This package offers a number of different functions for determining
global and generic identifiability of path diagrams / mixed
graphs and latent digraphs. The following sections highlight the primary ways in which the
package can be used. Much of the package's functionality can be accessed
through the wrapper function semID
.
To be able to implement the different algorithms described below we
created a MixedGraph class using the R.oo
package of
Bengtsson, H. (2003)The R.oo package - Object-Oriented Programming with References Using Standard R Code, Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), ISSN 1609-395X, Hornik, K.; Leisch, F. & Zeileis, A. (eds.) URL https://www.r-project.org/conferences/DSC-2003/Proceedings/Bengtsson.pdf
This class can make it much easier to represent a mixed graph and run experiments with them. For instance we can create a mixed graph, plot it, and test if there is a half-trek system between two sets of vertices very easily:
> # Mixed graphs are specified by their directed adjacency matrix L and
> # bidirected adjacency matrix O.
> library(SEMID)
> L = t(matrix(
+ c(0, 1, 0, 0, 0,
+ 0, 0, 0, 1, 1,
+ 0, 0, 0, 1, 0,
+ 0, 1, 0, 0, 1,
+ 0, 0, 0, 1, 0), 5, 5))
>
> O = t(matrix(
+ c(0, 0, 0, 0, 0,
+ 0, 0, 1, 0, 1,
+ 0, 0, 0, 1, 0,
+ 0, 0, 0, 0, 0,
+ 0, 0, 0, 0, 0), 5, 5)); O=O+t(O)
>
> # Create the mixed graph object corresponding to L and O
> g = MixedGraph(L, O)
>
> # Plot the mixed graph
> g$plot()
>
> # Test whether or not there is a half-trek system from the nodes
> # 1,2 to 3,4
> g$getHalfTrekSystem(c(1,2), c(3,4))
$systemExists
[1] TRUE
$activeFrom
[1] 1 2
See the documentation for the MixedGraph class ?MixedGraph
for more
information.
Drton, Foygel, and Sullivant (2011) showed that there exist if and
only if graphical conditions for testing whether or not the parameters
in a mixed graph are globally identifiable. This criterion can be
accessed through the function graphID.globalID
.
Drton, M., Foygel, R., and Sullivant, S. (2011) Global identifiability of linear structural equation models. Ann. Statist. 39(2): 865-886.
There still do not exist any 'if and only if' graphical conditions for testing whether or not certain parameters in a mixed graph are generically identifiable. There do, however, exist some necessary and some sufficient conditions which work for a large collection of graphs.
Until recently, criterions for generic identifiability, like the half-trek criterion of Foygel, Draisma, and Drton (2012), had to show that all edges incoming to a node where generically identifiable simultaenously and thus, if any single such edge incoming to a node was generically nonidentifiable, the criterion would fail. The recent work of Weihs, Robeva, Robinson, et al. (2017) develops new criteria that are able to identify subsets of edges coming into a node substantially improving upon prior methods at the cost of computational efficiency. We list both the older algorithms (available in prior versions of this package) and the newer algorithms below.
htcID
- implements the half-trek critierion ofFoygel, Rina; Draisma, Jan; Drton, Mathias. Half-trek criterion for generic identifiability of linear structural equation models. Ann. Statist. 40 (2012), no. 3, 1682--1713. doi:10.1214/12-AOS1012.
and is an updated version of the (deprecated) function
graphID.htcID
.
ancestralID
- implements the ancestor decomposition techniques ofDrton, M., and Weihs, L. (2016) Generic Identifiability of Linear Structural Equation Models by Ancestor Decomposition. Scand J Statist, 43: 1035–1045. doi: 10.1111/sjos.12227.
and is an updated version of the (deprecated) function
graphID.ancestralID
. This new version of the function works also on
cyclic graphs by using an updated version of the Tian decomposition.
edgewiseID
- an edgewise identification algorithm strictly
improving upon the half-trek criterion.
edgewiseTSID
- an edgewise identification algorithm leveraging
trek-separation relations to identify even more edges than
edgewiseID
, this has the downside being computationally expensive.
generalGenericID
- an generic identification algorithm template
that allows you to mix and match different identification techniques
to find the right balance between computational efficiency and
exhaustiveness of the procedure. See the below examples for more
details.
graphID.nonHtcID
- the best known test for whether any edges in a
mixed graph are generically non-identifiabile. This comes Foygel,
Draisma, and Drton (2012).Lets use a few of the above functions to check the generic identifiability of parameters in a mixed graph.
> library(SEMID)
> # Mixed graphs are specified by their directed adjacency matrix L and
> # bidirected adjacency matrix O.
> L = t(matrix(
+ c(0, 1, 1, 0, 0,
+ 0, 0, 1, 1, 1,
+ 0, 0, 0, 1, 0,
+ 0, 0, 0, 0, 1,
+ 0, 0, 0, 0, 0), 5, 5))
>
> O = t(matrix(
+ c(0, 0, 0, 1, 0,
+ 0, 0, 1, 0, 1,
+ 0, 0, 0, 0, 0,
+ 0, 0, 0, 0, 0,
+ 0, 0, 0, 0, 0), 5, 5)); O=O+t(O)
>
> # Create a mixed graph object
> graph = MixedGraph(L, O)
>
> # Without using decomposition techniques we can't identify all nodes
> # just using the half-trek criterion
> htcID(graph, tianDecompose = F)
Call: htcID(mixedGraph = graph, tianDecompose = F)
Mixed Graph Info.
# nodes: 5
# dir. edges: 7
# bi. edges: 3
Generic Identifiability Summary
# dir. edges shown gen. identifiable: 1
# bi. edges shown gen. identifiable: 0
Generically identifiable dir. edges:
1->2
Generically identifiable bi. edges:
None
>
> # The edgewiseTSID function can show that all edges are generically
> # identifiable without proprocessing with decomposition techniques
> edgewiseTSID(graph, tianDecompose = F)
Call: edgewiseTSID(mixedGraph = graph, tianDecompose = F)
Mixed Graph Info.
# nodes: 5
# dir. edges: 7
# bi. edges: 3
Generic Identifiability Summary
# dir. edges shown gen. identifiable: 7
# bi. edges shown gen. identifiable: 3
Generically identifiable dir. edges:
1->2, 1->3, 2->3, 2->4, 3->4, 2->5, 4->5
Generically identifiable bi. edges:
1<->4, 2<->3, 2<->5
>
> # The above shows that all edges in the graph are generically identifiable.
> # See the help of edgewiseTSID to find out more information about what
> # else is returned by edgewiseTSID.
Using the generalGenericId
method we can also mix and match
different identification strategies. Lets say we wanted to first try
to identify everything using the half-trek criterion but then, if
there are still things that cant be shown generically identifiable, we
want to use the edgewise criterion by limiting the edgesets it looks
at to be a small size. We can do this as follows:
> library(SEMID)
> # Lets first define some matrices for a mixed graph
> L = t(matrix(
+ c(0, 1, 0, 0, 0,
+ 0, 0, 0, 1, 1,
+ 0, 0, 0, 1, 0,
+ 0, 1, 0, 0, 1,
+ 0, 0, 0, 1, 0), 5, 5))
>
> O = t(matrix(
+ c(0, 0, 0, 0, 0,
+ 0, 0, 1, 0, 1,
+ 0, 0, 0, 1, 0,
+ 0, 0, 0, 0, 0,
+ 0, 0, 0, 0, 0), 5, 5)); O=O+t(O)
>
> # Create a mixed graph object
> graph = MixedGraph(L, O)
>
> # Now lets define an "identification step" function corresponding to
> # using the edgewise identification algorithm but with subsets
> # controlled by 1.
> restrictedEdgewiseIdentifyStep <- function(mixedGraph,
+ unsolvedParents,
+ solvedParents,
+ identifier) {
+ return(edgewiseIdentifyStep(mixedGraph, unsolvedParents,
+ solvedParents, identifier,
+ subsetSizeControl = 1))
+ }
>
> # Now we run an identification algorithm that iterates between the
> # htc and the "restricted" edgewise identification algorithm
> generalGenericID(graph, list(htcIdentifyStep,
+ restrictedEdgewiseIdentifyStep),
+ tianDecompose = F)
Call: generalGenericID(mixedGraph = graph, idStepFunctions = list(htcIdentifyStep,
restrictedEdgewiseIdentifyStep), tianDecompose = F)
Mixed Graph Info.
# nodes: 5
# dir. edges: 7
# bi. edges: 3
Generic Identifiability Summary
# dir. edges shown gen. identifiable: 2
# bi. edges shown gen. identifiable: 0
Generically identifiable dir. edges:
2->5, 4->5
Generically identifiable bi. edges:
None
>
> # We can do better (fewer unsolved parents) if we don't restrict the edgewise
> # identifier algorithm as much
> generalGenericID(graph, list(htcIdentifyStep, edgewiseIdentifyStep),
+ tianDecompose = F)
Mixed Graph Info.
# nodes: 5
# dir. edges: 7
# bi. edges: 3
Generic Identifiability Summary
# dir. edges shown gen. identifiable: 4
# bi. edges shown gen. identifiable: 0
Generically identifiable dir. edges:
2->4, 5->4, 2->5, 4->5
Generically identifiable bi. edges:
None
The latent-factor half-trek criterion (LF-HTC) by Barber, Drton, Sturma and Weihs (2022) is a sufficient criterion to check generic identifiability in directed graphical models with explicitly modeled latent variables. We created a LatentDigraph class to represent such graphs.
> # Latent digraphs are specified by their directed adjacency matrix L
> library(SEMID)
> L = matrix(c(0, 1, 0, 0, 0, 0,
+ 0, 0, 1, 0, 0, 0,
+ 0, 0, 0, 0, 0, 0,
+ 0, 0, 0, 0, 1, 0,
+ 0, 0, 0, 0, 0, 0,
+ 1, 1, 1, 1, 1, 0), 6, 6, byrow=TRUE)
> observedNodes = seq(1,5)
> latentNodes = c(6)
>
> # Create the latent digraph object corresponding to L
> g = LatentDigraph(L, observedNodes, latentNodes)
>
> # Plot latent digraph
> plot(g)
The function lfhtcID
implements the algorithm to check LF-HTC-identifiability as presented in
Barber, R. F., Drton, M., Sturma, N., and Weihs L. (2022). Half-Trek Criterion for Identifiability of Latent Variable Models. arXiv preprint arXiv:2201.04457.
The LF-HTC is applicable to all graphs where the latent nodes are source nodes.
> lfhtcID(g)
Call: lfhtcID(graph = g)
Latent Digraph Info
# observed nodes: 5
# latent nodes: 1
# total nr. of edges between observed nodes: 3
Generic Identifiability Summary
# nr. of edges between observed nodes shown gen. identifiable: 3
# gen. identifiable edges: 1->2, 2->3, 4->5
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