InducedGraphs: Graphs induced by marginalization or conditioning
In ggm: Graphical Markov Models with Mixed Graphs
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
Functions to find induced graphs after conditioning on a set of variables and marginalizing over another set.
Usage
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inducedCovGraph(amat, sel = rownames(amat), cond = NULL)
inducedConGraph(amat, sel = rownames(amat), cond = NULL)
inducedRegGraph(amat, sel = rownames(amat), cond = NULL)
inducedChainGraph(amat, cc=rownames(amat), cond = NULL, type="LWF")
inducedDAG(amat, order, cond = NULL)
Arguments
amat
a square Boolean matrix, the adjacency matrix of a directed
acyclic graph. The names of rows and of the columns
are the nodes of the DAG.
sel
a character vector representing a subset
of selected variables.
The elements of the vector must be a subset of
the names of the nodes i.e. of rownames(A).
By default sel is the set of the nodes of the DAG.
cond
a character vector representing the variables on which you
want to condition. cond must be disjoint from sel and
their union must be a subset of the set of nodes. The set difference
between the set of nodes and the union of sel and
cond are the variables over which we marginalize.
cond may be the null vector (the default), meaning that you want
to condition on the empty set.
cc
a list of character vectors specifying the chain components
for the chain graph.
type
a string indicating the interpretation of the chain
graph. It can be either "LWF" (Lauritzen, Wermuth, Frydenberg
interpretation), "AMP" (Andersson, Madigan, Perlman interpretation)
or "MRG" (Multivariate regression graph interpretation).
order
a character vector indicating the ordering of the vertices
of a DAG (left to right, past to future).
Details
Given a directed acyclic graph representing a set of
conditional independencies it is possible to obtain other graphs
of conditional independence implied after marginalizingover and
conditionig on sets of nodes. Such graphs are
the covariance graph, the concentration graph, the multivariate
regression graph and the chain graph with different interpretations (see Cox \& Wermuth, 1996, 2004).
Value
inducedCovGraph returns the adjacency matrix of the covariance
graph of the variables in set sel given the variables
in set cond, implied by the original directed acyclic graph
with adjacency matrix amat.
inducedConGraph returns the adjacency matrix of the concentration
graph of the variables in set sel given the variables
in set cond, implied by the original directed acyclic graph
with adjacency matrix amat.
inducedRegGraph returns the adjacency matrix of the
multivariate regression graph of the variables in set sel
given the variables in set cond, implied by the original
directed acyclic graph with adjacency matrix amat.
inducedChainGraph returns the adjacency matrix of the
chain graph for the variables in chain components
cc, given the variables in set cond, with
interpretation specified by string type, implied by the original
directed acyclic graph with adjacency matrix amat.
inducedDAG returns the adjacency matrix of the DAG with
the ordering order, implied by the original directed acyclic
graph with adjacency matrix amat.
Note
If sel is NULL the functions return the null matrix.
If cond is NULL, the conditioning set is empty and the
functions inducedConGraph and inducedCovGraph
return the overall induced covariance or concentration
matrices of the selected variables. If you do not specify sel
you cannot specify a non NULL value of cond.
Author(s)
Giovanni M. Marchetti
References
Cox, D. R. \& Wermuth, N. (1996). Multivariate
dependencies. London: Chapman \& Hall.
Wermuth, N. \& Cox, D.R. (2004). Joint response graphs
and separation induced by triangular systems. J.R. Statist. Soc. B,
66, Part 3, 687-717.
See Also
Examples
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## Define a DAG
dag <- DAG(a ~ x, c ~ b+d, d~ x)
dag
## Induced covariance graph of a, b, d given the empty set.
inducedCovGraph(dag, sel=c("a", "b", "d"), cond=NULL)
## Induced concentration graph of a, b, c given x
inducedConGraph(dag, sel=c("a", "b", "c"), cond="x")
## Overall covariance graph
inducedCovGraph(dag)
## Overall concentration graph
inducedConGraph(dag)
## Induced covariance graph of x, b, d given c, x.
inducedCovGraph(dag, sel=c("a", "b", "d"), cond=c("c", "x"))
## Induced concentration graph of a, x, c given d, b.
inducedConGraph(dag, sel=c("a", "x", "c"), cond=c("d", "b"))
## The DAG on p. 198 of Cox & Wermuth (1996)
dag <- DAG(y1~ y2 + y3, y3 ~ y5, y4 ~ y5)
## Cf. figure 8.7 p. 203 in Cox & Wermuth (1996)
inducedCovGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond="y1")
inducedCovGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond="y3")
inducedCovGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond="y5")
## Cf. figure 8.8 p. 203 in Cox & Wermuth (1996)
inducedConGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond="y1")
inducedConGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond="y3")
inducedConGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond="y5")
## Cf. figure 8.9 p. 204 in Cox & Wermuth (1996)
inducedCovGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond=NULL)
inducedCovGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond=NULL)
inducedCovGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond=NULL)
## Cf. figure 8.10 p. 204 in Cox & Wermuth (1996)
inducedConGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond=NULL)
inducedConGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond=NULL)
inducedConGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond=NULL)
## An induced regression graph
dag2 = DAG(Y ~ X+U, W ~ Z+U)
inducedRegGraph(dag2, sel="W", cond=c("Y", "X", "Z"))
## An induced DAG
inducedDAG(dag2, order=c("X","Y","Z","W"))
## An induced multivariate regression graph
inducedRegGraph(dag2, sel=c("Y", "W"), cond=c("X", "Z"))
## An induced chain graph with LWF interpretation
dag3 = DAG(X~W, W~Y, U~Y+Z)
cc = list(c("W", "U"), c("X", "Y", "Z"))
inducedChainGraph(dag3, cc=cc, type="LWF")
## ... with AMP interpretation
inducedChainGraph(dag3, cc=cc, type="AMP")
## ... with multivariate regression interpretation
cc= list(c("U"), c("Z", "Y"), c("X", "W"))
inducedChainGraph(dag3, cc=cc, type="MRG")
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
a x c b d
a 0 0 0 0 0
x 1 0 0 0 1
c 0 0 0 0 0
b 0 0 1 0 0
d 0 0 1 0 0
a b d
a 0 0 100
b 0 0 0
d 100 0 0
a b c
a 0 0 0
b 0 0 10
c 0 10 0
a x c b d
a 0 100 100 0 100
x 100 0 100 0 100
c 100 100 0 100 100
b 0 0 100 0 0
d 100 100 100 0 0
a x c b d
a 0 10 0 0 0
x 10 0 0 0 10
c 0 0 0 10 10
b 0 0 10 0 10
d 0 10 10 10 0
a b d
a 0 0 0
b 0 0 100
d 0 100 0
a x c
a 0 10 0
x 10 0 0
c 0 0 0
y2 y3 y4 y5
y2 0 100 100 100
y3 100 0 100 100
y4 100 100 0 100
y5 100 100 100 0
y1 y2 y4 y5
y1 0 100 0 0
y2 100 0 0 0
y4 0 0 0 100
y5 0 0 100 0
y1 y2 y3 y4
y1 0 100 100 0
y2 100 0 0 0
y3 100 0 0 0
y4 0 0 0 0
y2 y3 y4 y5
y2 0 10 0 0
y3 10 0 0 10
y4 0 0 0 10
y5 0 10 10 0
y1 y2 y4 y5
y1 0 10 0 0
y2 10 0 0 0
y4 0 0 0 10
y5 0 0 10 0
y1 y2 y3 y4
y1 0 10 10 0
y2 10 0 10 0
y3 10 10 0 0
y4 0 0 0 0
y2 y3 y4 y5
y2 0 0 0 0
y3 0 0 100 100
y4 0 100 0 100
y5 0 100 100 0
y1 y2 y4 y5
y1 0 100 100 100
y2 100 0 0 0
y4 100 0 0 100
y5 100 0 100 0
y1 y2 y3 y4
y1 0 100 100 100
y2 100 0 0 0
y3 100 0 0 100
y4 100 0 100 0
y2 y3 y4 y5
y2 0 0 0 0
y3 0 0 0 10
y4 0 0 0 10
y5 0 10 10 0
y1 y2 y4 y5
y1 0 10 0 10
y2 10 0 0 10
y4 0 0 0 10
y5 10 10 10 0
y1 y2 y3 y4
y1 0 10 10 0
y2 10 0 10 0
y3 10 10 0 10
y4 0 0 10 0
W
Y 1
X 1
Z 1
X Y Z W
X 0 1 0 1
Y 0 0 0 1
Z 0 0 0 1
W 0 0 0 0
Y W
X 1 0
Z 0 1
W U X Y Z
W 0 10 1 1 0
U 10 0 0 1 1
X 0 0 0 0 0
Y 0 0 0 0 10
Z 0 0 0 10 0
W U X Y Z
W 0 10 1 1 1
U 10 0 0 1 1
X 0 0 0 0 0
Y 0 0 0 0 10
Z 0 0 0 10 0
U Z Y X W
U 0 1 1 0 0
Z 0 0 100 0 0
Y 0 100 0 1 1
X 0 0 0 0 100
W 0 0 0 100 0
ggm documentation built on March 26, 2020, 7:49 p.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
Description
Functions to find induced graphs after conditioning on a set of variables and marginalizing over another set.
Usage
1 2 3 4 5 | inducedCovGraph(amat, sel = rownames(amat), cond = NULL)
inducedConGraph(amat, sel = rownames(amat), cond = NULL)
inducedRegGraph(amat, sel = rownames(amat), cond = NULL)
inducedChainGraph(amat, cc=rownames(amat), cond = NULL, type="LWF")
inducedDAG(amat, order, cond = NULL)
|
Arguments
amat |
a square Boolean matrix, the adjacency matrix of a directed acyclic graph. The names of rows and of the columns are the nodes of the DAG. |
sel |
a character vector representing a subset
of selected variables.
The elements of the vector must be a subset of
the names of the nodes i.e. of |
cond |
a character vector representing the variables on which you
want to condition. |
cc |
a list of character vectors specifying the chain components for the chain graph. |
type |
a string indicating the interpretation of the chain graph. It can be either "LWF" (Lauritzen, Wermuth, Frydenberg interpretation), "AMP" (Andersson, Madigan, Perlman interpretation) or "MRG" (Multivariate regression graph interpretation). |
order |
a character vector indicating the ordering of the vertices of a DAG (left to right, past to future). |
Details
Given a directed acyclic graph representing a set of conditional independencies it is possible to obtain other graphs of conditional independence implied after marginalizingover and conditionig on sets of nodes. Such graphs are the covariance graph, the concentration graph, the multivariate regression graph and the chain graph with different interpretations (see Cox \& Wermuth, 1996, 2004).
Value
inducedCovGraph returns the adjacency matrix of the covariance
graph of the variables in set sel given the variables
in set cond, implied by the original directed acyclic graph
with adjacency matrix amat.
inducedConGraph returns the adjacency matrix of the concentration
graph of the variables in set sel given the variables
in set cond, implied by the original directed acyclic graph
with adjacency matrix amat.
inducedRegGraph returns the adjacency matrix of the
multivariate regression graph of the variables in set sel
given the variables in set cond, implied by the original
directed acyclic graph with adjacency matrix amat.
inducedChainGraph returns the adjacency matrix of the
chain graph for the variables in chain components
cc, given the variables in set cond, with
interpretation specified by string type, implied by the original
directed acyclic graph with adjacency matrix amat.
inducedDAG returns the adjacency matrix of the DAG with
the ordering order, implied by the original directed acyclic
graph with adjacency matrix amat.
Note
If sel is NULL the functions return the null matrix.
If cond is NULL, the conditioning set is empty and the
functions inducedConGraph and inducedCovGraph
return the overall induced covariance or concentration
matrices of the selected variables. If you do not specify sel
you cannot specify a non NULL value of cond.
Author(s)
Giovanni M. Marchetti
References
Cox, D. R. \& Wermuth, N. (1996). Multivariate dependencies. London: Chapman \& Hall.
Wermuth, N. \& Cox, D.R. (2004). Joint response graphs and separation induced by triangular systems. J.R. Statist. Soc. B, 66, Part 3, 687-717.
See Also
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 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | ## Define a DAG
dag <- DAG(a ~ x, c ~ b+d, d~ x)
dag
## Induced covariance graph of a, b, d given the empty set.
inducedCovGraph(dag, sel=c("a", "b", "d"), cond=NULL)
## Induced concentration graph of a, b, c given x
inducedConGraph(dag, sel=c("a", "b", "c"), cond="x")
## Overall covariance graph
inducedCovGraph(dag)
## Overall concentration graph
inducedConGraph(dag)
## Induced covariance graph of x, b, d given c, x.
inducedCovGraph(dag, sel=c("a", "b", "d"), cond=c("c", "x"))
## Induced concentration graph of a, x, c given d, b.
inducedConGraph(dag, sel=c("a", "x", "c"), cond=c("d", "b"))
## The DAG on p. 198 of Cox & Wermuth (1996)
dag <- DAG(y1~ y2 + y3, y3 ~ y5, y4 ~ y5)
## Cf. figure 8.7 p. 203 in Cox & Wermuth (1996)
inducedCovGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond="y1")
inducedCovGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond="y3")
inducedCovGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond="y5")
## Cf. figure 8.8 p. 203 in Cox & Wermuth (1996)
inducedConGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond="y1")
inducedConGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond="y3")
inducedConGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond="y5")
## Cf. figure 8.9 p. 204 in Cox & Wermuth (1996)
inducedCovGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond=NULL)
inducedCovGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond=NULL)
inducedCovGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond=NULL)
## Cf. figure 8.10 p. 204 in Cox & Wermuth (1996)
inducedConGraph(dag, sel=c("y2", "y3", "y4", "y5"), cond=NULL)
inducedConGraph(dag, sel=c("y1", "y2", "y4", "y5"), cond=NULL)
inducedConGraph(dag, sel=c("y1", "y2", "y3", "y4"), cond=NULL)
## An induced regression graph
dag2 = DAG(Y ~ X+U, W ~ Z+U)
inducedRegGraph(dag2, sel="W", cond=c("Y", "X", "Z"))
## An induced DAG
inducedDAG(dag2, order=c("X","Y","Z","W"))
## An induced multivariate regression graph
inducedRegGraph(dag2, sel=c("Y", "W"), cond=c("X", "Z"))
## An induced chain graph with LWF interpretation
dag3 = DAG(X~W, W~Y, U~Y+Z)
cc = list(c("W", "U"), c("X", "Y", "Z"))
inducedChainGraph(dag3, cc=cc, type="LWF")
## ... with AMP interpretation
inducedChainGraph(dag3, cc=cc, type="AMP")
## ... with multivariate regression interpretation
cc= list(c("U"), c("Z", "Y"), c("X", "W"))
inducedChainGraph(dag3, cc=cc, type="MRG")
|
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
a x c b d
a 0 0 0 0 0
x 1 0 0 0 1
c 0 0 0 0 0
b 0 0 1 0 0
d 0 0 1 0 0
a b d
a 0 0 100
b 0 0 0
d 100 0 0
a b c
a 0 0 0
b 0 0 10
c 0 10 0
a x c b d
a 0 100 100 0 100
x 100 0 100 0 100
c 100 100 0 100 100
b 0 0 100 0 0
d 100 100 100 0 0
a x c b d
a 0 10 0 0 0
x 10 0 0 0 10
c 0 0 0 10 10
b 0 0 10 0 10
d 0 10 10 10 0
a b d
a 0 0 0
b 0 0 100
d 0 100 0
a x c
a 0 10 0
x 10 0 0
c 0 0 0
y2 y3 y4 y5
y2 0 100 100 100
y3 100 0 100 100
y4 100 100 0 100
y5 100 100 100 0
y1 y2 y4 y5
y1 0 100 0 0
y2 100 0 0 0
y4 0 0 0 100
y5 0 0 100 0
y1 y2 y3 y4
y1 0 100 100 0
y2 100 0 0 0
y3 100 0 0 0
y4 0 0 0 0
y2 y3 y4 y5
y2 0 10 0 0
y3 10 0 0 10
y4 0 0 0 10
y5 0 10 10 0
y1 y2 y4 y5
y1 0 10 0 0
y2 10 0 0 0
y4 0 0 0 10
y5 0 0 10 0
y1 y2 y3 y4
y1 0 10 10 0
y2 10 0 10 0
y3 10 10 0 0
y4 0 0 0 0
y2 y3 y4 y5
y2 0 0 0 0
y3 0 0 100 100
y4 0 100 0 100
y5 0 100 100 0
y1 y2 y4 y5
y1 0 100 100 100
y2 100 0 0 0
y4 100 0 0 100
y5 100 0 100 0
y1 y2 y3 y4
y1 0 100 100 100
y2 100 0 0 0
y3 100 0 0 100
y4 100 0 100 0
y2 y3 y4 y5
y2 0 0 0 0
y3 0 0 0 10
y4 0 0 0 10
y5 0 10 10 0
y1 y2 y4 y5
y1 0 10 0 10
y2 10 0 0 10
y4 0 0 0 10
y5 10 10 10 0
y1 y2 y3 y4
y1 0 10 10 0
y2 10 0 10 0
y3 10 10 0 10
y4 0 0 10 0
W
Y 1
X 1
Z 1
X Y Z W
X 0 1 0 1
Y 0 0 0 1
Z 0 0 0 1
W 0 0 0 0
Y W
X 1 0
Z 0 1
W U X Y Z
W 0 10 1 1 0
U 10 0 0 1 1
X 0 0 0 0 0
Y 0 0 0 0 10
Z 0 0 0 10 0
W U X Y Z
W 0 10 1 1 1
U 10 0 0 1 1
X 0 0 0 0 0
Y 0 0 0 0 10
Z 0 0 0 10 0
U Z Y X W
U 0 1 1 0 0
Z 0 0 100 0 0
Y 0 100 0 1 1
X 0 0 0 0 100
W 0 0 0 100 0
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