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Tetrad Single HTML Manual
Tetrad Manual
Last updated: 2/6/2024
Table of Contents
Introduction
Tetrad is a suite of software for the discovery, estimation, and
simulation of causal models. Some of the functions that you can perform
with Tetrad include, but are not limited to:
- Loading an existing data set, restricting
potential models using your a-priori causal knowledge, and searching for
a model that explains it using one of Tetrad’s causal search algorithms
- Loading an existing causal graph and existing data set, and
estimating a parameterized model from them
- Creating a new causal
graph, parameterizing a model from it, and simulating data from that
model
Tetrad allows for numerous types of data, graph, and model to be
input and output, and some functions may be restricted based on what
types of data or graph the user inputs. Other functions may simply not
perform as well on certain types of data.
All analysis in Tetrad is performed graphically using a box
paradigm, found in a sidebar to the left of the workspace. A box either
houses an object such as a graph or a dataset, or performs an operation
such as a search or an estimation. Some boxes require input from other
boxes in order to work. Complex operations are performed by stringing
chains of boxes together in the workspace. For instance, to simulate
data, you would input a graph box into a parametric model box, the PM box
into an instantiated model box, and finally the IM box into a simulation
box.
In order to use a box, click on it in the sidebar, then click
inside the workspace. This creates an empty box, which you can
be instantiated by double-clicking. Most boxes have multiple options
available on instantiation, which will be explained in further detail in
this manual.
In order to use one box as input to another, draw an arrow between
them by clicking on the arrow tool in the sidebar, and clicking and
dragging from the first box to the second in the workspace.
Starting 1/14/2024, we will compile Tetrad under JDK 17 and use language level 17.
Tetrad may be cited using the following reference: Ramsey, J. D.,
Zhang, K., Glymour, M., Romero, R. S., Huang, B., Ebert-Uphoff, I., ... &
Glymour, C. (2018). TETRAD—A toolbox for causal discovery. In 8th
International Workshop on Climate Informatics.
Graph
Box
The graph box can be used to create a new graph, or to copy or
edit a graph from another box.
Possible Parent Boxes of the Graph Box
- Another graph box
- A parametric model box
- An instantiated model box
- An estimator box
- A data box
- A simulation box
- A search box
- An updater box
- A regression box
Possible Child Boxes of the Graph Box
- Another graph box
- A compare box
- A parametric model
box
- A data box
- A simulation box
- A search
box
- A knowledge box
Creating a New Graph
When you first open a graph box with no parent, you will be
presented with several options for which kind of graph you would like to
create: a general graph, a directed acyclic graph (DAG), a structural
equation model (SEM)graph, or a time lag graph. Once you have selected
the type of graph you want to create, an empty graph box will open.
You can add variables to your graph by clicking on the variable
button on the left, then clicking inside the graph area. Add edges by
clicking on an edge type, then clicking and dragging from one variable to
another. Variables may be measured (represented by rectangular icons) or
latent (represented by elliptical icons). Edges may be directed,
undirected, bidirected, or uncertain (represented by circles at the ends
of an edge). Depending on the type of graph you choose to create, your
choice of edges may be limited.
DAGs allow only directed edges. If an edge would create a
cycle, it will not be accepted. A graph box containing a DAG can be used
as input for any parametric model box, and is the only kind of graph box
that can be used as input for a Bayes parametric model.
SEM graphs allow only directed and bidirected edges. A
graph box containing a SEM graph can be used as input to a SEM parametric
model or generalized SEM parametric model, where a bidirected edge
between two variables X and Y will be interpreted as X and Y having
correlated error terms.
Time lag graphs allow only directed edges. New variables
that you add will be initialized with a single lag. (The number of lags
in the graph may be changed under “Edit—Configuration…”) Edges from later
lags to earlier lags will not be accepted. Edges added within one lag
will automatically be replicated in later lags.
The general graph option allows all edge types and
configurations.
Creating a Random Graph
Instead of manually creating a new graph, you can randomly create
one. To do so, open up a new empty graph box and click on “Graph—Random
Graph.” This will open up a dialog box from which you can choose the type
of random graph you would like to create by clicking through the tabs at
the top of the window. Tetrad will randomly generate a DAG, a multiple
indicator model (MIM) graph, or a scale-free graph. Each type of graph is
associated with a number of parameters (including but not limited to the
number of nodes and the maximum degree) which you can set.
Once a graph has been randomly generated, you can directly edit it
within the same graph box by adding or removing any variables or edges
that that type of graph box allows. So, for instance, although you cannot
randomly generate a graph with bidirected edges, you can manually add
bidirected edges to a randomly generated DAG in a SEM graph box.
Random graph generation is not available for time lag graphs.
Loading a Saved Graph
If you have previously saved a graph from Tetrad, you can load it
into a new graph box by clicking “File—Load…,” and then clicking on the
file type of the saved graph. Tetrad can load graphs from XML, from text,
and from JSON files.
To save a graph to file, click “File—Save…,” then click on the
file type you would like to save your graph as. Tetrad can save graphs to
XML, text, JSON, R and dot files. (If you save your graph to R or dot,
you will not be able to load that file back into Tetrad.)
You can also save an image of your graph by clicking “File—Save
Graph Image…” Tetrad cannot load graphs from saved image files.
Copying a Graph
There are two ways to copy a graph.
To copy a graph from any box which contains one, first, create a
new graph box in the workspace, and draw an arrow from the box whose
graph you want to copy to the new graph box. When opened, the new graph
box will automatically contain a direct copy of the graph its parent box
contains.
Manipulating a Graph
If you create a graph box as a child of another box, you can also
choose to perform a graph manipulation on the parent graph. Your graph
box will then contain the manipulated version of the parent graph.
The available graph manipulations are:
Display Subgraphs
This option allows you to isolate a subgraph from the parent
graph. Add variables to the subgraph by highlighting the variable name in
the “Unselected” pane and clicking on the right arrow. The highlighted
variable will then show up in the “Selected” pane. (You may also define
which variables go in the “Selected” pane by clicking on the “Text
Input…” button and typing the variable names directly into the window.)
Choose the type of subgraph you want to display from the drop-down panel
below. Then click “Graph It!” and the resulting subgraph of the selected
variables will appear in the pane on the right. (Some types of subgraph,
such as “Markov Blanket,” will include unselected variables if they are
part of the subgraph as defined on the selected variables. So, for
instance, an unselected variable that is in the Markov blanket of a
selected variable will appear in the Markov Blanket subgraph. Edges
between unselected variables will not be shown.) For large or very dense
graphs, it may take a long time to isolate and display subgraphs.
The types of subgraphs that can be displayed are:
- Subgraph (displays the selected nodes and all
edges between them)
- Adjacents (displays the selected nodes and
all edges between them, as well as nodes adjacent to the selected nodes)
- Adjacents of adjacents (displays the selected nodes and all
edges between them, as well as nodes adjacent to the selected nodes and
nodes adjacent to adjacencies of the selected nodes)
- Adjacents
of adjacents of adjacents (displays the selected nodes and all edges
between them, as well as nodes adjacent to the selected nodes, nodes
adjacent to adjacencies of the selected nodes, and nodes adjacent to
adjacencies of adjacencies of the selected nodes)
- Markov
Blankets (displays the selected nodes and all edges between them, as well
as the Markov blankets of each selected node)
- Treks (displays
the selected nodes, with an edge between each pair if and only if a trek
exists between them in the full graph)
- Trek Edges (displays the
selected nodes, and any treks between them, including nodes not in the
selected set if they are part of a trek)
- Paths (displays the
selected nodes, with an edge between each pair if and only if a path
exists between them in the full graph)
- Path Edges (displays the
selected nodes, and any paths between them, including nodes not in the
selected set if they are part of a path)
- Directed Paths
(displays the selected nodes, with a directed edge between each pair if
and only if a directed path exists between them in the full graph)
- Directed Path Edges (displays the selected nodes, and any directed
paths between them, including nodes not in the selected set if they are
part of a path)
- Y Structures (displays any Y structures
involving at least two of the selected nodes)
- Pag_Y Structures
(displays any Y PAGs involving at least two of the selected nodes)
- Indegree (displays the selected nodes and their parents)
- Outdegree (displays the selected nodes and their children)
- Degree (displays the selected nodes and their parents and
children)
Choose Random DAG in CPDAG
If given a CPDAG as input, this chooses a random DAG from the
Markov equivalence class of the CPDAG to display. The resulting DAG
functions as a normal graph box.
Choose Zhang MAG in PAG
If given a partial ancestral graph (PAG) as input, this chooses a
mixed ancestral graph (MAG) from the equivalence class of the PAG
to display using Zhang's method. The resulting MAG functions as a
normal graph box.
Show DAGs in CPDAG
If given a CPDAG as input, this displays all DAGs in the CPDAG’s
Markov equivalence class. Each DAG is displayed in its own tab. Most
graph box functionality is not available in this type of graph box, but
the DAG currently on display can be copied by clicking “Copy Selected
Graph.”
Generate CPDAG from DAG
If given a DAG as input, this displays the CPDAG of the Markov
equivalence class to which the parent graph belongs. The resulting CPDAG
functions as a normal graph box.
Generate PAG from DAG
Converts an input graph from partial ancestral to directed acyclic
format. The resulting DAG functions as a normal graph box.
Generate PAG from tsDAG
Converts an input graph from partial ancestral to time series DAG
format. The resulting DAG functions as a normal graph box.
Make Bidirected Edges Undirected
Replaces all bidirected edges in the input graph with undirected
edges.
Make Undirected Edges Bidirected
Replaces all undirected edges in the input graph with bidirected
edges.
Make All Edges Undirected
Replaces all edges in the input graph with undirected edges.
Generate Complete Graph
Creates a completely connected, undirected graph from the
variables in the input graph.
Extract Structure Model
Isolates the subgraph of the input graph involving all and only
latent variables.
Other Graph Box Functions
Edges and Edge Type Frequencies
At the bottom of the graph box, the Edges and Edge Type
Frequencies section provides an accounting of every edge in the graph,
and how certain Tetrad is of its type. The first three columns contain a
list, in text form, of all the edges in the graph. The columns to the
right are all blank in manually constructed graphs, user-loaded graphs,
and graphs output by searches with default settings. They are only filled
in for graphs that are output by searches performed with bootstrapping.
In those cases, the fourth column will contain the percentage of
bootstrap outputs in which the edge type between these two variables
matches the edge type in the final graph. All the columns to the right
contain the percentages of the bootstrap outputs that output each
possible edge type.
For more information on bootstrap searches, see the Search Box
section of the manual.
Layout
You can change the layout of your graph by clicking on the
“Layout” tab and choosing between several common layouts. You can also
rearrange the layout of one graph box to match the layout of another
graph box (so long as the two graphs have identical variables) by
clicking “Layout—Copy Layout” and “Layout—Paste Layout.” You do not need
to a highlight the graph in order to copy the layout.
Graph Properties
Clicking on “Graph—Graph Properties” will give you a text box
containing the following properties of your graph:
- Number of nodes
- Number of latent
nodes
- Number of adjacencies
- Number of directed edges (not in 2-cycles)
- Number of bidirected edges
- Number of undirected edges
- Max degree
- Max indegree
- Max outdegree
- Average degree
- Density
- Number of latents
- Cyclic/Acyclic
Paths
Clicking on “Graph—Paths” opens a dialog box that allows you to
see all the paths between any two variables. You can specify whether you
want to see only adjacencies, only directed paths, only semidirected
paths, or all treks between the two variables of interest, and the
maximum length of the paths you are interested in using drop boxes at the
top of the pane. To apply those settings, click “update.”
Correlation
You can automatically correlate or uncorrelated exogenous
variables under the Graph tab.
Highlighting
You can highlight bidirected edges, undirected edges, and latent
nodes under the Graph tab.
Compare
Box
The compare box compares two or more graphs.
Possible Parent Boxes of the Compare box:
- A graph box
- An instantiated model
box
- An estimator box
- A simulation box
- A search
box
- A regression box
Possible Child Boxes of the Compare box:
- None
Edgewise Comparisons
An edgewise comparison compares two graphs, and gives a textual
list of the edges which must be added to or taken away from one to make
it identical to the other.
Take, for example, the following two graphs. The first is the
reference graph, the second is the graph to be compared to it.
When the Edgewise Comparison box is opened, a comparison like this
appears:
You may choose (by a menu in the upper left part of the box) whether the
graph being compared is the original DAG, or the CPDAG of the original
DAG, of the PAG of the original DAG
When the listed changes have been made to the second graph, it
will be identical to the first graph.
Stats List Graph Comparisons
A stats list graph comparison tallies up and presents statistics
for the differences and similarities between a true graph and a reference
graph. Consider the example used in the above section; once again, we’ll
let graph one be the true graph. Just as above, when the graphs are input
to the tabular graph compare box, we must specify which of the graphs is
the reference graph, and whether it contains latent variables. When the
comparison is complete, the following window results:
You may choose (by a menu in the upper left part of the box) whether the
graph being compared is the original DAG, or the CPDAG of the original
DAG, of the PAG of the original DAG
The first columns gives an abbreviation for the statistic; the
second columns gives a definition of the statistic. The third columns
gives the statistic value.
Misclassifications
A misclassification procedure organizes a graph comparison by edge
type. The edge types (undirected, directed, uncertain, partially
uncertain, bidirected, and null) are listed as the rows and columns of a
matrix, with the true graph edges as the row headers and the target graph
edges as the column headers. If, for example, there are three pairs of
variables that are connected by undirected edges in the reference graph,
but are connected by directed edges in the estimated graph, then there
will be a 3 in the (undirected, directed) cell of the matrix. An
analogous method is used to represent endpoint errors. For example:
Graph Intersections
A graph intersection compares two or more graphs in the same
comparison. It does so by ranking adjacencies (edges without regard to
direction) and orientations based on how many of the graphs they appear
in. In an n-graph comparison, it first lists any adjacencies found in all
n graphs. Then it lists all adjacencies found in n – 1 graphs, then
adjacencies found in n – 2 graphs, and so on.
After it has listed all adjacencies, it lists any orientations
that are not contradicted among the graphs, again in descending order of
how many graphs the orientation appears in. An uncontradicted orientation
is one on which all graphs either agree or have no opinion. So if the
edge X Y appears in all n graphs, it will be listed first. If the edge
X Z appears in n – 1 graphs, it will be listed next, but only if the
nth graph doesn’t contradict it—that is, only if the edge Z X does not
appear in the final graph. If the undirected edge Z – X appears in the
final graph, the orientation X Z is still considered to be
uncontradicted.
Finally, any contradicted orientations (orientations that the
graphs disagree on) are listed.
Independence Facts Comparison
Rather than comparing edges or orientation, this option directly
compares the implied dependencies in two graphs. When you initially open
the box, you will see the following window:
The drop-down menu allows you to choose which variables you want
to check the dependence of. If you select more than two variables, any
subsequent variables will be considered members of the conditioning set.
So, if you select variables X1, X2, and X3, in that order, the box will
determine whether X1 is independent of X2, conditional on X3, in each of
the graphs being compared. When you click “List,” in the bottom right of
the window, the results will be displayed in the center of the window:
Edge Weight Similarity Comparisons
Edge weight (linear coefficient) similarity comparisons compare
two linear SEM instantiated models. The output is a score equal to the
sum of the squares of the differences between each corresponding edge
weight in each model. Therefore, the lower the score, the more similar
the two graphs are. The score has peculiarities: it does not take account
of the variances of the variables, and may therefore best be used with
standardized models; the complete absence of an edge is scored as 0—so a
negative coefficient compares less well with a positive coefficient than
does no edge at all.
Consider, for example, an edge weight similarity comparison
between the following two SEM IMs:
When they are input into an edge weight similarity comparison, the
following window results:
This is, unsurprisingly, a high score; the input models have few
adjacencies in common, let alone similar parameters.
Model Fit
A model fit comparison takes a simulation box and a search box
(ideally, a search that has been run on the simulated data in the
simulation box), and provides goodness-of-fit statistics, including a
Student’s t statistic and p value for each edge, for the output graph and
the data, as well as estimating the values of any parameters. It looks
and functions identically to the estimator box, but unlike the estimator
box, it takes the search box directly as a parent, without needing to
isolate and parameterize the graph output by the search.
Markov Check
The Markov Checker checks to see whether the Markov Condition is satisfied
for a given graph. A simple version of the Markov Condition states that
for any variable X, X is independent of all non-descendants of X given
X’s parents. The Markov Checker will output all such implied independences
and their p-values; these p-values should be distributed as U(0, 1) if the
Markov Condition is satisfied, so violations can often be detected by
plotting a histogram of these p-values or doing an Anderson-Darling test
or a Kolmogorov-Smirnov test to see if the hypothesis that they are drawn
from a U(0, 1) distribution can be rejected. This sort of check is actually
more general, since graphical implicaitons of separation are not limited to
directect acyclic graphs (DAGs) but can be inferred from many types of
graphs, including CPDAGs, MAGs, ADMGs, and PAGs.
Instructions for using the Markov Checker are included in the box itself,
in the "Help" tab.
IDA Check
The IDA Checker check loops through all pairs of variable (X, Y) and
calculates the IDA minimum effect for each X on Y, for a linear CPDAG
model. The IDA minimum effect is the minimum effect of X on Y, regressing
Y on S U {X} for all possible parent sets of X in the CPDAG. This gives
a range of effects, and one then see whether the true effect of X on Y
(as calculated from the true SEM IM) falls within this range. If it does
not, then the IDA minimum effect is not consistent with the true effect.
The IDA check table gives information to help the user assess these results
along with several summary statistics. Further instructions for using the
IDA Checker are included in the box itself, in the "Help" tab.
Algcomparison
The Algcomparison (Algorithm comparison) tool allows the user to compare the
results of multiple searches. The user can select one or more simulations,
one or more algorithm (with selected test and/or score), and one or more
table columns representing columns of parameter values or else statistics
that are calculated based on the comparisons done. For most of these values,
the use can specify multiple options for values, and the tool will iterate
over all sensible combinations of these values and output a table of results.
Full results, including all simulated dataset, all true graphs, all estimated
graphs, and all timing results, are saved to the hard drive.
Further instructions for using the Algcomparison tool are included in the
box itself, in the "Help" tab.
Parametric Model Box
The parametric model box takes a nonparameterized input graph and
creates a causal model.
Possible Parent Boxes of the Parametric Model Box:
- A graph box
- Another parametric model
box
- An instantiated model box
- An estimator box
- A data box
- A simulation box
- A search box
- A
regression box
Possible Child Boxes of the Parametric Model Box:
- A graph box
- Another parametric model
box
- An instantiated model box
- An estimator box
- A data box
- A simulation box
- A search box
- A
knowledge box
Bayes Parametric Models
A Bayes parametric model takes as input a DAG. Bayes PMs represent
causal structures in which all the variables are categorical.
Bayes PMs consist of three components: the graphical
representation of the causal structure of the model; for each named
variable, the number of categories which that variable can assume; and
the names of the categories associated with each variable.
You may either manually assign categories to the variables or have
Tetrad assign them at random. If you choose to manually create a Bayes
PM, each variable will initially be assigned two categories, named
numerically. If you choose to have Tetrad assign the categories, you can
specify a minimum and maximum number of categories possible for any given
variable. You can then manually edit the number of categories and
category names.
Take, for example, the following DAG:
One possible random Bayes PM that Tetrad might generate from the
above DAG, using the default settings, looks like this:
To view the number and names of the categories associated with
each variable, you can click on that variable in the graph, or choose it
from the drop-down menu on the right. In this graph, X1 and X2 each have
three categories, and the rest of the variables have four categories. The
categories are named numerically by default.
The number of categories associated with a particular variable can
be changed by clicking up or down in the drop-down menu on the right.
Names of categories can be changed by overwriting the text already
present.
Additionally, several commonly-used preset variable names are
provided under the “Presets” tab on the right. If you choose one of these
configurations, the number of categories associated with the current
variable will automatically be changed to agree with the configuration
you have chosen. If you want all the categories associated with a
variable to have the same name with a number appended (e.g., x1, x2, x3),
choose the “x1, x2, x3…” option under Presets.
You can also copy category names between variables in the same
Bayes PM by clicking on “Transfer—Copy categories” and “Transfer—Paste
categories.”
SEM Parametric Models
The parametric model of a structural equation model (SEM) will
take any type of graph as input, as long as the graph contains only
directed and bidirected edges. SEM PMs represent causal structures in
which all variables are continuous.
A SEM PM has two components: the graphical causal structure of the
model, and a list of parameters used in a set of linear equations that
define the causal relationships in the model. Each variable in a SEM PM
is a linear function of a subset of the other variables and of an error
term drawn from a Normal distribution.
Here is an example of a SEM graph and the SEM PM that Tetrad
creates from it:
You can see the error terms in the model by clicking
“Parameters—Show Error Terms.” In a SEM model, a bidirected edge
indicates that error terms are correlated, so when error terms are
visible, the edge between X1 and X2 will instead run between their error
terms.
To change a parameter’s name or starting value for estimation,
double-click on the parameter in the window.
Generalized SEM Parametric Models
A generalized SEM parametric model takes as input any type of
graph, as long as the graph contains only directed edges. (The
generalized SEM PM cannot currently interpret bidirected edges.) Like a
SEM PM, it represents causal structures in which all variables are
continuous. Also like a SEM PM, a generalized SEM PM contains two
components: the graphical causal structure of the model, and a set of
equations representing the causal structure of the model. Each variable
in a generalized SEM PM is a function of a subset of the other variables
and an error term. By default, the functions are linear and the error
terms are drawn from a Normal distribution (as in a SEM PM), but the
purpose of a generalized SEM PM is to allow editing of these
features.
Here is an example of a general graph and the default generalized
SEM PM Tetrad creates using it:
You can view the error terms by clicking “Tools: Show Error
Terms.”
The Variables tab contains a list of the variables and the
expressions that define them, and a list of the error terms and the
distributions from which their values will be drawn. Values will be drawn
independently for each case if the model is instantiated (see IM box) and
used to simulate data (see data box).
The Parameters tab contains a list of the parameters and the
distributions from which they are drawn. When the model in instantiated
in the IM box, a fixed value of each parameter will be selected according
to the specified distribution.
To edit an expression or parameter, double-click on it (in any
tab). This will open up a window allowing you to change the function that
defines the variable or distribution of the parameter.
For instance, if you double-click on the expression next to X1
(b1*X5+E_X1), the following window opens:
The drop-down menu at the top of the window lists valid operators
and functions. You could, for example, change the expression from linear
to quadratic by replacing b1*X5+E_X1 with b1*X5^2+E_X1. You can also form
more complicated expressions, using, for instance, exponential or sine
functions. If the expression you type is well-formed, it will appear in
black text; if it is invalid, it will appear in red text. Tetrad will not
accept any invalid changes.
Parameters are edited in the same way as expressions.
If you want several expressions or parameters to follow the same
non-linear model, you may wish to use the Apply Templates tool. This
allows you to edit the expressions or parameters associated with several
variables at the same time. To use the Apply Templates tool, click
“Tools: Apply Templates….” This will open the following window:
You can choose to edit variables, error terms, or parameters by
clicking through the “apply to” radio buttons. If you type a letter or
expression into the “starts with” box, the template you create will apply
only to variables, error terms, or parameters which begin with that
letter for expression. For example, in the given generalized PM, there
are two types of parameters: the standard deviations s1-s6 and the edge
weights b1-b7. If you click on the “Parameters” radio button and type “b”
into the “Starts with” box, only parameters b1-b7 will be affected by the
changes you make.
The “Type Template” box itself works in the same way that the
“Type Expression” box works in the “Edit Expression” window, with a few
additions. If you scroll through the drop-down menu at the top of the
window, you will see the options NEW, TSUM, and TPROD. Adding NEW to a
template creates a new parameter for every variable the template is
applied to. TSUM means “sum of the values of this variable’s parents,”
and TPROD means “product of the values of this variable’s parents.” The
contents of the parentheses following TSUM and TPROD indicate any
operations which should be performed upon each variable in the sum or
product, with the dollar sign ($) functioning as a wild card. For
example, in the image above, TSUM(NEW(b)*$) means that, for each parent
variable of the variable in question, a new “b” will be created and
multiplied by the parent variable’s value, and then all the products
will be added together.
Instantiated Model Box
The instantiated model (IM) box takes a parametric model and
assigns values to the parameters.
Possible Parent Boxes of the Instantiated Model Box:
- A parametric model box
- Another
instantiated model box
- An estimator box
- A simulation
box
- An updater box
Possible Child Boxes of the Instantiated Model Box:
- A graph box
- A compare box
- A
parametric model box
- Another instantiated model box
- An
estimator box
- A simulation box
- A search box
- An
updater box
- A classify box
- A knowledge box
Bayes Instantiated Models
A Bayes IM consists of a Bayes parametric model with defined
probability values for all variables. This means that, conditional on the
values of each of its parent variables, there is a defined probability
that a variable will take on each of its possible values. For each
assignment of a value to each of the parents of a variable X, the
probabilities of the several values of X must sum to 1.
You can manually set the probability values for each variable, or
have Tetrad assign them randomly. If you choose to have Tetrad assign
probability values, you can manually edit them later.
Here is an example of a Bayes PM and its randomly created
instantiated model:
In the model above, when X4 and X5 are both 0, the probability
that X5 is 0 is 0.0346, that X5 is 1 is 0.4425, and that X5 is 2 is
0.5229. Since X5 must be 0, 1, or 2, those three values must add up to
one, as must the values in every row.
To view the probability values of a variable, either double click
on the variable in the graph or choose it from the drop-down menu on the
right. You can manually set a given probability value by overwriting the
text box. Be warned that changing the value in one cell will delete the
values in all the other cells in the row. Since the values in any row
must sum to one, if all the cells in a row but one are set, Tetrad
will automatically change the value in the last cell to make the sum
correct. For instance, in the above model, if you change the first row
such that the probability that X5 = 0 is 0.5000 and the probability that
X5 = 1 is 0.4000, the probability that X5 = 2 will automatically be set
to 0.1000.
If you right-click on a cell in the table (or two-finger click on
Macs), you can choose to randomize the probabilities in the row
containing that cell, randomize the values in all incomplete rows in the
table, randomize the entire table, or randomize the table of every
variable in the model. You can also choose to clear the row or table.
Dirichlet Instantiated Models
A Dirichlet instantiated model is a specialized form of a Bayes
instantiated model. Like a Bayes IM, a Dirichlet IM consists of a Bayes
parametric model with defined probability values. Unlike a Bayes IM,
these probability values are not manually set or assigned randomly.
Instead, the pseudocount is manually set or assigned uniformly, and the
probability values are derived from it. The pseudocount of a given value
of a variable is the number of data points for which the variable takes
on that value, conditional on the values of the variable’s parents, where
these numbers are permitted to take on non-negative real values. Since we
are creating models without data, we can set the pseudocount to be any
number we want. If you choose to create a Dirichlet IM, a window will
open allowing you to either manually set the pseudocounts, or have Tetrad
set all the pseudocounts in the model to one number, which you
specify.
Here is an example of a Bayes PM and the Dirichlet IM which Tetrad
creates from it when all pseudocounts are set to one:
In the above model, when X2=0 and X6=0, there is one (pseudo) data
point at which X4=0, one at which X4=1, and one at which X4=2. There are
three total (pseudo) data points in which X2=0 and X6=0. You can view the
pseudocounts of any variable by clicking on it in the graph or choosing
it from the drop-down menu at the top of the window. To edit the value of
a pseudocount, double-click on it and overwrite it. The total count of a
row cannot be directly edited.
From the pseudocounts, Tetrad determines the conditional
probability of a category. This estimation is done by taking the
pseudocount of a category and dividing it by the total count for its row.
For instance, the total count of X4 when X2=0 and X6=0 is 3. So the
conditional probability of X4=0 given that X2=0 and X6=0 is 1/3. The
reasoning behind this is clear: in a third of the data points in which X2
and X6 are both 0, X4 is also 0, so the probability that X4=0 given that
X2 and X6 also equal 0 is probably one third. This also guarantees that
the conditional probabilities for any configuration of parent variables
add up to one, which is necessary.
To view the table of conditional probabilities for a variable,
click the Probabilities tab. In the above model, the Probabilities tab
looks like this:
SEM Instantiated Models
A SEM instantiated model is a SEM parametric model in which the
parameters and error terms have defined values. It assumes that
relationships between variables are linear, and that error terms have
Gaussian distributions. If you choose to create a SEM IM, the following
window will open:
Using this box, you can specify the ranges of values from which
you want coefficients, covariances, and variances to be drawn for the
parameters in the model. In the above box, for example, all linear
coefficients will be between -1.0 and 1.0. If you uncheck “symmetric
about zero,” they will only be between 0.0 and 1.0.
Here is an example of a SEM PM and a SEM IM generated from it
using the default settings:
You can now manually edit the values of parameters in one of two
ways. Double-clicking on the parameter in the graph will open up a small
text box for you to overwrite. Or you can click on the Tabular Editor
tab, which will show all the parameters in a table which you can edit.
The Tabular Editor tab of our SEM IM looks like this:
In the Tabular Editor tab of a SEM estimator box (which functions
similarly to the SEM IM box), the SE, T, and P columns provide statistics
showing how robust the estimation of each parameter is. Our SEM IM,
however, is in an instantiated model box, so these columns are empty.
The Implied Matrices tab shows matrices of relationships between
variables in the model. In the Implied Matrices tab, you can view the
covariance or correlation matrix for all variables (including latents) or
just measured variables. In our SEM IM, the Implied Matrices tab looks
like this:
You can choose the matrix you wish to view from the drop-down menu
at the top of the window. Only half of any matrix is shown, because in a
well-formed acyclic model, the matrices should be symmetric. The cells in
the Implied Matrices tab cannot be edited.
In an estimator box, the Model Statistics tab provides goodness of
fit statistics for the SEM IM which has been estimated. Our SEM IM,
however, is in an instantiated model box, so no estimation has occurred,
and the Model Statistics tab is empty.
Standardized SEM Instantiated Models
A standardized SEM instantiated model consists of a SEM parametric
model with defined values for its parameters. In a standardized SEM IM,
each variable (not error terms) has a Normal distribution with 0 mean and
unit variance. The input PM to a standardized SEM IM must be acyclic.
Here is an example of an acyclic SEM PM and the standardized SEM
IM which Tetrad creates from it
To edit a parameter, double-click on it. A slider will open at the
bottom of the window (shown above for the edge parameter between X1 and
X2). Click and drag the slider to change the value of the parameter, or
enter the specific value you wish into the box. The value must stay
within a certain range in order for the variables in the model to
remain standard Normal (N(0, 1)), so if you attempt to overwrite
the text box on the bottom right with a value outside the listed range,
Tetrad will not allow it. That is, given that the variables are all
distributed as N(0, 1), there is a limited range in which each parameter
may be adjusted; these ranges vary parameter by parameter, given
the values of the other parameters. In a standardized SEM IM, error
terms are not considered parameters and cannot be edited, but you can
view them by clicking Parameters: Show Error Terms.
It is possible to make a SEM IM with a time lag graph, even with latent
variables. This does not work for other types of models, such as
Bayes IMs or for mixed data (for which no IM is currently available--
though mixed data can be simulated in the Simulate box with an appropriate
choice of simulation model). Standardization for time lag model
is not currently available.
The Implied Matrices tab works in the same way that it does in a
normal SEM IM.
Generalized SEM Instantiated Models
A generalized SEM instantiated model consists of a generalized SEM
parametric model with defined values for its parameters. Since the
distributions of the parameters were specified in the SEM PM, Tetrad does
not give you the option of specifying these before it creates the
instantiated model.
Here is an example of a generalized SEM PM and its generalized SEM
IM:
Note that the expressions for X6 and X2 are not shown, having been
replaced with the words “long formula.” Formulae over a certain
length—the default setting is 25 characters—are hidden to improve
visibility. Long formulae can be viewed in the Variables tab, which lists
all variables and their formulae. You can change the cutoff point for
long formulae by clicking Tools: Formula Cutoff.
If you double-click on a formula in either the graph or the
Variables tab, you can change the value of the parameters in that
formula.
Data Box
The data box stores or manipulates data sets.
Possible Parent Boxes of the Data Box
- A graph box
- An estimator box
- Another data box
- A simulation box
- A regression
box
Possible Child Boxes of the Data Box
- A graph box
- A parametric model box
- Another data box
- An estimator box
- A simulation
box
- A search box
- A classify box
- A regression
box
- A knowledge box
Using the Data Box:
The data box stores the actual data sets from which causal
structures are determined. Data can be loaded into the data box from a
preexisting source, manually filled in Tetrad, or simulated from an
instantiated model.
Loading Data
Data sets loaded into Tetrad may be categorical, continuous,
mixed, or covariance data.
General Tabular Data
To load data, create a data box with no parent. When you double-click
it, an empty data window will appear:
Click "File -> Load Data" and select the text file or files that
contain your data. The following window will appear:
The text of the source file appears in the Data Preview window.
Above, there are options to describe your file, so that Tetrad can load
it correctly. If you are loading categorical, continuous, or mixed data
values, select the “Tabular Data” button. If you are loading a covariance
matrix, select “Covariance Data.” Note that if you are loading a
covariance matrix, your text file should contain only the lower half of
the matrix, as Tetrad will not accept an entire matrix.
Below the file type, you can specify a number of other details
about your file, including information about the type of data
(categorical/continuous/mixed), metadata JSON file, delimiter between
data values, variable names, and more. If your data is mixed (some
variables categorical, and some continuous), you must specify the maximum
number of categories discrete variables in your data can take on. All
columns with more than that number of values will be treated as
continuous; the others will be treated as categorical. If you do not list
the variable names in the file, you should uncheck “First row variable
names.” If you provide case IDs, check the box for the appropriate column
in the “Case ID column to ignore” area. If the case ID column is labeled,
provide the name of the label; otherwise, the case ID column should be
the first column, and you should check “First column.”
Below this, you can specify your comment markers, quote
characters, and the character which marks missing data values. Tetrad
will use that information to distinguish continuous from discrete
variables. You may also choose more files to load (or remove files that
you do not wish to load) in the “Files” panel on the lower left.
Metadata JSON File
Metadata is optional in general data
handling. But it can be very helpful if you want to overwrite the data
type of given variable column. And the metadata MUST be a JSON file
like the following example.
{
"domains": [
{
"name": "raf",
"discrete": false
},
{
"name": "mek",
"discrete": true
}
]
}
You can specify the name and data type for each variable.
Variables that are not in the metadata file will be treated as domain
variables and their data type will be the default data type when reading
in columns described previously.
When you are satisfied with your description of your data, click
“Validate” at the bottom of the window. Tetrad will check that your file
is correctly formatted. If it is, you will receive a screen telling you
that validation has passed with no error. At this point, you can revisit
the settings page, or click “Load” to load the data.
You can now save this data set to a text file by clicking File:
Save Data.
In addition to loading data from a file, you can manually enter
data values and variable names by overwriting cells in the data
table.
Covariance Data
Covariance matrices loaded into Tetrad
should be ascii text files. The first row contains the sample size, the
second row contains the names of the variables. The first two rows are
followed by a lower triangular matrix. For example:
1000
X1 X2 X3 X4 X5 X6
1.0000
0.0312 1.0000
-0.5746 0.4168 1.0000
-0.5996 0.4261 0.9544 1.0000
0.8691 0.0414 -0.4372 -0.4487 1.0000
0.6188 0.0427 -0.1023 -0.0913 0.7172 1.0000
Categorical, continuous, or mixed data should also be an ascii
text file, with columns representing variables and rows representing
cases. Beyond that, there is a great deal of flexibility in the layout:
delimiters may be commas, colons, tabs, spaces, semicolons, pipe symbols,
or whitespace; comments and missing data may be marked by any symbol you
like; there may be a row of variable names or not; and case IDs may be
present or not. There should be no sample size row. For example:
X1 X2 X3 X4 X5
-3.0133 1.0361 0.2329 2.7829 -0.2878
0.5542 0.3661 0.2480 1.6881 0.0775
3.5579 -0.7431 -0.5960 -2.5502 1.5641
-0.0858 1.0400 -0.8255 0.3021 0.2654
-0.9666 -0.5873 -0.6350 -0.1248 1.1684
-1.7821 1.8063 -0.9814 1.8505 -0.7537
-0.8162 -0.6715 0.3339 2.6631 0.9014
-0.3150 -0.5103 -2.2830 -1.2462 -1.2765
-4.1204 2.9980 -0.3609 4.8079 0.6005
1.4658 -1.4069 1.7234 -1.7129 -3.8298
Handling Tabular Data with Interventional Variables
This is an advanced topic for datasets that contain interventional
(i.e., experimental) variables. We model a single intervention using two
variables: status variable and value variable. Below is a sample dataset,
in which `raf`, `mek`, `pip2`, `erk`, `atk` are the 5 domain variables,
and `cd3_s` and `cd3_v` are an interventional pair (status and value
variable respectively). `icam` in another intervention variable, but it's
a combined variable that doesn't have status.
raf mek pip2 erk akt cd3_s cd3_v icam
3.5946 3.1442 3.3429 2.81 3.2958 0 1.2223 *
3.8265 3.2771 3.2884 3.3534 3.7495 0 2.3344 *
4.2399 3.9908 3.0057 3.2149 3.7495 1 0 3.4423
4.4188 4.5304 3.157 2.7619 3.0819 1 3.4533 1.0067
3.7773 3.3945 2.9821 3.4372 4.0271 0 4.0976 *
And the sample metadata JSON file looks like this:
{
"interventions": [
{
"status": {
"name": "cd3_s",
"discrete": true
},
"value": {
"name": "cd3_v",
"discrete": false
}
},
{
"status": null,
"value": {
"name": "icam",
"discrete": false
}
}
],
"domains": [
{
"name": "raf",
"discrete": false
},
{
"name": "mek",
"discrete": false
}
]
}
Each intervention consists of a status variable and value
variable. There are cases that you may have a combined interventional
variable that doesn't have the status variable. In this case, just use
`null`. The data type of each variable can either be discrete or
continuous. We use a boolean flag to indicate the data type. From the
above example, we only specified two domain variables in the metadata
JSON, any variables not specified in the metadata will be treated as
domain variables.
Manipulating Data
The data box can also be used to manipulate data sets that have
already been loaded or simulated. If you create a data box as the child
of another box containing a data set, you will be presented with a list
of operations that can be performed on the data. The available data
manipulations are:
Discretize Dataset
This operation allows you to make some or all variables in a data
set discrete. If you choose it, a window will open.
When the window first opens, no variables are selected, and the
right side of the window appears blank; in this case, we have already
selected X1 ourselves. In order to discretize a variable, Tetrad assigns
all data points within a certain range to a category. You can tell Tetrad
to break the range of the dataset into approximately even sections
(Evenly Distributed Intervals) or to break the data points themselves
into approximately even chunks (Evenly Distributed Values). Use the
scrolling menu to increase or decrease the number of categories to
create. You can also rename categories by overwriting the text boxes on
the left, or change the ranges of the categories by overwriting the text
boxes on the right. To discretize another variable, simply select it from
the left. If you want your new data set to include the variables you did
not discretize, check the box at the bottom of the window.
You may discretize multiple variables at once by selecting
multiple variables. In this case, the ranges are not shown, as they will
be different from variable to variable.
Convert Numerical Discrete to Continuous
If you choose this option, any discrete variables with numerical
category values will be treated as continuous variables with real values.
For example, “1” will be converted to “1.0.”
Calculator
The Calculator option allows you to add and edit relationships
between variables in your data set, and to add new variables to the data
set.
In many ways, this tool works like the Edit Expression window in a
generalized SEM parametric model. To edit the formula that defines a
variable (which will change that variable’s values in the table) type
that variable name into the text box to the left of the equals sign. To
create a new variable, type a name for that variable into the text box to
the left of the equals sign. Then, in the box on the right, write the
formula by which you wish to define a new variable in place of, or in
addition to, the old variable. You can select functions from the
scrolling menu below. (For an explanation of the meaning of some the
functions, see the section on generalized SEM models in the Parametric
Model Box chapter.) To edit or create several formulae at once, click the
“Add Expression” button, and another blank formula will appear. To delete
a formula, check the box next to it and click the “Remove Selected
Expressions” button.
When you click “Save” a table will appear listing the data. Values
of variables whose formulae you changed will be changed, and any new
variables you created will appear with defined values.
Merge Deterministic Interventional Variables
This option looks for pairs of interventional variables (currently
only discrete variables) that are deterministic and merges them into one
combined variable. For domain variables that are fully determined,
we'll add an attribute to them. Later in the knowledge box (Edges and
Tiers), all the interventional variables (both status and value
variables) and the fully-determined domain variables will be
automatically put to top tier. And all other domain variables will be
placed in the second tier.
Merge Datasets
This operation takes two or more data boxes as parents and creates
a data box containing all data sets in the parent boxes. Individual data
sets will be contained in their own tabs in the resulting box.
Convert to Correlation Matrix
This operation takes a tabular data set and outputs the lower half
of the correlation matrix of that data set.
Convert to Covariance Matrix
This operation takes a tabular data set and outputs the lower half
of the covariance matrix of that data set.
Inverse Matrix
This operation takes a covariance or correlation matrix and
outputs its inverse. (Note: The output will not be acceptable in Tetrad
as a covariance or correlation matrix, as it is not lower
triangular.)
Simulate Tabular from Covariance
This operation takes a covariance matrix and outputs a tabular
data set whose covariances comply with the matrix.
Difference of Covariance Matrices
This operation takes two covariance matrices and outputs their
difference. The resulting matrix will be a well-formatted Tetrad
covariance matrix data set.
Sum of Covariance Matrices
This operation takes two covariance matrices and outputs their
sum. The resulting matrix will be a well-formatted Tetrad covariance
matrix data set.
Average of Covariance Matrices
This operation takes two or more covariance matrices and outputs
their average. The resulting matrix will be a well-formatted Tetrad
covariance matrix data set.
Convert to Time Lag Data
This operation takes a tabular data set and outputs a time lag
data set, in which each variable is recorded several times over the
course of an experiment. You can specify the number of lags in the data.
Each contains the same data, shifted by one “time unit.” For instance, if
the original data set had 1000 cases, and you specify that the time lag
data set should contain two lags, then the third stage variable values
will be those of cases 1 to 998, the second stage variable values will be
those of cases 2 to 999, and the first stage variable values will be
those of cases 3 to 1000.
Convert to Time Lag Data with Index
This operation takes a tabular data set and outputs a time lag
data set in the same manner as “Convert to Time Lag Data,” then adds an
index variable.
Convert to AR Residuals
This operation is performed on a time lag data set. Tetrad
performs a linear regression on each variable in each lag with respect to
each of the variables in the previous lag, and derives the error terms.
The output data set contains only the error terms.
Whiten
Takes a continuous tabular data set and converts it to a data set
whose covariance matrix is the identity matrix.
Nonparanormal Transform
Takes a continuous tabular data set and increases its Gaussianity,
using a nonparanormal transformation to smooth the variables. (Note: This
operation increases only marginal Gaussianity, not the joint, and in
linear systems may eliminate information about higher moments that can
aid in non-Gaussian orientation procedures.)
Convert to Residuals
The input for this operation is a directed acyclic graph (DAG) and
a data set. Tetrad performs a linear regression on each variable in the
data set with respect to all the variables that the graph shows to be
its parents, and derives the error terms. The output data set contains
only the error terms.
Standardize Data
This operation manipulates the data in your data set such that
each variable has 0 mean and unit variance.
Remove Cases with Missing Values
If you choose this operation, Tetrad will remove any row in which
one or more of the values is missing.
Replace Missing Values with Column Mode
If you choose this operation, Tetrad will replace any missing
value markers with the most commonly used value in the column.
Replace Missing Values with Column Mean
If you choose this operation, Tetrad will replace any missing
value markers with the average of all the values in the column.
Replace Missing Values with Regression Predictions: If you choose this
operation, Tetrad will perform a linear regression on the data in order
to estimate the most likely value of any missing value.
Replace Missing Values by Extra Category
This operation takes as input a discrete data set. For every
variable which has missing values, Tetrad will create an extra category
for that variable (named by default “Missing”) and replace any missing
data markers with that category.
Replace Missing with Random
For discrete data, replaces missing values at random from the list
of categories the variable takes in other cases. For continuous data,
finds the minimum and maximum values of the column (ignoring the missing
values) and picks a random number from U(min, max)
Inject Missing Data Randomly
If you choose this operation, Tetrad will replace randomly
selected data values with a missing data marker. You can set the
probability with which any particular value will be replaced (that is,
approximately the percentage of values for each variable which will be
replaced with missing data markers).
Bootstrap Sample
This operation draws a random subset of the input data set (you
specify the size of the subset) with replacement (that is, cases which
appear once in the original data set can appear multiple times in the
subset). The resulting data set can be used along with similar subsets to
achieve more accurate estimates of parameters.
Split by Cases
This operation allows you to split a data set into several smaller
data sets. When you choose it, a window opens.
If you would like the subsets to retain the ordering they had in
the original set, click “Original Order.” Otherwise, the ordering of the
subsets will be assigned at random. You can also increase and decrease
the number of subsets created, and specify the range of each subset.
Permute Rows
This operation randomly reassigns the ordering of a data set’s
cases.
First Differences
This operation takes a tabular data set and outputs the first
differences of the data (i.e., if X is a variable in the original data
set and X’ is its equivalent in the first differences data set, X’1 = X2
– X1). The resulting data set will have one fewer row than the
original.
Concatenate Datasets
This operation takes two or more datasets and concatenates. The
parent datasets must have the same number of variables.
Copy Continuous Variables
This operation takes as input a data set and creates a new data
set containing only the continuous variables present in the original.
Copy Discrete Variables
This operation takes as input a data set and creates a new data
set containing only the discrete variables present in the original.
Remove Selected Variables
Copy Selected Variables
As explained above, you can select an entire column in a data set
by clicking on the C1, C2, C3, etc… cell above the column. To select
multiple columns, press and hold the “control” key while clicking on the
cells. Once you have done so, you can use the Copy Selected Variables
tool to create a data set in which only those columns appear.
Remove Constant Columns
This operation takes a data set as input, and creates a data set
which contains all columns in the original data set except for those with
constant values (such as, for example, a column containing nothing but
2’s).
Randomly Reorder Columns
This operation randomly reassigns the ordering of a data set’s
variables.
Manually Editing Data
Under the Edit tab, there are several options to manipulate data.
If you select a number of cells and click “Clear Cells,” Tetrad will
replace the data values in the selected cells with a missing data marker.
If you select an entire row or column and click “Delete selected rows or
columns,” Tetrad will delete all data values in the row or column, and
the name of the row or column. (To select an entire column, click on the
category number above it, labeled C1, C2, C3, and so on. To select an
entire row, click on the row number to the left of it, labeled 1, 2, 3,
and so on.) You can also copy, cut, and paste data values to and from
selected cells. You can choose to show or hide category names, and if you
click on “Set Constants Col to Missing,” then in any column in which the
variable takes on only one value (for example, a column in which every
cell contains the number 2) Tetrad will set every cell to the missing
data marker.
Under the Tools tab, the Calculator tool allows you to add or edit
relationships between variables in the graph. For more information on how
the Calculator tool works, see “Manipulating Data” section above.
Data Information
Under the Tools tab, there are options to view information about your data in several
formats.
The Plot Matrix tool shows a grid of scatter plots and histograms for selected variables.
This may be used for continuous, discrete, or mixtures of continuous and discrete data.
To select which variables to include in the rows and columns of the grid, click the variable
lists to the right of the tool. To select multiple variables in these lists, use the shift
or control keys when clicking; shift-click select ranges, whereas control-click will select
additional single variables.
Histograms show the data distribution for a variable, with the width of each bar
representing a range of values and the height of each bar representing how many data
points fall into that range.
Scatter plots show a plot of variables taken two at a time. They plot values of one
variable's values against another variable's values, point by point, and allow one
to see the distribution of points for the pair of variables.
If viewing a grid of plots, one wishes to view a single plot in this grid, double-click on
the desired plot, and it will be magnified so that it is in the only plot viewed.
Double-click on the magnified plot to return to the grid.
The “Settings” menu contains some tools to control the output. One may add regression lines
to the scatter plots or select the number of bins to include in the histograms.
Finally, one may condition on ranges of variables or particular discrete values by
selecting “Edit Conditioning Variables and Ranges.” This brings up a dialog that lets one
add conditioning variables with particular ranges for continuous variables or values for
discrete values. For continuous ranges, one may pick “Above Average,” “Below Average,” or
“In n-tile” (where n is specified) or give a particular range manually. One may add as many
conditions as one prefers; when one clicks “OK,” all plots will be updated to reflect these
conditioning choices.
The Q-Q Plot tool is a test for normality of distribution.
If a variable has a distribution which is approximately Normal,
its Q-Q plot should appear as a straight line with a positive slope. You
can select the variable whose Q-Q plot you wish to view from the
drop-down menu on the right.
The Normality Tests tool gives a text box with the results of the
Kolmogorov and Anderson Darling Tests for normality for each variable.
The Descriptive Statistics tool gives a text box with statistical
information such as the mean, median, and variance of each variable.
Estimator
Box
The estimator box takes as input a data box (or simulation box)
and a parametric model box and estimates, tests, and outputs an
instantiated model for the data. Except for the EM Bayes
estimator, Tetrad estimators do not accept missing values. If your data
set contains missing values, the missing values can be interpolated or
removed using the data box. (Note that missing values are allowed in
various Tetrad search procedures; see the section on the search box.)
Possible Parent Boxes of the Estimator Box:
- A parametric model box
Possible Child Boxes of the Estimator Box:
- A graph box
- A simulation box
- An
updater box
ML Bayes Estimations
Bayes nets are acyclic graphical models parameterized by the
conditional probability distribution of each variable on its parents'
values, as in the instantiated model box. When the model contains no
latent variables, the joint distribution of the variables equals the
product of the distributions of the variables conditional on their
respective parents. The maximum likelihood (ML) estimate of the joint
probability distribution under a model is the product of the
corresponding frequencies in the sample.
The ML Bayes estimator, because it estimates Bayes IMs, works only
on models with discrete variables. The model estimated must not include
latent variables, and the input data set must not include missing data
values. A sample estimate looks like this:
The Model tab works exactly as it does in a Bayes instantiated
model. The Model Statistics tab provides the p-value for a chi square
test of the model, degrees of freedom, the chi square value, and the
Bayes Information Criterion (BIC) score of the model. Note that BIC
is calculated as 2L -
Dirichlet Estimations
A Dirichlet estimate estimates a Bayes instantiated model using a
Dirichlet distribution for each category. In a Dirichlet estimate, the
probability of each value of a variable (conditional on the values of the
variable’s parents) is estimated by adding together a prior pseudo count
(which is 1, by default, of cases and the number of cases in which the
variable takes that value in the data, and then dividing by the total
number of cases in the pseudocounts and in the data with that
configuration of values of parent variables. The default prior
pseudo-count can be changed inside the box. (For a full explanation of
pseudocounts and Dirichlet estimate, see the section on Dirichlet
instantiated models.)
The Dirichlet estimator in TETRAD does not work if the input data
set contains missing data values.
EM Bayes Estimations
The EM Bayes estimator takes the same input and gives the same
output as the ML Bayes estimator, but is designed to handle data sets
with missing data values, and input models with latent variables.
SEM Estimates
A SEM estimator estimates the values of parameters for a SEM
parametric model. SEM estimates do not work if the input data set
contains missing data values. A sample output looks like this:
Tetrad provides five parameter optimizers: RICF,( Drton, M., &
Richardson, T. S. (2004, July). Iterative conditional fitting for
Gaussian ancestral graph models. In Proceedings of the 20th conference
on Uncertainty in artificial intelligence (pp. 130-137). AUAI Press).
expectation-maximization (EM), regression, Powell Journal of
Econometrics 25 (1984) 303-325) and random search. Accurate regression
estimates assume that the input parametric model is a DAG, and that its
associated statistics are based on a linear, Gaussian model. The EM
optimizer has the same input constraints as regression, but can handle
latent variables.
Tetrad also provides two scores that can be used in estimation:
feasible generalized the least squares (FGLS) and Full Information Maximum
Likelihood (FML).
If the graph for the SEM is a DAG, and we may assume that the SEM
is linear with Gaussian error terms, we use multilinear regression to
estimate coefficients and residual variances. Otherwise, we use a
standard maximum likelihood fitting function (see Bollen, Structural
Equations with Latent Variables, Wiley, 1989, pg. 107) to minimize the
distance between (a) the covariance over the variables as implied by the
coefficient and error covariance parameter values of the model and (b)
the sample covariance matrix. Following Bollen, we denote this function
Fml; it maps points in parameter values space to real numbers, and, when
minimized, yields the maximum likelihood estimation point in parameter
space.
In either case, a Fml value may be obtained for the maximum
likelihood point in parameter space, either by regression or by direct
minimization of the Fml function itself. The value of Fml at this minimum
(maximum likelihood) point, multiplied by N - 1 (where N is the sample
size), yields a chi square statistics (ch^2) for the model, which when
referred to the chi square table with appropriate degrees of freedom,
yields a model p value. The degrees of freedom (dof) in this case is
equal to the m(m-1)/2 - f, where m is the number of measured variables,
and f is the number of free parameters, equal to the number of
coefficient parameters plus the number of covariance parameters. (Note
that the degrees of freedom many be negative, in which case estimation
should not be done.) The BIC score is calculated as ch^2 - dof *
log(N), so "higher is better.".
You can change which score optimizer Tetrad uses by choosing them
from the drop-down menus at the bottom of the window and clicking
“Estimate Again.”
The Tabular Editor and Implied Matrices tabs function exactly as
they do in the instantiated model box, but in the estimator box, the last
three columns of the table in the Tabular Editor tab are filled in. The
SE, T, and P columns provide the standard errors, t statistics, and p
values of the estimation.
The Model Statistics tab provides the degrees of freedom, chi
square, p value, comparative fit index (CFI), root-mean-square error of
approximation (RMSEA) and BIC score of a test of the model. It should be
noted that while these test statistics are standard, they are not in
general correct. See Mathias Drton, 2009, Likelihood ratio tests and
singularities. Annals of Statistics 37(2):979-1012.
arXiv:math.ST/0703360. Note also that BIC is calculated as 2L - k ln N,
so "higher is better."
When the EM algorithm is used with latent variable models, we
recommend multiple random restarts. The number of restarts can be set in
the lower right hand corner of the Estimator Box.
Generalized Estimator
A generalized graphical model may have non-linear relations and
non-Gaussian distributions. These models are automatically estimated by
the Powell method, which seeks a maximum likelihood solution.
Updater
Box
The updater box takes an instantiated model as input, and, given
information about the values of parameters in that model, updates the
information about the values and relationships of other parameters.
The Updater allows the user to specify values of variables as
“Evidence.” The default is that the conditional probabilities (Bayes net
models; categorical variables) or conditional means (SEM models;
continuous variables) are computed. For any variable for which evidence
is specified, the user can click on “Manipulated,” in which case the
Updater will calculate the conditional probabilities or conditional means
for other variables when the evidence variables are forced to have their
specified values. In manipulated calculations, all connections into a
measured variable are discarded, the manipulated variables are treated as
independent of their causes in the graph, and probabilities for variables
that are causes of the manipulated variables are unchanged.
There are four available updater algorithms in Tetrad: the
approximate updater, the row summing exact updater, and the Junction Tree Updater,
and the SEM updater. All except for
the SEM updater function only when given Bayes instantiated models as
input; the SEM updater functions when given a SEM instantiated model as
input. None of the updaters work on cyclic models.
Possible Parent Boxes of the Updater Box:
- An instantiated model box
- An estimator
box
Possible Child Boxes of the Updater Box:
- An instantiated model box (Note that the
instantiated model will have the updated parameters)
Approximate Updater
The approximated updater is a fast but inexact algorithm. It
randomly draws a sample data set from the instantiated model and
calculates the conditional frequency of the variable to be estimated.
Take, for example, the following instantiated model:
When it is input into the approximate updater, the following
window results:
If we click “Do Update Now” now, without giving the updater any
evidence, the right side of the screen changes to show us the marginal
probabilities of the variables.
The blue lines, and the values listed across from them, indicate
the probability that the variable takes on the given value in the input
instantiated model. The red lines indicate the probability that the
variable takes on the given value, given the evidence we’ve added to the
updater.
Since we have added no evidence to the updater, the red and blue
lines are very similar in length. To view the marginal probabilities for
a variable, either click on the variable in the graph to the left, or
choose it from the scrolling menu at the top of the window. At the
moment, they should all be very close to the marginal probabilities taken
from the instantiated model.
Now, we’ll return to the original window. We can do so by clicking
“Edit Evidence” under the Evidence tab. Suppose we know that X1 takes on
the value 1 in our model, or suppose we merely want to see how X1 taking
that value affects the values of the other variables. We can click on the
box that says “1” next to X1. When we click “Do Update Now,” we again get
a list of the marginal probabilities for X1.
Now that we have added evidence, the “red line” marginal
probabilities have changed; for X1, the probability that X1=1 is 1,
because we’ve told Tetrad that that is the case. Likewise, the
probabilities that X1=0 and X1=2 are both 0.
Now, let’s look at the updated marginal probabilities for X2, a
parent of X1.
The first image is the marginal probabilities before we added the
evidence that X1=1. The second image is the updated marginal
probabilities. They have changed; in particular, it has become much more
likely that X2=0.
Under the Mode tab, we can change the type of information that the
updater box gives us. The mode we have been using so far is “Marginals
Only (Multiple Variables).” We can switch the mode to “In-Depth
Information (Single Variable).” Under this mode, when we perform the
update, we receive more information (such as log odds and joints, when
supported; joint probabilities are not supported by the approximate
updater), but only about the variable which was selected in the graph
when we performed the update. To view information about a different
variable, we must re-edit the evidence with that variable selected.
If the variable can take one of several values, or if we know the
values of more than one variable, we can select multiple values by
pressing and holding the Shift key and then making our selections. For
instance, in the model above, suppose that we know that X1 can be 1 or 2,
but not 0. We can hold the Shift key and select the boxes for 1 and 2,
and when we click “Do Update Now,” the marginal probabilities for X2 look
like this:
Since X1 must be 1 or 2, the updated probability that it is 0 is
now 0. The marginal probabilities of X2 also change:
The updated marginal probabilities are much closer to their
original values than they were when we knew that X1 was 1.
Finally, if we are arbitrarily setting the value of a
variable—that is, the values of its parents have no effect on its
value—we can check the “Manipulated” box next to it while we are we
editing evidence, and the update will reflect this information.
Note that multiple values cannot be selected for evidence for SEM
models.
Row Summing Exact Updater
The row summing exact updater is a slower but more accurate
updater than the approximate updater. The complexity of the algorithm
depends on the number of variables and the number of categories each
variable has. It creates a full exact conditional probability table and
updates from that. Its window functions exactly as the approximate
updater does, with two exceptions: in “Multiple Variables” mode, you can
see conditional as well as marginal probabilities, and in “Single
Variable” mode, you can see joint values.
Junction Tree Exact Updater
The Junction Tree exact updater is another exact learning
algorithm. Its window functions exactly as the approximate updater down,
with one exception: in “Multiple Variables” mode, you can see conditional
as well as marginal probabilities.
SEM Updater
The SEM updater does not deal with marginal probabilities;
instead, it estimates means.
When it is input to the SEM updater, the following window results:
Suppose we know that the mean of X1 is .5. When we enter that
value into the text box on the left and click “Do Update Now,” the model
on the right updates to reflect that mean, changing the means of both X1
and several other variables. In the new model, the means of X2, X4, and
X5 will all have changed. If we click the “Manipulated” check box as
well, it means that we have arbitrarily set the mean of X1 to .5, and
that the value of its parent variable, X4, has no effect on it. The
graph, as well as the updated means, changes to reflect this.
The rest of the window has the same functionality as a SEM
instantiated model window, except as noted above.
Knowledge
Box
The knowledge box takes as input a graph or a data set and imposes
additional constraints onto it, to aid with search.
Possible Parent Boxes of the Knowledge Box:
- A graph box
- A parametric model box
- An instantiated model box
- A data box
- A simulation
box
- A search box
- Another knowledge box
Possible Child Boxes of the Knowledge Box:
- A search box
- Another knowledge box
Tiers and Edges
The tiers and edges option allows you to sort variables into
groupings that can or cannot affect each other. It also allows you to
manually add forbidden and required edges one at a time.
Tiers
The tiers tab for a graph with ten variables looks like this:
Tiers separate your variables into a timeline. Variables in
higher-numbered tiers occur later than variables in lower-numbered tiers,
which gives Tetrad information about causation. For example, a variable
in Tier 3 could not possibly be a cause of a variable in Tier 1.
To place a variable in a tier, click on the variable in the “Not
in tier” box, and then click on the box of the tier. If you check the
“Forbid Within Tier” box for a tier, variables in that tier will not be
allowed to be causes of each other. To increase or decrease the number of
tiers, use the scrolling box in the upper right corner of the window.
You can quickly search, select and place variables in a tier using
the Find button associated with each tier. Enter a search string into the
Find dialogue box using asterisks as wildcard indicators. E.g., "X1*"
would find and select variables X1 and X10.
You can also limit the search such that edges from one tier only
are added to the next immediate tier e.g., if Tier 1 "Can cause only next
tier" is checked then edges from variables in Tier 1 to variables in Tier
3 are forbidden.
Handling of Interventional Variables in Tiers
If you have annotated your variables with interventional status
and interventional value tags using a metadata JSON file (see Data Box
section) the Tiers and Edges panel will automatically place these
variables in Tier 1. If you have information about the effects of the
intervention variables you can use the groups tab to indicate this.
Groups
The groups tab for a graph with four variables looks like this:
In the groups tab, you can specify certain groups of variables
which are forbidden or required to cause other groups of variables. To
add a variable to the “cause” section of a group, click on the variable
in the box at the top, and then click on the box to the left of the
group’s arrow. To add a variable to the “effect” section of a group,
click on the variable in the box at the top, and then click on the box to
the right of the group’s arrow. You can add a group by clicking on one of
the buttons at the top of the window, and remove one by clicking the
“remove” button above the group’s boxes.
Edges
The edges tab for a graph with four variables looks like this:
In the edges tab, you can require or forbid individual causal
edges between variables. To add an edge, click the type of edge you’d
like to create, and then click and drag from the “cause” variable to the
“effect” variable.
You can also use this tab to see the effects of the knowledge you
created in the other tabs by checking and unchecking the boxes at the
bottom of the window. You can adjust the layout to mimic the layout of
the source (by clicking “source layout”) or to see the variables in their
timeline tiers (by clicking “knowledge layout”).
Forbidden Graph
If you use a graph as input to a knowledge box with the “Forbidden
Graph” operation, the box will immediately add all edges in the parent
graph as forbidden edges. It will otherwise work like a Tiers and Edges
box.
Required Graph
If you use a graph as input to a knowledge box with the “Required
Graph” operation, the box will immediately add all edges in the parent
graph as required edges. It will otherwise work like a Tiers and Edges
box.
Measurement Model
This option allows you to build clusters for a measurement model.
When first opened, the window looks like this:
You can change the number of clusters using the text box in the
upper right hand corner. To place a variable in a cluster, click and drag
the box with its name into the cluster pane. To move multiple variables
at once, shift- or command-click on the variables, and (without releasing
the shift/command button or the mouse after the final click) drag. In the
search boxes, these variables will be assumed to be children of a common
latent cause.
Simulation
Box
The simulation box takes a graph, parametric model, or
instantiated model and uses it to simulate a data set.
Possible Parent Boxes of the Simulation Box
- A graph box
- A parametric model box
- An instantiated model box
- An estimator box
- A data
box
- Another simulation box
- A search box
- An
updater box
- A regression box
Possible Child Boxes of the Simulation Box
- A graph box
- A compare box
- A
parametric model box
- An instantiated model box
- An
estimator box
- A data box
- Another simulation box
- A search box
- A classify box
- A regression box
- A knowledge box
Using the Simulator Box
When you first open the simulation box, you will see some
variation on this window:
The “True Graph” tab contains the graph from which data is
simulated.
The Simulation Box with no Input
Because it has no input box to create constraints, a parentless
simulation box offers the greatest freedom for setting the graph type,
model type, and parameters of your simulated data. In particular, it is
the only way that the simulation box will allow you to create a random
graph or graphs within the box. (If you are simulating multiple data
sets, and want to use a different random graph for each one, you can
select “Yes” under “Yes if a different graph should be used for each
run.”) You can choose the type of graph you want Tetrad to create from
the “Type of Graph” drop-down list.
Random Forward DAG
This option creates a DAG by randomly adding forward edges (edges
that do not point to a variable’s ancestors) one at a time. You can
specify graph parameters such as number of variables, maximum and minimum
degrees, and connectedness.
Erdos Renyi DAG
This option creates a DAG by randomly adding edges with a given edge
probability. The graph is then oriented as a DAG by choosing a
causal order.
Scale Free DAG
This option creates a DAG whose variable’s degrees obey a power
law. You can specify graph parameters such as number of variables, alpha,
beta, and delta values.
Cyclic, constructed from small loops
This option creates a cyclic graph. You can specify graph
parameters such as number of variables, maximum and average degrees, and
the probability of the graph containing at least one cycle.
It is very important when dealing with cyclic models to realize
that the potential exists always to instantiate these models with
coefficients that are too large. Always, to keep simulations from
"exploding" ("diverging"--i.e., having simulation values that
tend to infinity over time), it is necessary to make sure that
coefficient values are relatively small, usually less than 1.
One can tell whether a model will produce simulations that diverge
in value by testing the eigenvalues of the covariance matrix
of the data. If any of these eigenvalues are greater than 1,
the potential exists for the simulation to "explode" toward
infinity over time.
Random One Factor MIM
This option creates a one-factor multiple indicator model. You
can specify graph parameters such as number of latent nodes, number of
measurements per latent, and number of impure edges.
Random Two Factor MIM
This option creates a two-factor multiple indicator model. You
can specify graph parameters such as number of latent nodes, number of
measurements per latent, and number of impure edges.
In addition to the graph type, you can also specify the type of
model you would like Tetrad to simulate.
Bayes net
Simulates a Bayes instantiated model. You can specify model
parameters including maximum and minimum number of categories for each
variable.
Structural Equation Model
Simulates a SEM instantiated model. You can specify model
parameters including coefficient, variance, and covariance ranges.
Linear Fisher Model
Simulates data using a linear Markov 1 DBN without concurrent
edges. The Fisher model suggests that shocks should be applied at
intervals and the time series be allowed to move to convergence between
shocks. This simulation has many parameters that can be adjusted, as
indicated in the interface. The ones that require some explanation are as
follows.
- Low end of coefficient range, high end of
coefficient range, low end of variance range, high end of variance range.
Each variable is a linear function of the parents of the variable (in the
previous time lag) plus Gaussian noise. The coefficients are drawn
randomly from U(a, b) where a is the low end of the coefficient range and
b is the high end of the coefficient range. Here, a < b. The Gaussian
noise is drawn uniformly from U(c, d), where c is the low end of the
variance range and d is the high end of the variance range. Here, c < d.
- Yes, if negative values should be considered. If no, only
positive values will be recorded. This should not be used for large
numbers of variables, since it is more difficult to find cases with all
positive values when the number of variables is large.
- Percentage of discrete variables. The model generates continuous
data, but some or all of the variables may be discretized at random. The
user needs to indicate the percentage of variables (randomly chosen that
one wishes to have discretized. The default is zero—i.e., all continuous
variables.
- Number of categories of discrete variables. For the
variables that are discretized, the number of categories to use to
discretize each of these variables.
- Sample size. The number of
records to be simulated.
- Interval between shocks. The number of
time steps between shocks in the model.
- Interval between data
recordings. The data are recorded every so many steps. If one wishes to
allow to completely converge between steps (i.e., produce equilibrium
data), set this interval to some large number like 20 and set the
interval between shocks likewise to 20 Other values can be used, however.
- Epsilon for convergence. Even if you set the interval between
data recordings to a large number, you can specify an epsilon such that
if all values of variables differ from their values one time step back by
less than epsilon, the series will be taken to have converged, and the
remaining steps between data recordings will be skipped, the data point
being recorded at convergence.
Lee & Hastie
This is a model for simulating mixed data (data with both
continuous and discrete variables. The model is given in Lee J, Hastie T.
2013, Structure Learning of Mixed Graphical Models, Journal of Machine
Learning Research 31: 388-396. Here, mixtures of continuous and discrete
variables are treated as log-linear.
- Percentage of discrete variables. The model
generates continuous data, but some or all of the variables may be
discretized at random. The user needs to indicate the percentage of
variables (randomly chosen that one wishes to have discretized. The
default is zero—i.e., all continuous variables.
- Number of
categories of discrete variables. For the variables that are discretized,
the number of categories to use to discretize each of these variables.
- Sample size. The number of records to be simulated.
Time Series
This is a special simulation for representing time series.
Concurrent edges are allowed. This can take a Time Series Graph as input,
in which variables in the current lag are written as functions of the
parents in the current and previous lags.
- Sample size. The number of records to be
simulated.
Boolean Glass
The instantiated model used to simulate the data will be
re-parameterized for each run of the simulation.
The Simulation Box with a Graph Input
If you input a graph, you will be able to simulate any kind of
model, with any parameters. But the model will be constrained by the
graph you have input (or the subgraph you choose in the “True Graph”
tab.) Because of this, if you create a simulation box with a graph as a
parent, you will not see the “Type of Graph” option.
The Simulation Box with a Parametric Model Input
At the time of writing, a simulation box with a parametric model
input acts as though the PM’s underlying graph had been input into the
box.
The Simulation Box with an Instantiated Model Input
If you input an instantiated model, your only options will be the
sample size of your simulation and the number of data sets you want to
simulate; Tetrad will simulate every one of them based on the parameters
of the IM. The model will not be re-parameterized for each run of the
simulation.
Search
Box
The search box takes as input a data set (in either a data or
simulation box) and optionally a knowledge box, and searches for causal
explanations represented by directed graphs. The result of a search is
not necessarily—and not usually—a unique graph, but an object such as a
CPDAG that represents a set of graphs, usually a Markov Equivalence
class. More alternatives can be found by varying the parameters of search
algorithms.
Possible Parent Boxes of the Search Box
- A graph box
- A parametric model box
- An instantiated model box
- An estimator box
- A data
box
- A simulation box
- Another search box
- A
regression box
- A knowledge box
Possible Child Boxes of the Simulation Box
- A graph box
- A compare box
- A
parametric model box
- A simulation box
- Another search
box
- A knowledge box
Using the Search Box
For more information that in included about the search algorithms than
is included in the text below, please see our
Javadocs for the Search package.
Using the search box requires you to select an algorithm
(optionally select a test/score), confirm/change search parameters and
finally run the search.
The search box first asks what algorithm, statistical tests and/or
scoring functions you would like to use in the search. The upper left
panel allows you to filter for different types of search algorithms with
the results of filtering appearing in the middle panel. Selecting a
particular algorithm will update the algorithm description on the right
panel.
Choosing the correct algorithm for your needs is an important
consideration. Tetrad provides over 30 search algorithms (and more are
added all the time) each of which makes different assumptions about
the input data, uses different parameters, and produces different kinds
of output. For instance, some algorithms produce Markov blankets or
CPDAGs, and some produce full graphs; some algorithms work best with
Gaussian or non-Gaussian data; some algorithms require an alpha value,
some require a penalty discount, and some require both or neither. You
can narrow down the list using the “Algorithm filter" panel, which allows
you to limit the provided algorithms according to whichever factor is
important to you.
Depending on the datatype used as input for the search (i.e.,
continuous, discrete, or mixed data) and algorithm selected, the lower
left panel will display available statistical tests (i.e., tests of
independence) and Bayesian scoring functions.
After selecting the algorithm and desired test/score, click on
"Set parameters" which will allow you to confirm/change the parameters of
the search.
After optionally changing any search parameters, click on "Run
Search and Generate Graph" which will execute the search.
Notably there are some experimental algorithms available in this box.
To see these, select File->Settings->Enable Experimental.
Search Algorithms
PC
Description
PC algorithm (Spirtes and Glymour, Social Science Computer
Review, 1991) is a CPDAG search which assumes that the underlying
causal structure of the input data is acyclic, and that no two
variables are caused by the same latent (unmeasured) variable. In
addition, it is assumed that the input data set is either entirely
continuous or entirely discrete; if the data set is continuous, it is
assumed that the causal relation between any two variables is linear,
and that the distribution of each variable is Normal. Finally, the
sample should ideally be i.i.d.. Simulations show that PC and several
of the other algorithms described here often succeed when these
assumptions, needed to prove their correctness, do not strictly hold.
The PC algorithm will sometimes output double-headed edges. In the
large sample limit, double-headed edges in the output indicate that
the adjacent variables have an unrecorded common cause, but PC tends
to produce false positive double-headed edges on small samples.
The PC algorithm is correct whenever decision procedures for
independence and conditional independence are available. The
procedure conducts a sequence of independence and conditional
independence tests, and efficiently builds a CPDAG from the results
of those tests. As implemented in TETRAD, PC is intended for
multinomial and approximately Normal distributions with i.i.d. data.
The tests have an alpha value for rejecting the null hypothesis,
which is always a hypothesis of independence or conditional
independence. For continuous variables, PC uses tests of zero
correlation or zero partial correlation for independence or
conditional independence respectively. For discrete or categorical
variables, PC uses either a chi square or a g square test of
independence or conditional independence (see Causation, Prediction,
and Search for details on tests). In either case, the tests require
an alpha value for rejecting the null hypothesis, which can be
adjusted by the user. The procedures make no adjustment for multiple
testing. (For PC, CPC, FCI, all testing searches.)
The PC algorithm as given in Causation, Prediction and Search (Spirtes,
Glymour, and Scheines, 2000) comes with three heuristics designed
to reduce dependence on the order of the variables. The heuristic PC-1
simple sorts the variables in alphabetical order. The heuristic PC-2
and PC-3 sort edges by their p-values in the search. PP-3 further sorts
parents of nodes in reverse order by the p-values of the conditional
independence facts used to removed edges in the search. Please see
Causation, Prediction, and Search for more details for these
heuristics.
Note that it is possible for PC with some choices of parameters to output
bidirected edges or cycles, or to imply cycles by the Meek rules, against
assumption. Bidirected edges can be prevented by choosing an appropriate collider
conflict rule. Cycles can be prevented by setting the guaranteeCpdag
parameter to 'true' ('Yes). This parameter has two effects. First, it prevents
the orientation of any collider that would create a cycle in the graph.
Second, whenever the final Meek rules attempt to directed an undirected edge
as a directed edge, if this orientation would create a cycle, the edge
is oriented in reverse, adding one or more new (arbitrary) unshielded triples
to the graph. When this happens, the behavior is logged.
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
The algorithm effectively takes conditional independence facts as
input. Thus, it will work for any type of data for which a conditional
independence facts are known. In the interface, it will work for linear,
Gaussian data (the Fisher Z test), discrete multinomial data the Chi
Square test) and mixed multinomial/Gaussian data (the Conditional
Gaussian test).
Output Format
The graph outputs a CPDAG. This is an equivalence class of
directed acyclic graphs (DAGs). Each DAG in the equivalence class has all
the adjacencies (and no more) of the CPDAG. Each oriented edge in the
CPDAG is so oriented in each of the DAG in the equivalence class.
Unoriented edges in the equivalence class cannot be oriented by
conditional independence facts. For example, if the model is X->Y->Z, the
output will be X—Y—Z. There are no collider in this model, so the
algorithm will not detect one. Since there are no colliders, the Meek
cannot orient additional edges. If the model were X<-Y<-Z, the output
would also be X—Y—Z; this model is in the same equivalence class as
X->Y->Z. The model X->Y<-Z would be its own equivalence class, since the
collider in this model can be oriented. See Spirtes et al. (2000) for
more details.
Parameters
The CPC Algorithm
Description
The CPC (Conservative PC) algorithm (Ramsey et al., ??)
modifies the collider orientation step of PC to make it more
conservative—that is, to increase the precision of collider
orientations at the expense of recall. It does this as follows. Say
you want to orient X—Y—Z as a collider or a noncollider; the PC
algorithm looks at variables adjacent to X or variables adjacent to Z
to find a subset S such that X is independent of Z conditional on S.
The CPC algorithm considers all possible such sets and records the
set on which X is conditionally independent of Z. If all of these
sets contain Y, it orients X—Y—Z as a noncollider. If none of them
contains Z, if orient X—Y—Z as a collider. If some contain Z but
other don’t, it marks it as ambiguous, with an underline. Thus, the
output is ambiguous between CPDAGs; in order to get a specific CPDAG
out of the output, one needs first to decide whether the underlined
triples are colliders or noncolliders and then to apply the
orientation rules in Meek (1997).
The PC algorithm is correct whenever decision procedures for
independence and conditional independence are available. The
procedure conducts a sequence of independence and conditional
independence tests, and efficiently builds a CPDAG from the results
of those tests. As implemented in TETRAD, PC is intended for
multinomial and approximately Normal distributions with i.i.d. data.
The tests have an alpha value for rejecting the null hypothesis,
which is always a hypothesis of independence or conditional
independence. For continuous variables, PC uses tests of zero
correlation or zero partial correlation for independence or
conditional independence respectively. For discrete or categorical
variables, PC uses either a chi square or a g square test of
independence or conditional independence (see Causation, Prediction,
and Search for details on tests). In either case, the tests require
an alpha value for rejecting the null hypothesis, which can be
adjusted by the user. The procedures make no adjustment for multiple
testing. (For PC, CPC, FCI, all testing searches.)
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
Same as for PC.
Output Format
An e-CPDAG (extended CPDAG), consistent of directed and undirected
edges where some of the triple may have been marked with underlines to
indicate ambiguity, as above. It may be that bidirected edges are
oriented as X->Y<->X<-W if two adjacent colliders are oriented; this is
not ruled out.
Parameters
The PcMax Algorithm
Description
Similar in spirit to CPC but orients all unshielded triples
using maximum likelihood conditioning sets. The idea is as follows.
The adjacency search is the same as for PC, but colliders are
oriented differently. Let X—Y—Z be an unshielded triple (X not
adjacent to Z) and find all subsets S from among the adjacents of X
or the adjacents of Z such that X is independent of Z conditional on
S. However, instead of using the CPC rule to orient the triple,
instead just list the p-values for each of these conditional
independence judgments and pick the set S’ that yields the highest
such p-value. Then orient X->Y<-Z if S does not contain Y and X—Y—Z
otherwise. This orients all unshielded triples. It’s possible (though
rare) that adjacent triples both be oriented as 2-cycles,
X->Y<->Z<-W. If this happens, pick one of the other of these triples
or orient as a collider, arbitrarily. This guarantees that the
resulting graph will be a CPDAG.
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
Same as for PC.
Output Format
Same as PC, a CPDAG.
Parameters
alpha, depth, useMaxPOrientationHeuristic, maxPOrientationMaxPathLength
The FGES Algorithm
Description
FGES is an optimized and parallelized version of an algorithm
developed by Meek [Meek, 1997] called the Greedy Equivalence Search
(GES). The algorithm was further developed and studied by Chickering
[Chickering, 2002]. GES is a Bayesian algorithm that heuristically
searches the space of CBNs and returns the model with the highest
Bayesian score it finds. In particular, GES starts its search with
the empty graph. It then performs a forward stepping search in which
edges are added between nodes in order to increase the Bayesian
score. This process continues until no single edge addition increases
the score. Finally, it performs a backward stepping search that
removes edges until no single edge removal can increase the score.
More information is available here and here. The reference
is Ramsey et al., 2017.
The algorithm requires a decomposable score—that is, a score
that for the entire DAG model is a sum of logged scores of each
variables given its parents in the model. The algorithms can take all
continuous data (using the SEM BIC score), all discrete data (using
the BDeu score) or a mixture of continuous and discrete data (using
the Conditional Gaussian score); these are all decomposable scores.
Note that in all case, BIC is calculated as 2L - k ln N, so "higher
is better."
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Note: It is possible to run FGES followed some non-Gaussian
orientation algorithm like FASK-pairwise or R3 or RSkew. To do this,
see the FASK algorithm. There one can select an algorithm to
estimate adjacencies and a pairwise algorithm to estimate orientations.
This is for the linear, non-Gaussian case, where such pairwise
algorithms are effective.
Input Assumptions
Data that’s all continuous, all discrete, or a mixture of
continuous and discrete variables. Continuous variables will be assumed
to be linearly associated; discrete variable will be assumed to be
associated by multinomial conditional probability tables. Continuous
variables for the mixed case will be assumed to be jointly Gaussian.
Output Format
A CPDAG, same as PC.
The IMaGES Algorithm
Description
Adjusts the selected score for FGES to
allow for multiple datasets as input. The linear, Gaussian BIC scores
for each data set are averaged at each step of the algorithm,
producing a model for all data sets that assumes they have the same
graphical structure across dataset. Note that BIC is calculated
as 2L - k ln N, so "higher is better."
Input Assumptions
A set of datasets consistent with the chosen score with the same variables and sample
sizes.
Output Format
A CPDAG, interpreted as a common model for all datasets.
Parameters
All the parameters from FGES are available for IMaGES.
Additionally:
The IMaGES-BOSS Algorithm
Description
Wraps the IMaGES algorithm for continuous variables. This version uses the BOSS algorithm in place of
FGES.
Requires that the parameter 'randomSelectionSize' be set to indicate how many datasets should be taken at a
time (randomly). This cannot be given multiple values.
Input Assumptions
A set of datasets consistent with the chosen score with the same variables and sample
sizes.
Output Format
A CPDAG, interpreted as a common model for all datasets.
Parameters
All the parameters from FGES are available for IMaGES.
Additionally:
The FCI Algorithm
Description
The FCI algorithm is a constraint-based algorithm that takes
as input sample data and optional background knowledge and in the
large sample limit outputs an equivalence class of CBNs that
(including those with hidden confounders) that entail the set of
conditional independence relations judged to hold in the population.
It is limited to several thousand variables, and on realistic sample
sizes it is inaccurate in both adjacencies and orientations. FCI has
two phases: an adjacency phase and an orientation phase. The
adjacency phase of the algorithm starts with a complete undirected
graph and then performs a sequence of conditional independence tests
that lead to the removal of an edge between any two adjacent
variables that are judged to be independent, conditional on some
subset of the observed variables; any conditioning set that leads to
the removal of an adjacency is stored. After the adjacency phase, the
resulting undirected graph has the correct set of adjacencies, but
all the edges are unoriented. FCI then enters an orientation phase
that uses the stored conditioning sets that led to the removal of
adjacencies to orient as many of the edges as possible. See [Spirtes,
1993].
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
The data are continuous, discrete, or mixed.
Output Format
A partial ancestral graph (see Spirtes et al., 2000).
Parameters
All the parameters from FCI are below.
depth, maxPathLength, completeRuleSetUsed
The FCI-Max Algorithm
Description
The FCI-Max algorithm simply changes the first collider
orientation rule in FCI to use the PC-Max orientation.
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
The data are continuous, discrete, or mixed.
Output Format
A partial ancestral graph (see Spirtes et al., 2000).
Parameters
All the parameters from FCI are below.
depth, maxPathLength, completeRuleSetUsed
The CFCI Algorithm
Description
Adjusts FCI (see) to use conservative orientation as in CPC (see). Because the collider orientation is
conservative,
there may be ambiguous triples; these may be retrieved using that accessor method.
This class is configured to respect knowledge of forbidden and required edges, including knowledge of
temporal
tiers.
Input Assumptions
Data for which a conditional independence test is available.
Output Format
An adjustment to a partial ancestral graph (PAG), where ambiguous colliders are marked with underlings. See
Spirtes et al., 2000. Also, see the CPC algorithm.
Parameters
All the parameters from FCI are available for RFCI.
Additionally:
depth, maxPathLength, completeRuleSetUsed
The RFCI Algorithm
Description
A modification of the FCI algorithm in which some expensive
steps are finessed and the output is somewhat differently
interpreted. In most cases this runs faster than FCI (which can be
slow in some steps) and is almost as informative. See Colombo et al.,
2012.
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
Data for which a conditional independence test is available.
Output Format
A partial ancestral graph (PAG). See Spirtes et al., 2000.
Parameters
All the parameters from FCI are available for RFCI.
Additionally:
depth, maxPathLength, completeRuleSetUsed
The PAG-Sampling RFCI Algorithm
Description
A modification of the RFCI algorithm which does probabilistic bootstrap
sampling with respect to the RFCI PAG output.For discrete data only.
Parameters are: (a) Number of search probabilistic models, (b) boostrap
ensemble method to use (see bootstrapping), (c) maximum size of
conditioning set (depth), (d) maximum length of any discriminating path
(a property for RFCI). This must use the probabilistic test, which must
be selected. Parameters for the probabilistic test are (d) independence
cutoff threshold, default 0.5, (e) prior equivalent sample size, and
(f) whether the cutoff in (d) is used in the independence test calculation;
if not, then a coin flip is used (probability 0.5).
Input Assumptions
A discrete dataset.
Output Format
A partial ancestral graph (PAG). See Spirtes et al., 2000.
The GFCI Algorithm
Description
GFCI is a combination of the FGES [FGES, 2016] algorithm
and the FCI algorithm [Spirtes, 1993] that improves upon the accuracy
and efficiency of FCI. In order to understand the basic methodology
of GFCI, it is necessary to understand some basic facts about the
FGES and FCI algorithms. The FGES algorithm is used to improve the
accuracy of both the adjacency phase and the orientation phase of FCI
by providing a more accurate initial graph that contains a subset of
both the non-adjacencies and orientations of the final output of FCI.
The initial set of nonadjacencies given by FGES is augmented by FCI
performing a set of conditional independence tests that lead to the
removal of some further adjacencies whenever a conditioning set is
found that makes two adjacent variables independent. After the
adjacency phase of FCI, some of the orientations of FGES are then
used to provide an initial orientation of the undirected graph that
is then augmented by the orientation phase of FCI to provide
additional orientations. A verbose description of GFCI can be found
here (discrete
variables) and here (continuous variables).
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
Same as for FCI.
Output Format
Same as for FCI.
Parameters
Uses all of the parameters of FCI (see Spirtes et al., 1993) and
FGES (see FGES, 2016).
The GRaSP-FCI Algorithm
Description
GRaSP-FCI is wrapper around the GRaSP algorithm that replaces
the FGES step in GFCI with the more accurate GRaSP algorithm, which
reasons by considering permutations of variables. The second collider
orientation step in GFCI is also done using permutation reasoning,
leaving the discriminating path rule as the only rule that requires a
"raw" conditional independence judgment. Ultimately this independence
judgment can be decided using the same scoring apparatus as the other
permutation steps, so ultimately GRaSP-FCI can be treated as a scoring
algorithm.
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
Same as for FCI.
Output Format
Same as for FCI.
Parameters
Uses all of the parameters of FCI (see Spirtes et al., 1993) and
GRaSP.
The LV-Lite Algorithm
Description
LV-Lite starts with BOSS or GRaSP and get a valid order and a DAG/CPDAG. It sets the initial state of the
estimted PAG to this CPDAG and orients as o-o and then copy all of the unshielded colliders from the CPDAG
to the PAG. It then performs a testing modified from the step in GFCI that is more efficient. In GFCI, an
edge x *-* y is removed from the graph is there is a subset S of adj(x) \ {y} or adj(y) \ {x} such that x
_||_ y | S. For LV-Lite we instead search outward from x for a set S using a novel procedure that blocks all
existing paths in the estimated PAG either unconditionally or conditionally. Bounds may be placed on the
lengths of the blocked paths to consider and the depth (maximum |S|). We also allow bounds on the time
spent on any testing step. As edges are removed in this way, any additional colliders are oriented given
these sepsets S. We then run the final FCI orientation rules, where for the discriminating path rule we use
the same "out of x" sepset finding idea. We optionally allow the user to request that a legal PAG be
returned using a novel procedure; this guarantee has been extended to all latent variable algorithms in
Tetrad that return partial ancestral graphs (PAGs).
LV-Lite, along with BFCI, can produce highly accurate models in simulation. LV-Lite, in particular, is highly
scalable. A paper describing BFCI and LV-Lite in more detail is planned. We make both BFCI and LV-Lite
non-experimental for this version since requests have been made to use them.
Note: If one wants to analyze time series data using this algorithm, one may set the time lag parameter to a
value greater than 0, which will automatically apply the time lag transform. The same goes for any algorithm
that has this parameter available in the interface.
Input Assumptions
Same as for FCI.
Output Format
Same as for FCI.
Parameters
Uses all of the parameters of FCI (see Spirtes et al., 1993) and
GRaSP.
The BFCI Algorithm
Description
Uses BOSS in place of FGES for the initial step in the GFCI algorithm. This tends to produce a accurate PAG
than GFCI as a result, for the latent variables case. This is a simple substitution; the reference for GFCI
is here:
J.M. Ogarrio and P. Spirtes and J. Ramsey, "A Hybrid Causal Search Algorithm for Latent Variable Models,"
JMLR 2016. Here, BOSS has been substituted for FGES.
For BOSS only a score is needed, but there are steps in GFCI that require a test, so for this method, both a
test and a score need to be given. The reference for BOSS is:
Andrews, B., Ramsey, J., Sanchez Romero, R., Camchong, J., & Kummerfeld, E. (2023). Fast scalable and
accurate discovery of dags using the best order score search and grow shrink trees. Advances in Neural
Information Processing Systems, 36, 63945-63956.
This class is configured to respect knowledge of forbidden and required edges, including knowledge of
temporal tiers.
LV-Lite, along with BFCI, can produce highly accurate models in simulation. LV-Lite, in particular, is highly
scalable. A paper describing BFCI and LV-Lite in more detail is planned. We make both BFCI and LV-Lite
non-experimental for this version since requests have been made to use them.
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
Input Assumptions
Same as for FCI.
Output Format
Same as for FCI.
Parameters
Uses all of the parameters of FCI (see Spirtes et al., 1993) and
GRaSP.
The SP-FCI Algorithm
Description
SP-FCI is wrapper around the SP algorithm that replaces
the FGES step in GFCI with the more accurate SP algorithm, which
reasons by considering permutations of variables. This uses the
same method for wrapping SP with an FCI method similar to GFCI
as GRaSP-FCI. The difference is that SP considers every permutation
of the variables and so necessarily return a frugal DAG. This
can only be used for very small models, of up to about 8 or 9
variables, because of the super-exponential step of considering
every permutation of the variables. The second collider
orientation step in GFCI is done using permutation reasoning,
leaving the discriminating path rule as the only rule that requires a
"raw" conditional independence judgment. Ultimately this independence
judgment can be decided using the same scoring apparatus as the other
permutation steps, so ultimately GRaSP-FCI can be treated as a scoring
algorithm.
Input Assumptions
Same as for FCI.
Output Format
Same as for FCI.
Parameters
Uses all of the parameters of FCI (see Spirtes et al., 1993) and
SP.
The FCI-IOD Algorithm
Description
Runs FCI on multiple datasets using the IOD pooled dataset independence test. The reference is here:
Tillman, R., & Spirtes, P. (2011, June). Learning equivalence classes of acyclic models with latent and
selection variables from multiple datasets with overlapping variables. In Proceedings of the Fourteenth
International Conference on Artificial Intelligence and Statistics (pp. 3-15). JMLR Workshop and Conference
Proceedings.
The SvarFCI Algorithm
Description
The SvarFCI algorithm is a version of FCI for time series
data. See the FCI documentation for a description of the FCI
algorithm, which allows for unmeasured (hidden, latent) variables in
the data-generating process and produces a PAG (partial ancestral
graph). svarFCI takes as input a “time lag data set,” i.e., a data
set which includes time series observations of variables X1, X2, X3,
..., and their lags X1:1, X2:1, X3:1, ..., X1:2, X2:2,X3:2, ... and
so on. X1:n is the nth-lag of the variable X1. To create a time lag
data set from a standard tabular data set (i.e., a matrix of
observations of X1, X2, X3, ...), use the “create time lag data”
function in the data manipulation toolbox. The user will be prompted
to specify the number of lags (n), and a new data set will be created
with the above naming convention. The new sample size will be the old
sample size minus n.
Since this algorithm specifically requires time series data,
one must set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform.
Input Assumptions
The (continuous) data has been generated by a time series, and the
"Convert to Time Lag" converter in the Data box has been used to format
the data as a time lag dataset. (Manual formatting of the data will not
work for this.
Output Format
A PAG over the input variables with stated number of lags.
Parameters
The SvarGFCI Algorithm
Description
SvarGFCI uses a BIC score to search for a skeleton. Thus, the
only user-specified parameter is an optional “penalty score” to bias
the search in favor of more sparse models. See the description of the
GES algorithm for discussion of the penalty score. For the
traditional definition of the BIC score, set the penalty to 1.0. The
orientation rules are the same as for FCI. As is the case with
SvarFCI, SvarFCI will automatically respect the time order of the
variables and impose a repeating structure. Firstly, it puts lagged
variables in appropriate tiers so, e.g., X3:2 can cause X3:1 and X3
but X3:1 cannot cause X3:2 and X3 cannot cause either X3:1 or X3:2.
Also, it will assume that the causal structure is the same across
time, so that if the edge between X1 and X2 is removed because this
increases the BIC score, then also the edge between X1:1 and X2:1 is
removed, and so on for additional lags if they exist. When some edge
is removed as the result of a score increase, all similar (or
“homologous”) edges are also removed. Note that BIC is calculated
as 2L - k ln N, so "higher is better."
Since this algorithm specifically requires time series data,
one must set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform.
Input Assumptions
The (continuous) data has been generated by a time series, and the
"Convert to Time Lag" converter in the Data box has been used to format
the data as a time lag dataset. (Manual formatting of the data will not
work for this.
Output Format
A PAG over the input variables with stated number of lags.
Parameters
Uses all of the parameters of FCI (see Spirtes et al., 1993) and
FGES (see FGES, 2016).
Input Assumptions
The (continuous) data has been generated by a time series.
Output Format
A PAG over the input variables with stated number of lags.
Parameters
Uses the parameters of IMaGES.
The CCD (Cyclic Causal Discovery) Algorithm
Description
CCD assumes that the data are causally sufficient (no latent variables)
though possibly with directed cycles. No background knowledge is permitted.
It generates a "cyclic PAG". See Richardson, T. S. (2013). A discovery
algorithm for directed cyclic graphs. arXiv preprint arXiv:1302.3599,
for details. Note that the output graph contains circle endpoints
as with a latent variable PAG, but these are interpreted differently.
CCD reasons about cyclic (feedback) models using conditional
independence facts alone, as with PC or FCI, so is general in this
sense.
Input Assumptions
Data from a possibly cyclic (feedback) model without latent variables for
which an independence test is available.
Output Format
Same as for FCI.
A cyclic PAG--see reference above.
Parameters
Cutoff for alpha, maximum size of conditioning set,
Yes if orient away from arrow rule should be applied,
Yes if verbose output should be printed.
The FGES-MB Algorithm
Description
This is a restriction of the FGES algorithm to union of edges
over the combined Markov blankets of a set of targets, including the
targets. In the interface, just one target may be specified. See
Ramsey et al., 2017 for details. In the general case, finding the
graph over the Markov blanket variables of a target (including the
target) is far faster than finding the CPDAG for all the
variables.
Input Assumptions
The same as FGES
Output Format
A graph over a selected group of nodes that includes the target
and each node in the Markov blanket of the target. This will be the same
as if FGES were run and the result restricted to just these variables, so
some edges may be oriented in the returned graph that may not have been
oriented in a CPDAG over the selected nodes.
Parameters
Uses the parameters of FGES (see FGES, 2016).
The PC-MB Algorithm
Description
PC-MB. Similar to FGES-MB (see FGES,
2016) but using PC as the basic search instead of FGES. The rules of
the PC search are restricted to just the variables in the Markov
blanket of a target T, including T; the result is a graph that is a
CPDAG over these variables.
Input Assumptions
Same as for PC
Output Format
A CPDAG over a selected group of nodes that includes the target
and each node in the Markov blanket of the target.
Parameters
Uses the parameters of PC.
The FAS Algorithm
Description
This is just the adjacency search of the PC algorithm,
included here for times when just the adjacency search is needed, as
when one is subsequently just going to orient variables pairwise.
Input Assumptions
Same as for PC
Output Format
An undirected graph over the variables of the input dataset. In
particular, parents of a variables are not married by FAS, so the
resulting graph is not a Markov random field. For example, if X->Y<-Z,
the output will be X—Y—Z with X—Z. The parents of Y will be joined by an
undirected edge, morally, only if they are joined by a trek in the true
graph.
Parameters
The MGM Algorithm
Description
Need reference. Finds a Markov random field (with parents
married) for a dataset in which continuous and discrete variables are
mixed together. For example, if X->Y<-Z, the output will be X—Y—Z
with X—Z. The parents of Y will be joined by an undirected edge,
morally, even though this edge does not occur in the true model.
Input Assumptions
Data are mixed.
Output Format
A Markov random field for the data.
Parameters
mgmParam1, mgmParam2, mgmParam3
The SP Algorithm
Description
SP (Sparsest Permutation, Raskutti, G., & Uhler, C. (2018). Learning
directed acyclic graph models based on sparsest permutations.
Stat, 7(1), e183) searches for model satisfying the SMR (frugality)
assumption for small models of up to about 9 variables.
The algorithms works searching over all possible permutations
of the variables and building DAGs for them in the same way as the
GRaSP algorithm. Two ways of building DAGs are considered, one
independence-based, due to Raskutti and Uhler and a score-based
method, using the Grow-Shrink (GS) method (Margaritis and Thrun, 1999).
If the Pearl method is selected, an independence test will be used;
if the GS method is selected, a score will be used, so both a test
and a score need to be supplied so that this choice can be made.
Input Assumptions
Causal sufficiency
Output Format
The CPDAG equivalence class of DAGs representing estimated
possible causal structures over the set of variables.
Parameters
graspDepth, graspNonSingularDepth graspSingularDepth graspOrderedAlg verbose numStarts
The GRaSP Algorithm
Description
GRaSP (Greedy Relations of Sparsest Permutation) is an
algorithm that generalizes and extends the GSP (Greedy Sparsest
Permutation) algorithm. Is generalizes specifically the algorithms
TSP (or which GSP is a bounded and iterated version) and ESP (Solus
et al., 2017) by allowing those algorithms to equivalently be run by
selecting particular parameterizations of GRaSP, where ESP enlarges
the search space of GSP. The implementation given for ESP renders
that algorithm tractable. By choosing other parameterizations of
GRaSP, it is possible to further enlarge the search space even over
and above that of ESP to render the algorithm significantly more
accurate. In all cases, these algorithms come back quickly enough to
analyze accurately sparse problems of up to 300 variables and denser
problems with up to about 100 variables.
The parameters are as
follows:
graspDepth - This controls the overall recursion depth
for a depth first search.
graspNonsingularDepth - This controls the
depth at which nonsingular tucks are explored.
graspSingularDepth
- This controls the depth at which singular tucks are considered.
numRestarts - By default 1; if > 1, additional random restarts are
done, and the best of these results is returned.
TSP corresponds
to singular depth 0, nonsingular depth 0. ESP corresponds to singular
depth > 0, nonsingular depth = 0. GRaSP corresponds to singular depth >
0, nonsingular depth > 0. In each case, an ordering option is available
to find the best permutations from lower levels before proceeding to
higher levels.
The algorithm works by building DAGs given
permutations in ways similar to those described in Raskutti and Uhler
(ref?) and Solus et al. Two ways of building DAGs are considered, one
independence-based, due to Raskutti and Uhler and a score-based
method, using the Grow-Shrink (GS) method (Margaritis and Thrun, 1999).
If the Pearl method is selected, an independence test will be used;
if the GS method is selected, a score will be used, so both a test
and a score need to be supplied so that this choice can be made.
We recommend
that the user turn on logging to watch the progress of the algorithm;
this helps with larger searches especially.
Knowledge of
forbidden edges may be used with GRaSP; currently knowledge of
required edges is not implemented.
Dimitris Margaritis and
Sebastian Thrun. Bayesian network induction via local neighborhoods.
Advances in neural information processing systems, 12, 1999.
Raskutti, G., & Uhler, C. (2018). Learning directed acyclic graph
models based on sparsest permutations. Stat, 7(1),
Solus,
L., Wang, Y., Matejovicova, L., & Uhler, C. (2017). Consistency
guarantees for permutation-based causal inference algorithms. arXiv
preprint arXiv:1702.03530.
Lam, W. Y., Andrews, B., & Ramsey, J. (2022, August). Greedy relaxations
of the sparsest permutation algorithm. In Uncertainty in Artificial
Intelligence (pp. 1052-1062). PMLR.
Input Assumptions
Causal sufficiency
Output Format
The CPDAG equivalence class of DAGs representing estimated
possible causal structures over the set of variables.
Parameters
graspDepth, graspNonSingularDepth graspSingularDepth graspOrderedAlg verbose numStarts
The BOSS Algorithm
Description
BOSS (Best Order Score Search) is an algorithm that, like GRaSP,
generalizes and extends the GSP (Greedy Sparsest Permutation) algorithm.
It has been tested to 1000 variables with an average degree of 20
and gives near perfect precisions and recalls for N = 10,000 (with
recall that drop to 0.9 for N = 1000). The algorithm works by building DAGs given
permutations in ways similar to those described in Raskutti and Uhler
(ref?) and Solus et al.
The parameters are as
follows:
useBes - True if the final BES (Backward Equivalence Search)
step is used from the GES (Greedy Equivalence Search) algorithm.
This step is needed for correctness but for large models, since usually
nearly all edges are oriented in the CPDAG, it is heuristically not needed.
numStarts - The number of times the algorithm should be re-run from different
random starting permutations. The model with the most optimal BIC score will
be selected.
allowInternalRandomness - If true, the algorithm allow the algorithm to
use certain heuristic random steps. This can improve performance, but
may make the algorithm non-deterministic.
timeLag - This creates a time-series model automatically with a certain
number of lags.
Knowledge of forbidden edges and required edges may be used with this
algorithm. Also, knowledge of tiers may be used. If tiered knowledge is
supplied, the algorithm will analyze the tiers in order, so that the
time required for the algorithm is linear in the number of tiers.
Dimitris Margaritis and
Sebastian Thrun. Bayesian network induction via local neighborhoods.
Advances in neural information processing systems, 12, 1999.
Raskutti, G., & Uhler, C. (2018). Learning directed acyclic graph
models based on sparsest permutations. Stat, 7(1), e183.
Solus,
L., Wang, Y., Matejovicova, L., & Uhler, C. (2017). Consistency
guarantees for permutation-based causal inference algorithms. arXiv
preprint arXiv:1702.03530.
Lam, W. Y., Andrews, B., & Ramsey, J. (2022, August). Greedy relaxations
of the sparsest permutation algorithm. In Uncertainty in Artificial
Intelligence (pp. 1052-1062). PMLR.
Input Assumptions
Causal sufficiency
Output Format
The CPDAG equivalence class of DAGs representing estimated
possible causal structures over the set of variables.
Parameters
graspDepth, graspNonSingularDepth graspSingularDepth graspOrderedAlg verbose numStarts
The BPC/Mimbuild Algorithm
Description
Searches for causal structure over latent variables, where the
true models are Multiple Indicator Models (MIM’s) as described in the
Graphs section. The idea is this. There is a set of latent
(unmeasured) variables over which a directed acyclic model has been
defined. Then for each of these latent L there are 3 (preferably 4)
or more measures of that variable—that is, measured variables that
are all children of L. Under these conditions, one may define tetrad
constraints (see Spirtes et al., 2000). There is a theorem to the
effect that if certain CPDAGs of these tetrad constraints hold, there
must be a latent common cause of all of them (the Tetrad
Representation Theorem, see Silva et al., 2003, where the BPC (“Build
Pure Clusters”) algorithm is defined and discussed.) Moreover, once
one has such a “measurement model,” once can estimate a covariance
matrix over the latent variables that are parents of the measures and
use some algorithm such as PC or GES to estimate a CPDAG over the
latents. The algorithm to run PC or GES on this covariance matrix is
called MimBuild (“MIM” is the graph, Multiple Indicator Model;
“Build” means build). In this way, one may recover causal structure
over the latents. The more measures one has for each latent, the
better the result is, generally. The larger the sample size the
better. One important issue is that the algorithm is sensitive to
so-called “impurities”—that is, causal edges among the measured
variables, or between measured variables and unintended latent. The
algorithm will in effect remove one measure in each impure pair from
consideration.
The algorithm to run PC or GES on this covariance matrix is
called MimBuild (“MIM” is the graph, Multiple Indicator Model;
“Build” means build). MimBUILD is an optional choice inside FOFC
In this way, one may recover causal structure over the latents.
The more measures one has for each latent the better the result
is, generally.
A reference for the Wishart and Delta Tetrad tests is: Bollen,
K. A., & Ting, K. F. (1993). Confirmatory tetrad analysis.
Sociological methodology, 147-175. The original reference for
the Delta Tetrad test is Bollen, K. A. (1990). Outlier screening
and a distribution-free test for vanishing tetrads. Sociological
Methods & Research, 19(1), 80-92.
Input Assumptions
Continuous data, a collection of measurements in the above sense,
excluding the latent variables (which are to be learned).
Output Format
For BPC, a measurement model, in the above sense. This is
represented as a clustering of variables; it may be inferred that there
is a single latent for each output cluster. For MimBuild, a CPDAG over
the latent variables, one for each cluster.
Parameters
The FOFC/MIMBUILD Algorithm
Description
Searches for causal structure over latent variables, where the
true models are Multiple Indicator Models (MIM’s) as described in the
Graphs section. The idea is this. There is a set of latent
(unmeasured) variables over which a directed acyclic model has been
defined, Then for each of these latent L there are 3 (preferably 4)
or more measures of that variable—that is, measured variables that
are all children of L. Under these conditions, one may define tetrad
constraints (see Spirtes et al., 2000). There is a theorem to the
effect that if certain CPDAGs of these tetrad constraints hold, there
must be a latent common cause of all of them (the Tetrad
Representation Theorem). The FOFC (Find One Factor Clusters) takes
advantage of this fact. The basic idea is to build up clusters one at
a time by adding variables that keep them pure in the sense that all
relevant tetrad constraints still hold. There are different ways of
going about this. One could try to build one cluster up as far as
possible, then remove all of those variables from the set, and try to
make another cluster using the remaining variables (SAG, Seed and
Grow). Or one can try in parallel to grow all possible clusters and
then choose among the grown clusters using some criterion such as
cluster size (GAP, Grow and Pick). In general, GAP is more accurate.
The result is a clustering of variables. Once one has such a
“measurement model, one can estimate (using the ESTIMATOR box) a
covariance matrix over the latent variables that are parents of the
measures and use some algorithm such as PC or GES to estimate a CPDAG
over the latent variables. The algorithm to run PC or GES on this
covariance matrix is called MimBuild (“MIM” is the graph, Multiple
Indicator Model; “Build” means build). MimBUILD is an optional choice
inside FOFC In this way, one may recover causal structure over the
latents. The more measures one has for each latent the better the
result is, generally. At least 3 measured indicator variables are
needed for each latent variable. The larger the sample size the
better. One important issue is that the algorithm is sensitive to
so-called “impurities”—that is,causal edges among the measured
variables, or between measured variables and multiple latent
variables. The algorithm will in effect remove one measure in each
impure pair from consideration.
Note that for FOFC, a test is done for each final cluster to see
whether the variables in the cluster are all mutually
dependent. In the interface, in order to see teh results of this
test, one needs to open the logging window. See the Logging menu.
The FTFC Algorithm
Description
FTFC (Find Two Factor Clusters) is similar to FOFC, but
instead of each cluster having one latent that is the parent of all
the measure in the cluster, it instead has two such latents. So
each measure has two latent parents; these are two “factors.”
Similarly to FOFC, constraints are checked for, but in this case, the
constraints must be sextad constraints, and more of them must be
satisfied for each pure cluster (see Kummerfeld et al., 2014). Thus,
the number of measures in each cluster, once impure edges have been
taken into account, must be at least six, preferably more.
Input Assumptions
Continuous data over the measures with at least six variable
variables in each cluster once variables involve in impure edges have
been removed.
Output Format
A clustering of measures. It may be assumed that each cluster has
at least two factors and that the clusters are pure.
Parameters
The ICA LiNGAM Algorithm
Description
ICA LiNGAM (Shimizu et al., 2006) was one of the
first of the algorithms that assumed linearity among the variables and
non-Gaussianity of error term, and still one of the best for smaller
models, for the basic algorithm, implemented here. The idea is to use the
Independent Components Analysis (ICA) algorithm to check all permutations
of the variables to find one that is a causal order—that is, one in which
earlier variables can cause later variables but not vice-versa. The
method is clever. First, since we assume the model is a directed acyclic
graph (DAG), there must be some permutation of the variables for which
the main diagonal of the inverse of the weight matrix contains no zeros.
This gives us a permuted estimate of the weight matrix. Then we look for
a permutation of this weight matrix that is lower triangular. There must
be one, since the model is assumed to be a DAG. But a lower triangular
weight matrix just gives a causal order, so we’re done.
In the
referenced paper, we implement Algorithm A, which is described above.
Once one has a causal order the only thing one needs to do is to
eliminate the extra edges. For this, we use the causal order to define
knowledge of tiers and run FGES.
Our implementation of LiNGAM has one parameter, penalty
discount, used for the FGES adjacency search. The method as
implemented does not scale much beyond 10 variables, because it is
checking every permutation of all the variables (twice). The
implementation of ICA we use is FastIca (Hyvärinen et al., 2004).
Shimizu, S., Hoyer, P. O., Hyvärinen, A., & Kerminen, A.
(2006). A linear non-Gaussian acyclic model for causal discovery.
Journal of Machine Learning Research, 7(Oct), 2003-2030.
Hyvärinen, A., Karhunen, J., & Oja, E. (2004). Independent
component analysis (Vol. 46). John Wiley & Sons.
Direct LiNGAM is an implementation of said algorithms as specified in this reference:
Shimizu, S., Inazumi, T., Sogawa, Y., Hyvarinen, A., Kawahara, Y., Washio, T., ... & Hoyer, P. (2011).
DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model. Journal of
Machine Learning Research-JMLR, 12(Apr), 1225-1248.
This requires no parameters. This algorithm is applicable to data generated by linear, non-Gaussian
models.
The ICA LiNG-D Algorithm
Description
Implements the ICA-LiNG-D algorithm as well as a number of ancillary methods for LiNG-D and LiNGAM. The
reference is here:
Lacerda, G., Spirtes, P. L., Ramsey, J., & Hoyer, P. O. (2012). Discovering cyclic causal models by
independent components analysis. arXiv preprint arXiv:1206.3273.
ICA-LING-D is a method for estimating a possible cyclic linear models graph from a dataset. It is based on
the assumption that the data are generated by a linear model with non-Gaussian noise. The method is based on
the following assumptions:
- The data are generated by a linear model with non-Gaussian noise.
- The noise is independent across variables.
- The noises for all but possibly one variable are non-Gaussian.
- There is no unobserved confounding.
lower bound of the absolute value of the coefficients in the W matrix.
Under these assumptions, the method estimates a matrix W such that WX = e, where X is the data matrix, e is
a matrix of noise, and W is a matrix of coefficients. The matrix W is then used to estimate a matrix B Hat,
where B Hat is the matrix of coefficients in the linear model that generated the data. The graph is then
estimated by finding edges in B Hat.
We use the N Rooks algorithm to find alternative diagonals for permutations of the W matrix. The parameter
that N Rooks requires is a threshold for entries in W to be included in possible diagonals, which is the
lowest number in absolute value that a W matrix entry can take to be part of a diagonal; the implied
permutations is the permutations that permutes rows so that these combination lies along the their
respective diagonals in W, which are then scaled, and the separate satisfactory B Hat matrices reported.
ICA-LiNG-D, which takes this W as an impute, is a method for estimating a directed graph (DG) from a
dataset. The graph is estimated by finding a
permutation of the columns of the dataset so that the resulting matrix has a strong diagonal. This
permutation is then used to estimate a DG. The method is an relaxation of LiNGAM, which estimates a DAG from
a dataset using independent components analysis (ICA). LiNG-D is particularly useful when the underlying
data may have multiple consistent cyclic models.
This class is not configured to respect knowledge of forbidden and required edges.
The FASK Algorithm
Description
FASK learns a linear model in which all the
variables are skewed.
The idea is as follows. First, FAS-stable is run on the data,
producing an undirected graph. We use the BIC score as a conditional
independence test with a specified penalty discount c. This yields
undirected graph G0 . The reason FAS-stable works for sparse cyclic
models where the linear coefficients are all less than 1 is that
correlations induced by long cyclic paths are statistically judged as
zero, since they are products of multiple coefficients less than 1.
Then, each of the X − Y adjacencies in G0 is oriented as a 2-cycle X
+= Y , or X → Y , or X ← Y . Taking up each adjacency in turn, one
tests to see whether the adjacency is a 2-cycle by testing if the
difference between corr(X, Y ) and corr(X, Y |X > 0), and corr(X, Y )
and corr(X, Y |Y > 0), are both significantly not zero. If so, the
edges X → Y and X ← Y are added to the output graph G1 . If not, the
Left-Right orientation is rule is applied: Orient X → Y in G1, if
(E(X Y |X > 0)/ E(X 2|X > 0)E(Y 2 |X > 0) − E(X Y |Y > 0)/ E(X 2 |Y >
0)E(Y 2|Y > 0)) > 0; otherwise orient X ← Y . G1 will be a fully
oriented graph. For some models, where the true coefficients of a
2-cycle between X and Y are more or less equal in magnitude but
opposite in sign, FAS-stable may fail to detect an edge between X and
Y when in fact a 2-cycle exists. In this case, we check explicitly
whether corr(X, Y |X > 0) and corr(X, Y |Y > 0) differ by more than a
set amount of 0.3. If so, the adjacency is added to the graph and
oriented using the aforementioned rules.
We include pairwise orientation rule RSkew, Skew, and Tanh
from Hyvärinen, A., & Smith, S. M. (2013). Pairwise likelihood ratios
for estimation of non-Gaussian structural equation models. Journal of
Machine Learning Research, 14(Jan), 111-152, so in some
configurations FASK can be made to implement an algorithm that has
been called in the literature "Pairwise LiNGAM"--this is intentional;
we do this for ease of comparison. You'll get this configuration if
you choose one of these pairwise orientation rules, together with the
FAS with orientation alpha and two-cycle threshold set to zero and
skewness threshold set to 1, for instance.
See Sanchez-Romero R, Ramsey JD, Zhang K, Glymour MR, Huang B,
Glymour C. Causal discovery of feedback networks with functional
magnetic resonance imaging. Network Neuroscience 2018.
Input Assumptions
Continuous, linear data in which all the variables are
skewed.
Output Format
A fully directed, potentially cyclic, causal graph.
The DirectLiNGAM Algorithm
Description
Implements the Direct-LiNGAM algorithm. The reference is here:
S. Shimizu, T. Inazumi, Y. Sogawa, A. Hyvärinen, Y. Kawahara, T. Washio, P. O. Hoyer and K. Bollen.
DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model. Journal of
Machine Learning Research,
12(Apr): 1225–1248, 2011.
A. Hyvärinen and S. M. Smith. Pairwise likelihood ratios for estimation of non-Gaussian structural
evaluation models.
Journal of Machine Learning Research 14:111-152, 2013.
The DAGMA Algorithm
Description
Implements the DAGMA algorithm. The reference is here:
Bello, K., Aragam, B., & Ravikumar, P. (2022). Dagma: Learning dags via m-matrices and a log-determinant
acyclicity characterization. Advances in Neural Information Processing Systems, 35, 8226-8239.
Input Assumptions
Same as FASK.
Output Format
Same as FASK.
The FASK-Vote Algorithm
Description
FASK-Vote is a metascript that learns a
model from a list of datasets in a method similar to IMaGES (see). For
adjacencies, it uses FAS-Stable with the voting-based score from IMaGES
used as a test (using all the datasets, standardized), producing a
single undirected graph G. It then orients each edge X--Y in G for each
dataset using the FASK (see) left-right rule and orient X->Y if that rule
orients X--Y as such in at least half of the datasets. The final graph is
returned.
For FASK, See Sanchez-Romero R, Ramsey JD, Zhang K, Glymour
MR, Huang B, Glymour C. Causal discovery of feedback networks with
functional magnetic resonance imaging. Network Neuroscience 2018.
Input Assumptions
Same as FASK.
Output Format
Same as FASK.
Orientation Algorithms (R3, RSkew, Skew)
Description
This is an algorithm that orients an edge X--Y for continuous
variables based on non-Gaussian information. This rule in particular
uses an entropy calculation to make the orientation. Note that if the
variables X and Y are both Gaussian, and the model is linear, it is
not possible to orient the edge X--Y pairwise; any attempt to do so
would result in random orientation. But if X and Y are non-Gaussian,
the orientation is fairly easy. This rule is similar to Hyvarinen and
Smith's (2013) EB rule, but using Anderson Darling for the measure of
non-Gaussianity, to somewhat better effect. See Ramsey et al. (2012).
This is an algorithm that orients an edge X--Y for continuous
variables based on non-Gaussian information. This rule in particular
uses a skewness to make the orientation. Note that if the variables X
and Y are both Gaussian, and the model is linear, it is not possible
to orient the edge X--Y pairwise; any attempt to do so would result
in random orientation. But if X and Y are non-Gaussian, in particular
in this case, if X and Y are skewed, the orientation is relatively
straightforward. See Hyvarinen and Smith (2013) for details.
The Skew rule is differently motivated from the RSkew rule
(see), though they both appeal to the skewness of the variables.
This is an algorithm that orients an edge X--Y for continuous
variables based on non-Gaussian information. This rule in particular
uses a skewness to make the orientation. Note that if the variables X
and Y are both Gaussian, and the model is linear, it is not possible
to orient the edge X--Y pairwise; any attempt to do so would result
in random orientation. But if X and Y are non-Gaussian, in particular
in this case, if X and Y are skewed, the orientation is relatively
straightforward. See Hyvarinen and Smith (2013) for details.
The RSkew rule is differently motivated from the Skew rule
(see), though they both appeal to the skewness of the variables.
This is an algorithm that orients an edge X--Y for continuous
variables based on non-Gaussian information. This rule in particular
uses the FASK pairwise rule to make the orientation. Note that if the
variables X and Y are both Gaussian, and the model is linear, it is
not possible to orient the edge X--Y pairwise; any attempt to do so
would result in random orientation. But if X and Y are non-Gaussian,
in particular in this case, if X and Y are skewed, the orientation is
relatively straightforward. See Hyvarinen and Smith (2013) for
details.
The FASK-PW rule appeals to skewness in a different way than
Skew and RSkew.
Input Assumptions
Continuous data in which the variables are non-Gaussian.
Non-Gaussianity can be assessed using the Anderson-Darling score, which
is available in the Data box.
Output Format
Orients all the edges in the input graph using the selected
score.
Parameters
The BOSS-LiNGAM Algorithm
Description
Implements the BOSS-LiNGAM algorithm which first finds a CPDAG for the variables and then uses a
non-Gaussian orientation method to orient the undirected edges. The reference is as follows:
Hoyer et al., "Causal discovery of linear acyclic models with arbitrary distributions," UAI 2008.
The test for normality used for residuals is Anderson-Darling, following 'ad.test' in the nortest package of
R. The default alpha level is 0.05--that is, p values from AD below 0.05 are taken to indicate
nongaussianity.
It is assumed that the CPDAG is the result of a CPDAG search such as PC or GES. In any case, it is important
that the residuals be independent for ICA to work.
This may be replaced by a more general algorithm that allows alternatives for the CPDAG search and for the
the non-Gaussian orientation method.
This class is not configured to respect knowledge of forbidden and required edges.
Note: If one wants to analyze time series data using this algorithm,
one may set the time lag parameter to a value greater than 0, which
will automatically apply the time lag transform. The same goes for any
algorithm that has this parameter available in the interface.
The CStaR Algorithm
Description
The CStaR algorithm (Causal Stability Ranking, Stekhoven, D. J., Moraes, I., Sveinbjörnsson, G., Hennig,
L.,
Maathuis, M. H., & Bühlmann, P. 2012. Causal stability ranking. Bioinformatics, 28(21), 2819-2823)
calculates lower bounds on estimated parameters for the causally sufficient case. It first runs a CPDAG
algorithm and then for X->Y locally, about Y, finds all possible orientations of the edges in the CPDAG
and
does an estimation for each of these, and finds their lower bound. In the interface, all nodes that are
found in the top bracket more than 50% of the time are marked as into that target node.
The procedure is best used, however, as a command line procedure, in py-causal, rpy-causal, or Causal
Command. Upon running the algorithm (even in the interface), a directory of result files will be
produced as
a record, including the dataset used, the possible causes and effects used, all the CPDAGs used and
the
tables of their IDA effects, and the CStaR output table itself.
Parameters.
Algorithm. This is the algorithm to use to calculate bootstrapped CPDAGs. Current options are PC Stable,
FGES, BOSS, or Restricted BOSS. For large datasets, we recommend Restricted BOSS, which calculates
variables
with marginal effect on one of the targets and then runs BOSS over this restricted set.
Results Output Path. A default is “cstar-out”, which will place result-files in a subdirectory of the
current
directory named path = “cstar-out”.[n], where n is the first index for which no such directory exists.
If a
directory already exists at the path, then any information available in path directory will be used to
generate results in the path-.[n] directory.
Number of Sub-samples. CStaR finds CPDAGs over sub-sampled data of size n / 2; this specifies how many
sub-samples to use.
Minimum effect size. This allows a shorter table to be produced. It this is set to a value m > 0, then
only
records with PI > m will be displayed.
Target Names. A list of names of variables (comma or space separated) can be given that are considered
possible effects. These will be excluded from the list of possible causes, which will be all other
variables
in the dataset.
Top Bracket. The CStaR algorithm tries to find possible causes that regularly sort into the top set of
variables by minimum IDA effect. This gives the number q of variables to include in the top bracket,
where 1
<= q <= # possible causes.
Parallelized. Yes, if the search should be parallelized, no if not. Default no.
The main results of the algorithm is table in the format of Table 1 in Stekhoven et al. here is the
beginning
of one such table.
# Potential Causes = 19
# Potential Effects = 1
Top Bracket ('q') = 10
Index Cause Effect PI Effect PCER
1 X15 X20 0.9000 0.0691 0.0035
2 X4 X20 0.8800 0.0888 0.0146
3 X18 X20 0.7200 0.0451 0.0567
4 X17 X20 0.7000 0.0765 0.1108
5 X19 X20 0.6800 0.0559 0.1924
6 X1 X20 0.6600 0.0635 0.3116
7 X3 X20 0.6600 0.0438 0.4242
Here, the number of possible causes and the number of possible effects is listed, as well as the top
bracket
(‘q’) used to generate the table. For each record considered, its cause and effect (e.g., X15 to X20).
The
percentage of times this cause/effect pair ended up in the top bracket is given as PI, and the minimum
IDA
effect size for it is given in Effect. The Per Comparison Error Rate (which can only be calculated for
PI >
0.5) is given in PCER. See the Stekhoven paper for details.
Input Assumptions
Same as for PC.
Statistical Tests
All the below tests do testwise deletion as a default way of
dealing with missing values. For testwise deletion, if a test, say, I(X,
Y | Z), is done, columns for X, Y, and Z are scanned for missing values.
If any row occurs in which X, Y, or Z is missing, that row is deleted
from the data for those three variables. So if a different test, I(R, W |
Q, T) is done, different rows may be stricken from the data. That is, the
deletion is done testwise. For a useful discussion of the testwise
deletion condition, see for instance Tu, R., Zhang, C., Ackermann, P.,
Mohan, K., Kjellström, H., & Zhang, K. (2019, April). Causal discovery in
the presence of missing data. In The 22nd International Conference on
Artificial Intelligence and Statistics (pp. 1762-1770). PMLR. For all of
these tests, if no data are missing, the behavior will be as if testwise
deletion were not being done.
BDeu Test
This is a test based on the BDeu score given
in Heckerman, D., Geiger, D., & Chickering, D. M. (1995). Learning
Bayesian networks: The combination of knowledge and statistical data.
Machine learning, 20(3), 197-243, used as a test. This gives a
score for any two variables conditioned on any list of others which is
more positive for distributions which are more strongly dependent. The
test for X _||_ Y | Z compares two different models, X conditional on Y,
and X conditional on Y and Z; the scores for the two models are
subtracted, in that order. If the difference is negative, independence is
inferred.
Fisher Z Test
Fisher Z judges independence if the
conditional correlation is cannot statistically be distinguished from
zero. Primarily for the linear, Gaussian case.
Parameters
SEM BIC Test
This uses the SEM BIC Score to create a
test for the linear, Gaussian case, where we include an additional
penalty term, which is commonly used. We call this the penalty
discount. So our formulas has BIC = 2L - ck log N,where L is the
likelihood, c the penalty discount (usually greater than or equal to 1),
and N the sample size. Since the assumption is that the data are
distributed as Gaussian, this reduces to BIC = -n log sigma - ck ln N,
where sigma is the standard deviation of the linear residual obtained by
regressing a child variable onto all of its parents in the model.
Mixed Variable Polynomial Likelihood Ratio Test
Performs a test of conditional independence X _||_ Y | Z1...Zn where all variables are either continuous or
discrete.
This test is valid for both ordinal and non-ordinal discrete searchVariables.
Andrews, B., Ramsey, J., & Cooper, G. F. (2018). Scoring Bayesian networks of mixed variables.
International
journal of data science and analytics, 6, 3-18.
Parameters
d
Generalized Information Criterion Scores
This is a set of generalized information criterion (GIC) scores, used as tests,
based on the paper, Kim, Y., Kwon, S., & Choi, H. (2012). Consistent model selection
criteria on high dimensions. The Journal 0of Machine Learning Research,
13(1), 1037-1057. One needs to select which lambda to use in place of the
usual lambda for the linear, Gaussian BIC score. A penalty discount parameter
may also be specified, though this is by default for these scores equal
to 1 (since the lambda choice is essentially picking a penalty discount
for you).
L
MAG SEM BIC Test
This gives a BIC score (used as a test here) for a Mixed Ancestral Graph (MAG).
Note that BIC is calculated as 2L - k ln N, so "higher is better."
Probabilistic Test
The Probabilistic Test applies a Bayesian
method to derive the posterior probability of an independence constraint
R = (X⊥Y|Z) given a dataset D. This is intended for use with datasets
with discrete variables. It can be used with constraint-based algorithms
(e.g., PC and FCI). Since this test provides a probability for each
independence constraint, it can be used stochastically by sampling based
on the probabilities of the queried independence constraints to obtain
several output graphs. It can also be used deterministically by using a
fixed decision threshold on the probabilities of the queried independence
constraints to generate a single output graph.
Parameters
noRandomlyDeterminedIndependence
cutoffIndTest priorEquivalentSampleSize
Conditional Correlation Independence (CCI) Test
CCI ("Conditional Correlation Independence")
is a fairly general independence test—not completely general, but general
for additive noise models—that is, model in which each variable is equal
to a (possibly nonlinear) function of its parents, plus some additive
noise, where the noise may be arbitrarily distributed. That is, X =
f(parent(X)) + E, where f is any function and E is noise however
distributed; the only requirement is that there be the “+” in the formula
separating the function from the noise. The noise can’t for instance, be
multiplicative, e.g., X = f(parent(X)) x E. The goal of the method is to
estimate whether X is independent of Y given variables Z, for some X, Y,
and Z. It works by calculating the residual of X given Z and the residual
of Y given Z and looking to see whether those two residuals are
independent. This test may be used with any constraint-based algorithm
(PC, FCI, etc.).
Parameters
alpha, numBasisFunctions, kernelType, kernelMultiplier, basisType, kernelRegressionSampleSize
Chi Square Test
This is the usual Chi-Square test for
discrete variables; consult an introductory statistics book for details
for the unconditional case, where you're just trying, e.g., to determine
if X and Y are independent. For the conditional case, the test proceeds
as in Fienberg, S. E. (2007). The analysis of cross-classified
categorical data, Springer Science & Business Media, by identifying
and removing from consideration zero rows or columns in the conditional
tables and judging dependence based on the remaining rows and columns.
Parameters
M-Separation Test
This is the usual test of m-separation, a
property of graphs, not distributions. It's not really a test, but it can
be used in place of a test of the true graph is known. This is a way to
find out, for constraint-based algorithms, or even for some score-based
algorithms like FGES, what answer the algorithm would give if all the
statistical decisions made are correct. Just draw an edge from the true
graph to the algorithm--the m-separation option will appear, and you can
then just run the search as usual. Note that D-Separation and M-separation
use the same algorithm; we uniformly call the algorithm "M-Separation" for
clarity. D-Separation is M-Separation applied to DAGs.
Discrete BIC Test
This is a BIC score for the discrete
case, used as a test. The likelihood is judged by the multinomial tables
directly, and this is penalized as is usual for a BIC score. The only
surprising thing perhaps is that we use the formula BIC = 2L - k ln N,
where L is the likelihood, k the number of parameters, and N the sample
size, so "higher is better." In the case of independence, the BIC
score will be negative, since the likelihood will be zero, and this will
be penalized. The test yields a p-value; we simply use alpha - p as the
score, where alpha is the cutoff for rejecting the null hypothesis of
independence. This is a number that is positive for dependent cases and
negative for independent cases.
Parameters
penaltyDiscount, structurePrior
G Square Test
This is completely parallel to the
Chi-Square statistic, using a slightly different method for estimating
the statistic. The alternative statistic is still distributed as
chi-square in the limit. In practice, this statistic is more or less
indistinguishable in most cases from Chi-Square. For an explanation, see
Spirtes, P., Glymour, C. N., Scheines, R., Heckerman, D., Meek, C.,
Cooper, G., & Richardson, T. (2000). Causation, prediction, and
search. MIT press.
Parameters
Kernel Conditional Independence (KCI) Test
KCI ("Kernel Conditional Independence") is a
general independence test for model in which X = f(parents(X), eY); here,
eY does not need to be additive; it can stand in any functional
relationships to the other variables. The variables may even be discrete.
The goal of the method is to estimate whether X is independent of Y given
Z, completely generally. It uses the kernel trick to estimate this. As a
result of using the kernel trick, the method is complex in the direction
of sample size, meaning that it may be very slow for large samples. Since
it’s slow, individual independence results are always printed to the
console so the user knows how far a procedure has gotten. This test may
be used with any constraint-based algorithm (PC, FCI, etc.)
Parameters
alpha, kciUseApproximation, kernelMultiplier, kciNumBootstraps, thresholdForNumEigenvalues, kciEpsilon
Conditional Gaussian Likelihood Ratio Test
Conditional Gaussian Test is a likelihood
ratio test based on the conditional Gaussian likelihood function. This is
intended for use with datasets where there is a mixture of continuous and
discrete variables. It is assumed that the continuous variables are
Gaussian conditional on each combination of values for the discrete
variables, though it will work fairly well even if that assumption does
not hold strictly. This test may be used with any constraint-based
algorithm (PC, FCI, etc.). See Andrews, B., Ramsey, J., & Cooper, G.
F. (2018). Scoring Bayesian networks of mixed variables. International
journal of data science and analytics, 6(1), 3-18.
Degenerate Gaussian Likelihood Ratio Test
may be used for the case where there is a mixture of discrete and
Gaussian variables. Calculates a likelihood ratio based on likelihood
that is calculated using a conditional Gaussian assumption. See Andrews,
B., Ramsey, J., & Cooper, G. F. (2019). Learning high-dimensional
directed acyclic graphs with mixed data-types. Proceedings of machine
learning research, 104, 4.
Parameters
Parameters
Resampling
Most TETRAD searches can be performed with resampling. This option
is available on the Set Parameters screen. When it is selected, the
search will be performed multiple times on randomly selected subsets of
the data, and the final output graph will be the result of a voting
procedure among all the graphs. These subsets may be selected with
replacement (bootstrapping) or without. There are also options for the
user to set the size of the subset, and the number of resampling runs.
The default number of resampling runs is zero, in which case no
resampling will be performed.
For each potential edge in the final output graph, the individual
sampled graphs may contain a directed edge in one direction, the other
direction, a bidirected edge, an uncertain edge, or no edge at all. The
voting procedure reconciles all of these possible answers into a single
final graph, and the "ensemble method," which can be set by the user in
the parameter settings screen, determines how it will do that.
The four available ensemble methods are Preserved, Highest, and
Majority. Preserved tends to return the densest graphs, then Highest, and
finally Majority returns the sparsest. The Preserved ensemble method
ensures that an edge that has been found by some portion of the
individual sample graphs is preserved in the final graph, even if the
majority of sample graphs returned [no edge] as their answer for that
edge. So the voting procedure for Preserved is to return the edge
orientation that the highest percentage of sample graphs returned, other
than [no edge]. The Highest ensemble method, on the other hand, simply
returns the edge orientation which the highest proportion of sample
graphs returned, even if that means returning [no edge]. And the Majority
method requires that at least 50 percent of the sample graphs agree on an
edge orientation in order to return any edge at all. If the highest
proportion of sample graphs agree on, for instance, a bidirected edge,
but only 40 percent of them do so, then the Majority ensemble method will
return [no edge] for that edge.
One small point of clarification. In the bootstrapping, for a --> edge,
there are three columns. These are to capture various types of --> edges
that may be found in estimated PAGs--whether the --> edge is visible and/or
definitely direct. For estimated CPDAGs, these three columns may be summed.
Similarly, for <-- edges.
Scoring Functions
Like the tests, above, all the below tests do testwise deletion
as a default way of dealing with missing values. For testwise deletion,
if a score, say, score(X | Y, Z), is done, columns for X, Y, and Z are
scanned for missing values. If any row occurs in which X, Y, or Z is
missing, that row is deleted from the data for those three variables. So
if a different test, score(R | W, Q, T) is done, different rows may be
stricken from the data. That is, the deletion is done testwise. For a
useful discussion of the testwise deletion condition, see for instance
Tu, R., Zhang, C., Ackermann, P., Mohan, K., Kjellström, H., & Zhang, K.
(2019, April). Causal discovery in the presence of missing data. In The
22nd International Conference on Artificial Intelligence and Statistics
(pp. 1762-1770). PMLR. For all of these tests, if no data are missing,
the behavior will be as if testwise deletion were not being done.
BDeu Score
This is the BDeu score given in Heckerman,
D., Geiger, D., & Chickering, D. M. (1995). Learning Bayesian networks:
The combination of knowledge and statistical data. Machine learning,
20(3), 197-243. This gives a score for any two variables conditioned
on any list of others which is more positive for distributions which are
more strongly dependent.
Parameters
Conditional Gaussian BIC Score
Conditional Gaussian BIC Score may be
used for the case where there is a mixture of discrete and Gaussian
variables. Calculates a BIC score based on likelihood that is calculated
using a conditional Gaussian assumption. See Andrews, B., Ramsey, J., &
Cooper, G. F. (2018). Scoring Bayesian networks of mixed variables.
International journal of data science and analytics, 6(1), 3-18.
Note that BIC is calculated as 2L - k ln N, so "higher is better."
Parameters
Degenerate Gaussian BIC Score may be used
for the case where there is a mixture of discrete and Gaussian variables.
Calculates a BIC score based on likelihood that is calculated using a
conditional Gaussian assumption. See Andrews, B., Ramsey, J., & Cooper,
G. F. (2019). Learning high-dimensional directed acyclic graphs with
mixed data-types. Proceedings of machine learning research, 104, 4.
Note that BIC is calculated as 2L - k ln N, so "higher is better."
Parameters
M-separation Score
This uses m-separation to make something
that acts as a score if you know the true graph. A score in Tetrad, for
FGES, say, is a function that for X and Y conditional on Z, returns a
negative number if X _||_ Y | Z and a positive number otherwise. So to
get this behavior in no u certain terms, we simply return -1 for
independent cases and +1 for dependent cases. Works like a charm. This
can be used for FGES to check what the ideal behavior of the algorithm
should be. Simply draw an edge from the true graph to the search box,
select FGES, and search as usual.
Discrete BIC Score
This is a BIC score for the discrete
case. The likelihood is judged by the multinomial tables directly, and
this is penalized as is usual for a BIC score. The only surprising thing
perhaps is that we use the formula BIC = 2L - k ln N, where L is the
likelihood, k the number of parameters, and N the sample size, instead of
the usual L + k / 2 ln N. So higher BIC scores will correspond to greater
dependence. In the case of independence, the BIC score will be negative,
since the likelihood will be zero, and this will be penalized.
SEM BIC Score
This is specifically a BIC score for the
linear, Gaussian case, where we include an additional penalty term, which
is commonly used. We call this the penalty discount. So our
formulas has BIC = 2L - ck log N, where L is the likelihood, c the
penalty discount (usually greater than or equal to 1), and N the sample
size. Since the assumption is that the data are distributed as Gaussian,
this reduces to BIC = -n log sigma - ck ln N, where sigma is the standard
deviation of the linear residual obtained by regressing a child variable
onto all of its parents in the model.
Parameters
EBIC Score
This is the Extended BIC (EBIC) score of
Chen and Chen (Chen, J., & Chen, Z. (2008). Extended Bayesian information
criteria for model selection with large model spaces. Biometrika, 95(3),
759-771.). This score is adapted to score-based search in high
dimensions. There is one parameter, gamma, which takes a value between 0
and 1; if it's 0, the score is standard BIC. A value of 0.5 or 1 is
recommended depending on how many variables there are per sample.
GIC Scores
This is a set of generalized information criterion (GIC) scores based on
the paper, Kim, Y., Kwon, S., & Choi, H. (2012). Consistent model selection
criteria on high dimensions. The Journal 0of Machine Learning Research,
13(1), 1037-1057. One needs to select which lambda to use in place of the
usual lambda for the linear, Gaussian BIC score. A penalty discount parameter
may also be specified, though this is by default for these scores equal
to 1 (since the lambda choice is essentially picking a penalty discount
for you).
Mixed Variable Polynomial Scores
Implements a mixed variable polynomial BIC score. The reference is here:
Andrews, B., Ramsey, J., & Cooper, G. F. (2018). Scoring Bayesian networks of mixed variables.
International journal of data science and analytics, 6, 3-18.
Poisson Prior Score
This is likelihood score attenuated by the log
of the Poisson distribution. It has one parameter, lambda, from the Poisson
distribution, which acts as a structure prior.
Zhang-Shen Bound Score
Uses Theorem 1 from Zhang, Y., & Shen, X. (2010). Model selection procedure for
high‐dimensional data. Statistical Analysis and Data Mining: The ASA Data
Science Journal, 3(5), 350-358, to make a score that controls false positives.
The is one parameter, the "risk bound", a number between 0 and 1 (a bound on
false positive risk probability).
Search Parameters
numThreads
- Short Description: The number of threads (>= 1) to use for the search
- Long Description: The number of threads to use for the search.
- Default Value: 1
- Lower
Bound: 1
- Upper Bound: 1000000
- Value Type:
Integer
bootstrappingNumThreads
- Short Description: The number of threads (>= 1) to use for the bootstrapping
- Long Description:
This is the number of threads for the bootstrapping itself. The number
of threads that each algorithm uses is set by the individual algorithm.
- Default Value: 1
- Lower
Bound: 1
- Upper Bound: 1000000
- Value Type:
Integer
Note: You must specify the "Value Type" of each parameter, and
the value type must be one of the following: Integer, Long, Double, String,
Boolean.
sampleStyle
- Short Description: Sample style: 1 = Subsample, 2 = Bootstrap
- Long Description: Sample style: 1 = Subsample, 2 = Bootstrap
- Default Value: 1
- Lower
Bound: 1
- Upper Bound: 2
- Value Type:
Integer
minSampleSizePerCell
- Short Description: For conditional Gaussian, the minimum sample size per cell/span>
- Long Description: For conditional Gaussian, the minimum sample size per cell
- Default Value: 4
- Lower
Bound: 2
- Upper Bound: 100000000
- Value Type:
Integer
removeEffectNodes
- Short Description: True if effect nodes should bre removed from possible causes
- Long Description: True if effect nodes should be removed from possible causes
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
piThr
- Short Description: A fixed threshold for calculating E[V] and PCER
- Long Description: A fixed threshold, default 0.5
- Default Value: 0.6
- Lower
Bound: 0
- Upper Bound: 1
- Value Type:
Double
cstarCpdagAlgorithm
- Short Description: Algorithm: 1 = PC Stable, 2 = FGES, 3 = BOSS, 4 = Restricted BOSS
- Long Description: The CPDAG algorithm to use: 1 = PC Stable, 2 = FGES, 3 = BOSS, 4 = Restricted BOSS
- Default Value: 4
- Lower
Bound: 1
- Upper Bound: 4
- Value Type:
Integer
trimmingStyle
- Short Description: Trimming Style: 1 = None, 2 = Adjacencies, 3 = MB DAG, 4 = Semidirected paths
- Long Description: 'Adjacencies' trims to the adjacencies the targets, MB DAGs to the Union(MB(targets)) U targets,
semidirected trims to nodes with semidirected paths to the targets.
- Default Value: 3
- Lower
Bound: 1
- Upper Bound: 4
- Value Type:
Integer
numberOfExpansions
- Short Description: Number of expansions of the algorithm away from the target
- Long Description: Each expansion iterates to concentrically more variables
- Default Value: 2
- Lower
Bound: 1
- Upper Bound: 1000
- Value Type:
Integer
lambda1
- Short Description: lambda1
- Long Description: Tuning parameter for DAGMA
- Default Value: 0.05
- Lower
Bound: 0
- Upper Bound: Infinity
- Value Type:
Double
cpdag
- Short Description: True if a CPDAG should be returned, false if a DAG
- Long Description: The algorithm returns a DAG; if this is
set to True, this DAG is converted to a CPDAG
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
wThreshold
- Short Description: wThreshold
- Long Description: Tuning parameter for DAGMA
- Default Value: 0.1
- Lower
Bound: 0
- Upper Bound: Infinity
- Value Type:
Double
addOriginalDataset
- Short Description: Yes, if adding the original dataset
as another bootstrapping
- Long Description: Select “Yes” here to include an
extra run using the original dataset for improved accuracy.
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
saveBootstrapGraphs
- Short Description: Yes if individual bootstrapping
graphs should be saved
- Long Description: Bootstrapping provides a summary
over individual search graphs; select Yes here if these individual
graphs should be saved
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
alpha
- Short Description: Cutoff for p values (alpha) (min =
0.0)
- Long Description:
The cutoff, beyond which test results are judged as dependent, for a
statistical test of independence. Default 0.05. Higher alpha yields a
sparser graph.
- Default Value: 0.01
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double
applyR1
- Short Description: Yes if the orient away from arrow rule
should be applied
- Long Description: Set this parameter to “No” if a chain of
directed edges pointing in the same direction when only the first few
such orientations are justified based on the data.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
avgDegree
- Short Description: Average degree of graph (min =
1)
- Long Description:
The average degree of a graph is equal to 2E / V, where E is the
number of edges in the graph and V the number of variables (vertices)
in the graph, since each edge has two endpoints.
- Default Value: 2
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type:
Double
probabilityOfEdge
- Short Description: Probability of an adjacency being
included in the graph
- Long Description: Every possible adjacency in the
graph is included it the graph with this probability.
- Default Value: 0.05
- Lower
Bound: 0.0
- Upper Bound: 1.0
- Value Type:
Double
basisType
- Short Description: Basis type (1 = Polynomial, 2 =
Cosine)
- Long Description: For CCI, this determines which basis type
will be used (1 = Polynomial, 2 = Cosine)
- Default
Value: 2
- Lower
Bound: 1
- Upper
Bound: 2
- Value Type:
Integer
cciScoreAlpha
- Short Description: Cutoff for p values (alpha) (min =
0.0)
- Long Description: Alpha level (0 to 1)
- Default Value: 0.01
- Lower Bound:
0.0
- Upper Bound:
1.0
- Value Type:
Double
cgExact
- Short Description: Yes if the exact algorithm should be used
for continuous parents and discrete children
- Long
Description: For the conditional
Gaussian likelihood, if the exact algorithm is desired for discrete
children and continuous parents, set this parameter to “Yes”.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
coefHigh
- Short Description: High end of coefficient range (min =
0.0)
- Long Description:
Value m2 for coefficients drawn from U(-m2, -m1) U U(m1, m2).
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double
coefLow
- Short Description: Low end of coefficient range (min =
0.0)
- Long Description:
The parameter m1 for coefficients drawn from U(-m2, -m1) U U(m1,
m2).
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value
Type: Double
coefSymmetric
- Short Description: Yes if negative coefficient values
should be considered
- Long Description: Yes if coefficients should be drawn
from +/-(a, b); No if from +(a, b).
- Default Value:
true
- Lower
Bound:
- Upper
Bound:
- Value
Type: Boolean
colliderDiscoveryRule
- Short Description: Collider discovery: 1 = Lookup
from adjacency sepsets, 2 = Conservative (CPC), 3 = Max-P
- Long Description: One may look them up from
sepsets, as in the original PC, or estimate them conservatively, as
from the Conservative PC algorithm, or by choosing the sepsets with
the maximum p-value, as in PC-Max.
- Default Value:
1
- Lower Bound: 1
- Upper Bound:
3
- Value
Type: Integer
completeRuleSetUsed
- Short Description: Yes if the complete FCI rule set
should be used
- Long Description: No if the (simpler) final
orientation rules set due to P. Spirtes, guaranteeing arrow
completeness, should be used; yes if the (fuller) set due to J. Zhang,
should be used guaranteeing additional tail completeness.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
resolveAlmostCyclicPaths
- Short Description:
True just in case almost cyclic paths should be resolved in the
direction of the cycle.
- Long Description:
If true we resolved <-> edges as --> if there is a directed path x~~>y.
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
doDiscriminatingPathColliderRule
- Short Description: Yes if the discriminating path collider rule
should be done, No if not
- Long Description: Yes if the discriminating path collider
FCI rule (part of the final orientation, requiring an additional test)
should be done, No if not
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
doDiscriminatingPathTailRule
- Short Description: Yes if the discriminating path tail rule
should be done, No if not
- Long Description: Yes if the discriminating path tail
FCI rule (part of the final orientation, requiring an additional test)
should be done, No if not
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
concurrentFAS
- Short Description: Yes if a concurrent FAS should be
done
- Long Description: Yes if the version of the PC adjacency
search that uses concurrent processing should be used, no if
not.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
conflictRule
- Short Description: Collider conflicts: 1 = Prioritize existing
colliders, 2 =
Orient bidirected, 3 = Overwrite existing colliders
- Long Description: 1 if the
“overwrite” rule as introduced in the PCALG R package, 2 if all
collider conflicts using bidirected edges, or 3 if existing colliders
should be prioritized, ignoring subsequent conflicting
information.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 3
- Value Type: Integer
guaranteeCpdag
- Short Description:
Guarantee that the output is a legal CPDAG
- Long Description:
It is possible due to unfaithfulness for the Meek rules to output a
non-CPDAG; this parameter guarantees a CPDAG if set to 'Yes'.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
connected
- Short Description: Yes if graph should be
connected
- Long Description: Yes if a random graph should be generated
in which paths exists from every node to every other, no if
not.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
topBracket
- Short Description: Top bracket to look for causes in
- Long
Description: Top bracket, 'q'
- Default
Value: 5
- Lower Bound:
1
- Upper Bound: 500000
- Value Type: Integer
targets
- Short Description: Target names
(comma or space separated)
- Long Description: Target names (comma or space separated).
- Default Value:
- Lower Bound:
- Upper Bound:
- Value Type: String
fileOutPath
- Short Description: Results output path
- Long Description: Path to a directory in which results can be stored
- Default Value: cstar-out
- Lower Bound:
- Upper Bound:
- Value Type: String
selectionMinEffect
- Short Description: Minimum effect size for listing
effects in the CStaR table
- Long Description: Minimum effect size for listing
effects in the CStaR table
- Default Value: 0.0
- Lower
Bound: 0.0
- Upper Bound: 1.0
- Value Type:
Double
numSubsamples
- Short Description:
The number of subsamples to generate.
- Long Description:
CStaR works by generating subsamples and summarizing across them; this
specified the number of subsamples to generate. Must be >= 1.
effects in the CStaR table
- Default Value: 10
- Lower
Bound: 1
- Upper Bound: 100000
- Value Type:
Integer
numSub-samples
- Short Description: Number of
sub-samples
- Long Description: Number of sub-samples
- Default Value: 50
- Lower Bound:
1
- Upper Bound:
500000
- Value
Type: Integer
covHigh
- Short Description: High end of covariance range (min =
0.0)
- Long Description:
The parameter c2 for range +/-U(c1, c2) for covariance values, c1 >=
0.0
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value
Type: Double
covLow
- Short Description: Low end of covariance range (min =
0.0)
- Long Description:
The parameter c1 for range +/-U(c1, c2) for covariance values, c2 >=
c1
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value
Type: Double
covSymmetric
- Short Description: Yes if negative covariance values should
be considered
- Long Description: Usually covariance values are chosen
from +/-U(a, b) for some a, b, no if from +U(a, b).
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
cutoffConstrainSearch
- Short Description: Constraint-independence cutoff
threshold
- Long Description: null
- Default
Value: 0.5
- Lower Bound: 0.0
- Upper
Bound: 1.0
- Value Type: Double
cutoffDataSearch
- Short Description: Independence cutoff
threshold
- Long Description: null
- Default Value:
0.5
- Lower
Bound: 0.0
- Upper Bound: 1.0
- Value Type:
Double
cutoffIndTest
- Short Description: Independence cutoff
threshold
- Long Description: null
- Default Value:
0.5
- Lower
Bound: 0.0
- Upper
Bound: 1.0
- Value
Type: Double
dataType
- Short Description: "continuous" or "discrete"
- Long Description: For a mixed data
type simulation, if this is set to “continuous” or “discrete”, all
variables are taken to be of that sort. This is used as a
double-check to make sure the percent discrete is set
appropriately.
- Default Value: categorical
- Lower Bound:
- Upper Bound:
- Value Type: String
depth
- Short Description: Maximum size of conditioning set ('depth', unlimited =-1)
- Long Description:
The depth of search for algorithms like the PC adjacency search,
which is the maximum size of any conditioning set considered. In
order to express that no limit should be imposed, use the value
-1.
- Default Value: -1
- Lower Bound: -1
- Upper Bound: 2147483647
- Value Type: Integer
determinismThreshold
- Short Description: Threshold for judging a
regression of a variable onto its parents to be deterministic (min =
0.0)
- Long Description: When regressing a child variable
onto a set of parent variables, one way to test for determinism is to
test how close to singular the data is; this gives a threshold for
this. The default value is 0.1.
- Default Value: 0.1
- Lower
Bound: 0.0
- Upper Bound: Infinity
- Value
Type: Double
differentGraphs
- Short Description: Yes if a different graph should be
used for each run
- Long Description: If ‘Yes’ a new random graph is chosen
for each run; if ‘No’, the same graph is always used.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
mb
- Short Description: Find Markov blanket(s)
- Long Description: Looks for the graph over the Markov blanket(s) and target(s) if true
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
discretize
- Short Description: Yes if continuous variables should be
discretized when child is discrete
- Long Description:
Yes if for the conditional Gaussian
likelihood, when scoring X->D where X is continuous and D discrete,
one should to simply discretize X for just those cases. If no, the
integration will be exact.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
calculateEuclidean
- Short Description: Yes if the Euclidean norm squared
should be calculated (slow), No if not
- Long
Description: The generalized
information criterion is defined with an information term that take a
Euclidean norm squares; there can be calculated directly.
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
takeLogs
- Short Description: Yes logs should be taken, No if not
- Long Description: The
formula for the score allows a log to be taken optionally in the
information term.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
doColliderOrientation
- Short Description: Yes if unshielded collider
orientation should be done
- Long Description: Please see the description of
this algorithm in Thomas Richardson and Peter Spirtes in Chapter 7 of
Computation, Causation, & Discovery by Glymour and Cooper eds.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
doFgesFirst
- Short Description: Yes if FGES should be done as an initial
step
- Long Description: For BOSS, for some cases, doing FGES as
an initial step can reduce the maximum permutation size needed to
solve a problem.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
errorsNormal
- Short Description: Yes if errors should be Normal; No if
they should be abs(Normal) (i.e., non-Gaussian)
- Long
Description: A “quick and dirty”
way to generate linear, non-Gaussian data is to set this parameter to
“No”; then the errors will be sampled from a Beta
distribution.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
skewEdgeThreshold
- Short Description: Threshold for including additional
edges detectable by skewness
- Long Description: For FASK, this includes an
adjacency X—Y in the model if |corr(X, Y | X > 0) – corr(X, Y | Y >
0)| exceeds some threshold. The default for this threshold is
0.3.
- Default Value: 0.3
- Lower Bound:
0.0
- Upper
Bound: Infinity
- Value Type: Double
twoCycleScreeningThreshold
- Short Description: Upper bound for
|left-right| to count as 2-cycle. (Set to zero to turn off
pre-screening.)
- Long Description: 2-cycles are screened by
looking to see if the left-right rule returns a difference smaller
than this threshold. To turn off the screening, set this to
zero.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper
Bound: Infinity
- Value Type: Double
orientationAlpha
- Short Description: Alpha threshold used for
orientation (where necessary). ('0' turns this off.)
- Long Description: Used for
orienting 2-cycles and testing for zero edges.
- Default Value: 0.0
- Lower Bound:
0.0
- Upper
Bound: 1.0
- Value Type: Double
faskDelta
- Short Description: For FASK v1, the bias for orienting
with negative coefficients ('0' means no bias.)
- Long Description: The bias procedure for v1
is given in the published description.
- Default
Value: 0.0
- Lower
Bound: -Infinity
- Upper Bound: Infinity
- Value Type: Double
faskLeftRightRule
- Short Description: The left right rule: 1 = FASK v1, 2
= FASK v2, 3 = RSkew, 4 = Skew, 5 = Tanh
- Long
Description: The FASK left
right rule v2 is default, but two other (related) left-right rules
are given for relation to the literature, and the v1 FASK rule is
included for backward compatibility.
- Default Value:
2
- Lower
Bound: 1
- Upper Bound: 5
- Value Type:
Integer
faskAssumeLinearity
- Short Description: Linearity assumed
- Long Description: True
if a linear, non-Gaussian, additive model is assume; false if a
nonlinear, non-Gaussian, additive model is assumed.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
faskNonempirical
- Short Description: Variables should be assumed to have
positive skewness
- Long Description: If false (default), each variable is
multiplied by the sign of its skewness in the left-right rule.
- Default Value: false
- Lower
Bound:
- Upper
Bound:
- Value
Type: Boolean
acceptanceProportion
- Short Description: Acceptance Proportion
- Long Description: An edge occurring in this
proportion of individual FASK graphs will appear in the final
graph.
- Default Value: 0.5
,.
- Lower
Bound: 0.0
- Upper Bound: 1.0
- Value Type:
Double
faskAdjacencyMethod
- Short Description: Non-skewness Adjacencies: 1 = FAS
Stable, 2 = FGES, 3 = External Graph, 4 = None
- Long
Description: This is the
method FASK will use to find non-skewness adjacencies. For External
graph, an external graph must be supplied.
- Default
Value: 1
- Lower Bound: 1
- Upper Bound:
4
- Value
Type: Integer
faithfulnessAssumed
- Short Description: Yes if (one edge) faithfulness
should be assumed
- Long Description: Assumes that if X _||_ Y, by an
independence test, then X _||_ Y | Z for nonempty Z.
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
pcHeuristic
- Short Description:
Heuristics to stabilize skeleton: 0 = None, 1 = Heuristic 1, 2 = Heuristic 2, 3 = Heuristic 3
- Long Description:
NONE = no heuristic, PC-1 = sort nodes alphabetically;
PC-1 = sort edges by p-value; PC-3 = additionally sort edges in reverse order
using p-values of associated independence facts. See CPS.
- Default Value: 0
- Lower Bound: 0
- Upper Bound: 3
- Value Type: Integer
fasRule
- Short Description: Adjacency search: 1 = PC, 2 = PC-Stable, 3 =f
Concurrent PC-Stable
- Long Description: For variants of PC, one may select either to
use the usual PC adjacency search, or the procedure from the
PC-Stable algorithm (Diego and Maathuis), or the latter using a
concurrent algorithm.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 3
- Value Type: Integer
fastIcaA
- Short Description: Fast ICA 'a' parameter.
- Long Description: This is the 'a'
parameter of Fast ICA. (See Hyvarinen, A. (2001); it ranges between 1
and 2; we use a default of 1.1.
- Default Value: 1.1
- Lower Bound: 1.0
- Upper Bound: 2.0
- Value Type: Double
fastIcaMaxIter
- Short Description: The maximum number of optimization
iterations.
- Long Description: This is the maximum number if
iterations of the optimization procedure of ICA. (See Hyvarinen, A.
(2001). It's an integer greater than 0; we use a default of
2000.
- Default Value: 2000
- Lower Bound:
1
- Upper Bound:
500000
- Value
Type: Double
fastIcaTolerance
- Short Description: Fast ICA tolerance parameter.
- Long Description: This is the tolerance parameter of
Fast ICA. (See Hyvarinen, A. (2001); we use a default of 1e-6.
- Default Value: 1e-6
- Lower Bound:
0.0
- Upper
Bound: 1000.0
- Value Type: Double
thresholdBHat
- Short Description: Threshold on the B Hat matrix.
- Long Description: The estimated B matrix
is thresholded by setting small entries less than this threshold
to zero.
- Default Value: 0.1
- Lower Bound:
0.0
- Upper
Bound: Infinity
- Value Type: Double
guaranteeAcyclic
- Short Description: True if the output should be
guaranteed to be acyclic
- Long Description: The estimated B matrix
is further thresholded by setting small coefficients to zero
until an acyclic model is produced.
- Default Value: true
- Lower Bound:
- Upper
Bound:
- Value Type: Boolean
thresholdW
- Short Description: Threshold on the W matrix.
- Long Description: The estimated W matrix
is thresholded by setting small entries less than this threshold
to zero.
- Default Value: 0.1
- Lower Bound:
0.0
- Upper
Bound: Infinity
- Value Type: Double
fisherEpsilon
- Short Description: Epsilon where |xi.t - xi.t-1| <
epsilon, criterion for convergence
- Long Description:
This is a parameter for the
linear Fisher option. The idea of Fisher model (for the linear case)
is to shock the system every so often and let it converge by applying
the rules of transformation (that is, the linear model) repeatedly
until convergence.
- Default Value: 0.001
- Lower Bound:
4.9E-324
- Upper
Bound: 1.7976931348623157E308
- Value Type: Double
generalSemErrorTemplate
- Short Description: General function for error
terms
- Long Description: This template specifies how
distributions for error terms are to be generated. For help in
constructing such templates, see the Generalized SEM PM model.
- Default Value: Beta(2, 5)
- Lower Bound:
- Upper
Bound:
- Value Type: String
generalSemFunctionTemplateLatent
- Short Description: General function
template for latent variables
- Long Description:
This template
specifies how equations for latent variables are to be generated. For
help in constructing such templates, see the Generalized SEM PM
model.
- Default Value:
TSUM(NEW(B)*$)/>
- Lower Bound:
- Upper Bound:
- Value Type: String
generalSemFunctionTemplateMeasured
- Short Description: General function
template for measured variables
- Long Description:
This
template specifies how equations for measured variables are to be
generated. For help in constructing such templates, see the
Generalized SEM PM model.
- Default Value: TSUM(NEW(B)*$>
- Lower Bound:
- Upper Bound:
- Value Type: String
generalSemParameterTemplate
- Short Description: General function for
parameters
- Long Description: This template specifies
how distributions for parameter terms are to be generated. For help
in constructing such templates, see the Generalized SEM PM
model.
- Default Value: Split(-1.0, -0.5, 0.5,
1.0)
- Lower Bound:
- Upper
Bound:
- Value
Type: String
imagesMetaAlg
- Short Description: IMaGES "meta" algorithm. 1 = FGES, 2 = BOSS-Tuck
- Long
Description: Sets the meta algorithm to be optimized using the IMaGES (average BIC) score.
- Default Value:
1
- Lower Bound: 1
- Upper Bound: 5
- Value Type: Integer
ia
- Short Description: IA parameter (GLASSO)
- Long
Description: Sets the maximum number of
iterations of the optimization loop.
- Default Value:
true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
includeNegativeCoefs
- Short Description: Yes if negative coefficients
should be included in the model
- Long Description:
One may include positive
coefficients, negative coefficients, or both, in the model. To
include negative coefficients, set this parameter to “Yes”.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
includeNegativeSkewsForBeta
- Short Description: Yes if negative skew
values should be included in the model, if Beta errors are
chosen
- Long Description: Yes if negative skew
values should be included in the model, if Beta errors are
chosen.
- Default Value: true
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
includePositiveCoefs
- Short Description: Yes if positive coefficients
should be included in the model
- Long Description:
Yes if We may include
positive coefficients, should be included in the model, no if
not.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
includePositiveSkewsForBeta
- Short Description: Yes if positive skew
values should be included in the model, if Beta errors are
chosen
- Long Description: Yes if positive skew
values should be included in the model, if Beta errors are
chosen.
- Default Value: true
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
usePseudoinverse
- Short Description:
Yes if the pseudoinverse should be used to avoid exceptions
- Long Description:
Yes if the pseudoinverse should be used in place of the regular
inverse in calculating residual variances to avoid singularity
exceptions
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
ablationLeaveOutTuckingStep
- Short Description:
ABLATION: Yes, if the tucking step should be left out for the LV-Lite procedure
- Long Description:
Allowing tucks can sometimes lead to lower arrowhead accuracies,
even they are theoretically correct.
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
ablationLeaveOutScoringStep
- Short Description:
ABLATION: Yes, if the scoring step should be left out for the LV-Lite procedure
- Long Description:
If true, this leaves out the scoring steps and being the algorithm with
a complete nondirected graph (which is Markov).
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
ablationLeaveOutTestingSteps
- Short Description:
ABLATION: Yes, if the testing steps should be left out for the LV-Lite procedure
- Long Description:
If true, this leaves out all testing steps from the algorithm and bases
the result on just the scoring steps.
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
sepsetFinderMethod
- Short Description:
The method to use for finding sepsets, 1 = Greedy, 2 = Min-p, 3 = Max-p (default).
- Long Description:
The method to use for finding sepsets, 1 = Greedy, 2 = Min-p, 3 = Max-p (default).
- Default Value: 3
- Lower Bound: 1
- Upper
Bound: 3
- Value
Type: Integer
lvLiteStartsWith
- Short Description:
The algorithm to find the initial CPDAG: 1 = BOSS, 2 = GRaSP
- Long Description:
The algorithm to find the initial CPDAG: 1 = BOSS, 2 = GRaSP
- Default Value: 1
- Lower Bound: 1
- Upper
Bound: 2
- Value
Type: Integer
extraEdgeRemovalStep
- Short Description:
The extra edge removal step to use: 1 = LV_LITE, 2 = Greedy, 3 = Max P, 4 = Min P
- Long Description:
The extra edge removal step to use: 1 = LV_LITE, 2 = Greedy, 3 = Max P, 4 = Min P
- Default Value: 1
- Lower Bound: 1
- Upper
Bound: 4
- Value
Type: Integer
maxBlockingPathLength
- Short Description: Maximum path length to block in
extra-edge removal step, >= 3
- Long Description:
In the extra edge removal step, we build conditioning sets based on the
current PAG to attempt to remove adjacencies from the graph, by
blocking paths from x to y of up to this length.
- Default
Value: 5
- Lower Bound: 3
- Upper
Bound: 2147483647
- Value Type: Integer
maxSepsetSize
- Short Description: For testing steps in LV-Lite, the
maximum conditioning set size
- Long Description:
In the extra edge removal step, we build conditioning sets based on the
current PAG to attempt to remove adjacencies from the graph, by
blocking paths from x to y of up to this length. This is the maximum
size these sets are allowed to grow to.
- Default
Value: 8
- Lower Bound: 0
- Upper
Bound: 2147483647
- Value Type: Integer
maxScoreDrop
- Short Description:
Maximum score drop for the process triples step
- Long Description:
In orienting unshielded colliders by examining triples of nodes,
the score is permitted to drop by this much.
- Default Value: 5
- Lower Bound: 0
- Upper
Bound: Infinity
- Value
Type: Double
guaranteePag
- Short Description:
Guarantee that the output is a legal PAG
- Long Description:
Repairs errors in PAGs due to almost cyclic paths or non-maximalities.
This comes with a certain risk; errors in PAGs indicate that the PAG
assumptions were not met; the user may wish to consider why before
selecting this option.
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
removeAlmostCycles
- Short Description:
Yes if almost-cycles should be removed from the PAG.
- Long Description:
When x <-> y, x ~~> y, removes any unshielded triples into x and
rebuilds the PAG.
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
ablationLeaveOutFinalOrientation
- Short Description:
ABLATION: Leave out final orientation step.
- Long Description:
If true, the final orientation step of the algorithm is not performed.
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
intervalBetweenRecordings
- Short Description: Interval between data
recordings for the linear Fisher model (min = 1)
- Long Description:
- Default
Value: 10
- Lower
Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer
intervalBetweenShocks
- Short Description: Interval between shocks (R. A.
Fisher simulation model) (min = 1)
- Long Description:
This is a parameter for
the linear Fisher option. This sets the number of step between
shocks.
- Default Value: 10
- Lower
Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer
ipen
- Short Description: IPEN parameter (GLASSO)
- Long
Description: This sets the maximum number
of iterations of the optimization loop.
- Default
Value: false
- Lower
Bound:
- Upper Bound:
- Value Type: Boolean
is
- Short Description: IS parameter (GLASSO)
- Long
Description: Sets the maximum number of
iterations of the optimization loop.
- Default Value:
false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
itr
- Short Description: ITR parameter (GLASSO)
- Long
Description: Sets the maximum number of
iterations of the optimization loop.
- Default Value:
false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
kciAlpha
- Short Description: Cutoff for p values (alpha) (min =
0.0)
- Long Description: Alpha level (0 to 1)
- Default
Value: 0.05
- Lower
Bound: 0.0
- Upper
Bound: 1.0
- Value
Type: Double
kciCutoff
- Short Description: Cutoff
- Long Description:
Cutoff for p-values.
- Default Value: 6
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type:
Integer
kciEpsilon
- Short Description: Epsilon for Proposition 5, a small
positive number
- Long Description: See Zhang, K., Peters, J., Janzing, D., &
Schölkopf, B. (2012). Kernel-based conditional independence test and
application in causal discovery.. This parameter is the epsilon for
Proposition 5, a small positive number.
- Default
Value: 0.001
- Lower Bound: 0.0
- Upper Bound: Infinity
- Value Type:
Double
kciNumBootstraps
- Short Description: Number of bootstraps for Theorems 4
and Proposition 5 for KCI
- Long Description: This parameter is the number of
bootstraps for Theorems 4 from Zhang, K., Peters, J., Janzing, D., &
Schölkopf, B. (2012) and Proposition 5, a positive integer.
- Default Value: 5000
- Lower Bound:
1
- Upper
Bound: 2147483647
- Value Type: Integer
kciUseApproximation
- Short Description: Use the approximate Gamma
approximation algorithm
- Long Description: Referring to Zhang, K., Peters, J.,
Janzing, D., & Schölkopf, B. (2012), if this parameter is set to
‘Yes’, the Gamma approximation algorithm is used; if no, the exact
procedure is used.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type: Boolean
kernelMultiplier
- Short Description: Bowman and Azzalini (1997) default
kernel bandwidths should be multiplied by...
- Long
Description: For the
conditional correlation algorithm. Bowman, A. W., & Azzalini, A.
(1997), give a formula for default optimal kernel widths. We allow
these defaults to be multiplied by this factor, to capture more or
less than this optimal signal.
- Default Value: 1.0
- Lower Bound:
4.9E-324
- Upper Bound: Infinity
- Value
Type: Double
kernelRegressionSampleSize
- Short Description: Minimum sample size to use
per conditioning for kernel regression
- Long
Description: The
smallest set of nearest data points on which to allow a judgment to
be based for a nonlinear regression.
- Default Value:
100
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer
minCountPerCell
- Short Description:
Minimum expected count per cell for nonempty Chi-Square tables.
- Long Description:
The minimum count per cell in any nonempty conditional Chi-Square table
This is real-valued and must be > 0.0; the default is 1.0. No matter
what this value is, if a row is empty, it will be removed from the
table.
- Default Value: 1.0
- Lower
Bound: 0.0
- Upper Bound: Infinity
- Value Type:
Double
kernelType
- Short Description: Kernel type (1 = Gaussian, 2 =
Epinechnikov)
- Long Description: For CCI, this determines which kernel type
will be used (1 = Gaussian, 2 = Epinechnikov).
- Default Value: 2
- Lower Bound: 1
- Upper Bound: 2
- Value Type: Integer
kernelWidth
- Short Description: Kernel width
- Long
Description: A larger kernel width
means that more information will be taken into account but possibly
less focused information.
- Default Value: 1.0
- Lower Bound: 4.9E-324
- Upper Bound:
Infinity
- Value
Type: Double
latentMeasuredImpureParents
- Short Description: Number of Latent -->
Measured impure edges
- Long Description: It is possible for
structural nodes to have as children measured variables that are
children of other structural nodes. These edges in the graph will be
considered impure.
- Default Value: 0
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer
lowerBound
- Short Description: Lower bound cutoff threshold
- Long Description: null
- Default Value: 0.3
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double
maxCategories
- Short Description: Maximum number of categories (min =
2)
- Long Description: The maximum number of categories to be
used for randomly generated discrete variables. The default is 2.
This needs to be greater or equal to than the minimum number of
categories.
- Default Value: 3
- Lower Bound: 2
- Upper Bound: 2147483647
- Value Type:
Integer
maxCorrelation
- Short Description: Maximum absolute correlation
considered
- Long Description: For the Nandy rule, the absolute max
correlation r. For the standard BIC or high-dimensional rule, the
maximum absolute residual correlation.
- Default Value:
1.0
- Lower
Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double
maxDegree
- Short Description: The maximum degree of the graph (min =
-1)
- Long Description: An upper bound on the maximum degree of any
node in the graph. If no limit is to be placed on the maximum degree,
use the value -1.
- Default Value: 1000
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type:
Integer
maxDistinctValuesDiscrete
- Short Description: The maximum number of
distinct values in a column for discrete variables (min = 0)
- Long Description: Discrete variables will be
simulated using any number of categories from 2 up to this maximum.
If set to 0 or 1, discrete variables will not be generated.
- Default Value: 0
- Lower
Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer
maxIndegree
- Short Description: Maximum indegree of graph (min =
1)
- Long Description: An upper bound on the maximum indegree of
any node in the graph. If no limit is to be placed on the maximum
degree, use the value -1.
- Default Value: 1000
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type:
Integer
zsMaxIndegree
- Short Description: Maximum indegree of true graph (min = 0)
- Long Description: This is the maximum number of parents one expects any node to have in the true model.
- Default Value: 4
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer
maxIterations
- Short Description: The maximum number of iterations the
algorithm should go through orienting edges
- Long
Description: In orienting, this
algorithm may go through a number of iterations, conditioning on more
and more variables until orientations are set. This sets that
number.
- Default Value: 15
- Lower Bound:
0
- Upper Bound:
2147483647
- Value
Type: Integer
maxOutdegree
- Short Description: Maximum outdegree of graph (min =
1)
- Long Description: An upper bound on the maximum outdegree
of any node in the graph. If no limit is to be placed on the maximum
degree, use the value -1.
- Default Value: 1000
- Lower Bound:
1
- Upper Bound:
2147483647
- Value
Type: Integer
maxPOrientationMaxPathLength
- Short Description: Maximum path length for
the unshielded collider heuristic for max P (min = 0)
- Long Description: For the Max P
“heuristic” to work, it must be the case that X and Z are only weakly
associated—that is, that paths between them are not too short. This
bounds the length of paths for this purpose.
- Default
Value: 3
- Lower Bound: 0
- Upper
Bound: 2147483647
- Value Type: Integer
maxPathLength
- Short Description: The maximum length for any
discriminating path. -1 if unlimited (min = -1)
- Long
Description: See Spirtes,
Glymour, and Scheines (2000) for the definition of discrimination
path. Finding discriminating paths can be expensive. This sets the
maximum length of such paths that the algorithm tries to find.
- Default Value: -1
- Lower Bound:
-1
- Upper Bound:
2147483647
- Value
Type: Integer
maxit
- Short Description: MAXIT parameter (GLASSO) (min = 1)
- Long Description: Sets the maximum
number of iterations of the optimization loop.
- Default Value: 10000
- Lower Bound: 1
- Upper
Bound: 2147483647
- Value
Type: Integer
meanHigh
- Short Description: High end of mean range (min =
0.0)
- Long Description:
The default is for there to be no shift in mean, but shifts from a
minimum value to a maximum value may be specified. The minimum must
be less than or equal to this maximum.
- Default
Value: 1.0
- Lower
Bound: 0.0
- Upper
Bound: 1.7976931348623157E308
- Value Type: Double
meanLow
- Short Description: Low end of mean range (min = 0.0)
- Long Description: The default is
for there to be no shift in mean, but shifts from a minimum value to
a maximum value may be specified. The minimum must be greater than or
equal to this minimum.
- Default Value: 0.5
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value
Type: Double
measuredMeasuredImpureAssociations
- Short Description: Number of
Measured <-> Measured impure edges
- Long Description:
It is
possible for measures from two different structural nodes to be
confounded. These confounding (bidirected) edges will be considered
to be impure.
- Default Value: 0
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer
measuredMeasuredImpureParents
- Short Description: Number of Measured -->
Measured impure edges
- Long Description: It is possible for
measures from two different structural nodes to have directed edges
between them. These edges will be considered to be impure.
- Default Value: 0
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer
measurementModelDegree
- Short Description: Number of measurements per
Latent
- Long Description: Each structural node in the
MIM will be created to have this many measured children.
- Default Value: 5
- Lower
Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer
measurementVariance
- Short Description: Additive measurement noise
variance (min = 0.0)
- Long Description: If the value is greater than
zero, independent Gaussian noise will be added with mean zero and the
given variance to each variable in the simulated output.
- Default Value: 0.0
- Lower
Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double
mgmParam1
- Short Description: MGM tuning parameter #1 (min =
0.0)
- Long Description: The MGM algorithm has three internal tuning
parameters, of which this is one.
- Default Value:
0.1
- Lower Bound:
0.0
- Upper Bound:
1.7976931348623157E308
- Value Type: Double
mgmParam2
- Short Description: MGM tuning parameter #2 (min =
0.0)
- Long Description: The MGM algorithm has three internal tuning
parameters, of which this is one.
- Default Value:
0.1
- Lower Bound:
0.0
- Upper Bound:
1.7976931348623157E308
- Value Type: Double
mgmParam3
- Short Description: MGM tuning parameter #3 (min =
0.0)
- Long Description: The MGM algorithm has three internal tuning
parameters, of which this is one.
- Default Value:
0.1
- Lower Bound:
0.0
- Upper Bound:
1.7976931348623157E308
- Value Type: Double
minCategories
- Short Description: Minimum number of categories (min =
2)
- Long Description: The minimum number of categories to be
used for randomly generated discrete variables. The default is
2.
- Default Value: 3
- Lower Bound: 2
- Upper Bound: 2147483647
- Value Type:
Integer
noRandomlyDeterminedIndependence
- Short Description: Yes, if using the
cutoff threshold for the independence test.
- Long
Description: null
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
numBasisFunctions
- Short Description: Number of functions to use in
(truncated) basis
- Long Description: This parameter specifies how many
of the most significant basis functions to use as a basis.
- Default Value: 30
- Lower Bound:
1
- Upper
Bound: 2147483647
- Value
Type: Integer
numBscBootstrapSamples
- Short Description: The number of bootstrappings
drawing from posterior dist. (min = 1)
- Long
Description: The number
of bootstrappings drawing from posterior dist. (min = 1)
- Default Value: 50
- Lower
Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer
numCategories
- Short Description: Number of categories for discrete
variables (min = 2)
- Long Description: The number of categories to be used for
randomly generated discrete variables. The default is 4; the minimum
is 2.
- Default Value: 4
- Lower Bound: 2
- Upper Bound: 2147483647
- Value Type:
Integer
numCategoriesToDiscretize
- Short Description: The number of categories
used to discretize continuous variables, if necessary (min =
2)
- Long Description: In case the exact algorithm
is not used for discrete children and continuous parents is not used,
this parameter gives the number of categories to use for this
second (discretize) backup copy of the continuous variables.
- Default Value: 3
- Lower
Bound: 2
- Upper Bound: 2147483647
- Value Type: Integer
numLags
- Short Description: The number of lags in the time lag
model
- Long Description:
A time lag model may take variables from previous time steps into
account. This determines how many steps back these relevant variables
might go.
- Default Value: 1
- Lower Bound: -2147483648
- Upper Bound:
2147483647
- Value Type:
Integer
numLatents
- Short Description: Number of additional latent variables (min
= 0)
- Long Description: The number of additional latent
variables to include in the datasets
- Default Value:
0
- Lower Bound:
0
- Upper Bound:
2147483647
- Value
Type: Integer
numMeasures
- Short Description: Number of measured variables (min =
1)
- Long Description: The number of measured (recorded in data)
variables to include in the dataset.
- Default Value:
10
- Lower Bound:
1
- Upper Bound:
2147483647
- Value
Type: Integer
numRandomizedSearchModels
- Short Description: The number of search
probabilistic model (min = 1)
- Long Description:
The number of search
probabilistic model (min = 1)
- Default Value: 10
- Lower
Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer
numRuns
- Short Description: Number of runs (min = 1)
- Long
Description: An analysis(randomly pick
graph, randomly simulate a dataset, run an algorithm on it, look at
the result) may be run over and over again this many times.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer
probRemoveColumn
- Short Description: Probability of randomly removing a column from a dataset
- Long
Description:
For testing algorithms with overlapping variables, columns may be removed
from datasets with this probability.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double
numStructuralEdges
- Short Description: Number of structural
edges
- Long Description: This is a parameter for generating
random multiple indicator models (MIMs). A structural edge is an edge
connecting two structural nodes.
- Default Value:
3
- Lower
Bound: -2147483648
- Upper
Bound: 2147483647
- Value
Type: Integer
numStructuralNodes
- Short Description: Number of structural
nodes
- Long Description: This is a parameter for generating
random multiple indicator models (MIMs). A structural node is one of
the latent variables in the model; each structural node has a number
of child measured variables.
- Default Value: 3
- Lower Bound:
-2147483648
- Upper Bound: 2147483647
- Value
Type: Integer
numberResampling
- Short Description: The number of bootstraps/resampling
iterations (min = 0)
- Long Description: For bootstrapping, the number of
bootstrap iterations that should be done by the algorithm, with
results summarized.
- Default Value: 0
- Lower Bound:
0
- Upper
Bound: 2147483647
- Value Type: Integer
numStarts
- Short Description: The number of restarts, random after the
first (default 1)
- Long Description: The number of times the algorithm should
be started from different initializations. By default, the algorithm
will be run through at least once using the initialized parameters
(zero random restarts).
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type:
Integer
useBes
- Short Description: True if the optional BES step should be used
- Long Description: This algorithm can use the backward equivalence search
from the GES algorithm as one of its steps.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
otherPermMethod
- Short Description: 1 = RCG, 2 = GSP, 3 = ESP, 4 =
SP
- Long Description: RCG (Random Carnival Game); GSP
("Greedy SP") GSP using tucking ESP ("Edge SP") is from Solus et al.
SP ("Sparsest Permutation") Raskutti and Uhler
- Default Value: 1
- Lower Bound:
1
- Upper Bound:
5
- Value Type:
Integer
bossAlg
- Short Description: Picks the BOSS algorithm type, BOSS1 or BOSS2
- Long
Description: 1 = BOSS1, 2 = BOSS2, 3 = BOSS3
- Default Value: 1
- Lower Bound:
1
- Upper Bound:
3
- Value Type:
Integer
graspCheckCovering
- Short Description: Yes if covering of edges should
be checked (GASP), no if not (GRASP)
- Long
Description: An edge X is
covered if Parents(X) = Parents(Y) \ {X}. Not checking covering
expands the search space.
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
graspForwardTuckOnly
- Short Description: Yes if only forward tucks
should be checked, no if also reverse tucks should be checked.
- Long Description: A forward tuck for X->Y moves Y
to the before position of X in the permutation. A reverse tuck moves
Y to after the position of X in the permutation. Including reverse
tucks expands the search space.
- Default Value:
false
- Lower Bound:
- Upper Bound:
- Value
Type: Boolean
graspBreakAfterImprovement
- Short Description: Yes if depth first search
returns after first improvement, No for depth first traversal.
- Long Description: Exploring the full list in
every DFS call is equivalent to what we've been calling the Random
Carnival Game procedure (RCG).
- Default Value: true
- Lower Bound:
- Upper
Bound:
- Value Type: Boolean
graspOrderedAlg
- Short Description: Yes if earlier GRaSP stages should
be performed before later stages
- Long Description:
GRaSP has three stages; these
can be performed separately or in order; by default Yes.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
graspUseScore
- Short Description: Yes if the score should be used for MB
calculations, no if the test should be used instead.
- Long Description: In either
case, compositional graphoid axioms are assumed by the Grow-Shrink
algorithm.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
graspUseRaskuttiUhler
- Short Description: Yes to use Raskutti and Uhler's
DAG-building method (test), No to use Grow-Shrink (score).
- Long Description:
Raskutti and Uhler's method adds and edge X->Y if Y ~_||_ X |
Prefix(Y, pi) \ {X}. Grow-Shrink adds an edge X->Y if X is in the
Markov blanket of Y where the variable set is restricted to Prefix(Y,
pi).
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
useDataOrder
- Short Description: Yes just in case data variable
order should be used for the first initial permutation.
- Long Description: In
either case, if multiple starting points are used, taking the best
scoring model from among these, subsequent starting points will all
be random shuffles.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type: Boolean
allowInternalRandomness
- Short Description: Allow randomness
inside algorithm
- Long Description: This allows
variables orders to be shuffled in certain sports to avoid local optima
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
graspUseVpScoring
- Short Description: No sure
- Long
Description: Not sure
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type: Boolean
graspAlg
- Short Description: 1 = GRaSP1, 2 = GRaSP2, 3 = esp, 4 =
GRaSP4, 5 = GRaSP4
- Long Description: Which version of GRaSP (temp parameter)
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 5
- Value Type: Integer
graspDepth
- Short Description: Recursion depth (for GRaSP)
- Long
Description: This is the depth of
recursion for the depth first search.
- Default
Value: 3
- Lower
Bound: 0
- Upper
Bound: 2147483647
- Value Type: Integer
graspSingularDepth
- Short Description: Recursion depth for singular
tucks
- Long Description: This is the depth of recursion
for the singular tucks.
- Default
Value: 1
- Lower Bound: 0
- Upper Bound:
2147483647
- Value Type: Integer
graspNonSingularDepth
- Short Description: Recursion depth for nonsingular
tucks
- Long Description: This is the depth of recursion
at which multiple tucks may be considered per score improvement
- Default Value: 1
- Lower
Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer
graspToleranceDepth
- Short Description: Recursion depth for tolerance
tucks
- Long Description: This is the maximum number of
non-greedy tucks in depth first order --that is, tucks where the
score is allowed to decrease rather than increase.
- Default Value: 0
- Lower Bound:
0
- Upper
Bound: 2147483647
- Value
Type: Integer
timeout
- Short Description: Timeout (best graph returned, -1 = no
timeout)
- Long Description: The algorithm will time out at approximately
this number of seconds from when it started and return the final
graph found at that point.
- Default Value: -1
- Lower Bound: -1
- Upper Bound: 2147483647
- Value Type: Integer
testTimeout
- Short Description:
Timeout for tests in milliseconds, or -1 if no timeout.
- Long Description:
Timeout for tests in milliseconds, or -1 if no timeout.
- Default Value: -1
- Lower Bound: -1
- Upper Bound: 9223372036854775807
- Value Type: Long
- Short Description:
Yes if the algorithm should try
moving variables pairwise
- Long Description: In some cases, two moves are required
simultaneously to get an orientation right in the final step. This is
not generally needed when optimizing using BIC or for large
models.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
recursive
- Short Description: Yes if the algorithm should proceed
recursively, no if not
- Long Description: Where recursive or nonrecursive variants of
an algorithm are available, this selects which one to use.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
orientTowardMConnections
- Short Description: Yes if Richardson's step C
(orient toward d-connection) should be used
- Long
Description: Please
see the description of this algorithm in Thomas Richardson and Peter
Spirtes in Chapter 7 of Computation, Causation, & Discovery by
Glymour and Cooper eds.
- Default Value: true
- Lower Bound:
- Upper
Bound:
- Value Type: Boolean
orientVisibleFeedbackLoops
- Short Description: Yes if visible feedback
loops should be oriented
- Long Description: Please see the description
of this algorithm in Thomas Richardson and Peter Spirtes in Chapter 7
of Computation, Causation, & Discovery by Glymour and Cooper
eds.
- Default Value: true
- Lower Bound:
- Upper
Bound:
- Value Type: Boolean
outputRBD
- Short Description: Constraint Scoring: Yes: Dependent
Scoring, No: Independent Scoring.
- Long Description:
Constraint Scoring: Yes: Dependent
Scoring, No: Independent Scoring.
- Default Value:
true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
precomputeCovariances
- Short Description: True if covariance matrix should
be precomputed for tabular continuous data
- Long Description:
For more than 5000 variables or so, set this to false in order to calculate covariances
on the fly from data.
- Default Value:
true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
penaltyDiscount
- Short Description: Penalty discount (min =
0.0)
- Long Description: The parameter c added to a modified
BIC score of the form 2L – c k ln N, where L is the likelihood, k the
number of degrees of freedom, and N the sample size. Higher c yield
sparser graphs.
- Default Value: 2.0
- Lower Bound:
0.0
- Upper
Bound: 1.7976931348623157E308
- Value Type: Double
penaltyDiscountZs
- Short Description: Penalty discount (min =
0.0)
- Long Description: The parameter c added to a modified
BIC score of the form 2L – c k lambda, where L is the likelihood, k the
number of degrees of freedom, and lambda the choice of GIC lambda. Higher c yield
sparser graphs.
- Default Value: 1.0
- Lower Bound:
0.0
- Upper
Bound: 1.7976931348623157E308
- Value Type: Double
zSRiskBound
- Short Description: Risk bound
- Long Description:
This is the probability of getting the true model if a correct model is discovered. Could underfit.
- Default Value: 0.1
- Lower Bound: 0
- Upper Bound: 1
- Value Type: Double
ebicGamma
- Short Description: EBIC Gamma (0-1)
- Long
Description: The gamma parameter for
Extended BIC (Chen and Chen). In [0, 1].
- Default
Value: 0.8
- Lower
Bound: 0.0
- Upper
Bound: 1.0
- Value
Type: Double
trueErrorVariance
- Short Description: True error variance
- Long Description: The
true error variance of the model, assuming this is the same for all
variables.
- Default Value: 1.0
- Lower Bound:
0.0
- Upper
Bound: 1.7976931348623157E308
- Value Type: Double
correlationThreshold
- Short Description: Correlation
Threshold
- Long Description: The algorithm will complain if
correlations are found that are greater than this in absolute
value.
- Default Value: 1
- Lower
Bound: 0
- Upper Bound: 1
- Value Type:
Double
manualLambda
- Short Description: Lambda (manually set)
- Long Description: The manually
set lambda for GIC--the default is 10, though this should be set by
the user to a good value.
- Default Value: 10.0
- Lower Bound:
0.0
- Upper Bound:
1.7976931348623157E308
- Value Type: Double
errorThreshold
- Short Description: Error Threshold
- Long
Description: Adjusts the
threshold for judging conditional dependence.
- Default Value: 0.5
- Lower Bound:
0.0
- Upper
Bound: 1
- Value
Type: Double
parallelized
- Short Description:
Yes if the search should be parallelized
- Long Description: This search is capable of being
parallelized; select yes if the search should be parallelized,
not if it should be run in a single thread
- Default Value:
false
- Lower Bound:
- Upper Bound:
- Value
Type: Boolean
percentDiscrete
- Short Description: Percentage of discrete variables (0 -
100) for mixed data
- Long Description: For a mixed data type simulation,
specifies the percentage of variables that should be simulated
(randomly) as discrete. The rest will be taken to be continuous. The
default is 0—i.e. no discrete variables.
- Default
Value: 50.0
- Lower Bound: 0.0
- Upper Bound:
100.0
- Value
Type: Double
percentResampleSize
- Short Description: The percentage of resample size
(min = 10%)
- Long Description: This parameter specifies the
percentage of records in the bootstrap (as a percentage of the total
original sample size of the data being bootstrapped).
- Default Value: 100
- Lower
Bound: 10
- Upper Bound: 100
- Value
Type: Integer
possibleMsepDone
- Short Description: Yes if the possible d-sep search
should be done
- Long Description: This algorithm has a possible d-sep
path search, which can be time-consuming. See Spirtes, Glymour, and
Scheines (2000) for details.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
probCycle
- Short Description: The probability of adding a cycle to the
graph
- Long Description: Sets the probability that any particular
set of 3, 4, or 5 of nodes will be used to form a cycle in the
graph.
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double
probTwoCycle
- Short Description: The probability of creating a 2-cycles
in the graph (0 - 1)
- Long Description: Any edge X*-*Y may be replaced with a
2-cycle (feedback loop) between X and Y with this probability.
- Default Value: 0.0
- Lower Bound:
0.0
- Upper Bound:
1.0
- Value Type:
Double
randomSelectionSize
- Short Description: The number of datasets that
should be taken in each random sample
- Long
Description: The number of
dataset that should be taken in each random sample of
datasets.
- Default Value: 1
- Lower Bound:
-2147483648
- Upper Bound: 2147483647
- Value
Type: Integer
randomizeColumns
- Short Description: Yes if the order of the columns in
each dataset should be randomized
- Long Description:
In the real world where
unfaithfulness is an issue the order of variables in the data may for
some algorithms affect the output. For testing purposes, if Yes, the
data columns are randomly re-ordered.
- Default Value:
true
- Lower
Bound:
- Upper
Bound:
- Value
Type: Boolean
guaranteeIid
- Short Description: Recursive simulation is used for acyclic models; if not should i.i.d. be assumed?
- Long Description:
For cyclic models, the Fisher simulation model is used, which is a time series. Selecting 'Yes' here
guarantees that a new data point starts from a new shock without influence from the previous time step.
- Default Value:
true
- Lower
Bound:
- Upper
Bound:
- Value
Type: Boolean
rcitNumFeatures
- Short Description: The number of random features to
use
- Long Description:
- Default Value: 10
- Lower Bound:
1
- Upper Bound:
2147483647
- Value Type: Integer
resamplingEnsemble
- Short Description: Ensemble method: Preserved (1),
Highest (2), Majority (3)
- Long Description: Preserved = keep the highest frequency
edges; Highest = keep the highest frequency edges but ignore the no edge
case if maximal; Majority = keep edges only if their frequency is
greater than 0.5.
- Default Value: 1
- Lower Bound:
1
- Upper
Bound: 3
- Value Type: Integer
resamplingWithReplacement
- Short Description: Yes, if sampling with
replacement (bootstrapping)
- Long Description: Yes if resampling can be
done with replacement, No if not. or without replacement. If with
replacement, it is possible to have more than one copy of some of the
records in the original dataset being included in the
bootstrap.
- Default Value: true
- Lower Bound:
- Upper
Bound:
- Value Type: Boolean
priorEquivalentSampleSize
- Short Description: Prior equivalent sample
size (min = 1.0)
- Long Description: This sets the prior
equivalent sample size. This number is added to the sample size for
each conditional probability table in the model and is divided
equally among the cells in the table.
- Default
Value: 10.0
- Lower Bound: 1.0
- Upper Bound:
1.7976931348623157E308
- Value Type:
Double
sampleSize
- Short Description: Sample size (min = 1)
- Long
Description: Determines now many
records should be generated for the data. The minimum number of
records is 1; the default is set to 1000.
- Default
Value: 1000
- Lower
Bound: 1
- Upper
Bound: 2147483647
- Value Type: Integer
saveLatentVars
- Short Description: Save latent variables.
- Long Description: Yes if one
wishes to have values for latent variables saved out with the rest of
the data; No if only data for the measured variables should be
saved.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type:
Boolean
scaleFreeAlpha
- Short Description: For scale-free graphs, the parameter
alpha (min = 0.0)
- Long Description: We use the algorithm for generating
scale free graphs described in B. Bollobas,C. Borgs, J. Chayes, and
O. Riordan (2003). Please see this article for a description of the
parameters.
- Default Value: 0.05
- Lower Bound:
0.0
- Upper
Bound: 1.0
- Value Type: Double
scaleFreeBeta
- Short Description: For scale-free graphs, the parameter
beta (min = 0.0)
- Long Description: We use the algorithm for generating
scale free graphs described in B. Bollobas,C. Borgs, J. Chayes, and
O. Riordan (2003). Please see this article for a description of the
parameters.
- Default Value: 0.9
- Lower Bound:
0.0
- Upper Bound:
1.0
- Value Type:
Double
scaleFreeDeltaIn
- Short Description: For scale-free graphs, the parameter
delta_in (min = 0.0)
- Long Description: We use the algorithm for generating
scale free graphs described in B. Bollobas,C. Borgs, J. Chayes, and
O. Riordan (2003). Please see this article for a description of the
parameters.
- Default Value: 3
- Lower Bound:
-2147483648
- Upper Bound: 2147483647
- Value
Type: Integer
scaleFreeDeltaOut
- Short Description: For scale-free graphs, the
parameter delta_out (min = 0.0)
- Long Description:
We use the algorithm for
generating scale free graphs described in B. Bollobas,C. Borgs, J.
Chayes, and O. Riordan (2003). Please see this article for a
description of the parameters.
- Default Value: 3
- Lower Bound:
-2147483648
- Upper Bound: 2147483647
- Value
Type: Integer
seed
- Short Description: Seed for pseudorandom number generator (-1 = off)
- Long Description: The seed is the initial value of the
internal state of the pseudorandom number generator. A value of -1
skips setting a new seed.
- Default Value: -1
- Lower Bound:
-1
- Upper
Bound: 9223372036854775807
- Value Type: Long
selfLoopCoef
- Short Description: The coefficient for the self-loop
(default 0.0)
- Long Description: For simulating time series data, each
variable depends on itself one time-step back with a linear edge that
has this coefficient.
- Default Value: 0.0
- Lower Bound:
0.0
- Upper Bound:
Infinity
- Value
Type: Double
semBicRule
- Short Description: Lambda: 1 = Chickering, 2 = Nandy
- Long Description: The
Chickering Rule uses the difference of BIC scores to add or remove
edges. The Nandy et al. rule uses a single calculation of a partial
correlation in place of the likelihood difference.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 2
- Value Type: Integer
semGicRule
- Short Description: Lambda: 1 = ln n, 2 = pn^1/3, 3 = 2 ln
pn, 4 = 2(ln pn + ln ln pn), 5 = ln ln n ln pn, 6 = ln n ln pn, 7 =
Manual
- Long Description: The rule used for calculating the lambda
term of the score. We follow Kim, Y., Kwon, S., & Choi, H. (2012) and
articles referenced therein. For high-dimensional data.
- Default Value: 4
- Lower Bound: 1
- Upper Bound: 7
- Value Type: Integer
semBicStructurePrior
- Short Description: Structure Prior for SEM BIC
(default 0)
- Long Description: Structure prior; default is 0
(turned off); may be any positive number otherwise
- Default Value: 0
- Lower
Bound: 0
- Upper Bound: Infinity
- Value
Type: Double
poissonLambda
- Short Description: Lambda parameter for the Poisson distribution
(> 0)
- Long Description: Lambda parameter for the Poisson distribution
- Default Value: 1
- Lower
Bound: 1e-10
- Upper Bound: Infinity
- Value
Type: Double
skipNumRecords
- Short Description: Number of records that should be
skipped between recordings (min = 0)
- Long
Description: Data recordings are
made every this many steps.
- Default Value: 0
- Lower Bound:
0
- Upper Bound:
2147483647
- Value Type: Integer
stableFAS
- Short Description:
Yes if the Colombo et al. 'stable' FAS should be done, to
avoid skeleton order dependency
- Long Description: If Yes, the "stable" version of the PC
adjacency search is used, which for k > 0 fixes the graph for depth k
+ 1 to that of the previous depth k.
- Default Value:
true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
standardize
- Short Description: Yes if the data should be
standardized
- Long Description: Yes if each variable in the data should
be standardized to have mean zero and variance 1.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
structurePrior
- Short Description: Structure prior coefficient (min =
0.0)
- Long Description: The default number of parents for any
conditional probability table. Higher weight is accorded to tables
with about that number of parents. The prior structure weights are
distributed according to a binomial distribution.
- Default Value: 0.0
- Lower Bound:
0.0
- Upper Bound:
1.7976931348623157E308
- Value Type: Double
symmetricFirstStep
- Short Description: Yes if the first step for
FGES should do scoring for both X->Y and Y->X
- Long
Description: If Yes, scores
for both X->Y and X<-Y will be calculated and the higher score
used.
- Default Value: false
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
targetName
- Short Description: Target variable name
- Long
Description: The name of the target
variables--for Markov blanket searches, this is the name of the
variable for which one wants the Markov blanket or Markov blanket
graph.
- Default Value:
- Lower Bound:
- Upper Bound:
- Value Type: String
thr
- Short Description: THR parameter (GLASSO) (min = 0.0)
- Long Description: Sets the maximum
number of iterations of the optimization loop.
- Default Value: 1.0E-4
- Lower Bound: 0.0
- Upper
Bound: 1.7976931348623157E308
- Value Type: Double
thresholdForNumEigenvalues
- Short Description: Threshold to determine
how many eigenvalues to use--the lower the more (0 to 1)
- Long Description: Referring to Zhang, K.,
Peters, J., Janzing, D., & Schölkopf, B. (2012), this parameter is
the threshold to determine how many eigenvalues to use--the lower the
more (0 to 1).
- Default Value: 0.001
- Lower Bound: 0.0
- Upper
Bound: Infinity
- Value Type: Double
thresholdNoRandomConstrainSearch
- Short Description: Yes, if using the
cutoff threshold for the meta-constraints independence test (stage
2).
- Long Description: Yes, if using the
cutoff threshold for the meta-constraints independence test (stage
2).
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
thresholdNoRandomDataSearch
- Short Description: Yes, if using the cutoff
threshold for the constraints independence test (stage 1).
- Long Description: null
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
twoCycleAlpha
- Short Description: Alpha orienting 2-cycles (min =
0.0)
- Long Description: The alpha level of a T-test used to
determine where 2-cycles exist in the graph. A value of zero turns
off 2-cycle detection.
- Default Value: 0.0
- Lower Bound:
0.0
- Upper Bound:
1.0
- Value Type:
Double
timeLimit
- Short Description: Time limit
- Long
Description: T-Separation requires a
time limit. Default 1000.
- Default Value: 1000.0
- Lower Bound:
0.0
- Upper Bound:
1.7976931348623157E308
- Value Type: Double
adjustOrientations
- Short Description: Yes, if the orientation adjustment
step should be included
- Long Description: Yes, if the orientation adjustment
step should be included
- Default Value: false
- Lower
Bound:
g
- Upper Bound:
- Value Type:
Boolean
upperBound
- Short Description: Upper bound cutoff threshold
- Long Description: null
- Default Value: 0.7
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double
useCorrDiffAdjacencies
- Short Description: Yes if adjacencies from
conditional correlation differences should be used
- Long Description:
FASK can use adjacencies X—Y where |corr(X,Y|X>0) – corr(X,Y|Y>0)| >
threshold. This expression will be nonzero only if there is a path
between X and Y; heuristically, if the difference is greater than,
say, 0.3, we infer an adjacency.
- Default Value:
true
- Lower Bound:
- Upper Bound:
- Value
Type: Boolean
useFasAdjacencies
- Short Description: Yes if adjacencies from the FAS
search (correlation) should be used
- Long
Description: Determines
whether adjacencies found by conditional correlation should be
included in the final model.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type: Boolean
semImSimulationType
- Short Description: Yes if recursive simulation, No
if reduced form simulation
- Long Description: Determines the type of simulation
done. If recursive, the graph must be a DAG in causal order. "Reduced
form" means X = (I - B)^-1 e, which requires a possibly large matrix
inversion.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
simulationErrorType
- Short Description: 1 = Usual LG SEM, 2 =
U(lb, ub), 3 = Exp(lambda), 4 = Gumbel(mu,
beta), 5 = Gamma(shape, scale)
- Long Description: Exogenous error type
- Default Value: 1
- Lower Bound:
1
- Upper
Bound: 5
- Value Type: Integer
simulationParam1
- Short Description: Indep error parameter
#1
- Long Description: Exogenous error parameter
#1
- Default Value: 0.0
- Lower Bound:
-1000
- Upper
Bound: 1000
- Value Type: Double
simulationParam2
- Short Description: Indep error parameter #2, if
used
- Long Description: Exogenous error parameter
#2
- Default Value: 1.0
- Lower Bound:
-1000
- Upper
Bound: 1000
- Value Type: Double
ess
- Short Description: Yes if the equivalent sample size should be used
in place of N
- Long Description: We calculate the equivalent sample size by
assuming that all record are equally correlated
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value Type: Boolean
timeLag
- Short Description: A time lag for time series data,
automatically applied (zero if none)
- Long Description: Automatically applies the time lag
transform to the data, creating additional lagged variables. If
zero, no time lag is applied. A positive integer
- Default Value: 0
- Lower Bound: 0
- Upper
Bound: 2147483647
- Value Type: Integer
useGap
- Short Description: Yes if the GAP algorithms should be used. Not
if the SAG algorithm should be used
- Long
Description: True if one should first
find all possible initial sets, grows these out, and then picks a
non-overlapping such largest sets from these. Not if one should grow
pure clusters one at a time, excluding variables found in earlier
clusters.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
useMaxPHeuristic
- Short Description: Yes if the max P
heuristic should be used to orient unshielded colliders
- Long Description: The max P heuristic orients
X*-*Y*-*Z by considering whether Y is in the subset S of variables
adjacent to X or Z that maximizes the p-value of X _||_ Y | S.
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
useMaxPOrientationHeuristic
- Short Description: Yes if the heuristic for
orienting unshielded colliders for max P should be used
- Long Description: Another way to do the
orientation if X and Z are only weakly dependent, is to simply see
whether the p-value for X _||_ Z | Y is greater than the p-value for
X _||_ Z. The purpose is to speed up the search.
- Default Value: false
- Lower Bound:
- Upper
Bound:
- Value
Type: Boolean
useSkewAdjacencies
- Short Description: Yes if adjacencies based on
skewness should be used
- Long Description: FASK can use adjacencies X—Y where
|corr(X,Y|X>0) – corr(X,Y|Y>0)| > threshold. This expression will be
nonzero only if there is a path between X and Y; heuristically, if
the difference is greater than, say, 0.3, we infer an adjacency. To
see adjacencies included for this reason, set this parameter to
“Yes”. Sanchez-Romero, Ramsey et al., (2018) Network
Neuroscience.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
useWishart
- Short Description: Yes if the Wishart test should be used. Not
if the Delta test should be used
- Long Description:
This is a parameter for the FOFC
(Find One Factor Clusters) algorithm. There are two tests implemented
there for testing for tetrads being zero, Wishart and Delta. This
parameter picks which of these tests should be used: ‘Yes’ for Wishart
and ‘No’ for Delta.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
checkType
- Short Description:
Model significance check type: 1 = Significance, 2 = Clique, 3 = None
- Long Description:
Model significance check type: 1 = Significance, 2 = Clique, 3 = None
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 3
- Value
Type: Integer
varHigh
- Short Description: High end of variance range (min =
0.0)
- Long Description:
The parameter 'b' for drawing independent variance values, from +U(a,
b).
- Default Value: 3.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value
Type: Double
significanceChecked
- Short Description:
True if the significance of the cluster should be checked.
- Long Description:
True if the significance of clusters should be checked, false if not.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value
Type: Boolean
varLow
- Short Description: Low end of variance range (min =
0.0)
- Long Description:
The parameter 'a' for drawing independent variance values, from +U(a,
b).
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value
Type: Double
verbose
- Short Description: Yes if verbose output should be printed or
logged
- Long Description: If this parameter is set to ‘Yes’, extra
(“verbose”) output will be printed if available giving some details
about the step-by-step operation of the algorithm.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
meekVerbose
- Short Description: Yes if verbose output for Meek rule
applications should be printed or logged
- Long
Description: If this parameter is
set to ‘Yes’, Meek rule applications will be printed out to the
log.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
useScore
- Short Description: Yes if the score should be used; no if the
test should be used
- Long Description: BOSS can run either from a score or a test;
this lets you choose which.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
outputCpdag
- Short Description: Yes if CPDAG should be output, no if a
DAG.
- Long Description: BOSS can output a DAG or the CPDAG of the
DAG.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
cacheScores
- Short Description: Yes score results should be cached, no if
not
- Long Description: Caching scores can use a lot of
memory.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean
verbose
- Short Description: Yes if the (MimBuild)
structure model should be included in the output graph
- Long Description:
FOFC proper yields a measurement model--that is, a set of pure
children for each of the discovered latents. One can estimate the
structure over the latents (the structure model) using Mimbuild. This
structure model is included in the output if this parameter is set to
Yes.
- Default Value: true
- Lower
Bound:
- Upper Bound:
- Value Type:
Boolean
pathsMaxNumSets
- Short Description:
The maximum number of adjustment sets to output
- Long
Description:
There may be too many legal adjustments to sets to output; this places
a bound on how many to output. These will be listed in order of
increasing size.
- Default Value:
4
- Lower
Bound: 0
- Upper Bound: 100000
- Value Type:
Integer
pathsMaxDistanceFromEndpoint
- Short Description:
The maximum distance of an allowable node from the endpoint of a path
for adjustment
- Long
Description:
In order to give guidance to which adjustment sets to report, this
parameter lets one give a maximum distance from the endpoint of a
path for a node to be included in an adjustment set.
- Default Value:
3
- Lower
Bound: 0
- Upper Bound: 100000
- Value Type:
Integer
pathsNearWhichEndpoint
- Short Description:
1 = near source, 2 = near target, 3 = near either
- Long
Description:
Adjustment sets may be found near the source, near the target, or
near either.
- Default Value:
1
- Lower
Bound: 1
- Upper Bound: 3
- Value Type:
Integer
pathsMaxLengthAdjustment
- Short Description:
The maximum length of a backdoor path to consider for adjustment.
- Long
Description:
The maximum length of a backdoor path to consider for finding an
adjustment set. Amenable paths of any length are considered.
- Default Value:
8
- Lower
Bound: 2
- Upper Bound: 100000
- Value Type:
Integer
pathsMaxLength
- Short Description:
The maximum length of a path to report
- Long
Description:
Since paths may be long, especially for large graphs, this parameter
allows one to limit the length of a path to report. It must be at least
2.
- Default Value:
8
- Lower
Bound: 2
- Upper Bound: 100000
- Value Type:
Integer
Regression
Box
The regression box performs regression on variables in a data set,
in an attempt to discover causal correlations between them. Both linear
and regression are available.
Possible Parent Boxes of the Regression Box
- A data box
- A simulation box
Possible Child Boxes of the Instantiated Model Box:
- A graph box
- A compare box
- A
parametric model box
- A data box
- A simulation box
- A search box
Multiple Linear Regression
Linear regression is performed upon continuous data sets. If you
have a categorical data set upon which you would like to perform linear
regression, you can make it continuous using the data manipulation
box.
Take, for example, a data set with the following underlying causal
structure:
When used as input to the linear regression box, the following
window results:
To select a variable as the response variable, click on it in the
leftmost box, and then click on the top right-pointing arrow. If you
change your mind about which variable should be the response variable,
simply click on another variable and click on the arrow again.
To select a variable as a predictor variable, click on it in the
leftmost box, and then click on the second right-pointing arrow. To
remove a predictor variable, click on it in the predictor box and then
click on the left-pointing arrow.
Clicking “Sort Variables” rearranges the variables in the
predictor box so that they follow the same order they did in the leftmost
box. The alpha value in the lower left corner is a threshold for
independence; the higher it is set, the less discerning Tetrad is when
determining the independence of two variables.
When we click “Execute,” the results of the regression appear in
the box to the right. For each predictor variable, Tetrad lists the
standard error, t value, and p value, and whether its correlation with
the response variable is significant.
The Output Graph tab contains a graphical model of the information
contained in the Model tab. For the case in which X4 is the response
variable and X1, X2, and X3 are the predictors, Tetrad finds that only X1
is significant, and the output graph looks like this:
Comparison to the true causal model shows that this correlation
does exist, but that it runs in the opposite direction.
Logistic Regression
Logistic regression may be run on discrete, continuous, or mixed
data sets; however, the response variable must be binary. In all other
ways, the logistic regression box functions like the linear regression
box.
Appendices
An Introduction to
PAGs
Peter Spirtes
The output of the FCI algorithm [Spirtes, 2001] is a partial
ancestral graph (PAG), which is a graphical object that represents a set
of causal Bayesian networks (CBNs) that cannot be distinguished by the
algorithm. Suppose we have a set of cases that were generated by random
sampling from some CBN. Under the assumptions that FCI makes, in the
large sample limit of the number of cases, the PAG returned by FCI is
guaranteed to include the CBN that generated the data.
An example of a PAG is shown in Figure 2. This PAG represents the
pair of CBNs in Figure 1a and 1b (where measured variables are in boxes
and unmeasured variables are in ovals), as well as an infinite number of
other CBNs that may have an arbitrarily large set of unmeasured
confounders. Despite the fact that there are important differences
between the CBNs in Figure 1a and 1b (e.g., there is an unmeasured
confounder of X1 and X2 in Figure 1 b but not in Figure 1a), they share a
number of important features in common (e.g., in both CBNs, X2 is a
direct cause of X6, there is no unmeasured confounder of X2 and X6, and
X6 is not a cause of X2). It can be shown that every CBN that a PAG
represents shares certain features in common. The features that all CBNs
represented by a PAG share in common can be read off of the output PAG
according to the rules described next.
There are 4 kinds of edges that occur in a PAG: A -> B, A o-> B, A
o–o B, and A <-> B. The edges indicate what the CBNs represented by the
PAG have in common. A description of the meaning of each edge in a PAG is
given in Table A1.
Table A1: Types of edges in a PAG.
Edge type
Relationships that are present
Relationships that are
absent
A --> B
A is a cause of B. It may
be a direct or indirect because that may include other measured
variables. Also, there may be an unmeasured confounder of A and B.
B is not a cause of A.
A <-> B
There is an unmeasured variable (call it L) that is a cause of A
and B. There may be measured variables along the causal pathway from
L to A or from L to B.
A is not a cause of B. B is not a
cause of A.
A o-> B
Either A is a cause
of B, or there is an unmeasured variable that is a cause of A and B,
or both.
B is not a cause of A.
A o–o
B
Exactly one of the following holds: (a) A is a cause of B,
or (b) B is a cause of A, or (c) there is an unmeasured variable that
is a cause of A and B, or (d) both a and c, or (e) both b and c.
Table A1 is sufficient to understand the basic meaning of edge
types in PAGs. Nonetheless, it can be helpful to know the following
additional perspective on the information encoded by PAGs. Each edge has
two endpoints, one on the A side, and one on the B side. For example A
--> B has a tail at the A end, and an arrowhead at the B end. Altogether,
there are three kinds of edge endpoints: a tail "–", an arrowhead ">",
and a "o." Note that some kinds of combinations of endpoints never occur;
for example, A o– B never occurs. As a mnemonic device, the basic meaning
of each kind of edge can be derived from three simple rules that explain
what the meaning of each kind of endpoint is. A tail "–" at the A end of
an edge between A and B means "A is a cause of B"; an arrowhead ">" at
the A end of an edge between A and B means "A is not a cause of B"; and a
circle "o" at the A end of an edge between A and B means "can't tell
whether A is a cause of B". For example A --> B means that A is a
cause of B, and that B is not a cause of A in all the CBNs represented
by the PAG.
The PAG in Figure 2 shows examples of each type of
edge, and the CBNs. Figure 1. show some examples of what kinds of CBNs
can be represented by that PAG.
Figure 1. Two
CBNs that FCI (as well as FCI+, GFCI, and RFCI) cannot distinguish.
Figure
2. The PAG that represents the CBN s in both Figures 1a and 1b.
Arc Specializations in PAGs
This section describes two types of edge specializations that
provide additional information about the nature of an arc in a PAG.
One edge specialization is colored green and is called definitely visible. In a PAG P without selection bias, a
green (definitely visible) arc from A to B denotes that A and B do not
have a latent confounder. If an arc is not definitely visible
(represented as black) then A and B may have a latent confounder.
Another edge specialization is shown as bold and is called definitely direct. In a PAG P without selection bias, a bold
(definitely direct) arc from A to B denotes that A is a direct cause of
B, relative to the other measured variables. If an arc is not definitely
direct (represented as not bolded) then A may not be a direct cause of B,
in which case there may be one or more measured variables on every causal
path from A to B.
In the following examples, the DAG representing a causal process
is on the left, and the corresponding PAG is on the right. All variables
are observed except for latent variable L.
Example of an edge C ➔ D that is definitely visible (green) and definitely
direct (bold):
Example of an edge (C ➔ E) that
is definitely visible (green) and not definitely direct (not bold):
Example of an edge (F ➔ E) that is not definitely visible (black)
and not definitely direct (not bold):
It is conjectured that it is not possible for an edge to be
definitely direct (bold) and not definitely visible (black).
Solving Out of Memory Errors
By default, Java will
allocate the smaller option of 1/4 system memory or 1GB to the Java virtual
machine (JVM). If you run out of memory (heap memory space) running your
analyses you should increase the memory allocated to the JVM with the
following switch '-XmxXXG' where XX is the number of gigabytes of ram you
allow the JVM to utilize. To run Tetrad with more memory you need to
start it from the command line or terminal. For example to allocate 8
gigabytes of ram you would add -Xmx8G immediately after the java command
e.g., java -Xmx8G -jar tetrad-gui.jar.
Glossary of Terms
Adjacent
Two vertices in a graph are adjacent if there is
a directed, or undirected, or double-headed edge between them.
Degree
The total number of edges directed both into and
out of a vertex.
Indegree
The number of edges directed into a vertex.
Markov Blanket
In a variable set V, with joint
probability Pr, the Markov Blanket of a variable X in V is the smallest
subset M of V \ {X} such that X II V \ M | M. In a DAG model, the Markov
Blanket of X is the union of the set of direct causes (parents) of X, the
set of direct effects (children) of X, and the set of direct causes of
direct effects of X.
Markov Equivalent Graphs
Two directed acyclic graphs
(DAGS) are Markov Equivalent if they have the same adjacencies and for
every triple X – Y – Z of adjacent vertices, if X and Z are not adjacent,
X -> Y <- Z in both graphs or in neither graph.
Meek Orientation Rules
Rules for finding all directions
of edges implied by a CPDAG, consistent with any specified “knowledge”
constraints on directions. See https://arxiv.org/pdf/1302.
4972.pdf
Mixed Ancestral Graph (MAG)
An acyclic graph with
directed and undirected edges. Directed edges have the same
interpretation as in DAGs. Undirected edges represent common causes. See
Richardson, T. (2003). Markov properties for acyclic directed mixed
graphs. Scandinavian Journal of Statistics, 30(1), 145-157.
Multiple Indicator Model
A graphical model in which
unmeasured variables each have multiple measured effects. There may be
directed edges between unmeasured variables, but no directed edges from
measured variables to unmeasured variables are allowed.
Outdegree
The number of edges directed out of a
vertex.
Partial Ancestral Graph (PAG)
See PAG description in this
manual.
CPDAG
A graphical representation of a Markov Equivalence
Class or Classes, having both directed and undirected edges, with an
undirected edge indicating that for each possible direction of the edge,
there is a graph in the class or classes having that edge direction.
Scale Free Graph
A network in which the frequency of
nodes with degree k obeys a power law--the relation between log of degree
and log of frequency is roughly linear. See https://cs.brynmawr.edu/Courses/cs380/
spring2013/section02/slides/10_ScaleFreeNetworks.pdf.
Trek
A trek between X and Y is a directed path from X to
Y or from Y to X, or two directed paths from a third variable Z into X
and Y that do not intersect except at Z.