes classified as parent will always have children nodes,
and some may have parent but will still be classified as parent
because they have children.
The directed edges (arrows) represent conditional distributions.
At the end of this material we provide some
detailed definition and examples.
Suppose we have $k$ variables of interest,
and want to model their correlation.
These $k$ variables will be represented by
the children nodes and
labeled as $c_i$, $i\in{1,\ldots,m}$.
The parent nodes are then labeled as $p_j$, $j\in{1,\ldots,k}$.
The parent $p_1$ is the main parent, with no parent,
which is the root of the tree.
We have some examples in Figure 1 bellow.
tree1 <- treepcor(p1 ~ c1 + c2 - c3)
tree2 <- treepcor(p1 ~ p2 + c1 + c2,
p2 ~ -c3 + c4)
tree3a <- treepcor(p1 ~ p2 + p3 + c1 + c2,
p2 ~ -c3 + c4,
p3 ~ c5 + c6)
tree3b <- treepcor(p1 ~ p2 + c1 + c2,
p2 ~ p3 - c3 + c4,
p3 ~ c5 + c6)
par(mfrow = c(2, 2), mar = c(0, 0, 0, 0))
plot(tree1)
legend("topleft", "(A)", bty = "n")
plot(tree2)
legend("topleft", "(B)", bty = "n")
plot(tree3a)
legend("topleft", "(C)", bty = "n")
plot(tree3b)
legend("topleft", "(D)", bty = "n")
NOTE: The children variable c3 is with a
negative sign, assuming it has a negative correlation
with the other children variables.
If a graph is used to represent conditional
distributions, when computing the marginal distribution,
it induces a the covariance between two nodes to be
non-zero covariance as long as there is a path linking them
in the graph.
The same apply in our approach.
Moreover, the strength of the correlation depends on
how many parents, and their hierarchy,
there is in the path linking them.
The implied covariance
Let us represent the correlation between $a$ and $b$ as C(a,b).
In the model defined with graph (A),
|C($c_1$,$c_2$)| = |C($c_1$,$c_3$)| = |C($c_2$,$c_3$)|.
From graph (B)
|C($c_1$,$c_3$)| = |C($c_1,c_4$)| = |C($c_2$,$c_3$)| =
|C($c_2$,$c_4$)| $\leq$ |C($c_1$,$c_2$)| $\leq$ |C($c_3$,$c_4$)|.
With the models defined from the graphs (C) and (D) we have
|C($c_1$,$c_2$)|$\leq$|C($c_3$,$c_4$)|,
|C($c_1$,$c_2$)|$\leq$|C($c_5$,$c_6$)|, and
|C($c_1$,$c_3$)|=|C($c_1$,$c_4$)|=
|C($c_2$,$c_3$)|=|C($c_2$,$c_4$)| and
|C($c_1$,$c_5$)|=|C($c_1$,$c_6$)|=
|C($c_2$,$c_5$)|=|C($c_2$,$c_6$)|.
If we want to not fix the order between |C($c_3$,$c_4$)|
and |C($c_5$,$c_6$) we can use the model from graph (C),
whereas model (D) impose |C($c_3$,$c_4$)|$\leq$|C($c_5$,$c_6$)| and
the correlation strength between a children of $p_3$ to a
children of $p_1$ to be smaller than to a children of $p_2$, e.g.
|C($c_1$,$c_5$)|$\leq$|C($c_3$,$c_5$)|.
These results follow from the application of the
law of total expectation, variance and covariance.
Let us start by having Var($p_j$) = $\gamma_j^2$ and
the conditional variance for each children as
Var($c_i$|$p_{c_i}$)=1.
Thus, the marginal variance of $C_i$ is
equal the sum of the
variances of its ancestors plus 1.
Illustrative code examples
In the \textbf{\textsf{R}} environment we represent the parent
variables with the letter p along with an integer number
(p1, $\ldots$, pk) and the children variables with
the letter c along with an integer number (c1, $\ldots$, cm).
We adopt a simple way to specify the parent children representation.
We consider the ~ (tilde) to represent the directed link and
+ (plus) or - (minus) to append the descendant to a parent.
E.g.: p1 ~ p2 + c1 + c2, p2 ~ c3 - c4 .
Intial example
Let us consider a correlation model for three variables,
with one parameter,
that is the same absolute correlation between each pair
but the sign may differ.
This can be the case when these variables share the
same latent factor.
The parent is represented as p1,
and the children variables as c1, c2 and c3.
We consider that c3 will be negatively correlated
with c1 and c2.
For this case we define the tree as
tree1 <- treepcor(p1 ~ c1 + c2 - c3)
tree1
summary(tree1)
where the summary shows their relationship.
The number of children and parent variables
are obtained with
dim(tree1)
This tree can be visualized with
plot(tree1)
From this model definition we will use the methods
to build the correlation matrix.
First, we build the precision matrix structure
(that is not yet a precision matrix):
prec(tree1)
and we can use inform the log of $\gamma_1$,
which is the standard error for $p_1$, with:
q1 <- prec(tree1, theta = 0)
q1
We can obtain the correlation matrix, which is our
primarily interest, from the precision matrix.
However, also have a covariance method to be
directly applied with
vcov(tree1) ## assume theta = 0 (\gamma_1 = 1)
vcov(tree1, theta = 0.5) # \gamma_1^2 = exp(2 * 0.5) = exp(1)
cov1a <- vcov(tree1, theta = 0)
cov1a
from where we obtain the desired matrix with
c1 <- cov2cor(cov1a)
round(c1, 3)
Correlation matrix with two parameters
In this example, we model the correlation between four
variables using two parameters.
We consider c1 and c2 having the same parent, p1
and c3 and c4 having the second parent as parent.
We want to have the correlation between c3 and c4
higher than the correlation between c1 and c3.
This requires p2 to be children of p1.
The tree for this is set by
tree2 <- treepcor(
p1 ~ p2 + c1 + c2,
p2 ~ -c3 + c4)
dim(tree2)
tree2
summary(tree2)
which can be visualized by
plot(tree2)
We can use the drop1 method to remove the last parent with
drop1(tree2)
which align with the prior for the model
parameters as proposed in @sterrantino2025graphical.
However, the code to implement such a prior is
an internal C code and do not use this drop1
which is fully a R code.
In order to apply this for real data
we do recommend to use the \textbf{\textsf{R}}
as shown further ahead in this vignette.
We now have two parameters:
$\gamma_1^2$ the variance of $p_1$
and $\gamma_2^2$ the conditional variance of $p_2$.
For $\gamma_1 = \gamma_2 = 1$,
the precision matrix can be obtained with:
q2 <- prec(tree2, theta = c(0, 0))
q2
The correlation matrix can be obtained with
cov2 <- vcov(tree2, theta = c(0, 0))
cov2
c2 <- cov2cor(cov2)
round(c2, 3)
Playing with sign
We can change the sign at any edge of the graph.
The change in the edge of parent to children is
simpler to interpret, as we can see in the
covariance/correlation from the two examples.
Let us consider the second example but change the sign
between the parents and swap the sign
in both terms of the second equation:
tree2b <- treepcor(
p1 ~ -p2 + c1 + c2,
p2 ~ -c3 + c4)
tree2b
summary(tree2b)
This gives the precision matrix as
q2b <- prec(tree2b, theta = c(0, 0))
q2b
The covariance computed from the full precision
(and the correlation) between children is
the same as before
all.equal(solve(q2)[1:4, 1:4],
solve(q2b)[1:4, 1:4])
Therefore, allowing flexibility in an edge of parent to another parent is not useful and will only imply more complexity.
Therefore we will not consider it in the vcov method.
NOTE: The vcov applied to a treepcor does not
takes into account the sign between parent variables!
So, please use it with care.
In case we just want to compute the raw covariance,
assuming that V($p_j$)=1, to see the law of sum of
for the four examples in Figure 1 we have
vcov(tree1, raw = TRUE)
vcov(tree2, raw = TRUE)
tree3a <- treepcor(p1 ~ p2 + p3 + c1 + c2,
p2 ~ -c3 + c4,
p3 ~ c5 + c6)
vcov(tree3a, raw = TRUE)
tree3b <- treepcor(p1 ~ p2 + c1 + c2,
p2 ~ p3 - c3 + c4,
p3 ~ c5 + c6)
vcov(tree3b, raw = TRUE)
NOTE: The vcov method for a treepcor
without the theta argument assume
$\gamma_j=1$ and $\sigma_i=1$.
With not providing raw (or raw = FALSE) it
will standardize the raw matrix then
apply the $\sigma_i$.
The penalized complexity prior
To implement the prior proposed in @sterrantino2025graphical
we have used the cgeneric interface.
First, let us define the model and its parameters.
Let us consider the two parents example.
tree2
(np <- dim(tree2))
Define the standard deviation parameters for the
variables of interest
sigmas <- c(4, 2, 1, 0.5)
and the log of the parent variance parameters
thetal <- c(0, 1)
With that we can build the covariance with
theta1 <- c(log(sigmas), thetal)
Vg <- vcov(tree2, theta = theta1)
Vg
Now we can simulate some data
nrep <- 100
nd <- nrep * np[1]
xx <- matrix(rnorm(nd), nrep) %*% chol(Vg)
cov(xx)
theta.y <- log(10)
datar <- data.frame(
r = rep(1:nrep, np[1]),
i = rep(1:np[1], each = nrep),
y = rnorm(nd, 1 + xx, exp(-2*theta.y))
)
We now use the cgeneric interface
implemented by the \textbf{\textsf{INLAtools}} package.
For that, we can apply the cgeneric method to a
treepcor object.
The prior parameter is the param argument.
cmodel <- cgeneric(
tree2, lambda = 5,
sigma.prior.reference = rep(1, np[1]),
sigma.prior.probability = rep(0.05, np[1]))
The following code chunks require INLA installed.
haveINLA <- FALSE
Perform some checks (INLA is required):
graph(cmodel)
initial(cmodel)
prior(cmodel, theta = rep(0, sum(np)))
prior(cmodel, theta = rep(1, sum(np)))
np
prec(cmodel, theta = rep(0, sum(np)))
(Qc <- prec(cmodel, theta = theta1))
all.equal(Vg, as.matrix(solve(Qc)))
Model fit (if have INLA installed)
m1 <- y ~ f(i, model = cmodel, replicate = r)
pfix <- list(prec = list(initial = 10, fixed = TRUE))
fit <- inla(
formula = m1,
data = datar,
control.family = list(hyper = pfix)
)
fit$cpu.used
Compare the true covariance, the observed covariance
and the fitted covariance (from the posterior mode for theta)
round(Vg, 2)
round(cov(xx), 2)
round(Vfit <- vcov(tree2, theta = fit$mode$theta), 2)
The fitted correlation (from posterior mode):
round(Cfit <- vcov(tree2, theta = fit$mode$theta[np[1]+1:np[2]]), 2)
round(cov2cor(Vfit), 2)
Model definition details
We now define a set of rules to propose a model.
We assume a distribution for the main parent, $p_1$,
and conditional distributions for each of the
remaining $k-1$ parents and each of the $m$ children.
The arrows represent each of these conditional distributions.
Since we assume Gaussian distributions,
we just need to define the (conditional) mean and variance.
-
1st rule: For the main parent $p_1$,
the mean is zero and the variance is $\gamma_1^2$.
-
2nd rule: The conditional mean for $p_j$, $j=2$,...$k$,
is its parent and the conditional variance is $\gamma_j^2$.
-
3rd rule: The conditional mean of $c_i$, $i=1$, ..., $m$,
is $s_i$ times its parent and unit conditional variance,
where $s_i$ is assumed known, either "-1" or "+1".
We have $p$ parameters, one for each parent.
These conditional distributions build up a joint one
that will have a covariance matrix.
Since we are primarily interested in the correlations,
we will scale it by the implied variances and obtaining
a correlation matrix.
The marginal variances for each children are assigned
after this scaling.
As we will usually have $k<m$,
this represents a reduction on the number of
parameters, from $m(m+1)/2$ to $k+m$.
As an illustration, consider the graph (A) from Figure 1.
The joint distribution for ${c_1, c_2, c_3, p_1}$ is
[
\pi(c_1, c_2, c_3, p_1) \propto \mathrm{exp}
\left{-\frac{1}{2} \left[ \frac{p_1^2}{\gamma_1^2} + \frac{(c_1-s_1p_1)^2}{s_1^2} + \frac{(c_2-s_2p_1)^2}{s_2^2} + \frac{(c_3-s_3p_1)^2}{s_3^2} \right] \right}.
]
From this set of conditional distributions we obtain the precision matrix as
[
\left[ \begin {array}{cccc}
{{\it s_1}}^{-2}&0&0&-{{\it s_1}}^{-1}\
0&{{\it s_2}}^{-2}&0&-{{\it s_2}}^{-1}\
0&0&{{\it s_3}}^{-2}&-{{\it s_3}}^{-1}\
-{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&-{{\it s_3}}^{-1}&{\gamma_1}^{-2}+3
\end {array} \right].
]
The joint distribution for ${c_1, c_2, c_3, c_4, p_1, p_2}$,
from the DAG (B) in Figure 1, is proportional to
[
\mathrm{exp}
\left{-\frac{1}{2} \left[ \frac{p_1^2}{\gamma_1^2} + \frac{(p_2-p_1)^2}{\gamma_2^2} +
\frac{(c_1-s_1p_1)^2}{s_1^2} + \frac{(c_2-s_2p_1)^2}{s_2^2} +
\frac{(c_3-s_3p_2)^2}{s_3^2} + \frac{(c_4-s_4p_2)^2}{s_4^2} \right] \right}.
]
The precision matrix is
[
\left[ \begin {array}{cccccc}
{{\it s_1}}^{-2}&0&0&0&-{{\it s_1}}^{-1}&0\
0&{{\it s_2}}^{-2}&0&0&-{{\it s_2}}^{-1}&0\
0&0&{{\it s_3}}^{-2}&0&0&-{{\it s_3}}^{-1}\
0&0&0&{{\it s_4}}^{-2}&0&-{{\it s_4}}^{-1}\
-{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&0&0&
{\gamma_1}^{-2}+{\gamma_2}^{-2}+2&-{\gamma_2}^{-2}\
0&0&-{{\it s_3}}^{-1}&-{{\it s_4}}^{-1}&-{\gamma_2}^{-2}&{\gamma_2}^{-2}+2
\end {array} \right]
]
The models represented by graphs (C) and (D) in Figure 1 have both the same
number of children and parents.
The only difference is that parent $p_3$ is children to parent $p_1$ in (C)
whereas in (D) $p_3$ is children to parent $p_2$.
For the case when $p_1$ is parent to $p_3$ the
precision matrix is
[
\left[ \begin {array}{ccccccccc}
{{\it s_1}}^{-2}&0&0&0&0&0&-{{\it s_1}}^{-1}&0&0\
0&{{\it s_2}}^{-2}&0&0&0&0&-{{\it s_2}}^{-1}&0&0\
0&0&{{\it s_3}}^{-2}&0&0&0&0&-{{\it s_3}}^{-1}&0\
0&0&0&{{\it s_4}}^{-2}&0&0&0&-{{\it s_4}}^{-1}&0\
0&0&0&0&{{\it s_5}}^{-2}&0&0&0&-{{\it s_5}}^{-1} \
0&0&0&0&0&{{\it s_6}}^{-2}&0&0&-{{\it s_6}}^{-1}\
-{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&0&0&0&0&
{\gamma_1}^{-2}+{\gamma_2}^{-2}+{\gamma_3}^{-2} +2 &
-{\gamma_2}^{-2}&-{\gamma_3}^{-2}\
0&0&-{{\it s_3}}^{-1}&-{{\it s_4}}^{-1}&0&0&
-{\gamma_2}^{-2}&{\gamma_2}^{-2}+2&0\
0&0&0&0&-{{\it s_5}}^{-1}&-{{\it s_6}}^{-1}&
-{\gamma_3}^{-2}&0&{\gamma_3}^{-2}+2
\end {array} \right]
]
and the case when $p_2$ is parent to $p_3$ the last three rows of the precision matrix are
[
\left[ \begin {array}{ccccccccc}
-{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&0&0&0&0&{\gamma_1}^{-2}+{\gamma_2}^{-2}+2&-{\gamma_2}^{-2}
&0\ 0&0&-{{\it s_3}}^{-1}&-{{\it s_4}}^{-1}&0&0&-{\gamma_2}^{-2}&{\gamma_2}^{-2}+{\gamma_3}^{-2}+2&-{\gamma_3}^{-2}\ 0&0&0&0&-{{\it s_5}}^{-1}&-{
{\it s_6}}^{-1}&0&-{\gamma_3}^{-2}&{\gamma_3}^{-2}+2\end {array} \right]
]
and the first $6$ rows are equal the ones from the previous case.
Model properties
The conditional distributions defines a precision matrix
for all the $m+k$ variables in the model and a joint distribution.
Therefore we can obtain the implied covariance matrix
to learn the marginal properties, and correlations as well.
Since our primarily interest is in the correlation between the children variables, we consider its first $m$ rows and columns.
The model defined from the DAG (A) in Figure 1 for
${c_1, c_2, c_3, p_1}$ induces a covariance matrix equals to
[
\left[ \begin {array}{cccc}
( 1+{\gamma_1}^{2} ) {{\it s_1}}^{2} & {\it s_2}{\it s_1}{\gamma_1}^{2}&
{\it s_3}{\it s_1}{\gamma_1}^{2} & {\it s_1}{\gamma_1}^{2} \
{\it s_2}{\it s_1}{\gamma_1}^{2} & ( 1+{\gamma_1}^{2} ) {{\it s_2}}^{2}&
{\it s_3}{\it s_2}{\gamma_1}^{2} & {\it s_2}{\gamma_1}^{2} \
{\it s_3}{\it s_1}{\gamma_1}^{2} &{\it s_3}{\it s_2}{\gamma_1}^{2}&
( 1+{\gamma_1}^{2} ) {{\it s_3}}^{2} & {\it s_3}{\gamma_1}^{2} \
{\it s_1}{\gamma_1}^{2} & {\it s_2}{\gamma_1}^{2} &
{\it s_3}{\gamma_1}^{2}&{\gamma_1}^{2}
\end {array} \right]
]
and the C($c_i,c_j$) = sign($s_is_j$)$\gamma_1^2/(1+\gamma_1^2)$, for $i,j\in{1,2,3}$ and $i\neq j$.
The model defined from the DAG (B) in Figure 1 gives the covariance matrix
for ${c_1, c_2, c_3, c_4, p_1, p_2}$ as
[
\left[ \begin {array}{cccccc} \left( 1+{\gamma_1}^{2} \right) {{\it s_1}}^{2}&{\it s_2}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it
s_1}&{\gamma_1}^{2}{\it s_1}&{\gamma_1}^{2}{\it s_1}\ {\it s_2}\,{\gamma_1}^{2}{\it s_1}& \left( 1+{\gamma_1}^{2} \right) {{\it s_2}}^{2}&{\it s_3}\,{\gamma_1
}^{2}{\it s_2}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\gamma_1}^{2}{\it s_2}&{\gamma_1}^{2}{\it s_2}\ {\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{
2}{\it s_2}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_3}}^{2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\gamma_1}^{2}{\it s_3}& \left( {{
\it \gamma_2}}^{2}+{\gamma_1}^{2} \right) {\it s_3}\ {\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\, \left( {\gamma_2}^{2}+{{\it
\gamma_1}}^{2} \right) {\it s_3}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_4}}^{2}&{\gamma_1}^{2}{\it s_4}& \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}
\ {\gamma_1}^{2}{\it s_1}&{\gamma_1}^{2}{\it s_2}&{\gamma_1}^{2}{\it s_3}&{\gamma_1}^{2}{\it s_4}&{\gamma_1}^{2}&{\gamma_1}^{2}\ {\gamma_1
}^{2}{\it s_1}&{\gamma_1}^{2}{\it s_2}& \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}& \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}&{\gamma_1}^{2}&{\gamma_2}
^{2}+{\gamma_1}^{2}\end {array} \right]
]
from what the marginal properties can be learn.
The upper triangle of the children correlation matrix is
[
\left[ \begin {array}{ccc}
\frac{\textrm{sign}(s_1s_2){\gamma_1}^{2}}{{( {\gamma_1}^{2}+1 )}} &
\frac{\textrm{sign}(s_1 s_3){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}} &
\frac{\textrm{sign}(s_1 s_4){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}}\
& \frac{\textrm{sign}(s_2 s_3){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}} &
\frac{\textrm{sign}(s_2 s_4){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}}\
& & \frac{{\textrm{sign}(s_3 s_4) ( {\gamma_2}^{2}+{\gamma_1}^{2}) }}{
{( 1+{\gamma_2}^{2}+{\gamma_1}^{2})}}
\end{array} \right]
]
where C($s_i,s_j$) = sign($s_is_j$)$\frac{\gamma_1^2}{(\gamma_1^2+1)(1+\gamma_1^2+\gamma_2^2)}$
for $i\in{1,2}$ and $j\in{3,4}$, which defines a block with the same absolute value
and is the main structure that this model induced for the correlation matrix.
In the model given from graph (C), when $p_1$ is parent to $p_3$,
the covariance matrix of ${c_1,...,c_6}$ is
[ \scriptsize
\left[ \begin {array}{cccccc} \left( 1+{\gamma_1}^{2} \right) {{\it s_1}}^{2}&{\it s_2}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it
s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_1}\ {\it s_2}\,{\gamma_1}^{2}{\it s_1}& \left( 1+{\gamma_1}^{2} \right) {{\it s_2}}^{2}
&{\it s_3}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}\ {\it s_3}\,{{
\it \gamma_1}}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_2}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_3}}^{2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2}
\right) {\it s_3}&{\it s_5}\,{\gamma_1}^{2}{\it s_3}&{\it s_6}\,{\gamma_1}^{2}{\it s_3}\ {\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_2
}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_4}}^{2}&{\it s_5}\,{\gamma_1}^{2}{\it s_4}&{\it s_6}
\,{\gamma_1}^{2}{\it s_4}\ {\it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\,{\gamma_1}^{2}{\it s_3}&{\it s_5}\,{\gamma_1}^{2}
{\it s_4}& \left( 1+{\gamma_3}^{2}+{\gamma_1}^{2} \right) {{\it s_5}}^{2}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{\gamma_1}^{2} \right) \ {\it s_6}\,{{\it
\gamma_1}}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\,{\gamma_1}^{2}{\it s_3}&{\it s_6}\,{\gamma_1}^{2}{\it s_4}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{{\it
\gamma_1}}^{2} \right) & \left( 1+{\gamma_3}^{2}+{\gamma_1}^{2} \right) {{\it s_6}}^{2}\end {array} \right]
]
where the variances and covariance for $c_5$ and $c_6$ does not depends on $\gamma_2^2$.
For the case when $p_2$ is parent to $p_3$ the covariance of $c_1$, ..., $c_6$ is
[ \tiny
\left[ \begin {array}{cccccc} \left( 1+{\gamma_1}^{2} \right) {{\it s_1}}^{2}&{\it s_2}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it
s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_1}\ {\it s_2}\,{\gamma_1}^{2}{\it s_1}& \left( 1+{\gamma_1}^{2} \right) {{\it s_2}}^{2}
&{\it s_3}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}\ {\it s_3}\,{{
\it \gamma_1}}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_2}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_3}}^{2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2}
\right) {\it s_3}&{\it s_5}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_6}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}\ {\it
s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2}
\right) {{\it s_4}}^{2}&{\it s_5}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}&{\it s_6}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}\ {
\it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_5}\, \left( {\gamma_2}^{2}+{{\it
\gamma_1}}^{2} \right) {\it s_4}& \left( 1+{\gamma_3}^{2}+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_5}}^{2}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{\gamma_2}^{2}+{\gamma_1}
^{2} \right) \ {\it s_6}\,{\gamma_1}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_6
}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{\gamma_2}^{2}+{\gamma_1}^{2} \right) & \left( 1+{\gamma_3}^{2}+{\gamma_2
}^{2}+{\gamma_1}^{2} \right) {{\it s_6}}^{2}\end {array} \right]
]
where the variances and covariance for $c_5$ and $c_6$ now depends on $\gamma_2^2$.
For the correlations, we now compare C($c_i,c_j$), for $i\neq j$ and $i,j\in{3,4,5,6}$,
with these two different models.
The upper triangle of the correlation matrix for $c_3$, ..., $c_6$
obtained with the model where $p_1$ is parent to $p_3$ is
[
\left[ \begin {array}{ccc}
\textrm{sign}(s_3s_4)\frac{\gamma_1^2+\gamma_2^2}{1+\gamma_1^2+\gamma_2^2} &
\textrm{sign}(s_3s_5)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} &
\textrm{sign}(s_3s_6)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} \
& \textrm{sign}(s_4s_5)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} &
\textrm{sign}(s_4s_6)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} \
& & \textrm{sign}(s_5s_6)\frac{\gamma_1^2 + \gamma_3^2}{\sqrt{(1+\gamma_1^2+\gamma_3^2)}} \
\end{array} \right],
]
and with the model when $p_2$ is parent to $p_3$ it is
[
\left[ \begin {array}{ccc}
\textrm{sign}(s_3s_4)\frac{\gamma_1^2+\gamma_2^2}{1+\gamma_1^2+\gamma_2^2} &
\textrm{sign}(s_3s_5)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} &
\textrm{sign}(s_3s_6)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} \
& \textrm{sign}(s_4s_5)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} &
\textrm{sign}(s_4s_6)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} \
& & \textrm{sign}(s_5s_6)\frac{\gamma_1^2+\gamma_2^2+\gamma_3^2}{1+\gamma_1^2+\gamma_2^2+\gamma_3^2} \
\end{array} \right].
]
Comparing these results, when $p_1$ is parent to $p_3$ there is no contribution
from $\gamma_2^2$ in C($c_5,c_6$) and neither in C($c_i,c_j$), for $i\in{3,4}$,
whereas when $p_2$ is parent to $p_3$ there is.
From these examples we can list a set of properties as follows.
- Property 1: The marginal variance of $p_i$ is $\gamma_i^2$ plus the variance of its parent.
- Property 1a: For $p_1$ it is equal its conditional variance, $\gamma_1^2$.
- Property 1b: The marginal variance of $p_i$ is $\gamma_i^2$ plus
the conditional variance of its ancestors.
- Property 2: The marginal variance of $c_i$ is equal $s_i^2(1+V)$,
where $V$ is the variance of its parent.
- Property 3: The covariance between $c_i$ and $c_j$ is equal $s_is_jR$,
where $R$ is the variance of the top indexed parent in the path linking them.
- Property 3a: If $c_i$ and $c_j$ have a common parent,
$R$ is the variance of their parent.
- Property 4: C($c_i,c_j$)=sign($s_is_j$)R/$\sqrt{\gamma_{c_i}\gamma_{c_j}}$,
where $\gamma_{c_i}$ and $\gamma_{c_j}$ are the marginal variances of $c_i$ and $c_j$, respectively.
- Property 4a: If $c_i$ and $c_j$ have a common parent,
the correlation is sign($s_is_j$)$V/(1+V)$ where $V$ is the variance of their parent.
- Property 5: Let $k_0$ be the number of parents with no children variable as children,
$k_1$ be the number of parents with one children variable as children,
and $k_2$ be the number of parents with at least two children variables as children.
The number of distinct absolute correlation values is $k_2+(k_1-k_0)(k_1-k_0-1)/2$.
- Property 5a: If each parent variable has only one children variable as children,
this number is $k(k-1)/2$.
- Property 5b: The maximum number of distinct absolute correlation values is
$k(k+1)/2$, achieved if each parent variable is parent to at least two children variables.
This intuitive modeling approach gives way to build a
structured correlation matrix.
The number of distinct absolute correlation values is $k(k+1)/2$.
The sign assumed for each arrow ending at a children variable
allows the full range of the correlation values, from $-1$ to $1$.
Prior distributions can be considered for the marginal variances
and for the correlation matrix separately.
A further example
We now consider more variables and use two models, as shown in Figure 2 bellow.
The difference in on who is the parent to $p_4$.
We will use it to further illustrate the Property 2a.
t4a <- treepcor(
p1 ~ p2 + p3 + c1 + c2,
p2 ~ p4 + c3 + c4,
p3 ~ p5 + c5 + c6,
p4 ~ c7 + c8, p5 ~ c9 + c10)
t4b <- treepcor(p1 ~ p2 + p3 + c1 + c2,
p2 ~ c3 + c4, p3 ~ p4 + p5 + c5 + c6,
p4 ~ c7 + c8, p5 ~ c9 + c10)
par(mfrow = c(1, 2), mar = c(0, 0, 0, 0))
plot(t4a)
legend("topleft", "(E)", bty = "n")
plot(t4b)
legend("topleft", "(F)", bty = "n")
The model for the DAG in (E) gives a precision matrix whose last five columns are
[ \scriptsize
\left[ \begin {array}{ccccc} -{{\it s_1}}^{-1}&0&0&0&0\ -{{\it s_2}}^{-1}&0&0&0&0\ 0&-{{\it s_3}}^{-1}&0&0&0\ 0&-{{\it s_4}}^{-1}&0&0
&0\ 0&0&-{{\it s_5}}^{-1}&0&0\ 0&0&-{{\it s_6}}^{-1}&0&0\ 0&0&0&-{{\it s_7}}^{-1}&0\ 0&0&0&-{{\it s_8}}^{-1}&0
\ 0&0&0&0&-{{\it s_9}}^{-1}\ 0&0&0&0&-{{\it s_{10}}}^{-1}\ {\gamma_1}^{-2}+2+{\gamma_2}^{-2}+{\gamma_3}^{-2}&-{\gamma_2}^{-2}&-{{
\it \gamma_3}}^{-2}&0&0\ -{\gamma_2}^{-2}&{\gamma_2}^{-2}+2+{\gamma_4}^{-2}&0&-{\gamma_4}^{-2}&0\ -{\gamma_3}^{-2}&0&{\gamma_3}^{-2}+2+{\gamma_5}^
{-2}&0&-{\gamma_5}^{-2}\ 0&-{\gamma_4}^{-2}&0&{\gamma_4}^{-2}+2&0\ 0&0&-{\gamma_5}^{-2}&0&{\gamma_5}^{-2}+2\end {array} \right]
]
while the last five columns for the precision matrix for the model for DAG in (F) are
[ \scriptsize
\left[ \begin {array}{ccccc}
-{{\it s_1}}^{-1}&0&0&0&0\ -{{\it s_2}}^{-1}&0&0&0&0\ 0&-{{\it s_3}}^{-1}&0&0&0\ 0&-{{\it s_4}}^{-1}&0&0
&0\ 0&0&-{{\it s_5}}^{-1}&0&0\ 0&0&-{{\it s_6}}^{-1}&0&0\ 0&0&0&-{{\it s_7}}^{-1}&0\ 0&0&0&-{{\it s_8}}^{-1}&0
\ 0&0&0&0&-{{\it s_9}}^{-1}\ 0&0&0&0&-{{\it s_{10}}}^{-1}\ {\gamma_1}^{-2}+2+{\gamma_2}^{-2}+{\gamma_3}^{-2}&-{\gamma_2}^{-2}&-{{
\it \gamma_3}}^{-2}&0&0\ -{\gamma_2}^{-2}&{\gamma_2}^{-2}+2&0&0&0\ -{\gamma_3}^{-2}&0&{\gamma_3}^{-2}+2+{\gamma_4}^{-2}+{\gamma_5}^{-2}&-{\gamma_4
}^{-2}&-{\gamma_5}^{-2}\ 0&0&-{\gamma_4}^{-2}&{\gamma_4}^{-2}+2&0\ 0&0&-{\gamma_5}^{-2}&0&{\gamma_5}^{-2}+2
\end {array} \right]
]
To illustrate property about the top parent on a path, we consider
the upper triangle of the correlation matrix for the last four children variables, $c_7,...,c_{10}$.
For the model defined from (E) we have it as
[
\left[ \begin {array}{ccc}
{\frac { \textrm{sign}({\it s_7 s_8}) ( {\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} )}{( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} )}} &
\frac {\textrm{sign}({\it s_7 s_9}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}} &
\frac {\textrm{sign}({\it s_7 s_{10}}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}} \
& \frac {\textrm{sign}({\it s_8 s_9}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}} &
\frac {\textrm{sign}({\it s_8 s_{10}}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}}\
& & \frac { \textrm{sign}({\it s_{9} s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} \
\end {array} \right]
]
where C($c_i,c_j$), for $i\in{7,8}$ and $j\in{9,10}$
increase depends only on $\gamma_1^2$ and
depends on all $\gamma_j^2$ for $j={1,\ldots,5}$.
With the model defined from (F) we have
[ \
\left[ \begin {array}{ccc}
\frac {\textrm{sign}({\it s_7s_8}) ( {\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) }{( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) } &
\frac { \textrm{sign}({\it s_7s_9}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} &
\frac { \textrm{sign}({\it s_7s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} \
& \frac { \textrm{sign}({\it s_9s_9}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} &
\frac { \textrm{sign}({\it s_8s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} \
& & \frac {\textrm{sign}({\it s_9s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }{ ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}
\end {array} \right]
]
where now C($c_i,c_j$), for $i\in{7,8}$ and $j\in{9,10}$,
increases if $\gamma_1^2+\gamma_3^2$ (the variance of $p_3$) increases
and no longer depends on $\gamma_2^2$.
References
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knitr::opts_chunk$set( echo = TRUE, message = FALSE, warning = FALSE, fig.width = 9, fig.height = 5, out.width = '0.49\\textwidth', fig.align = 'center') library(graphpcor) chinclude = TRUE
This material illustrate and detail the model properties of the graphical approach to model correlations proposed in @sterrantino2025graphical. We first formalize the specific kind of graph used to define the model and provides an intuitive visualization. We use some examples for illustration and show how to implement the tree based correlation model proposed. We add details of the model definition at the end.
The directed tree model definition
A graph is defined from a set of nodes and edges, where the nodes are linked by the edges. A directed acyclic graph consider directed edges linking the nodes. In the later case there is no closed loop. Yet, a tree is a graph where there is only one possible path between a pair of nodes. Furthermore, the edges of a tree are directed edges. This also impose a hierarchy on the nodes.
The parents and children tree
We consider a particular tree definition where we have two different class of nodes to represent two kind of variables. We classify the nodes as parent or children. The nodes classified as children are leafs of the tree, which are nodes with an ancestor (parent) but with no children. The nodes classified as parent will always have children nodes, and some may have parent but will still be classified as parent because they have children. The directed edges (arrows) represent conditional distributions. At the end of this material we provide some detailed definition and examples.
Suppose we have $k$ variables of interest, and want to model their correlation. These $k$ variables will be represented by the children nodes and labeled as $c_i$, $i\in{1,\ldots,m}$. The parent nodes are then labeled as $p_j$, $j\in{1,\ldots,k}$. The parent $p_1$ is the main parent, with no parent, which is the root of the tree. We have some examples in Figure 1 bellow.
tree1 <- treepcor(p1 ~ c1 + c2 - c3) tree2 <- treepcor(p1 ~ p2 + c1 + c2, p2 ~ -c3 + c4) tree3a <- treepcor(p1 ~ p2 + p3 + c1 + c2, p2 ~ -c3 + c4, p3 ~ c5 + c6) tree3b <- treepcor(p1 ~ p2 + c1 + c2, p2 ~ p3 - c3 + c4, p3 ~ c5 + c6) par(mfrow = c(2, 2), mar = c(0, 0, 0, 0)) plot(tree1) legend("topleft", "(A)", bty = "n") plot(tree2) legend("topleft", "(B)", bty = "n") plot(tree3a) legend("topleft", "(C)", bty = "n") plot(tree3b) legend("topleft", "(D)", bty = "n")
NOTE: The children variable c3 is with a
negative sign, assuming it has a negative correlation
with the other children variables.
If a graph is used to represent conditional distributions, when computing the marginal distribution, it induces a the covariance between two nodes to be non-zero covariance as long as there is a path linking them in the graph. The same apply in our approach. Moreover, the strength of the correlation depends on how many parents, and their hierarchy, there is in the path linking them.
The implied covariance
Let us represent the correlation between $a$ and $b$ as C(a,b). In the model defined with graph (A), |C($c_1$,$c_2$)| = |C($c_1$,$c_3$)| = |C($c_2$,$c_3$)|. From graph (B) |C($c_1$,$c_3$)| = |C($c_1,c_4$)| = |C($c_2$,$c_3$)| = |C($c_2$,$c_4$)| $\leq$ |C($c_1$,$c_2$)| $\leq$ |C($c_3$,$c_4$)|.
With the models defined from the graphs (C) and (D) we have |C($c_1$,$c_2$)|$\leq$|C($c_3$,$c_4$)|, |C($c_1$,$c_2$)|$\leq$|C($c_5$,$c_6$)|, and |C($c_1$,$c_3$)|=|C($c_1$,$c_4$)|= |C($c_2$,$c_3$)|=|C($c_2$,$c_4$)| and |C($c_1$,$c_5$)|=|C($c_1$,$c_6$)|= |C($c_2$,$c_5$)|=|C($c_2$,$c_6$)|. If we want to not fix the order between |C($c_3$,$c_4$)| and |C($c_5$,$c_6$) we can use the model from graph (C), whereas model (D) impose |C($c_3$,$c_4$)|$\leq$|C($c_5$,$c_6$)| and the correlation strength between a children of $p_3$ to a children of $p_1$ to be smaller than to a children of $p_2$, e.g. |C($c_1$,$c_5$)|$\leq$|C($c_3$,$c_5$)|.
These results follow from the application of the law of total expectation, variance and covariance. Let us start by having Var($p_j$) = $\gamma_j^2$ and the conditional variance for each children as Var($c_i$|$p_{c_i}$)=1. Thus, the marginal variance of $C_i$ is equal the sum of the variances of its ancestors plus 1.
Illustrative code examples
In the \textbf{\textsf{R}} environment we represent the parent
variables with the letter p along with an integer number
(p1, $\ldots$, pk) and the children variables with
the letter c along with an integer number (c1, $\ldots$, cm).
We adopt a simple way to specify the parent children representation.
We consider the ~ (tilde) to represent the directed link and
+ (plus) or - (minus) to append the descendant to a parent.
E.g.: p1 ~ p2 + c1 + c2, p2 ~ c3 - c4 .
Intial example
Let us consider a correlation model for three variables,
with one parameter,
that is the same absolute correlation between each pair
but the sign may differ.
This can be the case when these variables share the
same latent factor.
The parent is represented as p1,
and the children variables as c1, c2 and c3.
We consider that c3 will be negatively correlated
with c1 and c2.
For this case we define the tree as
tree1 <- treepcor(p1 ~ c1 + c2 - c3) tree1 summary(tree1)
where the summary shows their relationship. The number of children and parent variables are obtained with
dim(tree1)
This tree can be visualized with
plot(tree1)
From this model definition we will use the methods to build the correlation matrix. First, we build the precision matrix structure (that is not yet a precision matrix):
prec(tree1)
and we can use inform the log of $\gamma_1$, which is the standard error for $p_1$, with:
q1 <- prec(tree1, theta = 0) q1
We can obtain the correlation matrix, which is our primarily interest, from the precision matrix. However, also have a covariance method to be directly applied with
vcov(tree1) ## assume theta = 0 (\gamma_1 = 1) vcov(tree1, theta = 0.5) # \gamma_1^2 = exp(2 * 0.5) = exp(1) cov1a <- vcov(tree1, theta = 0) cov1a
from where we obtain the desired matrix with
c1 <- cov2cor(cov1a) round(c1, 3)
Correlation matrix with two parameters
In this example, we model the correlation between four
variables using two parameters.
We consider c1 and c2 having the same parent, p1
and c3 and c4 having the second parent as parent.
We want to have the correlation between c3 and c4
higher than the correlation between c1 and c3.
This requires p2 to be children of p1.
The tree for this is set by
tree2 <- treepcor( p1 ~ p2 + c1 + c2, p2 ~ -c3 + c4) dim(tree2) tree2 summary(tree2)
which can be visualized by
plot(tree2)
We can use the drop1 method to remove the last parent with
drop1(tree2)
which align with the prior for the model
parameters as proposed in @sterrantino2025graphical.
However, the code to implement such a prior is
an internal C code and do not use this drop1
which is fully a R code.
In order to apply this for real data
we do recommend to use the \textbf{\textsf{R}}
as shown further ahead in this vignette.
We now have two parameters: $\gamma_1^2$ the variance of $p_1$ and $\gamma_2^2$ the conditional variance of $p_2$. For $\gamma_1 = \gamma_2 = 1$, the precision matrix can be obtained with:
q2 <- prec(tree2, theta = c(0, 0)) q2
The correlation matrix can be obtained with
cov2 <- vcov(tree2, theta = c(0, 0)) cov2 c2 <- cov2cor(cov2) round(c2, 3)
Playing with sign
We can change the sign at any edge of the graph. The change in the edge of parent to children is simpler to interpret, as we can see in the covariance/correlation from the two examples.
Let us consider the second example but change the sign between the parents and swap the sign in both terms of the second equation:
tree2b <- treepcor( p1 ~ -p2 + c1 + c2, p2 ~ -c3 + c4) tree2b summary(tree2b)
This gives the precision matrix as
q2b <- prec(tree2b, theta = c(0, 0)) q2b
The covariance computed from the full precision (and the correlation) between children is the same as before
all.equal(solve(q2)[1:4, 1:4], solve(q2b)[1:4, 1:4])
Therefore, allowing flexibility in an edge of parent to another parent is not useful and will only imply more complexity.
Therefore we will not consider it in the vcov method.
NOTE: The vcov applied to a treepcor does not
takes into account the sign between parent variables!
So, please use it with care.
In case we just want to compute the raw covariance, assuming that V($p_j$)=1, to see the law of sum of for the four examples in Figure 1 we have
vcov(tree1, raw = TRUE) vcov(tree2, raw = TRUE) tree3a <- treepcor(p1 ~ p2 + p3 + c1 + c2, p2 ~ -c3 + c4, p3 ~ c5 + c6) vcov(tree3a, raw = TRUE) tree3b <- treepcor(p1 ~ p2 + c1 + c2, p2 ~ p3 - c3 + c4, p3 ~ c5 + c6) vcov(tree3b, raw = TRUE)
NOTE: The vcov method for a treepcor
without the theta argument assume
$\gamma_j=1$ and $\sigma_i=1$.
With not providing raw (or raw = FALSE) it
will standardize the raw matrix then
apply the $\sigma_i$.
The penalized complexity prior
To implement the prior proposed in @sterrantino2025graphical
we have used the cgeneric interface.
First, let us define the model and its parameters.
Let us consider the two parents example.
tree2 (np <- dim(tree2))
Define the standard deviation parameters for the variables of interest
sigmas <- c(4, 2, 1, 0.5)
and the log of the parent variance parameters
thetal <- c(0, 1)
With that we can build the covariance with
theta1 <- c(log(sigmas), thetal) Vg <- vcov(tree2, theta = theta1) Vg
Now we can simulate some data
nrep <- 100 nd <- nrep * np[1] xx <- matrix(rnorm(nd), nrep) %*% chol(Vg) cov(xx) theta.y <- log(10) datar <- data.frame( r = rep(1:nrep, np[1]), i = rep(1:np[1], each = nrep), y = rnorm(nd, 1 + xx, exp(-2*theta.y)) )
We now use the cgeneric interface
implemented by the \textbf{\textsf{INLAtools}} package.
For that, we can apply the cgeneric method to a
treepcor object.
The prior parameter is the param argument.
cmodel <- cgeneric( tree2, lambda = 5, sigma.prior.reference = rep(1, np[1]), sigma.prior.probability = rep(0.05, np[1]))
The following code chunks require INLA installed.
haveINLA <- FALSE
Perform some checks (INLA is required):
graph(cmodel) initial(cmodel) prior(cmodel, theta = rep(0, sum(np))) prior(cmodel, theta = rep(1, sum(np))) np prec(cmodel, theta = rep(0, sum(np))) (Qc <- prec(cmodel, theta = theta1)) all.equal(Vg, as.matrix(solve(Qc)))
Model fit (if have INLA installed)
m1 <- y ~ f(i, model = cmodel, replicate = r) pfix <- list(prec = list(initial = 10, fixed = TRUE)) fit <- inla( formula = m1, data = datar, control.family = list(hyper = pfix) ) fit$cpu.used
Compare the true covariance, the observed covariance and the fitted covariance (from the posterior mode for theta)
round(Vg, 2) round(cov(xx), 2) round(Vfit <- vcov(tree2, theta = fit$mode$theta), 2)
The fitted correlation (from posterior mode):
round(Cfit <- vcov(tree2, theta = fit$mode$theta[np[1]+1:np[2]]), 2) round(cov2cor(Vfit), 2)
Model definition details
We now define a set of rules to propose a model. We assume a distribution for the main parent, $p_1$, and conditional distributions for each of the remaining $k-1$ parents and each of the $m$ children. The arrows represent each of these conditional distributions. Since we assume Gaussian distributions, we just need to define the (conditional) mean and variance.
-
1st rule: For the main parent $p_1$, the mean is zero and the variance is $\gamma_1^2$.
-
2nd rule: The conditional mean for $p_j$, $j=2$,...$k$, is its parent and the conditional variance is $\gamma_j^2$.
-
3rd rule: The conditional mean of $c_i$, $i=1$, ..., $m$, is $s_i$ times its parent and unit conditional variance, where $s_i$ is assumed known, either "-1" or "+1".
We have $p$ parameters, one for each parent. These conditional distributions build up a joint one that will have a covariance matrix. Since we are primarily interested in the correlations, we will scale it by the implied variances and obtaining a correlation matrix. The marginal variances for each children are assigned after this scaling. As we will usually have $k<m$, this represents a reduction on the number of parameters, from $m(m+1)/2$ to $k+m$.
As an illustration, consider the graph (A) from Figure 1. The joint distribution for ${c_1, c_2, c_3, p_1}$ is [ \pi(c_1, c_2, c_3, p_1) \propto \mathrm{exp} \left{-\frac{1}{2} \left[ \frac{p_1^2}{\gamma_1^2} + \frac{(c_1-s_1p_1)^2}{s_1^2} + \frac{(c_2-s_2p_1)^2}{s_2^2} + \frac{(c_3-s_3p_1)^2}{s_3^2} \right] \right}. ] From this set of conditional distributions we obtain the precision matrix as [ \left[ \begin {array}{cccc} {{\it s_1}}^{-2}&0&0&-{{\it s_1}}^{-1}\ 0&{{\it s_2}}^{-2}&0&-{{\it s_2}}^{-1}\ 0&0&{{\it s_3}}^{-2}&-{{\it s_3}}^{-1}\ -{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&-{{\it s_3}}^{-1}&{\gamma_1}^{-2}+3 \end {array} \right]. ]
The joint distribution for ${c_1, c_2, c_3, c_4, p_1, p_2}$, from the DAG (B) in Figure 1, is proportional to [ \mathrm{exp} \left{-\frac{1}{2} \left[ \frac{p_1^2}{\gamma_1^2} + \frac{(p_2-p_1)^2}{\gamma_2^2} + \frac{(c_1-s_1p_1)^2}{s_1^2} + \frac{(c_2-s_2p_1)^2}{s_2^2} + \frac{(c_3-s_3p_2)^2}{s_3^2} + \frac{(c_4-s_4p_2)^2}{s_4^2} \right] \right}. ] The precision matrix is [ \left[ \begin {array}{cccccc} {{\it s_1}}^{-2}&0&0&0&-{{\it s_1}}^{-1}&0\ 0&{{\it s_2}}^{-2}&0&0&-{{\it s_2}}^{-1}&0\ 0&0&{{\it s_3}}^{-2}&0&0&-{{\it s_3}}^{-1}\ 0&0&0&{{\it s_4}}^{-2}&0&-{{\it s_4}}^{-1}\ -{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&0&0& {\gamma_1}^{-2}+{\gamma_2}^{-2}+2&-{\gamma_2}^{-2}\ 0&0&-{{\it s_3}}^{-1}&-{{\it s_4}}^{-1}&-{\gamma_2}^{-2}&{\gamma_2}^{-2}+2 \end {array} \right] ]
The models represented by graphs (C) and (D) in Figure 1 have both the same number of children and parents. The only difference is that parent $p_3$ is children to parent $p_1$ in (C) whereas in (D) $p_3$ is children to parent $p_2$. For the case when $p_1$ is parent to $p_3$ the precision matrix is [ \left[ \begin {array}{ccccccccc} {{\it s_1}}^{-2}&0&0&0&0&0&-{{\it s_1}}^{-1}&0&0\ 0&{{\it s_2}}^{-2}&0&0&0&0&-{{\it s_2}}^{-1}&0&0\ 0&0&{{\it s_3}}^{-2}&0&0&0&0&-{{\it s_3}}^{-1}&0\ 0&0&0&{{\it s_4}}^{-2}&0&0&0&-{{\it s_4}}^{-1}&0\ 0&0&0&0&{{\it s_5}}^{-2}&0&0&0&-{{\it s_5}}^{-1} \ 0&0&0&0&0&{{\it s_6}}^{-2}&0&0&-{{\it s_6}}^{-1}\ -{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&0&0&0&0& {\gamma_1}^{-2}+{\gamma_2}^{-2}+{\gamma_3}^{-2} +2 & -{\gamma_2}^{-2}&-{\gamma_3}^{-2}\ 0&0&-{{\it s_3}}^{-1}&-{{\it s_4}}^{-1}&0&0& -{\gamma_2}^{-2}&{\gamma_2}^{-2}+2&0\ 0&0&0&0&-{{\it s_5}}^{-1}&-{{\it s_6}}^{-1}& -{\gamma_3}^{-2}&0&{\gamma_3}^{-2}+2 \end {array} \right] ] and the case when $p_2$ is parent to $p_3$ the last three rows of the precision matrix are [ \left[ \begin {array}{ccccccccc} -{{\it s_1}}^{-1}&-{{\it s_2}}^{-1}&0&0&0&0&{\gamma_1}^{-2}+{\gamma_2}^{-2}+2&-{\gamma_2}^{-2} &0\ 0&0&-{{\it s_3}}^{-1}&-{{\it s_4}}^{-1}&0&0&-{\gamma_2}^{-2}&{\gamma_2}^{-2}+{\gamma_3}^{-2}+2&-{\gamma_3}^{-2}\ 0&0&0&0&-{{\it s_5}}^{-1}&-{ {\it s_6}}^{-1}&0&-{\gamma_3}^{-2}&{\gamma_3}^{-2}+2\end {array} \right] ] and the first $6$ rows are equal the ones from the previous case.
Model properties
The conditional distributions defines a precision matrix for all the $m+k$ variables in the model and a joint distribution. Therefore we can obtain the implied covariance matrix to learn the marginal properties, and correlations as well. Since our primarily interest is in the correlation between the children variables, we consider its first $m$ rows and columns.
The model defined from the DAG (A) in Figure 1 for
${c_1, c_2, c_3, p_1}$ induces a covariance matrix equals to
[
\left[ \begin {array}{cccc}
( 1+{\gamma_1}^{2} ) {{\it s_1}}^{2} & {\it s_2}{\it s_1}{\gamma_1}^{2}&
{\it s_3}{\it s_1}{\gamma_1}^{2} & {\it s_1}{\gamma_1}^{2} \
{\it s_2}{\it s_1}{\gamma_1}^{2} & ( 1+{\gamma_1}^{2} ) {{\it s_2}}^{2}&
{\it s_3}{\it s_2}{\gamma_1}^{2} & {\it s_2}{\gamma_1}^{2} \
{\it s_3}{\it s_1}{\gamma_1}^{2} &{\it s_3}{\it s_2}{\gamma_1}^{2}&
( 1+{\gamma_1}^{2} ) {{\it s_3}}^{2} & {\it s_3}{\gamma_1}^{2} \
{\it s_1}{\gamma_1}^{2} & {\it s_2}{\gamma_1}^{2} &
{\it s_3}{\gamma_1}^{2}&{\gamma_1}^{2}
\end {array} \right]
]
and the C($c_i,c_j$) = sign($s_is_j$)$\gamma_1^2/(1+\gamma_1^2)$, for $i,j\in{1,2,3}$ and $i\neq j$.
The model defined from the DAG (B) in Figure 1 gives the covariance matrix for ${c_1, c_2, c_3, c_4, p_1, p_2}$ as [ \left[ \begin {array}{cccccc} \left( 1+{\gamma_1}^{2} \right) {{\it s_1}}^{2}&{\it s_2}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\gamma_1}^{2}{\it s_1}&{\gamma_1}^{2}{\it s_1}\ {\it s_2}\,{\gamma_1}^{2}{\it s_1}& \left( 1+{\gamma_1}^{2} \right) {{\it s_2}}^{2}&{\it s_3}\,{\gamma_1 }^{2}{\it s_2}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\gamma_1}^{2}{\it s_2}&{\gamma_1}^{2}{\it s_2}\ {\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{ 2}{\it s_2}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_3}}^{2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\gamma_1}^{2}{\it s_3}& \left( {{ \it \gamma_2}}^{2}+{\gamma_1}^{2} \right) {\it s_3}\ {\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\, \left( {\gamma_2}^{2}+{{\it \gamma_1}}^{2} \right) {\it s_3}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_4}}^{2}&{\gamma_1}^{2}{\it s_4}& \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4} \ {\gamma_1}^{2}{\it s_1}&{\gamma_1}^{2}{\it s_2}&{\gamma_1}^{2}{\it s_3}&{\gamma_1}^{2}{\it s_4}&{\gamma_1}^{2}&{\gamma_1}^{2}\ {\gamma_1 }^{2}{\it s_1}&{\gamma_1}^{2}{\it s_2}& \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}& \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}&{\gamma_1}^{2}&{\gamma_2} ^{2}+{\gamma_1}^{2}\end {array} \right] ] from what the marginal properties can be learn. The upper triangle of the children correlation matrix is [ \left[ \begin {array}{ccc} \frac{\textrm{sign}(s_1s_2){\gamma_1}^{2}}{{( {\gamma_1}^{2}+1 )}} & \frac{\textrm{sign}(s_1 s_3){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}} & \frac{\textrm{sign}(s_1 s_4){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}}\ & \frac{\textrm{sign}(s_2 s_3){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}} & \frac{\textrm{sign}(s_2 s_4){\gamma_1}^{2}} {\sqrt{( {\gamma_1}^{2}+1 )} \sqrt { ( 1+{\gamma_2}^{2}+{\gamma_1}^{2} )}}\ & & \frac{{\textrm{sign}(s_3 s_4) ( {\gamma_2}^{2}+{\gamma_1}^{2}) }}{ {( 1+{\gamma_2}^{2}+{\gamma_1}^{2})}} \end{array} \right] ] where C($s_i,s_j$) = sign($s_is_j$)$\frac{\gamma_1^2}{(\gamma_1^2+1)(1+\gamma_1^2+\gamma_2^2)}$ for $i\in{1,2}$ and $j\in{3,4}$, which defines a block with the same absolute value and is the main structure that this model induced for the correlation matrix.
In the model given from graph (C), when $p_1$ is parent to $p_3$, the covariance matrix of ${c_1,...,c_6}$ is [ \scriptsize \left[ \begin {array}{cccccc} \left( 1+{\gamma_1}^{2} \right) {{\it s_1}}^{2}&{\it s_2}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_1}\ {\it s_2}\,{\gamma_1}^{2}{\it s_1}& \left( 1+{\gamma_1}^{2} \right) {{\it s_2}}^{2} &{\it s_3}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}\ {\it s_3}\,{{ \it \gamma_1}}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_2}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_3}}^{2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_5}\,{\gamma_1}^{2}{\it s_3}&{\it s_6}\,{\gamma_1}^{2}{\it s_3}\ {\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_2 }&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_4}}^{2}&{\it s_5}\,{\gamma_1}^{2}{\it s_4}&{\it s_6} \,{\gamma_1}^{2}{\it s_4}\ {\it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\,{\gamma_1}^{2}{\it s_3}&{\it s_5}\,{\gamma_1}^{2} {\it s_4}& \left( 1+{\gamma_3}^{2}+{\gamma_1}^{2} \right) {{\it s_5}}^{2}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{\gamma_1}^{2} \right) \ {\it s_6}\,{{\it \gamma_1}}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\,{\gamma_1}^{2}{\it s_3}&{\it s_6}\,{\gamma_1}^{2}{\it s_4}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{{\it \gamma_1}}^{2} \right) & \left( 1+{\gamma_3}^{2}+{\gamma_1}^{2} \right) {{\it s_6}}^{2}\end {array} \right] ] where the variances and covariance for $c_5$ and $c_6$ does not depends on $\gamma_2^2$.
For the case when $p_2$ is parent to $p_3$ the covariance of $c_1$, ..., $c_6$ is [ \tiny \left[ \begin {array}{cccccc} \left( 1+{\gamma_1}^{2} \right) {{\it s_1}}^{2}&{\it s_2}\,{\gamma_1}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_1}\ {\it s_2}\,{\gamma_1}^{2}{\it s_1}& \left( 1+{\gamma_1}^{2} \right) {{\it s_2}}^{2} &{\it s_3}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}\ {\it s_3}\,{{ \it \gamma_1}}^{2}{\it s_1}&{\it s_3}\,{\gamma_1}^{2}{\it s_2}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_3}}^{2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_5}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_6}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}\ {\it s_4}\,{\gamma_1}^{2}{\it s_1}&{\it s_4}\,{\gamma_1}^{2}{\it s_2}&{\it s_4}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}& \left( 1+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_4}}^{2}&{\it s_5}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}&{\it s_6}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}\ { \it s_5}\,{\gamma_1}^{2}{\it s_1}&{\it s_5}\,{\gamma_1}^{2}{\it s_2}&{\it s_5}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_5}\, \left( {\gamma_2}^{2}+{{\it \gamma_1}}^{2} \right) {\it s_4}& \left( 1+{\gamma_3}^{2}+{\gamma_2}^{2}+{\gamma_1}^{2} \right) {{\it s_5}}^{2}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{\gamma_2}^{2}+{\gamma_1} ^{2} \right) \ {\it s_6}\,{\gamma_1}^{2}{\it s_1}&{\it s_6}\,{\gamma_1}^{2}{\it s_2}&{\it s_6}\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_3}&{\it s_6 }\, \left( {\gamma_2}^{2}+{\gamma_1}^{2} \right) {\it s_4}&{\it s_6}\,{\it s_5}\, \left( {\gamma_3}^{2}+{\gamma_2}^{2}+{\gamma_1}^{2} \right) & \left( 1+{\gamma_3}^{2}+{\gamma_2 }^{2}+{\gamma_1}^{2} \right) {{\it s_6}}^{2}\end {array} \right] ] where the variances and covariance for $c_5$ and $c_6$ now depends on $\gamma_2^2$.
For the correlations, we now compare C($c_i,c_j$), for $i\neq j$ and $i,j\in{3,4,5,6}$, with these two different models. The upper triangle of the correlation matrix for $c_3$, ..., $c_6$ obtained with the model where $p_1$ is parent to $p_3$ is [ \left[ \begin {array}{ccc} \textrm{sign}(s_3s_4)\frac{\gamma_1^2+\gamma_2^2}{1+\gamma_1^2+\gamma_2^2} & \textrm{sign}(s_3s_5)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} & \textrm{sign}(s_3s_6)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} \ & \textrm{sign}(s_4s_5)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} & \textrm{sign}(s_4s_6)\frac{\gamma_1^2}{\sqrt{(1+\gamma_1^2)(1+\gamma_1^2+\gamma_3^2)}} \ & & \textrm{sign}(s_5s_6)\frac{\gamma_1^2 + \gamma_3^2}{\sqrt{(1+\gamma_1^2+\gamma_3^2)}} \ \end{array} \right], ] and with the model when $p_2$ is parent to $p_3$ it is [ \left[ \begin {array}{ccc} \textrm{sign}(s_3s_4)\frac{\gamma_1^2+\gamma_2^2}{1+\gamma_1^2+\gamma_2^2} & \textrm{sign}(s_3s_5)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} & \textrm{sign}(s_3s_6)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} \ & \textrm{sign}(s_4s_5)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} & \textrm{sign}(s_4s_6)\frac{\gamma_1^2+\gamma_2^2}{\sqrt{(1+\gamma_1^2+\gamma_2^2)(1+\gamma_1^2+\gamma_3^2)}} \ & & \textrm{sign}(s_5s_6)\frac{\gamma_1^2+\gamma_2^2+\gamma_3^2}{1+\gamma_1^2+\gamma_2^2+\gamma_3^2} \ \end{array} \right]. ] Comparing these results, when $p_1$ is parent to $p_3$ there is no contribution from $\gamma_2^2$ in C($c_5,c_6$) and neither in C($c_i,c_j$), for $i\in{3,4}$, whereas when $p_2$ is parent to $p_3$ there is.
From these examples we can list a set of properties as follows.
- Property 1: The marginal variance of $p_i$ is $\gamma_i^2$ plus the variance of its parent.
- Property 1a: For $p_1$ it is equal its conditional variance, $\gamma_1^2$.
- Property 1b: The marginal variance of $p_i$ is $\gamma_i^2$ plus the conditional variance of its ancestors.
- Property 2: The marginal variance of $c_i$ is equal $s_i^2(1+V)$, where $V$ is the variance of its parent.
- Property 3: The covariance between $c_i$ and $c_j$ is equal $s_is_jR$, where $R$ is the variance of the top indexed parent in the path linking them.
- Property 3a: If $c_i$ and $c_j$ have a common parent, $R$ is the variance of their parent.
- Property 4: C($c_i,c_j$)=sign($s_is_j$)R/$\sqrt{\gamma_{c_i}\gamma_{c_j}}$, where $\gamma_{c_i}$ and $\gamma_{c_j}$ are the marginal variances of $c_i$ and $c_j$, respectively.
- Property 4a: If $c_i$ and $c_j$ have a common parent, the correlation is sign($s_is_j$)$V/(1+V)$ where $V$ is the variance of their parent.
- Property 5: Let $k_0$ be the number of parents with no children variable as children, $k_1$ be the number of parents with one children variable as children, and $k_2$ be the number of parents with at least two children variables as children. The number of distinct absolute correlation values is $k_2+(k_1-k_0)(k_1-k_0-1)/2$.
- Property 5a: If each parent variable has only one children variable as children, this number is $k(k-1)/2$.
- Property 5b: The maximum number of distinct absolute correlation values is $k(k+1)/2$, achieved if each parent variable is parent to at least two children variables.
This intuitive modeling approach gives way to build a structured correlation matrix. The number of distinct absolute correlation values is $k(k+1)/2$. The sign assumed for each arrow ending at a children variable allows the full range of the correlation values, from $-1$ to $1$. Prior distributions can be considered for the marginal variances and for the correlation matrix separately.
A further example
We now consider more variables and use two models, as shown in Figure 2 bellow. The difference in on who is the parent to $p_4$. We will use it to further illustrate the Property 2a.
t4a <- treepcor( p1 ~ p2 + p3 + c1 + c2, p2 ~ p4 + c3 + c4, p3 ~ p5 + c5 + c6, p4 ~ c7 + c8, p5 ~ c9 + c10) t4b <- treepcor(p1 ~ p2 + p3 + c1 + c2, p2 ~ c3 + c4, p3 ~ p4 + p5 + c5 + c6, p4 ~ c7 + c8, p5 ~ c9 + c10) par(mfrow = c(1, 2), mar = c(0, 0, 0, 0)) plot(t4a) legend("topleft", "(E)", bty = "n") plot(t4b) legend("topleft", "(F)", bty = "n")
The model for the DAG in (E) gives a precision matrix whose last five columns are
[ \scriptsize
\left[ \begin {array}{ccccc} -{{\it s_1}}^{-1}&0&0&0&0\ -{{\it s_2}}^{-1}&0&0&0&0\ 0&-{{\it s_3}}^{-1}&0&0&0\ 0&-{{\it s_4}}^{-1}&0&0
&0\ 0&0&-{{\it s_5}}^{-1}&0&0\ 0&0&-{{\it s_6}}^{-1}&0&0\ 0&0&0&-{{\it s_7}}^{-1}&0\ 0&0&0&-{{\it s_8}}^{-1}&0
\ 0&0&0&0&-{{\it s_9}}^{-1}\ 0&0&0&0&-{{\it s_{10}}}^{-1}\ {\gamma_1}^{-2}+2+{\gamma_2}^{-2}+{\gamma_3}^{-2}&-{\gamma_2}^{-2}&-{{
\it \gamma_3}}^{-2}&0&0\ -{\gamma_2}^{-2}&{\gamma_2}^{-2}+2+{\gamma_4}^{-2}&0&-{\gamma_4}^{-2}&0\ -{\gamma_3}^{-2}&0&{\gamma_3}^{-2}+2+{\gamma_5}^
{-2}&0&-{\gamma_5}^{-2}\ 0&-{\gamma_4}^{-2}&0&{\gamma_4}^{-2}+2&0\ 0&0&-{\gamma_5}^{-2}&0&{\gamma_5}^{-2}+2\end {array} \right]
]
while the last five columns for the precision matrix for the model for DAG in (F) are
[ \scriptsize
\left[ \begin {array}{ccccc}
-{{\it s_1}}^{-1}&0&0&0&0\ -{{\it s_2}}^{-1}&0&0&0&0\ 0&-{{\it s_3}}^{-1}&0&0&0\ 0&-{{\it s_4}}^{-1}&0&0
&0\ 0&0&-{{\it s_5}}^{-1}&0&0\ 0&0&-{{\it s_6}}^{-1}&0&0\ 0&0&0&-{{\it s_7}}^{-1}&0\ 0&0&0&-{{\it s_8}}^{-1}&0
\ 0&0&0&0&-{{\it s_9}}^{-1}\ 0&0&0&0&-{{\it s_{10}}}^{-1}\ {\gamma_1}^{-2}+2+{\gamma_2}^{-2}+{\gamma_3}^{-2}&-{\gamma_2}^{-2}&-{{
\it \gamma_3}}^{-2}&0&0\ -{\gamma_2}^{-2}&{\gamma_2}^{-2}+2&0&0&0\ -{\gamma_3}^{-2}&0&{\gamma_3}^{-2}+2+{\gamma_4}^{-2}+{\gamma_5}^{-2}&-{\gamma_4
}^{-2}&-{\gamma_5}^{-2}\ 0&0&-{\gamma_4}^{-2}&{\gamma_4}^{-2}+2&0\ 0&0&-{\gamma_5}^{-2}&0&{\gamma_5}^{-2}+2
\end {array} \right]
]
To illustrate property about the top parent on a path, we consider the upper triangle of the correlation matrix for the last four children variables, $c_7,...,c_{10}$. For the model defined from (E) we have it as [ \left[ \begin {array}{ccc} {\frac { \textrm{sign}({\it s_7 s_8}) ( {\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} )}{( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} )}} & \frac {\textrm{sign}({\it s_7 s_9}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}} & \frac {\textrm{sign}({\it s_7 s_{10}}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}} \ & \frac {\textrm{sign}({\it s_8 s_9}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}} & \frac {\textrm{sign}({\it s_8 s_{10}}) {\gamma_1}^{2}}{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_2}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )}}\ & & \frac { \textrm{sign}({\it s_{9} s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} \ \end {array} \right] ] where C($c_i,c_j$), for $i\in{7,8}$ and $j\in{9,10}$ increase depends only on $\gamma_1^2$ and depends on all $\gamma_j^2$ for $j={1,\ldots,5}$. With the model defined from (F) we have [ \ \left[ \begin {array}{ccc} \frac {\textrm{sign}({\it s_7s_8}) ( {\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) }{( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) } & \frac { \textrm{sign}({\it s_7s_9}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} & \frac { \textrm{sign}({\it s_7s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} \ & \frac { \textrm{sign}({\it s_9s_9}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} & \frac { \textrm{sign}({\it s_8s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2} ) }{\sqrt { ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_4}^{2} ) ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }} \ & & \frac {\textrm{sign}({\it s_9s_{10}}) ( {\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} ) }{ ( 1+{\gamma_1}^{2}+{\gamma_3}^{2}+{\gamma_5}^{2} )} \end {array} \right] ] where now C($c_i,c_j$), for $i\in{7,8}$ and $j\in{9,10}$, increases if $\gamma_1^2+\gamma_3^2$ (the variance of $p_3$) increases and no longer depends on $\gamma_2^2$.
References
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