fitAncestralGraph: Fitting of Gaussian Ancestral Graph Models
In ggm: Graphical Markov Models with Mixed Graphs
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
Iterative conditional fitting of Gaussian Ancestral Graph Models.
Usage
1
fitAncestralGraph(amat, S, n, tol = 1e-06)
Arguments
amat
a square matrix, representing the adjacency matrix of
an ancestral graph.
S
a symmetric positive definite matrix with row and col names, the
sample covariance matrix.
n
the sample size, a positive integer.
tol
a small positive number indicating the tolerance
used in convergence checks.
Details
In the Gaussian case, the models can be parameterized
using precision parameters, regression coefficients, and error
covariances (compare Richardson and Spirtes, 2002, Section 8). This
function finds the MLE L of the precision
parameters by fitting a concentration
graph model. The MLE B of the regression coefficients and
the MLE O of the error covariances are obtained by
iterative conditional fitting (Drton and Richardson, 2003, 2004). The
three sets of parameters are
combined to the MLE S of the covariance matrix by
matrix multiplication:
S = B^(-1) (L+O) B^(-t).
Note that in Richardson and Spirtes (2002), the matrices L
and O are defined as submatrices.
Value
Shat
the fitted covariance matrix.
Lhat
matrix of the fitted precisions associated with undirected
edges and vertices that do not have an arrowhead pointing at them.
Bhat
matrix of the fitted regression coefficients
associated to the directed edges. Precisely said Bhat
contains ones on the diagonal and the off-diagonal entry
(i,j) equals the negated MLE of the regression
coefficient for variable j in the regression of variable
i on its parents. Note that this (i,j) entry
in Bhat corresponds to a directed edge j -> i,
and thus to a one as (j,i) entry in the adjacency matrix.
Ohat
matrix of the error covariances and variances of the residuals
between regression equations associated with bi-directed edges and
vertices with an arrowhead pointing at them.
dev
the ‘deviance’ of the model.
df
the degrees of freedom.
it
the iterations.
Author(s)
Mathias Drton
References
Drton, M. and Richardson, T. S. (2003). A new algorithm for
maximum likelihood estimation in Gaussian graphical models for
marginal independence. Proceedings
of the Nineteenth Conference on Uncertainty in Artificial
Intelligence, 184-191.
Drton, M. and Richardson, T. S. (2004). Iterative Conditional Fitting
for Gaussian Ancestral Graph Models. Proceedings
of the 20th Conference on Uncertainty in Artificial Intelligence,
Department of Statistics, 130-137.
Richardson, T. S. and Spirtes, P. (2002). Ancestral Graph Markov
Models. Annals of Statistics. 30(4), 962-1030.
See Also
fitCovGraph, icf,
makeMG, fitDag
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
## A covariance matrix
"S" <- structure(c(2.93, -1.7, 0.76, -0.06,
-1.7, 1.64, -0.78, 0.1,
0.76, -0.78, 1.66, -0.78,
-0.06, 0.1, -0.78, 0.81), .Dim = c(4,4),
.Dimnames = list(c("y", "x", "z", "u"), c("y", "x", "z", "u")))
## The following should give the same fit.
## Fit an ancestral graph y -> x <-> z <- u
fitAncestralGraph(ag1 <- makeMG(dg=DAG(x~y,z~u), bg = UG(~x*z)), S, n=100)
## Fit an ancestral graph y <-> x <-> z <-> u
fitAncestralGraph(ag2 <- makeMG(bg= UG(~y*x+x*z+z*u)), S, n=100)
## Fit the same graph with fitCovGraph
fitCovGraph(ag2, S, n=100)
## Another example for the mathematics marks data
data(marks)
S <- var(marks)
mag1 <- makeMG(bg=UG(~mechanics*vectors*algebra+algebra*analysis*statistics))
fitAncestralGraph(mag1, S, n=88)
mag2 <- makeMG(ug=UG(~mechanics*vectors+analysis*statistics),
dg=DAG(algebra~mechanics+vectors+analysis+statistics))
fitAncestralGraph(mag2, S, n=88) # Same fit as above
Example output
Loading required package: igraph
Attaching package: 'igraph'
The following objects are masked from 'package:stats':
decompose, spectrum
The following object is masked from 'package:base':
union
Attaching package: 'ggm'
The following object is masked from 'package:igraph':
pa
$Shat
y x z u
y 2.930000 -1.4344254 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$Lhat
[,1] [,2] [,3] [,4]
[1,] 0 0.0000000 0.0000000 0
[2,] 0 0.6777235 -0.3430373 0
[3,] 0 -0.3430373 0.9104666 0
[4,] 0 0.0000000 0.0000000 0
$Bhat
y x z u
y 1.000000 0 0 0.0000000
x 0.489565 1 0 0.0000000
z 0.000000 0 1 0.9188294
u 0.000000 0 0 1.0000000
$Ohat
y x z u
y 0 0.0000000 0.0000000 0
x 0 0.6777235 -0.3430373 0
z 0 -0.3430373 0.9104666 0
u 0 0.0000000 0.0000000 0
$dev
[1] 21.57711
$df
[1] 3
$it
[1] 4
$Shat
y x z u
y 2.930000 -1.4344255 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$Lhat
[,1] [,2] [,3] [,4]
[1,] 0 0 0 0
[2,] 0 0 0 0
[3,] 0 0 0 0
[4,] 0 0 0 0
$Bhat
y x z u
y 1 0 0 0
x 0 1 0 0
z 0 0 1 0
u 0 0 0 1
$Ohat
y x z u
y 2.930000 -1.4344255 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$dev
[1] 21.57711
$df
[1] 3
$it
[1] 14
$Shat
y x z u
y 2.930000 -1.4344255 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$dev
[1] 21.57711
$df
[1] 3
$it
[1] 14
$Shat
mechanics vectors algebra analysis statistics
mechanics 305.68848 127.04336 53.15421 0.00000 0.00000
vectors 127.04336 172.84222 43.11507 0.00000 0.00000
algebra 53.15421 43.11507 88.39060 84.78364 92.65154
analysis 0.00000 0.00000 84.78364 220.38036 155.53553
statistics 0.00000 0.00000 92.65154 155.53553 297.75536
$Lhat
[,1] [,2] [,3] [,4] [,5]
[1,] 0 0 0 0 0
[2,] 0 0 0 0 0
[3,] 0 0 0 0 0
[4,] 0 0 0 0 0
[5,] 0 0 0 0 0
$Bhat
mechanics vectors algebra analysis statistics
mechanics 1 0 0 0 0
vectors 0 1 0 0 0
algebra 0 0 1 0 0
analysis 0 0 0 1 0
statistics 0 0 0 0 1
$Ohat
mechanics vectors algebra analysis statistics
mechanics 305.68848 127.04336 53.15421 0.00000 0.00000
vectors 127.04336 172.84222 43.11507 0.00000 0.00000
algebra 53.15421 43.11507 88.39060 84.78364 92.65154
analysis 0.00000 0.00000 84.78364 220.38036 155.53553
statistics 0.00000 0.00000 92.65154 155.53553 297.75536
$dev
[1] 31.97538
$df
[1] 4
$it
[1] 21
$Shat
mechanics vectors algebra analysis statistics
mechanics 3.056885e+02 1.270434e+02 53.15421 1.247322e-08 8.803090e-09
vectors 1.270434e+02 1.728422e+02 43.11507 5.183837e-09 3.658542e-09
algebra 5.315421e+01 4.311507e+01 88.39060 8.478364e+01 9.265154e+01
analysis 1.247322e-08 5.183837e-09 84.78364 2.203804e+02 1.555355e+02
statistics 8.803090e-09 3.658542e-09 92.65154 1.555355e+02 2.977554e+02
$Lhat
[,1] [,2] [,3] [,4] [,5]
[1,] 0 0 0.00000 0 0
[2,] 0 0 0.00000 0 0
[3,] 0 0 37.12009 0 0
[4,] 0 0 0.00000 0 0
[5,] 0 0 0.00000 0 0
$Bhat
mechanics vectors algebra analysis statistics
mechanics 1.0000000 0.0000000 0 0.0000000 0.0000000
vectors 0.0000000 1.0000000 0 0.0000000 0.0000000
algebra -0.1010961 -0.1751394 1 -0.2615174 -0.1745604
analysis 0.0000000 0.0000000 0 1.0000000 0.0000000
statistics 0.0000000 0.0000000 0 0.0000000 1.0000000
$Ohat
mechanics vectors algebra analysis statistics
mechanics 0 0 0.00000 0 0
vectors 0 0 0.00000 0 0
algebra 0 0 37.12009 0 0
analysis 0 0 0.00000 0 0
statistics 0 0 0.00000 0 0
$dev
[1] 31.97538
$df
[1] 4
$it
[1] 2
ggm documentation built on March 26, 2020, 7:49 p.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
Iterative conditional fitting of Gaussian Ancestral Graph Models.
Usage
1 | fitAncestralGraph(amat, S, n, tol = 1e-06)
|
Arguments
amat |
a square matrix, representing the adjacency matrix of an ancestral graph. |
S |
a symmetric positive definite matrix with row and col names, the sample covariance matrix. |
n |
the sample size, a positive integer. |
tol |
a small positive number indicating the tolerance used in convergence checks. |
Details
In the Gaussian case, the models can be parameterized using precision parameters, regression coefficients, and error covariances (compare Richardson and Spirtes, 2002, Section 8). This function finds the MLE L of the precision parameters by fitting a concentration graph model. The MLE B of the regression coefficients and the MLE O of the error covariances are obtained by iterative conditional fitting (Drton and Richardson, 2003, 2004). The three sets of parameters are combined to the MLE S of the covariance matrix by matrix multiplication:
S = B^(-1) (L+O) B^(-t).
Note that in Richardson and Spirtes (2002), the matrices L and O are defined as submatrices.
Value
Shat |
the fitted covariance matrix. |
Lhat |
matrix of the fitted precisions associated with undirected edges and vertices that do not have an arrowhead pointing at them. |
Bhat |
matrix of the fitted regression coefficients
associated to the directed edges. Precisely said |
Ohat |
matrix of the error covariances and variances of the residuals between regression equations associated with bi-directed edges and vertices with an arrowhead pointing at them. |
dev |
the ‘deviance’ of the model. |
df |
the degrees of freedom. |
it |
the iterations. |
Author(s)
Mathias Drton
References
Drton, M. and Richardson, T. S. (2003). A new algorithm for maximum likelihood estimation in Gaussian graphical models for marginal independence. Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence, 184-191.
Drton, M. and Richardson, T. S. (2004). Iterative Conditional Fitting for Gaussian Ancestral Graph Models. Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, Department of Statistics, 130-137.
Richardson, T. S. and Spirtes, P. (2002). Ancestral Graph Markov Models. Annals of Statistics. 30(4), 962-1030.
See Also
fitCovGraph, icf,
makeMG, fitDag
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ## A covariance matrix
"S" <- structure(c(2.93, -1.7, 0.76, -0.06,
-1.7, 1.64, -0.78, 0.1,
0.76, -0.78, 1.66, -0.78,
-0.06, 0.1, -0.78, 0.81), .Dim = c(4,4),
.Dimnames = list(c("y", "x", "z", "u"), c("y", "x", "z", "u")))
## The following should give the same fit.
## Fit an ancestral graph y -> x <-> z <- u
fitAncestralGraph(ag1 <- makeMG(dg=DAG(x~y,z~u), bg = UG(~x*z)), S, n=100)
## Fit an ancestral graph y <-> x <-> z <-> u
fitAncestralGraph(ag2 <- makeMG(bg= UG(~y*x+x*z+z*u)), S, n=100)
## Fit the same graph with fitCovGraph
fitCovGraph(ag2, S, n=100)
## Another example for the mathematics marks data
data(marks)
S <- var(marks)
mag1 <- makeMG(bg=UG(~mechanics*vectors*algebra+algebra*analysis*statistics))
fitAncestralGraph(mag1, S, n=88)
mag2 <- makeMG(ug=UG(~mechanics*vectors+analysis*statistics),
dg=DAG(algebra~mechanics+vectors+analysis+statistics))
fitAncestralGraph(mag2, S, n=88) # Same fit as above
|
Example output
Loading required package: igraph
Attaching package: 'igraph'
The following objects are masked from 'package:stats':
decompose, spectrum
The following object is masked from 'package:base':
union
Attaching package: 'ggm'
The following object is masked from 'package:igraph':
pa
$Shat
y x z u
y 2.930000 -1.4344254 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$Lhat
[,1] [,2] [,3] [,4]
[1,] 0 0.0000000 0.0000000 0
[2,] 0 0.6777235 -0.3430373 0
[3,] 0 -0.3430373 0.9104666 0
[4,] 0 0.0000000 0.0000000 0
$Bhat
y x z u
y 1.000000 0 0 0.0000000
x 0.489565 1 0 0.0000000
z 0.000000 0 1 0.9188294
u 0.000000 0 0 1.0000000
$Ohat
y x z u
y 0 0.0000000 0.0000000 0
x 0 0.6777235 -0.3430373 0
z 0 -0.3430373 0.9104666 0
u 0 0.0000000 0.0000000 0
$dev
[1] 21.57711
$df
[1] 3
$it
[1] 4
$Shat
y x z u
y 2.930000 -1.4344255 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$Lhat
[,1] [,2] [,3] [,4]
[1,] 0 0 0 0
[2,] 0 0 0 0
[3,] 0 0 0 0
[4,] 0 0 0 0
$Bhat
y x z u
y 1 0 0 0
x 0 1 0 0
z 0 0 1 0
u 0 0 0 1
$Ohat
y x z u
y 2.930000 -1.4344255 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$dev
[1] 21.57711
$df
[1] 3
$it
[1] 14
$Shat
y x z u
y 2.930000 -1.4344255 0.0000000 0.0000000
x -1.434425 1.3799680 -0.3430373 0.0000000
z 0.000000 -0.3430373 1.5943070 -0.7442518
u 0.000000 0.0000000 -0.7442518 0.8100000
$dev
[1] 21.57711
$df
[1] 3
$it
[1] 14
$Shat
mechanics vectors algebra analysis statistics
mechanics 305.68848 127.04336 53.15421 0.00000 0.00000
vectors 127.04336 172.84222 43.11507 0.00000 0.00000
algebra 53.15421 43.11507 88.39060 84.78364 92.65154
analysis 0.00000 0.00000 84.78364 220.38036 155.53553
statistics 0.00000 0.00000 92.65154 155.53553 297.75536
$Lhat
[,1] [,2] [,3] [,4] [,5]
[1,] 0 0 0 0 0
[2,] 0 0 0 0 0
[3,] 0 0 0 0 0
[4,] 0 0 0 0 0
[5,] 0 0 0 0 0
$Bhat
mechanics vectors algebra analysis statistics
mechanics 1 0 0 0 0
vectors 0 1 0 0 0
algebra 0 0 1 0 0
analysis 0 0 0 1 0
statistics 0 0 0 0 1
$Ohat
mechanics vectors algebra analysis statistics
mechanics 305.68848 127.04336 53.15421 0.00000 0.00000
vectors 127.04336 172.84222 43.11507 0.00000 0.00000
algebra 53.15421 43.11507 88.39060 84.78364 92.65154
analysis 0.00000 0.00000 84.78364 220.38036 155.53553
statistics 0.00000 0.00000 92.65154 155.53553 297.75536
$dev
[1] 31.97538
$df
[1] 4
$it
[1] 21
$Shat
mechanics vectors algebra analysis statistics
mechanics 3.056885e+02 1.270434e+02 53.15421 1.247322e-08 8.803090e-09
vectors 1.270434e+02 1.728422e+02 43.11507 5.183837e-09 3.658542e-09
algebra 5.315421e+01 4.311507e+01 88.39060 8.478364e+01 9.265154e+01
analysis 1.247322e-08 5.183837e-09 84.78364 2.203804e+02 1.555355e+02
statistics 8.803090e-09 3.658542e-09 92.65154 1.555355e+02 2.977554e+02
$Lhat
[,1] [,2] [,3] [,4] [,5]
[1,] 0 0 0.00000 0 0
[2,] 0 0 0.00000 0 0
[3,] 0 0 37.12009 0 0
[4,] 0 0 0.00000 0 0
[5,] 0 0 0.00000 0 0
$Bhat
mechanics vectors algebra analysis statistics
mechanics 1.0000000 0.0000000 0 0.0000000 0.0000000
vectors 0.0000000 1.0000000 0 0.0000000 0.0000000
algebra -0.1010961 -0.1751394 1 -0.2615174 -0.1745604
analysis 0.0000000 0.0000000 0 1.0000000 0.0000000
statistics 0.0000000 0.0000000 0 0.0000000 1.0000000
$Ohat
mechanics vectors algebra analysis statistics
mechanics 0 0 0.00000 0 0
vectors 0 0 0.00000 0 0
algebra 0 0 37.12009 0 0
analysis 0 0 0.00000 0 0
statistics 0 0 0.00000 0 0
$dev
[1] 31.97538
$df
[1] 4
$it
[1] 2
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