fci: Estimate a PAG with the FCI Algorithm
In pcalg: Methods for Graphical Models and Causal Inference
fci R Documentation
Estimate a PAG with the FCI Algorithm
Description
Estimate a Partial Ancestral Graph (PAG) from observational data, using
the FCI (Fast Causal Inference) algorithm, or from a combination of
data from different (e.g., observational and interventional) contexts,
using the FCI-JCI (Joint Causal Inference) extension.
Usage
fci(suffStat, indepTest, alpha, labels, p,
skel.method = c("stable", "original", "stable.fast"),
type = c("normal", "anytime", "adaptive"),
fixedGaps = NULL, fixedEdges = NULL,
NAdelete = TRUE, m.max = Inf, pdsep.max = Inf,
rules = rep(TRUE, 10), doPdsep = TRUE, biCC = FALSE,
conservative = FALSE, maj.rule = FALSE,
numCores = 1, selectionBias = TRUE,
jci = c("0","1","12","123"), contextVars = NULL,
verbose = FALSE)
Arguments
suffStat
sufficient statistics: A named list
containing all necessary elements for the conditional independence
decisions in the function indepTest.
indepTest
a function for testing conditional
independence. The function is internally called as
indepTest(x,y, S, suffStat), and tests conditional independence
of x and y given S. Here, x and
y are variables, and S is a (possibly empty) vector of
variables (all variables are denoted by their column numbers
in the adjacency matrix). suffStat is a list with all
relevant information, see above. The return value of
indepTest() is the p-value of the test for conditional
independence.
alpha
numeric significance level (in (0, 1)) for the
individual conditional independence tests.
labels
(optional) character vector of variable (or
“node”) names. Typically preferred to specifying p.
p
(optional) number of variables (or nodes). May be specified
if labels are not, in which case labels is set to
1:p.
skel.method
character string specifying method; the default,
"stable" provides an order-independent skeleton, see
skeleton.
type
character string specifying the version of the FCI
algorithm to be used. By default, it is "normal", and so the
normal FCI algorithm is called. If set to "anytime", the
‘Anytime FCI’ is called and m.max needs to be specified.
If set to "adaptive", the ‘Adaptive Anytime FCI’ is
called and m.max is not used. For more information, see Details.
fixedGaps
logical matrix of dimension p*p. If entry
[i,j] or [j,i] (or both) are TRUE, the edge i-j is
removed before starting the algorithm. Therefore, this edge is
guaranteed to be absent in the resulting graph.
fixedEdges
logical matrix of dimension p*p. If entry
[i,j] or [j,i] (or both) are TRUE, the edge i-j is
never considered for removal. Therefore, this edge is
guaranteed to be present in the resulting graph.
NAdelete
If indepTest returns NA and this option is
TRUE, the corresponding edge is deleted. If this option is
FALSE, the edge is not deleted.
m.max
Maximum size of the conditioning sets that are considered in the
conditional independence tests.
pdsep.max
Maximum size of Possible-D-SEP for which subsets are
considered as conditioning sets in the conditional independence
tests. If the nodes x and y are adjacent in the graph
and the size of \textrm{Possible-D-SEP}(x) \setminus \{x, y\}
is
bigger than pdsep.max, the edge is simply left in the graph.
Note that if pdsep.max is less than Inf, the final PAG
may be a supergraph of the one computed with pdsep.max = Inf,
because fewer tests may have been performed in the former.
rules
Logical vector of length 10 indicating which rules
should be used when directing edges. The order of the rules is taken
from Zhang (2008).
doPdsep
If TRUE, Possible-D-SEP is computed for all nodes,
and all subsets of Possible-D-SEP are considered as conditioning
sets in the conditional independence tests, if not defined otherwise
in pdsep.max. If FALSE, Possible-D-SEP
is not computed, so that the algorithm simplifies to the
Modified PC algorithm of Spirtes, Glymour and Scheines (2000, p.84).
biCC
If TRUE, only nodes on paths between nodes x
and y are considered to be in Possible-D-SEP(x) when
testing independence between x and y. Uses
biconnected components, biConnComp from RBGL.
conservative
Logical indicating if the unshielded triples
should be checked for ambiguity the second time when v-structures
are determined. For more information, see details.
maj.rule
Logical indicating if the unshielded triples should
be checked for ambiguity the second time when v-structures are
determined using a majority rule idea, which is less strict than the
standard conservative. For more information, see details.
numCores
Specifies the number of cores to be used for parallel
estimation of skeleton.
selectionBias
If TRUE, allow for selection bias.
If FALSE, selection bias is excluded by assumption and hence
rules R5-R7, as in Zhang (2008), are disabled.
jci
String specifying the JCI assumptions that are used.
It can be one of:
- "0"
No JCI assumption is made (default),
- "1"
JCI assumption 1 (no system variable causes any context
variable),
- "12"
JCI assumptions 1 and 2 (no system variable causes any
context variable, and no system variable is confounded with any
context variable),
- "123"
JCI assumptions 1, 2 and 3 (no system variable causes
any context variable, no system variable is confounded with any
context variable, and all context variables are confounded but
are not direct causes of each other).
For more information, see Mooij et al. (2020).
contextVars
Subset of variable indices {1,...,p} that will be
treated as context variables in the JCI extension of FCI.
verbose
If true, more detailed output is provided.
Details
This function is a generalization of the PC algorithm (see pc),
in the sense that it allows arbitrarily many latent and selection variables.
Under the assumption that the data are faithful to a DAG that includes all
latent and selection variables, the FCI algorithm (Fast Causal Inference
algorithm) (Spirtes, Glymour and Scheines, 2000) estimates the Markov
equivalence class of MAGs that describe the conditional independence
relationships between the observed variables. Under the assumption that the
data are \sigma-faithful to a simple (possibly cyclic) SCM that allows
for latent confounding (but selection bias is absent), the FCI algorithm
estimates the \sigma-Markov equivalence class of the DMGs (directed
mixed graphs) that describe the causal relations between the observed
variables and the conditional independence relationships in the observed
distribution through \sigma-separation (Mooij and Claassen, 2020). The
FCI-JCI (Joint Causal Inference) extension allows the algorithm to combine
data from different contexts, for example, observational and different types
of interventional data (Mooij et al., 2020).
FCI estimates a partial ancestral graph (PAG). The PAG
represents a Markov equivalence class of DAGs with latent and selection
variables in the acyclic case (Zhang, 2008), and a Markov equivalence class
of directed graphs with latent variables (but without selection variables)
in the cyclic \sigma-separation case.
A PAG contains the following types of edges: o-o, o–, o->, –>, <->, —.
The bidirected edges come from latent confounders,
and the undirected edges come from latent selection variables. The edges have
the following interpretation: (i) there is an edge between x
and y if and only if variables x and y are conditionally dependent
given S for all sets S consisting of all selection variables
and a subset of the observed variables (assuming the Markov property and
faithfulness); (ii) a tail x --* y at x on an edge between x and y means that
x is an ancestor of y or S; (iii) an arrowhead x <-* y at x on an edge between
x and y means that x is not an ancestor of y, nor of S;
(iv) a circle mark x o-* y at x on an edge between x and y means that there
exists both a graph in the Markov equivalence class where x is ancestor of
y or S, and one where x is not ancestor of y, nor of S. For further information
on the interpretation of PAGs see e.g. (Zhang, 2008) and (Mooij and Claassen, 2020).
The first part of the FCI algorithm is analogous to the PC algorithm. It
starts with a complete undirected graph and estimates an initial skeleton
using skeleton(*, method="stable") which produces an
initial order-independent skeleton, see skeleton for
more details. All edges of this skeleton are of
the form o-o. Due to the presence of hidden variables, it is no longer
sufficient to consider only subsets of the neighborhoods of nodes x
and y to decide whether the edge x o-o y should be removed.
Therefore, the initial skeleton may contain some superfluous edges.
These edges are removed in the next step of the algorithm which
requires some orientations. Therefore, the v-structures
are determined using the conservative method (see discussion on
conservative below).
After the v-structures have been oriented, Possible-D-SEP sets for each
node in the graph are computed at once. To decide whether edge
x o-o y should be removed, one performs conditional indepedence
tests of x and y given all possible subsets of Possible-D-SEP(x) and
of Possible-D-SEP(y). The edge is removed if a conditional
independence is found. This produces a fully order-independent final
skeleton as explained in Colombo and Maathuis (2014). Subsequently,
the v-structures are newly determined on the final skeleton (using
information in sepset). Finally, as many as possible undetermined edge
marks (o) are determined using (a subset of) the 10 orientation rules
given by Zhang (2008).
The “Anytime FCI” algorithm was introduced by Spirtes (2001). It
can be viewed as a modification of the FCI algorithm that only performs
conditional independence tests up to and including order m.max when
finding the initial skeleton, using the function skeleton, and
the final skeleton, using the function pdsep. Thus, Anytime FCI
performs fewer conditional independence tests than FCI. To use the
Anytime algorithm, one sets type = "anytime" and needs to
specify m.max, the maximum size of the conditioning sets.
The “Adaptive Anytime FCI” algorithm was introduced by Colombo
et. al (2012). The first part of the algorithm is identical to the normal
FCI described above. But in the second part when the final skeleton is
estimated using the function pdsep, the Adaptive Anytime
FCI algorithm only performs conditional independence tests up to and
including order m.max, where m.max is the maximum size of the
conditioning sets that were considered to determine the initial
skeleton using the function skeleton. Thus, m.max is chosen
adaptively and does not have to be specified by the user.
Conservative versions of FCI, Anytime FCI, and Adaptive Anytime FCI
are computed if conservative = TRUE is specified. After the
final skeleton is computed, all potential
v-structures a-b-c are checked in the following way. We test whether a
and c are independent conditioning on any subset of the neighbors of a
or any subset of the neighbors of c. When a subset makes a and c
conditionally independent, we call it a separating set. If b is in no
such separating set or in all such separating sets, no further action
is taken and the normal version of the FCI, Anytime FCI, or Adaptive
Anytime FCI algorithm is continued. If, however, b is in only some
separating sets, the triple a-b-c is marked ‘ambiguous’. If a is
independent of c given some S in the skeleton (i.e., the edge a-c
dropped out), but a and c remain dependent given all subsets of
neighbors of either a or c, we will call all triples a-b-c
‘unambiguous’. This is because in the FCI algorithm, the true separating set
might be outside the neighborhood of either a or c. An ambiguous
triple is not oriented as a v-structure. Furthermore, no further
orientation rule that needs to know whether a-b-c is a v-structure or
not is applied. Instead of using the conservative version, which is
quite strict towards the v-structures, Colombo and Maathuis (2014)
introduced a less strict version for the v-structures called majority
rule. This adaptation can be called using maj.rule = TRUE. In
this case, the triple a-b-c is marked as ‘ambiguous’ if and only if b
is in exactly 50 percent of such separating sets or no separating set
was found. If b is in less than 50 percent of the separating sets it
is set as a v-structure, and if in more than 50 percent it is set as a
non v-structure (for more details see Colombo and Maathuis,
2014). Colombo and Maathuis (2014) showed that with both these
modifications, the final skeleton and the decisions about the
v-structures of the FCI algorithm are fully order-independent.
Note that the order-dependence issues on the 10 orientation rules are
still present, see Colombo and Maathuis (2014) for more details.
The FCI-JCI extension of FCI was introduced by Mooij et. al (2020). It is an
implementation of the Joint Causal Inference (JCI) framework that
reduces causal discovery from several data sets corresponding to different
contexts (e.g., observational and different interventional settings, or data
measured in different labs or in different countries) to causal discovery
from the pooled data (treated as a single 'observational' data set). Two
types of variables are distinguished in the JCI framework: system variables
(describing aspects of the system in some environment) and context variables
(describing aspects of the environment of the system). Different
assumptions regarding the causal relations between context variables and
system variables can be made. The most common assumption (JCI Assumption 1:
'exogeneity') is that no system variable can affect any context variable.
The second, less common, assumption (JCI Assumption 2: 'complete randomized
context') is that there is no latent confounding between system and context.
The third assumption (JCI Assumption 3: 'generic context model') is in fact
a faithfulness assumption on the context distribution. The FCI-JCI extension
can be used by specifying the subset of the variables that are designated as
context variables with the contextVars argument, and by specifying
the combination of JCI assumptions that is used with the jci argument.
For the default values of these arguments, the JCI extension is not used.
The only difference between the FCI-JCI extension and the standard FCI
algorithm is that the background knowledge about the PAG implied by the
JCI assumptions is exploited at several points in the FCI-JCI algorithm to
enforce adjacencies between context variables (in case of JCI Assumption 3)
and to orient certain edges adjacent to context variables (in case of JCI
Assumptions 1, 2 or 3).
The current JCI framework assumes that no selection bias is present, and
therefore the FCI-JCI extension should be called with
selectionBias = FALSE. For more details on FCI-JCI, and the general
JCI framework, see Mooij et al. (2020).
Value
An object of class fciAlgo (see
fciAlgo) containing the estimated graph
(in the form of an adjacency matrix with various possible edge marks),
the conditioning sets that lead to edge removals (sepset) and several other
parameters.
Author(s)
Diego Colombo, Markus Kalisch (kalisch@stat.math.ethz.ch)
and Joris Mooij.
References
D. Colombo and M.H. Maathuis (2014). Order-independent
constraint-based causal structure learning. Journal of Machine
Learning Research 15 3741-3782.
D. Colombo, M. H. Maathuis, M. Kalisch, T. S. Richardson
(2012). Learning high-dimensional directed acyclic graphs with latent
and selection variables. Ann. Statist. 40, 294-321.
M.Kalisch, M. Maechler, D. Colombo, M. H. Maathuis,
P. Buehlmann (2012). Causal Inference Using Graphical Models with
the R Package pcalg. Journal of Statistical Software
47(11) 1–26, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v047.i11")}.
J. M. Mooij, S. Magliacane, T. Claassen (2020).
Joint Causal Inference from Multiple Contexts.
Journal of Machine Learning Research 21(99), 1-108.
J. M. Mooij and T. Claassen (2020).
Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles.
In Proc. of the 36th Conference on Uncertainty in Artificial Intelligence (UAI-20), 1159-1168.
P. Spirtes (2001). An anytime algorithm for causal inference. In
Proc. of the Eighth International Workshop on Artificial Intelligence
and Statistics 213-221. Morgan Kaufmann, San Francisco.
P. Spirtes, C. Glymour and R. Scheines (2000).
Causation, Prediction, and Search, 2nd edition, MIT Press,
Cambridge (MA).
P. Spirtes, C. Meek, T.S. Richardson (1999). In: Computation,
Causation and Discovery. An algorithm for causal
inference in the presence of latent variables and selection bias.
Pages 211-252. MIT Press.
T.S. Richardson and P. Spirtes (2002). Ancestral graph Markov models.
Annals of Statistics 30 962-1030.
J. Zhang (2008). On the completeness of orientation rules for
causal discovery in the presence of latent confounders and selection bias.
Artificial Intelligence 172 1873-1896.
See Also
fciPlus for a more efficient variation of FCI;
skeleton for estimating a skeleton
using the PC algorithm; pc for estimating a CPDAG using
the PC algorithm; pdsep for computing
Possible-D-SEP for each node and testing and adapting the graph
accordingly; qreach for a fast way of finding Possible-D-SEP
for a given node.
gaussCItest, disCItest,
binCItest, dsepTest and dsepAMTest
as examples for indepTest.
Examples
##################################################
## Example without latent variables
##################################################
set.seed(42)
p <- 7
## generate and draw random DAG :
myDAG <- randomDAG(p, prob = 0.4)
## find skeleton and PAG using the FCI algorithm
suffStat <- list(C = cov2cor(trueCov(myDAG)), n = 10^9)
res <- fci(suffStat, indepTest=gaussCItest,
alpha = 0.9999, p=p, doPdsep = FALSE)
##################################################
## Example with hidden variables
## Zhang (2008), Fig. 6, p.1882
##################################################
## create the graph g
p <- 4
L <- 1 # '1' is latent
V <- c("Ghost", "Max","Urs","Anna","Eva")
edL <- setNames(vector("list", length=length(V)), V)
edL[[1]] <- list(edges=c(2,4),weights=c(1,1))
edL[[2]] <- list(edges=3,weights=c(1))
edL[[3]] <- list(edges=5,weights=c(1))
edL[[4]] <- list(edges=5,weights=c(1))
g <- new("graphNEL", nodes=V, edgeL=edL, edgemode="directed")
## compute the true covariance matrix of g
cov.mat <- trueCov(g)
## delete rows and columns belonging to latent variable L
true.cov <- cov.mat[-L,-L]
## transform covariance matrix into a correlation matrix
true.corr <- cov2cor(true.cov)
## The same, for the following three examples
indepTest <- gaussCItest
suffStat <- list(C = true.corr, n = 10^9)
## find PAG with FCI algorithm.
## As dependence "oracle", we use the true correlation matrix in
## gaussCItest() with a large "virtual sample size" and a large alpha:
normal.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L],
verbose=TRUE)
## find PAG with Anytime FCI algorithm with m.max = 1
## This means that only conditioning sets of size 0 and 1 are considered.
## As dependence "oracle", we use the true correlation matrix in the
## function gaussCItest with a large "virtual sample size" and a large
## alpha
anytime.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L],
type = "anytime", m.max = 1,
verbose=TRUE)
## find PAG with Adaptive Anytime FCI algorithm.
## This means that only conditining sets up to size K are considered
## in estimating the final skeleton, where K is the maximal size of a
## conditioning set found while estimating the initial skeleton.
## As dependence "oracle", we use the true correlation matrix in the
## function gaussCItest with a large "virtual sample size" and a large
## alpha
adaptive.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L],
type = "adaptive",
verbose=TRUE)
## define PAG given in Zhang (2008), Fig. 6, p.1882
corr.pag <- rbind(c(0,1,1,0),
c(1,0,0,2),
c(1,0,0,2),
c(0,3,3,0))
## check if estimated and correct PAG are in agreement
all(corr.pag == normal.pag @ amat) # TRUE
all(corr.pag == anytime.pag @ amat) # FALSE
all(corr.pag == adaptive.pag@ amat) # TRUE
ij <- rbind(cbind(1:4,1:4), c(2,3), c(3,2))
all(corr.pag[ij] == anytime.pag @ amat[ij]) # TRUE
##################################################
## Joint Causal Inference Example
## Mooij et al. (2020), Fig. 43(a), p. 97
##################################################
# Encode MAG as adjacency matrix
p <- 8 # total number of variables
V <- c("Ca","Cb","Cc","X0","X1","X2","X3","X4") # 3 context variables, 5 system variables
# amat[i,j] = 0 iff no edge btw i,j
# amat[i,j] = 1 iff i *-o j
# amat[i,j] = 2 iff i *-> j
# amat[i,j] = 3 iff i *-- j
amat <- rbind(c(0,2,2,2,0,0,0,0),
c(2,0,2,0,2,0,0,0),
c(2,2,0,0,2,2,0,0),
c(3,0,0,0,0,0,2,0),
c(0,3,3,0,0,3,0,2),
c(0,0,3,0,2,0,0,0),
c(0,0,0,3,0,0,0,2),
c(0,0,0,0,2,0,3,0))
rownames(amat)<-V
colnames(amat)<-V
# Make use of d-separation oracle as "independence test"
indepTest <- dsepAMTest
suffStat<-list(g=amat,verbose=FALSE)
# Derive PAG that represents the Markov equivalence class of the MAG with the FCI algorithm
# (assuming no selection bias)
fci.pag <- fci(suffStat, indepTest, alpha = 0.5, labels = V,
verbose=TRUE, selectionBias=FALSE)
# Derive PAG with FCI-JCI, the Joint Causal Inference extension of FCI
# (assuming no selection bias, and all three JCI assumptions)
fcijci.pag <- fci(suffStat, indepTest, alpha = 0.5, labels = V,
verbose=TRUE, contextVars=c(1,2,3), jci="123", selectionBias=FALSE)
# Report results
cat('True MAG:\n')
print(amat)
cat('PAG output by FCI:\n')
print(fci.pag@amat)
cat('PAG output by FCI-JCI:\n')
print(fcijci.pag@amat)
# Read off causal features from the FCI PAG
cat('Identified absence (-1) and presence (+1) of ancestral causal relations from FCI PAG:\n')
print(pag2anc(fci.pag@amat))
cat('Identified absence (-1) and presence (+1) of direct causal relations from FCI PAG:\n')
print(pag2edge(fci.pag@amat))
cat('Identified absence (-1) and presence (+1) of pairwise latent confounding from FCI PAG:\n')
print(pag2conf(fci.pag@amat))
# Read off causal features from the FCI-JCI PAG
cat('Identified absence (-1) and presence (+1) of ancestral causal relations from FCI-JCI PAG:\n')
print(pag2anc(fcijci.pag@amat))
cat('Identified absence (-1) and presence (+1) of direct causal relations from FCI-JCI PAG:\n')
print(pag2edge(fcijci.pag@amat))
cat('Identified absence (-1) and presence (+1) of pairwise latent confounding from FCI-JCI PAG:\n')
print(pag2conf(fcijci.pag@amat))
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| fci | R Documentation |
Estimate a PAG with the FCI Algorithm
Description
Estimate a Partial Ancestral Graph (PAG) from observational data, using the FCI (Fast Causal Inference) algorithm, or from a combination of data from different (e.g., observational and interventional) contexts, using the FCI-JCI (Joint Causal Inference) extension.
Usage
fci(suffStat, indepTest, alpha, labels, p,
skel.method = c("stable", "original", "stable.fast"),
type = c("normal", "anytime", "adaptive"),
fixedGaps = NULL, fixedEdges = NULL,
NAdelete = TRUE, m.max = Inf, pdsep.max = Inf,
rules = rep(TRUE, 10), doPdsep = TRUE, biCC = FALSE,
conservative = FALSE, maj.rule = FALSE,
numCores = 1, selectionBias = TRUE,
jci = c("0","1","12","123"), contextVars = NULL,
verbose = FALSE)
Arguments
suffStat |
sufficient statistics: A named |
indepTest |
a |
alpha |
numeric significance level (in |
labels |
(optional) |
p |
(optional) number of variables (or nodes). May be specified
if |
skel.method |
character string specifying method; the default,
|
type |
character string specifying the version of the FCI
algorithm to be used. By default, it is |
fixedGaps |
|
fixedEdges |
logical matrix of dimension p*p. If entry
|
NAdelete |
If indepTest returns |
m.max |
Maximum size of the conditioning sets that are considered in the conditional independence tests. |
pdsep.max |
Maximum size of Possible-D-SEP for which subsets are
considered as conditioning sets in the conditional independence
tests. If the nodes |
rules |
Logical vector of length 10 indicating which rules should be used when directing edges. The order of the rules is taken from Zhang (2008). |
doPdsep |
If |
biCC |
If |
conservative |
Logical indicating if the unshielded triples should be checked for ambiguity the second time when v-structures are determined. For more information, see details. |
maj.rule |
Logical indicating if the unshielded triples should be checked for ambiguity the second time when v-structures are determined using a majority rule idea, which is less strict than the standard conservative. For more information, see details. |
numCores |
Specifies the number of cores to be used for parallel
estimation of |
selectionBias |
If |
jci |
String specifying the JCI assumptions that are used. It can be one of:
For more information, see Mooij et al. (2020). |
contextVars |
Subset of variable indices {1,...,p} that will be treated as context variables in the JCI extension of FCI. |
verbose |
If true, more detailed output is provided. |
Details
This function is a generalization of the PC algorithm (see pc),
in the sense that it allows arbitrarily many latent and selection variables.
Under the assumption that the data are faithful to a DAG that includes all
latent and selection variables, the FCI algorithm (Fast Causal Inference
algorithm) (Spirtes, Glymour and Scheines, 2000) estimates the Markov
equivalence class of MAGs that describe the conditional independence
relationships between the observed variables. Under the assumption that the
data are \sigma-faithful to a simple (possibly cyclic) SCM that allows
for latent confounding (but selection bias is absent), the FCI algorithm
estimates the \sigma-Markov equivalence class of the DMGs (directed
mixed graphs) that describe the causal relations between the observed
variables and the conditional independence relationships in the observed
distribution through \sigma-separation (Mooij and Claassen, 2020). The
FCI-JCI (Joint Causal Inference) extension allows the algorithm to combine
data from different contexts, for example, observational and different types
of interventional data (Mooij et al., 2020).
FCI estimates a partial ancestral graph (PAG). The PAG
represents a Markov equivalence class of DAGs with latent and selection
variables in the acyclic case (Zhang, 2008), and a Markov equivalence class
of directed graphs with latent variables (but without selection variables)
in the cyclic \sigma-separation case.
A PAG contains the following types of edges: o-o, o–, o->, –>, <->, —.
The bidirected edges come from latent confounders,
and the undirected edges come from latent selection variables. The edges have
the following interpretation: (i) there is an edge between x
and y if and only if variables x and y are conditionally dependent
given S for all sets S consisting of all selection variables
and a subset of the observed variables (assuming the Markov property and
faithfulness); (ii) a tail x --* y at x on an edge between x and y means that
x is an ancestor of y or S; (iii) an arrowhead x <-* y at x on an edge between
x and y means that x is not an ancestor of y, nor of S;
(iv) a circle mark x o-* y at x on an edge between x and y means that there
exists both a graph in the Markov equivalence class where x is ancestor of
y or S, and one where x is not ancestor of y, nor of S. For further information
on the interpretation of PAGs see e.g. (Zhang, 2008) and (Mooij and Claassen, 2020).
The first part of the FCI algorithm is analogous to the PC algorithm. It
starts with a complete undirected graph and estimates an initial skeleton
using skeleton(*, method="stable") which produces an
initial order-independent skeleton, see skeleton for
more details. All edges of this skeleton are of
the form o-o. Due to the presence of hidden variables, it is no longer
sufficient to consider only subsets of the neighborhoods of nodes x
and y to decide whether the edge x o-o y should be removed.
Therefore, the initial skeleton may contain some superfluous edges.
These edges are removed in the next step of the algorithm which
requires some orientations. Therefore, the v-structures
are determined using the conservative method (see discussion on
conservative below).
After the v-structures have been oriented, Possible-D-SEP sets for each
node in the graph are computed at once. To decide whether edge
x o-o y should be removed, one performs conditional indepedence
tests of x and y given all possible subsets of Possible-D-SEP(x) and
of Possible-D-SEP(y). The edge is removed if a conditional
independence is found. This produces a fully order-independent final
skeleton as explained in Colombo and Maathuis (2014). Subsequently,
the v-structures are newly determined on the final skeleton (using
information in sepset). Finally, as many as possible undetermined edge
marks (o) are determined using (a subset of) the 10 orientation rules
given by Zhang (2008).
The “Anytime FCI” algorithm was introduced by Spirtes (2001). It
can be viewed as a modification of the FCI algorithm that only performs
conditional independence tests up to and including order m.max when
finding the initial skeleton, using the function skeleton, and
the final skeleton, using the function pdsep. Thus, Anytime FCI
performs fewer conditional independence tests than FCI. To use the
Anytime algorithm, one sets type = "anytime" and needs to
specify m.max, the maximum size of the conditioning sets.
The “Adaptive Anytime FCI” algorithm was introduced by Colombo
et. al (2012). The first part of the algorithm is identical to the normal
FCI described above. But in the second part when the final skeleton is
estimated using the function pdsep, the Adaptive Anytime
FCI algorithm only performs conditional independence tests up to and
including order m.max, where m.max is the maximum size of the
conditioning sets that were considered to determine the initial
skeleton using the function skeleton. Thus, m.max is chosen
adaptively and does not have to be specified by the user.
Conservative versions of FCI, Anytime FCI, and Adaptive Anytime FCI
are computed if conservative = TRUE is specified. After the
final skeleton is computed, all potential
v-structures a-b-c are checked in the following way. We test whether a
and c are independent conditioning on any subset of the neighbors of a
or any subset of the neighbors of c. When a subset makes a and c
conditionally independent, we call it a separating set. If b is in no
such separating set or in all such separating sets, no further action
is taken and the normal version of the FCI, Anytime FCI, or Adaptive
Anytime FCI algorithm is continued. If, however, b is in only some
separating sets, the triple a-b-c is marked ‘ambiguous’. If a is
independent of c given some S in the skeleton (i.e., the edge a-c
dropped out), but a and c remain dependent given all subsets of
neighbors of either a or c, we will call all triples a-b-c
‘unambiguous’. This is because in the FCI algorithm, the true separating set
might be outside the neighborhood of either a or c. An ambiguous
triple is not oriented as a v-structure. Furthermore, no further
orientation rule that needs to know whether a-b-c is a v-structure or
not is applied. Instead of using the conservative version, which is
quite strict towards the v-structures, Colombo and Maathuis (2014)
introduced a less strict version for the v-structures called majority
rule. This adaptation can be called using maj.rule = TRUE. In
this case, the triple a-b-c is marked as ‘ambiguous’ if and only if b
is in exactly 50 percent of such separating sets or no separating set
was found. If b is in less than 50 percent of the separating sets it
is set as a v-structure, and if in more than 50 percent it is set as a
non v-structure (for more details see Colombo and Maathuis,
2014). Colombo and Maathuis (2014) showed that with both these
modifications, the final skeleton and the decisions about the
v-structures of the FCI algorithm are fully order-independent.
Note that the order-dependence issues on the 10 orientation rules are
still present, see Colombo and Maathuis (2014) for more details.
The FCI-JCI extension of FCI was introduced by Mooij et. al (2020). It is an
implementation of the Joint Causal Inference (JCI) framework that
reduces causal discovery from several data sets corresponding to different
contexts (e.g., observational and different interventional settings, or data
measured in different labs or in different countries) to causal discovery
from the pooled data (treated as a single 'observational' data set). Two
types of variables are distinguished in the JCI framework: system variables
(describing aspects of the system in some environment) and context variables
(describing aspects of the environment of the system). Different
assumptions regarding the causal relations between context variables and
system variables can be made. The most common assumption (JCI Assumption 1:
'exogeneity') is that no system variable can affect any context variable.
The second, less common, assumption (JCI Assumption 2: 'complete randomized
context') is that there is no latent confounding between system and context.
The third assumption (JCI Assumption 3: 'generic context model') is in fact
a faithfulness assumption on the context distribution. The FCI-JCI extension
can be used by specifying the subset of the variables that are designated as
context variables with the contextVars argument, and by specifying
the combination of JCI assumptions that is used with the jci argument.
For the default values of these arguments, the JCI extension is not used.
The only difference between the FCI-JCI extension and the standard FCI
algorithm is that the background knowledge about the PAG implied by the
JCI assumptions is exploited at several points in the FCI-JCI algorithm to
enforce adjacencies between context variables (in case of JCI Assumption 3)
and to orient certain edges adjacent to context variables (in case of JCI
Assumptions 1, 2 or 3).
The current JCI framework assumes that no selection bias is present, and
therefore the FCI-JCI extension should be called with
selectionBias = FALSE. For more details on FCI-JCI, and the general
JCI framework, see Mooij et al. (2020).
Value
An object of class fciAlgo (see
fciAlgo) containing the estimated graph
(in the form of an adjacency matrix with various possible edge marks),
the conditioning sets that lead to edge removals (sepset) and several other
parameters.
Author(s)
Diego Colombo, Markus Kalisch (kalisch@stat.math.ethz.ch) and Joris Mooij.
References
D. Colombo and M.H. Maathuis (2014). Order-independent constraint-based causal structure learning. Journal of Machine Learning Research 15 3741-3782.
D. Colombo, M. H. Maathuis, M. Kalisch, T. S. Richardson (2012). Learning high-dimensional directed acyclic graphs with latent and selection variables. Ann. Statist. 40, 294-321.
M.Kalisch, M. Maechler, D. Colombo, M. H. Maathuis, P. Buehlmann (2012). Causal Inference Using Graphical Models with the R Package pcalg. Journal of Statistical Software 47(11) 1–26, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v047.i11")}.
J. M. Mooij, S. Magliacane, T. Claassen (2020). Joint Causal Inference from Multiple Contexts. Journal of Machine Learning Research 21(99), 1-108.
J. M. Mooij and T. Claassen (2020). Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles. In Proc. of the 36th Conference on Uncertainty in Artificial Intelligence (UAI-20), 1159-1168.
P. Spirtes (2001). An anytime algorithm for causal inference. In Proc. of the Eighth International Workshop on Artificial Intelligence and Statistics 213-221. Morgan Kaufmann, San Francisco.
P. Spirtes, C. Glymour and R. Scheines (2000). Causation, Prediction, and Search, 2nd edition, MIT Press, Cambridge (MA).
P. Spirtes, C. Meek, T.S. Richardson (1999). In: Computation, Causation and Discovery. An algorithm for causal inference in the presence of latent variables and selection bias. Pages 211-252. MIT Press.
T.S. Richardson and P. Spirtes (2002). Ancestral graph Markov models. Annals of Statistics 30 962-1030.
J. Zhang (2008). On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias. Artificial Intelligence 172 1873-1896.
See Also
fciPlus for a more efficient variation of FCI;
skeleton for estimating a skeleton
using the PC algorithm; pc for estimating a CPDAG using
the PC algorithm; pdsep for computing
Possible-D-SEP for each node and testing and adapting the graph
accordingly; qreach for a fast way of finding Possible-D-SEP
for a given node.
gaussCItest, disCItest,
binCItest, dsepTest and dsepAMTest
as examples for indepTest.
Examples
##################################################
## Example without latent variables
##################################################
set.seed(42)
p <- 7
## generate and draw random DAG :
myDAG <- randomDAG(p, prob = 0.4)
## find skeleton and PAG using the FCI algorithm
suffStat <- list(C = cov2cor(trueCov(myDAG)), n = 10^9)
res <- fci(suffStat, indepTest=gaussCItest,
alpha = 0.9999, p=p, doPdsep = FALSE)
##################################################
## Example with hidden variables
## Zhang (2008), Fig. 6, p.1882
##################################################
## create the graph g
p <- 4
L <- 1 # '1' is latent
V <- c("Ghost", "Max","Urs","Anna","Eva")
edL <- setNames(vector("list", length=length(V)), V)
edL[[1]] <- list(edges=c(2,4),weights=c(1,1))
edL[[2]] <- list(edges=3,weights=c(1))
edL[[3]] <- list(edges=5,weights=c(1))
edL[[4]] <- list(edges=5,weights=c(1))
g <- new("graphNEL", nodes=V, edgeL=edL, edgemode="directed")
## compute the true covariance matrix of g
cov.mat <- trueCov(g)
## delete rows and columns belonging to latent variable L
true.cov <- cov.mat[-L,-L]
## transform covariance matrix into a correlation matrix
true.corr <- cov2cor(true.cov)
## The same, for the following three examples
indepTest <- gaussCItest
suffStat <- list(C = true.corr, n = 10^9)
## find PAG with FCI algorithm.
## As dependence "oracle", we use the true correlation matrix in
## gaussCItest() with a large "virtual sample size" and a large alpha:
normal.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L],
verbose=TRUE)
## find PAG with Anytime FCI algorithm with m.max = 1
## This means that only conditioning sets of size 0 and 1 are considered.
## As dependence "oracle", we use the true correlation matrix in the
## function gaussCItest with a large "virtual sample size" and a large
## alpha
anytime.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L],
type = "anytime", m.max = 1,
verbose=TRUE)
## find PAG with Adaptive Anytime FCI algorithm.
## This means that only conditining sets up to size K are considered
## in estimating the final skeleton, where K is the maximal size of a
## conditioning set found while estimating the initial skeleton.
## As dependence "oracle", we use the true correlation matrix in the
## function gaussCItest with a large "virtual sample size" and a large
## alpha
adaptive.pag <- fci(suffStat, indepTest, alpha = 0.9999, labels = V[-L],
type = "adaptive",
verbose=TRUE)
## define PAG given in Zhang (2008), Fig. 6, p.1882
corr.pag <- rbind(c(0,1,1,0),
c(1,0,0,2),
c(1,0,0,2),
c(0,3,3,0))
## check if estimated and correct PAG are in agreement
all(corr.pag == normal.pag @ amat) # TRUE
all(corr.pag == anytime.pag @ amat) # FALSE
all(corr.pag == adaptive.pag@ amat) # TRUE
ij <- rbind(cbind(1:4,1:4), c(2,3), c(3,2))
all(corr.pag[ij] == anytime.pag @ amat[ij]) # TRUE
##################################################
## Joint Causal Inference Example
## Mooij et al. (2020), Fig. 43(a), p. 97
##################################################
# Encode MAG as adjacency matrix
p <- 8 # total number of variables
V <- c("Ca","Cb","Cc","X0","X1","X2","X3","X4") # 3 context variables, 5 system variables
# amat[i,j] = 0 iff no edge btw i,j
# amat[i,j] = 1 iff i *-o j
# amat[i,j] = 2 iff i *-> j
# amat[i,j] = 3 iff i *-- j
amat <- rbind(c(0,2,2,2,0,0,0,0),
c(2,0,2,0,2,0,0,0),
c(2,2,0,0,2,2,0,0),
c(3,0,0,0,0,0,2,0),
c(0,3,3,0,0,3,0,2),
c(0,0,3,0,2,0,0,0),
c(0,0,0,3,0,0,0,2),
c(0,0,0,0,2,0,3,0))
rownames(amat)<-V
colnames(amat)<-V
# Make use of d-separation oracle as "independence test"
indepTest <- dsepAMTest
suffStat<-list(g=amat,verbose=FALSE)
# Derive PAG that represents the Markov equivalence class of the MAG with the FCI algorithm
# (assuming no selection bias)
fci.pag <- fci(suffStat, indepTest, alpha = 0.5, labels = V,
verbose=TRUE, selectionBias=FALSE)
# Derive PAG with FCI-JCI, the Joint Causal Inference extension of FCI
# (assuming no selection bias, and all three JCI assumptions)
fcijci.pag <- fci(suffStat, indepTest, alpha = 0.5, labels = V,
verbose=TRUE, contextVars=c(1,2,3), jci="123", selectionBias=FALSE)
# Report results
cat('True MAG:\n')
print(amat)
cat('PAG output by FCI:\n')
print(fci.pag@amat)
cat('PAG output by FCI-JCI:\n')
print(fcijci.pag@amat)
# Read off causal features from the FCI PAG
cat('Identified absence (-1) and presence (+1) of ancestral causal relations from FCI PAG:\n')
print(pag2anc(fci.pag@amat))
cat('Identified absence (-1) and presence (+1) of direct causal relations from FCI PAG:\n')
print(pag2edge(fci.pag@amat))
cat('Identified absence (-1) and presence (+1) of pairwise latent confounding from FCI PAG:\n')
print(pag2conf(fci.pag@amat))
# Read off causal features from the FCI-JCI PAG
cat('Identified absence (-1) and presence (+1) of ancestral causal relations from FCI-JCI PAG:\n')
print(pag2anc(fcijci.pag@amat))
cat('Identified absence (-1) and presence (+1) of direct causal relations from FCI-JCI PAG:\n')
print(pag2edge(fcijci.pag@amat))
cat('Identified absence (-1) and presence (+1) of pairwise latent confounding from FCI-JCI PAG:\n')
print(pag2conf(fcijci.pag@amat))
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