Package 'ggm'

Description

This package provides functions for defining, manipulating and fitting graphical Markov models with mixed graphs. It is intended as a contribution to the gR-project described by Lauritzen (2002).

Details

For a tutorial illustrating the new functions in the package ggm that deal with ancestral, summary and ribbonless graphs see Sadeghi and Marchetti (2012) in the references.

Functions

The main functions can be classified as follows:

• Functions for defining graphs (undirected, directed acyclic, ancestral and summary graphs): UG, DAG, makeMG, grMAT;

• Functions for doing graph operations (parents, boundary, cliques, connected components, fundamental cycles, d-separation, m-separation): pa, bd, conComp, fundCycles;

• Functions for testing independence statements and generating maximal graphs from non-maximal graphs: dSep, msep, Max;

• Function for finding covariance and concentration graphs induced by marginalization and conditioning: inducedCovGraph, inducedConGraph;

• Functions for finding multivariate regression graphs and chain graphs induced by marginalization and conditioning: inducedRegGraph, inducedChainGraph, inducedDAG;

• Functions for finding stable mixed graphs (ancestral, summary and ribbonless) after marginalization and conditioning: AG, SG, RG;

• Functions for fitting by ML Gaussian DAGs, concentration graphs, covariance graphs and ancestral graphs: fitDag, fitConGraph, fitCovGraph, fitAncestralGraph;

• Functions for testing several conditional independences: shipley.test;

• Functions for checking global identification of DAG Gaussian models with one latent variable (Stanghellini-Vicard's condition for concentration graphs, new sufficient conditions for DAGs): isGident, checkIdent;

• Functions for fitting Gaussian DAG models with one latent variable: fitDagLatent;

• Functions for testing Markov equivalences and generating Markov equivalent graphs of specific types: MarkEqRcg, MarkEqMag, RepMarDAG, RepMarUG, RepMarBG.

Authors

Giovanni M. Marchetti, Dipartimento di Statistica, Informatica, Applicazioni 'G. Parenti'. University of Florence, Italy

Mathias Drton, Department of Statistics, University of Washington, USA

Kayvan Sadeghi, Department of Statistics, Carnegie Mellon University, USA

Acknowledgements

Many thanks to Fulvia Pennoni for testing some of the functions, to Elena Stanghellini for discussion and examples and to Claus Dethlefsen and Jens Henrik Badsberg for suggestions and corrections. The function fitConGraph was corrected by Ilaria Carobbi. Helpful discussions with Steffen Lauritzen and Nanny Wermuth, are gratefully acknowledged. Thanks also to Michael Perlman, Thomas Richardson and David Edwards.

Giovanni Marchetti has been supported by MIUR, Italy, under grant scheme PRIN 2002, and Mathias Drton has been supported by NSF grant DMS-9972008 and University of Washington RRF grant 65-3010.

Author(s)

Maintainer: Giovanni M. Marchetti [email protected] (ORCID)

Authors:

• Giovanni M. Marchetti [email protected] (ORCID)

• Mathias Drton [email protected] (ORCID)

• Kayvan Sadeghi [email protected] (ORCID)

References

Lauritzen, S. L. (2002). gRaphical Models in R. R News, 3(2)39.

Sadeghi, K. and Marchetti, G.M. (2012). Graphical Markov models with mixed graphs in R. The R Journal, 4(2):65-73. https://journal.r-project.org/archive/2012/RJ-2012-015/RJ-2012-015.pdf

Description

Transforms the edgematrix() of a graph into the adjacency matrix.

Usage

adjMatrix(E)

Arguments

Details

Given the edge matrix E of a graph, this can be transformed into an adjacency matrix A with ⁠t(A) - diag(ncol(A)⁠.

Value

The adjacency matrix of the graph.

Author(s)

Giovanni M. Marchetti

See Also

edgematrix()

Examples

A <- DAG(y ~ x + z, z ~ u + v)
E <- edgematrix(A)
adjMatrix(E)

Description

AG generates and plots ancestral graphs after marginalization and conditioning.

Usage

AG(
  amat,
  M = c(),
  C = c(),
  showmat = TRUE,
  plot = FALSE,
  plotfun = plotGraph,
  ...
)

Arguments

Details

Support for graphNEL objects requires the graph package from Bioconductor, which is a suggested dependency. If the package is missing, passing a graphNEL object will trigger an informative error.

This function uses the internal functions AG() and Max().

Value

A matrix consisting of 4 different integers representing the $ij$-elements:

• 0 for a missing edge between $i$ and $j$,

• 1 for an arrow from $i$ to $j$,

• 10 for a full line between $i$ and $j$, and

• 100 for a bi-directed arrow between $i$ and $j$. These numbers are added when multiple edges of different types are present. The matrix is symmetric with respect to full lines and bi-directed arrows.

Author(s)

Kayvan Sadeghi

References

Richardson, T.S. and Spirtes, P. (2002). Ancestral graph Markov models. Annals of Statistics, 30(4), 962-1030.

Sadeghi, K. (2013). Stable mixed graphs. Bernoulli, 19(5B), 2330–2358.

See Also

MAG(), RG(), SG()

Examples

## The adjacency matrix of a DAG
ex <- matrix(c(
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0
), 16, 16, byrow = TRUE)
M <- c(3, 5, 6, 15, 16)
C <- c(4, 7)
AG(ex, M, C, plot = TRUE)

Description

Finds the set of edges of a graph. That is the set of undirected edges if the graph is undirected and the set of arrows if the graph is directed.

Usage

allEdges(amat)

Arguments

Value

A matrix with two columns. Each row of the matrix is a pair of indices indexing an edge of the graph. If the graph is undirected, only one of the pairs $(i,j), (j,i)$ is reported.

Author(s)

Giovanni M. Marchetti

See Also

cycleMatrix()

Examples

## A UG graph
allEdges(UG(~ y * v * k + v * k * d + y * d))

## A DAG
allEdges(DAG(u ~ h + o + p, h ~ o, o ~ p))

Description

Anger data

Usage

anger

Format

A covariance matrix for 4 variables measured on 684 female students.

X

anxiety state

Y

anger state

Z

anxiety trait

U

anger trait

Details

Trait variables are viewed as stable personality characteristics, and state variables denote behaviour in specific situations. See Cox and Wermuth (1996).

References

Cox, D. R. and Wermuth, N. (1996). Multivariate dependencies. London: Chapman and Hall.

Cox, D.R. and Wermuth, N. (1990). An approximation to maximum likelihood estimates in reduced models. 77(4), 747-761.

Examples

# Fit a chordless 4-cycle model 
data(anger) 
G = UG(~ Y*X + X*Z + Z*U + U*Y)
fitConGraph(G,anger, 684)

Description

Finds a basis set for the conditional independencies implied by a directed acyclic graph, that is a minimal set of independencies that imply all the other ones.

Usage

basiSet(amat)

Arguments

Details

Given a DAG and a pair of non adjacent nodes $(i,j)$ such that $j$ has higher causal order than $i$, the set of independency statements $i$ independent of $j$ given the union of the parents of both $i$ and $j$ is a basis set (see Shipley, 2000). This basis set has the property to lead to independent test statistics.

Value

A list of vectors representing several conditional independence statements. Each vector contains the names of two non adjacent nodes followed by the names of nodes in the conditioning (possibly empty) set.

Author(s)

Giovanni M. Marchetti

References

Shipley, B. (2000). A new inferential test for path models based on directed acyclic graphs. Structural Equation Modeling, 7(2), 206–218.

See Also

shipley.test(), dSep(), DAG()

Examples

## See Shipley (2000), Figure 2, p. 213
A <- DAG(x5 ~ x3 + x4, x3 ~ x2, x4 ~ x2, x2 ~ x1)
basiSet(A)

Description

Breadth-first search of a connected undirected graph.

Usage

bfsearch(amat, v = 1)

Arguments

Details

Breadth-first search is a systematic method for exploring a graph. The algorithm is taken from Aho, Hopcroft & Ullman (1983).

Value

A list with the following components:

• tree: the edge matrix of the resulting spanning tree.

• branches: a matrix with two columns, giving the indices of the branches of the spanning tree.

• chords: a matrix with two columns, giving the indices of the chords of the spanning tree.

Author(s)

Giovanni M. Marchetti

References

Aho, A.V., Hopcrtoft, J.E. & Ullman, J.D. (1983). Data structures and algorithms. Reading: Addison-Wesley.

Thulasiraman, K. & Swamy, M.N.S. (1992). Graphs: theory and algorithms. New York: Wiley.

See Also

UG(), findPath(), cycleMatrix()

Examples

# Finding a spanning tree of the butterfly graph
bfsearch(UG(~ a * b * o + o * u * j))
# Starting from another node
bfsearch(UG(~ a * b * o + o * u * j), v = 3)

Description

A marginal log-linear link is defined by $\eta = C (M \log p)$. See Bartolucci et al. (2007) for details.

Usage

binve(eta, C, M, G, maxit = 500, print = FALSE, tol = 1e-10)

Arguments

Value

A vector of probabilities p.

Note

From a Matlab function by Antonio Forcina, University of Perugia, Italy.

Author(s)

Antonio Forcina, Giovanni M. Marchetti mat.mlogit()

References

Bartolucci, F., Colombi, R. and Forcina, A. (2007). An extended class of marginal link functions for modelling contingency tables by equality and inequality constraints. Statist. Sinica, 17, 691-711.

Description

Block diagonal concatenation of input arguments.

Usage

blkdiag(...)

Arguments

Value

A block diagonal matrix diag(M1, M2, ...).

Author(s)

Giovanni M. Marchetti

See Also

base::diag(), blodiag()

Examples

X <- c(1, 1, 2, 2)
Z <- c(10, 20, 30, 40)
A <- factor(c(1, 2, 2, 2))
blkdiag(model.matrix(~ X + Z), model.matrix(~A))

Description

Split a vector x into a block diagonal matrix.

Usage

blodiag(x, blo)

Arguments

Value

A block-diagonal matrix with as many row as elements of blo and n columns. The vector x is split into length(blo)sub-vectors and these are the blocks of the resulting matrix.

Author(s)

Giovanni M. Marchetti

See Also

blkdiag(), diag(), fitmlogit().

Examples

blodiag(1:10, blo = c(2, 3, 5))
blodiag(1:8, blo = c(3, 4, 0, 1))
blodiag(1:4, blo = c(2, 0, 2))
blodiag(1:4, blo = c(2, 2, 0))

Description

Checks four sufficient conditions for identifiability of a Gaussian DAG model with one latent variable.

Usage

checkIdent(amat, latent)

Arguments

Details

Stanghellini and Wermuth (2005) give some sufficient conditions for checking if a Gaussian model that factorizes according to a DAG is identified when there is one hidden node over which we marginalize. Specifically, the function checks the conditions of Theorem 1, (i) and (ii) and of Theorem 2 (i) and (ii).

Value

A vector of length four, indicating if the model is identified according to the conditions of theorems 1 and 2 in Stanghellini & Wermuth (2005). The answer is TRUE if the condition holds and thus the model is globally identified or FALSE if the condition fails, and thus we do not know if the model is identifiable.

Author(s)

Giovanni M. Marchetti

References

Stanghellini, E. & Wermuth, N. (2005). On the identification of path-analysis models with one hidden variable. Biometrika, 92(2), 337-350.

See Also

isGident(), induced-graphs

Examples

# See DAG in Figure 4 (a) in Stanghellini & Wermuth (2005)
d <- DAG(y1 ~ y3, y2 ~ y3 + y5, y3 ~ y4 + y5, y4 ~ y6)
checkIdent(d, "y3") # Identifiable
checkIdent(d, "y4") # Not identifiable?

# See DAG in Figure 5 (a) in Stanghellini & Wermuth (2005)
d <- DAG(y1 ~ y5 + y4, y2 ~ y5 + y4, y3 ~ y5 + y4)
checkIdent(d, "y4") # Identifiable
checkIdent(d, "y5") # Identifiable

# A simple function to check identifiability for each node
is.ident <- function(amat) {
  # Check suff. conditions on each node of a DAG.
  p <- nrow(amat)
  # Degrees of freedom
  df <- p * (p + 1) / 2 - p - sum(amat == 1) - p + 1
  if (df <= 0) {
    warning(paste("The degrees of freedom are ", df))
  }
  a <- rownames(amat)
  for (i in a) {
    b <- checkIdent(amat, latent = i)
    if (TRUE %in% b) {
      cat("Node", i, names(b)[!is.na(b)], "\n")
    } else {
      cat("Unknown.\n")
    }
  }
}

Description

Finds the complementary graph of an undirected graph.

Usage

cmpGraph(amat, value = 1)

Arguments

Details

The complementary graph of an UG is the graph that has the same set of nodes and an undirected edge connecting $i$ and $j$ whenever there is not an $(i,j)$ edge in the original UG.

Value

The edge matrix of the complementary graph.

Author(s)

Giovanni M. Marchetti

References

Lauritzen, S. (2026). Graphical models 2nd ed. Oxford: Oxford University Press.

See Also

UG(), DAG()

Examples

## A chordless four-cycle
four <- UG(~ a * b + b * d + d * e + e * a)
four
cmpGraph(four)

Description

Finds the connectivity components of an undirected graph.

Usage

conComp(amat, method = 1)

Arguments

Value

An integer vector representing a partition of the set of nodes.

Author(s)

Giovanni M. Marchetti

References

#' Lauritzen, S. (2026). Graphical models 2nd ed. Oxford: Oxford University Press.

See Also

UG()

Examples

# three connected components
conComp(UG(~ a * c + c * d + e + g * o * u))
# one connected graph
conComp(UG(~ a * b + b * c + c * d + d * a))

Description

Computes a correlation matrix with ones along the diagonal, marginal correlations in the lower triangle and partial correlations given all remaining variables in the upper triangle.

Usage

correlations(x, placement = upper.tri)

Arguments

Value

A square correlation matrix with marginal correlations and partial correlations with the chosen placement.

Author(s)

Giovanni M. Marchetti

References

Cox, D. R. & Wermuth, N. (1996). Multivariate dependencies. London: Chapman & Hall.

See Also

parcor(), cor()

Examples

## See Table 6.1 in Cox & Wermuth (1996)
data(glucose)
correlations(glucose)

correlations(glucose, placement = lower.tri)

Description

Finds the matrix of fundamental cycles of a connected undirected graph.

Usage

cycleMatrix(amat)

Arguments

Details

All the cycles in an UG can be obtained from combination (ring sum) of the set of fundamental cycles. The matrix of fundamental cycles is a Boolean matrix having as rows the fundamental cycles and as columns the edges of the graph. If an entry is one then the edge associated to that column belongs to the cycle associated to the row.

Value

A Boolean matrix of the fundamental cycles of the undirected graph. If there is no cycle the function returns NULL.

Note

This function is used by isGident(). The row sum of the matrix gives the length of the cycles.

Author(s)

Giovanni M. Marchetti

References

Thulasiraman, K. & Swamy, M.N.S. (1992). Graphs: theory and algorithms. New York: Wiley.

See Also

UG(), findPath(), fundCycles(), isGident(), bfsearch()

Examples

## Three cycles
cycleMatrix(UG(~ a * b * d + d * e + e * a * f))
## No cycle
cycleMatrix(UG(~ a * b))
## two cycles: the first is even and the second is odd
cm <- cycleMatrix(UG(~ a * b + b * c + c * d + d * a + a * u * v))
apply(cm, 1, sum)

Description

A simple way to define a DAG by means of regression model formulae.

Usage

DAG(..., order = FALSE)

Arguments

Details

The DAG is defined by a sequence of recursive regression models. Each regression is defined by a model formula. For each formula the response defines a node of the graph and the explanatory variables the parents of that node. If the regressions are not recursive the function returns an error message.

Some authors prefer the terminology acyclic directed graphs (ADG).

Value

The adjacency matrix of the DAG, i.e., a square Boolean matrix of order equal to the number of nodes of the graph and a one in position $(i,j)$ if there is an arrow from $i$ to $j$ and zero otherwise. The rownames of the adjacency matrix are the nodes of the DAG.

If order = TRUE the adjacency matrix is permuted to have parents before children. This can always be done (in more than one way) for DAGs. The resulting adjacency matrix is upper triangular.

Note

The model formulae may contain interactions, but they are ignored in the graph.

Author(s)

G. M. Marchetti

References

Lauritzen, S. (1996). Graphical models 2nd ed. Oxford: Oxford University Press.

See Also

UG(), topSort(), edgematrix(), fitDag()

Examples

# A Markov chain
DAG(y ~ x, x ~ z, z ~ u)

# Another DAG
DAG(y ~ x + z + u, x ~ u, z ~ u)

# A DAG with an isolated node
DAG(v ~ v, y ~ x + z, z ~ w + u)

# There can be repetitions
DAG(y ~ x + u + v, y ~ z, u ~ v + z)

# Interactions are ignored
DAG(y ~ x * z + z * v, x ~ z)

# A cyclic graph returns an error!
## Not run: 
DAG(y ~ x, x ~ z, z ~ y)

## End(Not run)

## The order can be changed
DAG(y ~ z, y ~ x + u + v, u ~ v + z)

## If you want to order the nodes (topological sort of the DAG)
DAG(y ~ z, y ~ x + u + v, u ~ v + z, order = TRUE)

Description

Raw data on blood pressure, body mass and age on 44 female patients, and covariance matrix for derived variables.

Usage

derived

Format

A list containing a dataframe raw with 44 lines and 5 columns and a symmetric 4x4 covariance matrix S.

The following is the description of the variables in the dataframe raw

list("Sys")

Systolic blood pressure, in mm Hg

list("Dia")

Diastolic blood pressure, in mm Hg

list("Age")

Age of the patient, in years

list("Hei")

Height, in cm

list("Wei")

Weight, in kg

The following is the description of the variables for the covariance matrix S.

list("Y")

Derived variable Y=log(Sys/Dia)

list("X")

Derived variables X=log(Dia)

list("Z")

Body mass index Z=Wei/(Hei/100)^2

list("W")

Age

References

Wermuth N. and Cox D.R. (1995). Derived variables calculated from similar joint responses: some characteristics and examples. Computational Statistics and Data Analysis, 19, 223-234.

Examples

# A DAG model with a latent variable U
G = DAG(Y ~ Z + U, X ~ U + W, Z ~ W)

data(derived)

# The model fitted using the derived variables
out = fitDagLatent(G, derived$S, n = 44, latent = "U")

# An ancestral graph model marginalizing over U
H = AG(G, M = "U")

# The ancestral graph model fitted obtaining the 
# same result
out2 = fitAncestralGraph(H, derived$S, n = 44)

Description

Defines the adjacency matrix of a directed graph.

Usage

DG(...)

Arguments

Details

The directed graph is defined by a sequence of models formulae. For each formula the response defines a node of the graph and its parents. The graph contains no loops, but can have directed cycles.

Value

the adjacency matrix of the directed graph, i.e., a square Boolean matrix of order equal to the number of nodes of the graph and a one in position $(i,j)$ if there is an arrow from $i$ to $j$ and zero otherwise. The dimnames of the adjacency matrix are the labels for the nodes of the graph.

Author(s)

G. M. Marchetti

References

Lauritzen, S., L. (1996). Graphical models 2nd ed. Oxford: Oxford University Press.

See Also

DAG(), UG()

Examples

# A DAG
DG(y ~ x, x ~ z, z ~ u)

# A cyclic directed graph
DG(y ~ x, x ~ z, z ~ y)

# A graph with two arrows between two nodes
double <- DG(y ~ x, x ~ y)
double
plotGraph(double, directed = TRUE)

# There can be isolated nodes
DG(y ~ x, x ~ x)

Description

Computes faster the product of a diagonal matrix times a full matrix.

Usage

diagv(v, M)

Arguments

Details

Computes $N = D_v M$ where $D_v$ is diagonal avoiding the base::diag() operator.

Value

A matrix N.

See Also

base::diag().

Examples

v <- 1:1000
M <- matrix(runif(3000), 1000, 3)
dim(diagv(v, M))

Description

Draw a graph from its adjacency matrix representation.

Usage

drawGraph(
  amat,
  coor = NULL,
  adjust = FALSE,
  alpha = 1.5,
  beta = 3,
  lwd = 1,
  ecol = "blue",
  bda = 0.1,
  layout = igraph::layout_nicely
)

Arguments

Details

The function is a very simple tool useful for displaying small mixed graphs, with a rudimentary interface for moving nodes and edges of a given graph and adjusting the final plot. The function plotGraph() is more elaborate. Packages like visNetwork or DiagrammeR are better for advanced network graph rendering. See the references. Also one of the first interactive plots for graphs is the package dynamicGraph that is now archived by CRAN since 2021.

Value

A plot of the graph with a initial positioning of the nodes, as specified by coor and remains in a waiting state. The position of each node can be shifted by pointing and clicking (with the first mouse button) close to the node. When the mouse button is pressed the node which is closer to the selected point is moved to that position. Thus, one must be careful to click closer to the selected node than to any other node. The nodes can be moved to any position by repeating the previous operation. The adjustment process is terminated by pressing any mouse button other than the first. At the end of the process, the function returns invisibly the coordinates of the nodes. The coordinates may be used later to redisplay the graph.

Author(s)

Giovanni M. Marchetti

References

Richard Iannone (2026). DiagrammeR: Graph/Network Visualization. https://CRAN.R-project.org/package=DiagrammeR

Benoit Thieurmel (2025). visNetwork: Network Visualization using 'vis.js' Library. https://CRAN.R-project.org/package=visNetwork

#' Badsberg, J.H (2021). dynamicGraph: An Interactive Graphical Tool for Manipulating Graphs. https://CRAN.R-project.org/package=dynamicGraph

See Also

UG(), DAG(), makeMG(), plotGraph()

Examples

## A directed acyclic graph
d <- DAG(y1 ~ y2 + y6, y2 ~ y3, y3 ~ y5 + y6, y4 ~ y5 + y6)
## Not run: 
drawGraph(d)

## End(Not run)

## An undirected graph
g <- UG(~ giova * anto * armo + anto * arj * sara)
## Not run: 
drawGraph(d)

## End(Not run)

## An ancestral graph
ag <- makeMG(ug = UG(~ y0 * y1), dg = DAG(y4 ~ y2, y2 ~ y1), bg = UG(~ y2 * y3 + y3 * y4))
drawGraph(ag, adjust = FALSE)
drawGraph(ag, adjust = FALSE)

## A more complex example with coordinates: the UNIX evolution
xy <-
  structure(c(
    5, 15, 23, 25, 26, 17, 8, 6, 6, 7, 39, 33, 23, 49,
    19, 34, 13, 29, 50, 68, 70, 86, 89, 64, 81, 45, 64, 49, 64, 87,
    65, 65, 44, 37, 64, 68, 73, 85, 83, 95, 84, 0, 7, 15, 27, 44,
    37, 36, 20, 51, 65, 44, 64, 59, 73, 69, 78, 81, 90, 97, 89, 72,
    85, 74, 62, 68, 59, 52, 48, 43, 50, 34, 21, 18, 5, 1, 10, 2,
    11, 2, 1, 44
  ), .Dim = c(41, 2), .Dimnames = list(NULL, c(
    "x",
    "y"
  )))
Unix <- DAG(
  SystemV.3 ~ SystemV.2,
  SystemV.2 ~ SystemV.0,
  SystemV.0 ~ TS4.0,
  TS4.0 ~ Unix.TS3.0 + Unix.TS.PP + CB.Unix.3,
  PDP11.SysV ~ CB.Unix.3,
  CB.Unix.3 ~ CB.Unix.2,
  CB.Unix.2 ~ CB.Unix.1,
  Unix.TS.PP ~ CB.Unix.3,
  Unix.TS3.0 ~ Unix.TS1.0 + PWB2.0 + USG3.0 + Interdata,
  USG3.0 ~ USG2.0,
  PWB2.0 ~ Interdata + PWB1.2,
  USG2.0 ~ USG1.0,
  CB.Unix.1 ~ USG1.0,
  PWB1.2 ~ PWB1.0,
  USG1.0 ~ PWB1.0,
  PWB1.0 ~ FifthEd,
  SixthEd ~ FifthEd,
  LSX ~ SixthEd,
  MiniUnix ~ SixthEd,
  Interdata ~ SixthEd,
  Wollongong ~ SixthEd,
  SeventhEd ~ Interdata,
  BSD1 ~ SixthEd,
  Xenix ~ SeventhEd,
  V32 ~ SeventhEd,
  Uniplus ~ SeventhEd,
  BSD3 ~ V32,
  BSD2 ~ BSD1,
  BSD4 ~ BSD3,
  BSD4.1 ~ BSD4,
  EigthEd ~ SeventhEd + BSD4.1,
  NinethEd ~ EigthEd,
  Ultrix32 ~ BSD4.2,
  BSD4.2 ~ BSD4.1,
  BSD4.3 ~ BSD4.2,
  BSD2.8 ~ BSD4.1 + BSD2,
  BSD2.9 ~ BSD2.8,
  Ultrix11 ~ BSD2.8 + V7M + SeventhEd,
  V7M ~ SeventhEd
)
drawGraph(Unix, coor = xy, adjust = FALSE)
# dev.print(file="unix.fig", device=xfig) # Edit the graph with Xfig

Description

Determines if in a directed acyclic graph two set of nodes are d-separated by a third set of nodes.

Usage

dSep(amat, first, second, cond)

Arguments

Details

d-separation is a fundamental concept introduced by Pearl (1988).

Value

A logical value. Is TRUE if first and second are d-separated by cond.

Author(s)

Giovanni M. Marchetti

References

Pearl, J. (1988). Probabilistic reasoning in intelligent systems. San Mateo: Morgan Kaufmann.

Lauritzen, S. (2026). Graphical models, 2nd ed. Oxford: Oxford University Press.

See Also

DAG(), shipley.test(), msep(), induced-graphs

Examples

# Conditioning on a transition node
dSep(DAG(y ~ x, x ~ z), first = "y", second = "z", cond = "x")
# Conditioning on a collision node (collider)
dSep(DAG(y ~ x, y ~ z), first = "x", second = "z", cond = "y")
# Conditioning on a source node
dSep(DAG(y ~ x, z ~ x), first = "y", second = "z", cond = "x")
# Marginal independence
dSep(DAG(y ~ x, y ~ z), first = "x", second = "z", cond = NULL)
# The DAG defined on p.~47 of Lauritzen (1996)
dag <- DAG(g ~ x, h ~ x + f, f ~ b, x ~ l + d, d ~ c, c ~ a, l ~ y, y ~ b)
dSep(dag, first = "a", second = "b", cond = c("x", "y"))
dSep(dag, first = "a", second = c("b", "d"), cond = c("x", "y"))

Description

In some matrix computations the adjacency matrix of a graph is transformed into an "edge matrix". Briefly, if $B$ is the adjacency matrix of a DAG, the corresponding edge matrix $A$ is a binary matrix $A = (a_{ij})$ such that $a_{ij} = 1$ if and only if $i \to j$ or $i = j$ and $0$ otherwise. If the adjacency matrix is ordered according to a topological ordering the edge matrix is upper triangular with ones on the main diagonal. This order is the inverse of the topological order.

Usage

edgematrix(B, ord = FALSE)

Arguments

Value

• A: the edge matrix of the graph.

Author(s)

Giovanni M. Marchetti

References

Giovanni M. Marchetti and Nanny Wermuth (2009). Matrix representations and independencies in direct acyclic graphs, The Annals of Statistics, Vol. 37, No. 2, 961-978.

See Also

adjMatrix(), topSort()

Examples

B <- DAG(z ~ u + v, y ~ x + z)
edgematrix(B)
edgematrix(B, ord = TRUE)

Description

Find the essential graph from a given directed acyclic graph.

Usage

essentialGraph(dagx)

Arguments

Details

Converts a DAG into the Essential Graph. Is implemented by the algorithm by D.M. Chickering (1995).

Value

the adjacency matrix of the essential graph. Note that the essential graph is a mixed graph where the edges can be arrows and/or lines. Then, the elements of the adjacency matrix are coded as ⁠(0,1)⁠ if $i \rightarrow j$ and ⁠(1,0)⁠ if $i \leftarrow j$ and ⁠(10, 10)⁠ if $i - j$.

Author(s)

Giovanni M. Marchetti, from a MATLAB function by Tomas Kocka, AAU.

References

Chickering, D.M. (1995). A transformational characterization of equivalent Bayesian Network structures. Proceedings of Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QU, 87-98. Morgan Kaufmann.

See Also

makeMG() DAG(), induced-graphs.

Examples

dag <- DAG(U ~ Y + Z, Y ~ X, Z ~ X)
plotGraph(essentialGraph(dag))

Description

Finds one path between two nodes of a graph.

Usage

findPath(amat, st, en, path = c())

Arguments

Value

A vector of integers containing the sequence of nodes forming a path from st to en. If no path is found, it returns NULL. In some graphs (like spanning trees) there is only one path between two nodes.

Author(s)

Giovanni M. Marchetti, translating the original Python code (see references).

References

Python Software Foundation (2003). Python Patterns Implementing Graphs. https://www.python.org/doc/essays/graphs/.

See Also

fundCycles()

Examples

## A (single) path on a spanning tree
findPath(bfsearch(UG(~ a * b * c + b * d + d * e + e * c))$tree, st = 1, en = 5)

Description

Iterative conditional fitting of Gaussian Ancestral Graph Models.

Usage

fitAncestralGraph(amat, S, n, tol = 1e-06)

Arguments

Details

In the Gaussian case, the models can be parameterized using precision parameters, regression coefficients, and error covariances (see Richardson and Spirtes, 2002, Section 8). This function finds the MLE $\hat{\Lambda}$ of the precision parameters by fitting a concentration graph model. The MLE $\hat{B}$ of the regression coefficients and the MLE $\hat{\Omega}$ 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 $\hat{\Sigma}$ of the covariance matrix by matrix multiplication:

$\hat{\Sigma} = \hat{B}^{-1}(\hat{\Lambda}+\hat{\Omega})\hat{B}^{-T}$

Note that in Richardson and Spirtes (2002), the matrices $\Lambda$ and $\Omega$ are defined as submatrices.

Value

A list with the following components:

• 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, Bhat contains ones along 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 $j \rightarrow i$, 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 number of 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

# 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"))
)

## Fit an ancestral graph y -> x <-> z <- u
fitAncestralGraph(makeMG(dg = DAG(x ~ y, z ~ u), bg = UG(~ x * z)), S, n = 100)

# Fit an ancestral graph y <-> x <-> z <-> u
fitAncestralGraph(makeMG(bg = UG(~ y * x + x * z + z * u)), S, n = 100)

## Fit the same graph with fitCovGraph
fitCovGraph(makeMG(bg = UG(~ y * x + x * z + z * u)), S, n = 100)

# Another example for the mathematics marks data
data(marks)
S_marks <- var(marks)
mag1 <- makeMG(bg = UG(~ mechanics * vectors * algebra + algebra * analysis * statistics))
fitAncestralGraph(mag1, S_marks, n = 88)

Description

Fits a concentration graph (a covariance selection model).

Usage

fitConGraph(amat, S, n, cli = NULL, alg = 3, pri = FALSE, tol = 1e-06)

Arguments

Details

The algorithms for fitting concentration graph models by maximum likelihood are discussed in Speed and Kiiveri (1986). If the cliques are known the function uses the iterative proportional fitting algorithm described by Whittaker (1990, p. 184). If the cliques are not specified the function uses the algorithm by Hastie et al. (2009, p. 631ff).

Value

• Shat the fitted covariance matrix

• dev the deviance of the model

• df the degrees of freedom

• it the number of iterations

Author(s)

Giovanni M. Marchetti

References

Cox, D. R. & Wermuth, N. (1996). Multivariate dependencies. London: Chapman & Hall.

Hastie, T., Tibshirani, R. and Friedman, J. (2009). The elements of statistical learning. Springer Verlag: New York.

Speed, T.P. & Kiiveri, H (1986). Gaussian Markov distributions over finite graphs. Annals of Statistics, 14, 138-150.

Whittaker, J. (1990). Graphical models in applied multivariate statistics. Chichester: Wiley.

See Also

UG(), fitDag(), marks

Examples

# A model for the mathematics marks (Whittaker, 1990)
data(marks)
# A butterfly concentration graph
G <- UG(~ mechanics * vectors * algebra + algebra * analysis * statistics)
fitConGraph(G, cov(marks), nrow(marks))
# Using the cliques
cl <- list(c("mechanics", "vectors", "algebra"), c("algebra", "analysis", "statistics"))
fitConGraph(G, S = cov(marks), n = nrow(marks), cli = cl)

Description

Fits a Gaussian covariance graph model by maximum likelihood.

Usage

fitCovGraph(
  amat,
  S,
  n,
  alg = "icf",
  dual.alg = 2,
  start.icf = NULL,
  tol = 1e-06
)

Arguments

Details

A covariance graph is an undirected graph in which the variables associated to two non-adjacent nodes are marginally independent. The edges of these models are represented by bi-directed edges (Drton and Richardson, 2003) or by dashed lines (Cox and Wermuth, 1996).

By default, this function gives the ML estimates in the covariance graph model, by iterative conditional fitting (Drton and Richardson, 2003). Otherwise, the estimates from a dual likelihood estimator can be obtained (Kauermann, 1996; Edwards, 2000, section 7.4).

Value

• Shat the fitted covariance matrix

• dev the deviance of the fitted model

• df the degrees of freedom

• it' the number of iterations.

Author(s)

Mathias Drton

References

Cox, D. R. & Wermuth, N. (1996). Multivariate dependencies. London: Chapman & Hall.

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.

Kauermann, G. (1996). On a dualization of graphical Gaussian models. Scandinavian Journal of Statistics. 23, 105-116.

See Also

fitConGraph(), icf()

Examples

# Correlations among four strategies to cope with stress
#  for 72 students. Cox & Wermuth (1996), p. 73.

data(stress)

# A chordless 4-cycle covariance graph
G <- UG(~ Y * X + X * U + U * V + V * Y)

fitCovGraph(G, S = stress, n = 72)
fitCovGraph(G, S = stress, n = 72, alg = "dual")

Description

Fits linear recursive regressions with independent residuals specified by a DAG.

Usage

fitDag(amat, S, n)

Arguments

Details

fitDag checks if the order of the nodes in adjacency matrix is the same of S and if not it reorders the adjacency matrix to match the order of the variables in S. The nodes of the adjacency matrix may form a subset of the variables in S.

Value

• Shat the fitted covariance matrix

• Ahat a square matrix of the fitted regression coefficients. The entry Ahat[i,j] is minus the regression coefficient of variable i in the regression equation j. Thus there is a non zero partial regression coefficient Ahat[i,j] corresponding to each non zero value amat[j,i] in the adjacency matrix.

• Dhat a vector containing the partial variances of each variable given the parents.

• dev the deviance of the fitted model

• df the degrees of freedom.

Author(s)

Giovanni M. Marchetti

References

Cox, D. R. & Wermuth, N. (1996). Multivariate dependencies. London: Chapman & Hall.

See Also

DAG(), swp()

Examples

dag <- DAG(y ~ x + u, x ~ z, z ~ u)
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"))
)
fitDag(dag, S, 200)

Description

Fits by maximum likelihood a Gaussian DAG model where one of the nodes of the graph is latent and it is marginalised over.

Usage

fitDagLatent(
  amat,
  Syy,
  n,
  latent,
  norm = 1,
  seed,
  maxit = 9000,
  tol = 1e-06,
  pri = FALSE
)

Arguments

Details

For the EM algorithm used see Kiiveri (1987).

Value

• Shat the fitted covariance matrix of all the variables including the latent one. The latent variable is the last. If norm=1 then the variance of the latent variable is constrained to 1.

• Ahat a square matrix of the fitted regression coefficients. The entry Ahat[i,j] is minus the regression coefficient of variable i in the regression equation j. Thus there is a non zero partial regression coefficient Ahat[i,j] corresponding to each non zero value amat[j,i] in the adjacency matrix

• Dhat a vector containing the partial variances of each variable given the parents. If norm=2 then the partial variance of the latent variable is constrained to 1.

• dev the deviance of the fitted model

• df the degrees of freedom of the model

• it the number of EM algorithm iterations at convergence.

Author(s)

Giovanni M. Marchetti

References

Kiiveri,H. T. (1987). An incomplete data approach to the analysis of covariance structures. Psychometrika, 52, 4, 539-554.

Joreskog, K.G. and Goldberger, A.S. (1975). Estimation of a model with multiple indicators and multiple causes of a single latent variable. Journal of the American Statistical Association, 10, 631-639.

See Also

fitDag(), checkIdent()

Examples

# data from Joreskog and Goldberger (1975)
V <- matrix(c(
  1, 0.36, 0.21, 0.10, 0.156, 0.158,
  0.36, 1, 0.265, 0.284, 0.192, 0.324,
  0.210, 0.265, 1, 0.176, 0.136, 0.226,
  0.1, 0.284, 0.176, 1, 0.304, 0.305,
  0.156, 0.192, 0.136, 0.304, 1, 0.344,
  0.158, 0.324, 0.226, 0.305, 0.344, 1
), 6, 6)
nod <- c("y1", "y2", "y3", "x1", "x2", "x3")
dimnames(V) <- list(nod, nod)
dag <- DAG(y1 ~ z, y2 ~ z, y3 ~ z, z ~ x1 + x2 + x3, 
           x1 ~ x2 + x3, x2 ~ x3)
fitDagLatent(dag, V, n = 530, latent = "z", seed = 4564)
fitDagLatent(dag, V, n = 530, latent = "z", norm = 2,
              seed = 145)

Description

Fits a logistic regression model to multivariate binary responses. The explanatory variables can be binary and/or individual-level continuous covariates.

Usage

fitmlogit(..., C = c(), D = c(), mit = 100, ep = 1e-80, acc = 1e-04, daf)

Arguments

Details

See Evans and Forcina (2013).

Value

• LL The maximized log-likelihood.

• be The vector of the ML estimates of the parameters.

• S The estimated asymptotic covariance matrix.

• P The estimated cell probabilities for each individual.

Author(s)

Antonio Forcina, Giovanni M. Marchetti

References

Evans, R.J. and Forcina, A. (2013). Two algorithms for fitting constrained marginal models. Computational Statistics and Data Analysis, 66, 1-7.

See Also

stats::glm()

Examples

data(surdata)
out1 <- fitmlogit(A ~ X, B ~ Z, cbind(A, B) ~ X * Z, daf = surdata)
out1$beta
out2 <- fitmlogit(A ~ X, B ~ Z, cbind(A, B) ~ 1, daf = surdata)
out2$beta

Description

Finds the list of fundamental cycles of a connected undirected graph.

Usage

fundCycles(amat)

Arguments

Details

All the cycles in an UG can be obtained from combination (ring sum) of the set of fundamental cycles.

This function is used by cycleMatrix() and isGident().

Value

A list of matrices with two columns. Every component of the list is associated to a cycle. The cycle is described by a $k \times 2$ matrix whose rows are the edges of the cycle. If there is no cycle, the function returns NULL.

Author(s)

Giovanni M. Marchetti

References

Thulasiraman, K. & Swamy, M.N.S. (1992). Graphs: theory and algorithms. New York: Wiley.

See Also

UG(), findPath(), cycleMatrix() isGident(), bfsearch()

Examples

## Three fundamental cycles
fundCycles(UG(~ a * b * d + d * e + e * a * f))

Description

Data on glucose control of diabetes patients.

Usage

glucose

Format

A data frame with 68 observations on the following 8 variables.

Y

a numeric vector, Glucose control (glycosylated haemoglobin), values up to about 7 or 8 indicate good glucose control.

X

a numeric vector, a score for knowledge about the illness.

Z

a numeric vector, a score for fatalistic externality (mere chance determines what occurs).

U

a numeric vector, a score for social externality (powerful others are responsible).

V

a numeric vector, a score for internality (the patient is him or herself responsible).

W

a numeric vector, duration of the illness in years.

A

a numeric vector, level of education, with levels -1: at least 13 years of formal schooling, 1: less then 13 years.

B

a numeric vector, gender with levels -1: females, 1: males.

Details

Data on 68 patients with fewer than 25 years of diabetes. They were collected at the University of Mainz to identify psychological and socio-economic variables possibly important for glucose control, when patients choose the appropriate dose of treatment depending on the level of blood glucose measured several times per day.

The variable of primary interest is Y, glucose control, measured by glycosylated haemoglobin. X, knowledge about the illness, is a response of secondary interest. Variables Z, U and V measure patients' type of attribution, called fatalistic externality, social externality and internality. These are intermediate variables. Background variables are W, the duration of the illness, A the duration of formal schooling and B, gender. The background variables A and B are binary variables with coding -1, 1.

Source

Cox & Wermuth (1996), p. 229.

References

Cox, D. R. & Wermuth, N. (1996). Multivariate dependencies. London: Chapman & Hall.

Examples

data(glucose)
## See Cox & Wermuth (1996), Figure 6.3 p. 140
coplot(Y ~ W | A, data=glucose)

Description

grMAT converts graph objects to a mixed adjacency matrix.

Usage

grMAT(agr)

Arguments

Details

Support for graphNEL objects requires the graph package from Bioconductor, which is a suggested dependency. If the package is missing, passing a graphNEL object will trigger an informative error.

Value

A matrix consisting of 4 different integers representing the $ij$-elements:

• 0 for a missing edge between $i$ and $j$,

• 1 for an arrow from $i$ to $j$,

• 10 for a full line between $i$ and $j$, and

• 100 for a bi-directed arrow between $i$ and $j$. These numbers are added when multiple edges of different types are present. The matrix is symmetric with respect to full lines and bi-directed arrows.

Author(s)

Kayvan Sadeghi, Giovanni Marchetti

Examples

## Generating the adjacency matrix from a vector
exvec <- c(
  "b", 1, 2, "b", 1, 14, "a", 9, 8, "l", 9, 11, "a", 10, 8,
  "a", 11, 2, "a", 11, 10, "a", 12, 1, "b", 12, 14, "a", 13, 10, "a", 13, 12)
grMAT(exvec)

## Not run: 
## Example with graphNEL (requires the 'graph' package from Bioconductor)
if (requireNamespace("graph", quietly = TRUE)) {
  V <- c("a", "b", "c")
  g <- graph::graphNEL(nodes = V, edgemode = "undirected")
  grMAT(g)
 }

## End(Not run)

g <- c("a", 3, 1, "l", 1, 2, "*", 4, 4)
grMAT(g)

Description

Create a Boolean matrix from all combinations of binary variables

Usage

hypercube(n)

Arguments

Value

a matrix with $2^n$ rows and $n$ columns. Each row of the matrix is a TRUE/FALSE vector representing the coordinates of the vertices of a hypercube, ordered in inverse lexicographic order.

See Also

base::expand.grid()

Examples

hypercube(4)

Description

Finds the indicator matrix of the zeros of a matrix.

Usage

In(A)

Arguments

Details

The indicator matrix is a matrix of zeros and ones which has a zero element iff the corresponding element of A is (exactly) zero.

Value

a matrix of the same dimensions as A.

Author(s)

Giovanni M. Marchetti

References

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

DAG(), inducedCovGraph(), inducedConGraph(), In().

Examples

# A simple way to find the overall induced concentration graph
# The DAG on p. 198 of Cox & Wermuth (1996)
amat <- DAG(y1 ~ y2 + y3, y3 ~ y5, y4 ~ y5)
A <- edgematrix(amat)
In(crossprod(A))

Description

Functions to find induced graphs after conditioning on a set of variables and marginalizing over another set.

Usage

inducedChainGraph(amat, cc = rownames(amat), cond = NULL, type = "LWF")

inducedCovGraph(amat, sel = rownames(amat), cond = NULL)

inducedConGraph(amat, sel = rownames(amat), cond = NULL)

inducedRegGraph(amat, sel = rownames(amat), cond = NULL)

inducedDAG(amat, order, cond = NULL)

Arguments

Details

Given a directed acyclic graph representing a set of conditional independencies it is possible to obtain other graphs of conditional independence implied after marginalizing over 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 the 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 = NULL the functions return the null matrix. If cond = NULL, the conditioning set is empty and the functions inducedConGraph and inducedCovGraph return the overall induced covariance or concentration matrices of the selected variables. I f 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, 687-717.

See Also

DAG(), UG(), isAcyclic()

Examples

if (requireNamespace("ggm", quietly = TRUE)) {
  library(ggm)

  # 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")
}

Description

Checks if a given graph is acyclic.

Usage

isAcyclic(amat, method = 2)

Arguments

Value

a logical value, TRUE if the graph is acyclic and FALSE otherwise.

Author(s)

David Edwards, Giovanni M. Marchetti

References

Aho, A.V., Hopcroft, J.E. & Ullman, J.D. (1983). Data structures and algorithms. Reading: Addison-Wesley.

Examples

# A cyclic graph
d <- matrix(0, 3, 3)
rownames(d) <- colnames(d) <- c("x", "y", "z")
d["x", "y"] <- d["y", "z"] <- d["z", "x"] <- 1
## Test if the graph is acyclic
isAcyclic(d)
isAcyclic(d, method = 1)

Description

Check if it is an adjacency matrix of an ADMG.

Usage

isADMG(amat)

Arguments

Details

Checks if the following conditions must hold:

• no undirected edge meets an arrowhead

• no directed cycles

Value

A logical value, TRUE if it is an ADMG and FALSE otherwise.

Author(s)

Giovanni M. Marchetti, Mathias Drton

References

Richardson, T. S. and Spirtes, P. (2002). Ancestral Graph Markov Models. Annals of Statistics, 30(4), 962-1030.

See Also

makeMG(), isAG()

Examples

# Examples from Richardson and Spirtes (2002)
a1 <- makeMG(dg = DAG(a ~ b, b ~ d, d ~ c), 
             bg = UG(~ a * c))
isADMG(a1) # Not an AG. (a2) p.969

a2 <- makeMG(dg = DAG(b ~ a, d ~ c), 
             bg = UG(~ a * c + c * b + b * d)) 
             # Fig. 3 (b1) p.969
isADMG(a2)

Description

Check if it is an adjacency matrix of an ancestral graph.

Usage

isAG(amat)

Arguments

Details

Checks if the following conditions hold:

• no undirected edge meets an arrowhead

• no directed cycles

• spouses cannot be ancestors. For details see Richardson and Spirtes (2002).

Value

a logical value. TRUE if it is an ancestral graph and FALSE otherwise.

Author(s)

Giovanni M. Marchetti, Mathias Drton

References

Richardson, T. S. and Spirtes, P. (2002). Ancestral Graph Markov Models. Annals of Statistics, 30(4), 962-1030.

See Also

makeMG(), isADMG()

Examples

# Examples from Richardson and Spirtes (2002)
a1 <- makeMG(dg = DAG(a ~ b, b ~ d, d ~ c), 
             bg = UG(~ a * c))
suppressWarnings(isAG(a1)) # Not an AG. (a2) p. 969

a2 <- makeMG(dg = DAG(b ~ a, d ~ c), 
             bg = UG(~ a * c + c * b + b * d)) 
# Fig. 3 (b1) p.969
isAG(a2)

Description

Tests if an undirected graph is G-identifiable.

Usage

isGident(amat)

Arguments

Details

An undirected graph is said G-identifiable if every connected component of the complementary graph contains an odd cycle (Stanghellini and Wermuth, 2005). See also Tarantola and Vicard (2002).

Value

a logical value, TRUE if the graph is G-identifiable and FALSE if it is not.

Author(s)

Giovanni M. Marchetti UG(), cmpGraph(), cycleMatrix()

References

Stanghellini, E. & Wermuth, N. (2005). On the identification of path-analysis models with one hidden variable. Biometrika, 92(2), 337-350.

Stanghellini, E. (1997). Identification of a single-factor model using graphical Gaussian rules. Biometrika, 84, 241-244.

Tarantola, C. & Vicard, P. (2002). Spanning trees and identifiability of a single-factor model. Statistical Methods & Applications, 11, 139–152.

Vicard, P. (2000). On the identification of a single-factor model with correlated residuals. Biometrika, 87, 199-205.

Examples

# A not G-identifiable UG
G1 <- UG(~ a * b + u * v)
isGident(G1)
# G-identifiable UG
G2 <- UG(~ a + b + u * v)
isGident(G2)
# G-identifiable UG
G3 <- cmpGraph(UG(~ a * b * c + x * y * z))
isGident(G3)

Description

Computes the deviance of a Gaussian model given the concentration matrix, the sample covariance matrix, the sample size, and the number of variables.

Usage

likGau(K, S, n, k)

Arguments

Value

A numeric value representing the deviance of the Gaussian model.

Author(s)

Kayvan Sadeghi, Giovanni M. Marchetti

See Also

log, det, diag

Description

MAG generates and plots maximal ancestral graphs after marginalisation and conditioning.

Usage

MAG(
  amat,
  M = c(),
  C = c(),
  showmat = TRUE,
  plot = FALSE,
  plotfun = plotGraph,
  ...
)

Arguments

Details

Support for graphNEL objects requires the graph package from Bioconductor, which is a suggested dependency. If the package is missing, passing a graphNEL object will trigger an informative error.

This function uses the internal functions AG() and Max().

Value

A matrix consisting of 4 different integers representing the $ij$-elements:

• 0 for a missing edge between $i$ and $j$,

• 1 for an arrow from $i$ to $j$,

• 10 for a full line between $i$ and $j$, and

• 100 for a bi-directed arrow between $i$ and $j$. These numbers are added when multiple edges of different types are present. The matrix is symmetric with respect to full lines and bi-directed arrows.

Author(s)

Kayvan Sadeghi

References

Richardson, T. S. and Spirtes, P. (2002). Ancestral graph Markov models. Annals of Statistics, 30(4), 962-1030.

Sadeghi, K. (2013). Stable mixed graphs. Bernoulli, 19(5B), 2330–2358.

Sadeghi, K. and Lauritzen, S.L. (2014). Markov properties for loopless mixed graphs. Bernoulli, 20(2), 676-696.

See Also

AG(), Max(), MRG(), MSG()

Examples

# The adjacency matrix of a DAG
ex <- matrix(c(
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0
), 16, 16, byrow = TRUE)
M <- c(3, 5, 6, 15, 16)
C <- c(4, 7)
MAG(ex, M, C, plot = TRUE)

H <- matrix(c(0, 100, 1, 0, 
             100, 0, 100, 0, 
             0, 100, 0, 100, 
             0, 1, 100, 0), 4, 4)
Max(H)

Description

Defines a loopless mixed graph from directed, undirected and bidirected components.

Usage

makeMG(dg = NULL, ug = NULL, bg = NULL)

Arguments

Details

Note that the three adjacency matrices must have labels and may be defined using the functions DG (or 'DAG'), UG. The adjacency matrices of the undirected graphs may be just symmetric Boolean matrices.

Value

A square matrix obtained by combining the three graph components into an adjacency matrix of a mixed graph. The matrix consists of 4 different integers as an $ij$-element:

• 0 for a missing edge between $i$ and $j$,

• 1 for an arrow from $i$ to $j$,

• 10 for a full line between $i$ and $j$, and

• 100 for a bi-directed arrow between $i$ and $j$. These numbers are added to be associated with multiple edges of different types. The matrix is symmetric w.r.t full lines and bi-directed arrows.

Author(s)

Giovanni M. Marchetti, Mathias Drton, Kayvan Sadeghi

References

Richardson, T. S. and Spirtes, P. (2002). Ancestral Graph Markov Models. Annals of Statistics, 30(4), 962–1030.

See Also

DG(), UG(), DAG()

Examples

# Examples from Richardson and Spirtes (2002)
a1 <- makeMG(dg = DAG(a ~ b, b ~ d, d ~ c), 
             bg = UG(~ a * c))
suppressWarnings(isAG(a1)) # Not an AG. (a2) p.969

a2 <- makeMG(dg = DAG(b ~ a, d ~ c), 
bg = UG(~ a * c + c * b + b * d)) # Fig. 3 (b1) p.969
isAG(a2)

a3 <- makeMG(ug = UG(~ a * c), 
             dg = DAG(b ~ a, d ~ c), 
             bg = UG(~ b * d))   # Fig. 3 (b2) p.969
a3

a5 <- makeMG(bg = UG(~ alpha * beta + gamma * delta), 
             dg = DAG(alpha ~ gamma, delta ~ beta)) # Fig. 6 p. 973
a5

# Another Example
a4 <- makeMG(ug = UG(~ y0 * y1), 
             dg = DAG(y4 ~ y2, y2 ~ y1), 
             bg = UG(~ y2 * y3 + y3 * y4))
a4

# A mixed graphs with double edges
mg <- makeMG(
  dg = DG(Y ~ X, Z ~ W, W ~ Z, Q ~ X), 
  ug = UG(~ X * Q),
  bg = UG(~ Y * X + X * Q + Q * W + Y * Z)
)
mg

# Chronic pain data: a regression graph
chronic_pain <- makeMG(
     dg = DAG(Y ~ Za, Za ~ Zb + A, Xa ~ Xb,
              Xb ~ U + V, U ~ A + V, Zb ~ B, A ~ B), 
     bg = UG(~ Za * Xa + Zb * Xb)
     )
 chronic_pain

Description

Provides the contrast and marginalization matrices for the marginal parametrization of a probability vector.

Usage

marg.param(lev, type)

Arguments

Details

See Bartolucci, Colombi and Forcina (2007).

The function assumes that the vector of probabilities is in inverse lexicographic order. The interactions are returned in order of dimension, like e.g., 1, 2, 3, 12, 13, 23, 123.

Value

a list with slots

• C Matrix of constrasts (the first sum(lev)-length(r) elements are referred to univariate logits)

• M Marginalization matrix with elements 0 and 1

• G The design matrix for the corresponding log-linear model.

Author(s)

Francesco Bartolucci, Antonio Forcina. Giovanni M. Marchetti translated the Matlab function by Bartolucci and Forcina.

References

Bartolucci, F., Colombi, R. and Forcina, A. (2007). An extended class of marginal link functions for modelling contingency tables by equality and inequality constraints. Statistica Sinica, 17, 691-711.

See Also

mat.mlogit().

Examples

marg.param(c(3, 3), c("l", "g"))

Description

MarkEqMag determines whether two maximal ancestral graphs (MAGs) are Markov equivalent. The function checks whether the two graphs have the same skeleton and colliders with order.

Usage

MarkEqMag(amat, bmat)

Arguments

Details

@details Support for graphNEL objects requires the graph package from Bioconductor, which is a suggested dependency. If the package is missing, passing a graphNEL object will trigger an informative error.

Value

A character string: either "Markov Equivalent" or "NOT Markov Equivalent".

Author(s)

Kayvan Sadeghi

References

Ali, R.A., Richardson, T.S. and Spirtes, P. (2009) Markov equivalence for ancestral graphs. Annals of Statistics, 37(5B), 2808-2837.

See Also

MarkEqRcg(), msep()