Identifiability of Linear Structural Equation Models.
SEMID Package
Purpose
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.
The MixedGraph class
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.
Global Identifiability
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 globalID.
Drton, M., Foygel, R., and Sullivant, S. (2011) Global identifiability of linear structural equation models. Ann. Statist. 39(2): 865-886.
Generic Identiability of Parameters
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.
Sufficient Conditions
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 of
Foygel, 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.
ancestralID- implements the ancestor decomposition techniques of
Drton, 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.
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 thanedgewiseID, 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.
Necessary Conditions
graphID.nonHtcID- the best known test for whether any edges in a mixed graph are generically non-identifiabile; see Foygel, Draisma, and Drton (2012).
Examples
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
Identifiability of latent factor graphs
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. Ann. Statist. 50(6):3174--3196. doi:10.1214/22-AOS2221.
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