simulateInterventions {CompareCausalNetworks}  R Documentation 
Simulate data of a causal (possibly cyclic model) under interventions.
Description
Simulate data of a causal (possibly cyclic model) under interventions.
Usage
simulateInterventions(
n,
p,
df,
rhoNoise,
snrPar,
sparse,
doInterv,
numberInt,
strengthInt,
cyclic,
strengthCycle,
modelMis = FALSE,
modelMisPar = 1,
seed = 1
)
Arguments
n 
Number of observations. 
p 
Number of variables. 
df 
Degrees of freedom in tdistribution of noise and interventions. 
rhoNoise 
Correlation between noise terms to model hidden variabkes. Set to 0 for independent noise. 
snrPar 
Signaltonoise parameter: steers what proportion of the variance stems from
the signal resp.\ from the noise: The SNR is given by $SNR = (1 
sparse 
Probability that an entry 
doInterv 
Set to TRUE if interventions should be dointerventions; otherwise noise interventions (also called shift interventions) are generated. 
numberInt 
Total number of settings. 
strengthInt 
Regulates the strength of the interventions, see details. 
cyclic 
Set to TRUE is resulting graph should contain a cycle. 
strengthCycle 
Steers strength of feedback, see details. 
modelMis 
Add a model misspecification that applies 
modelMisPar 
Parameter steering the strength of the model misspecification. 
seed 
Random seed. 
Details
The adjacency matrix A
is generated as follows. Assume the variables
with indices {1, \ldots, p}
are causally ordered. For each edge from node
i
to node j
where i
precedes j
in the causal ordering,
we draw a sample from Bin(sparse
) to determine whether to add an edge
from node i
to node j
. After having sampled the nonzero entries
of A
in this fashion, we sample the coefficients from Unif(1,1).
As described below, the edge weights are later rescaled to achieve a specified
signaltonoise ratio. We exclude the possibility of A = 0
,
i.e. we resample until A
contains at least one nonzero entry.
Second, the interventions are generated as follows. numberInt
denotes the total
number of (interventional and observational) settings that are generated.
For each variable, we sample uniformly at random with replacement one setting
in which this variable is intervened on. In other words, each variable is
intervened on in exactly one setting. Hence it is possible that there are
settings where no interventions take place which then correspond to the
observational case. Similarly, there may be settings where interventions
are performed on multiple variables at once. After defining the settings,
we sample (uniformly at random with replacement) what setting each data point
belongs to. So for each setting we generate approximately the same number of
samples. In one generated data set, the interventions are all of the same
type, i.e. they are either all shift interventions (when doInterv = FALSE
)
or dointerventions (when doInterv = TRUE
). In both cases, an intervention
on X_j
is modelled by generating Z_j
as Z_j ~
strengthInt
* t
(dfNoise
).
If strengthInt
= 0, all interventional settings correspond to purely
observational data.
Third, the noise terms \epsilon
are generated by first sampling from
N(0,\Sigma)
where \Sigma_{i,i} = 1
and
\Sigma_{i,j} =
rhoNoise
. To steer the signaltonoise ratio,
we set the variance of the noise terms of all nodes except source nodes
to snrPar
where 0 <
snrPar
\le 1
. Stepping through the
variables in causal order, for each variable X_j
that has parents, we
uniformly rescale the edge weights \beta_{j,k}
for k = 1, \ldots, p
in the structural equation of variable X_j
such that the variance of
the sum \sum_{k=1}^p \beta_{j,k} X_k + \epsilon_j
is approximately
1 in the observational setting. In other words, the parameter snrPar
steers what proportion of the variance stems from the signal given by
\sum_{k=1}^p \beta_{j,k} X_k
and what proportion stems from the
noise \epsilon_j
. The signaltonoise ratio can then be computed
as SNR = (1snrPar
)/snrPar
.
Forth, a cycle is added to the causal graph if cyclic = TRUE
. If the
causal graph shall contain a cycle, we sample two nodes i
and j
such that adding an edge between them creates a cycle in the causal graph.
We then compute the largest possible coefficient for this edge such that the
cycle product is smaller than 1. Subsequently, we sample the sign of the
coefficient and set the magnitude by scaling the largest possible coefficient
by strengthCycle
where 0 <
strengthCycle
< 1
.
Fifth, we rescale the noise variables to obtain a t
distribution with
dfNoise
degrees of freedom. X
is then generated as
X = (IA)^{1}\epsilon
in the observational case; under a shift
interventions X
can be generated as X = (IA)^{1}(\epsilon + Z)
where the coordinates of Z
are only nonzero for the variables
that are intervened on. Under a dointervention on X_j
, \beta_{j,k}
for k = 1, \ldots, p
are set to 0 to yield A'
and \epsilon_j
is set to Z_j
to yield \epsilon_j'
. We then obtain X
as
X = (IA')^{1}\epsilon'
.
Lastly, if modelMis = TRUE
a model misspecification is added to the
data by marginally transforming all variables as tanh(modelMisPar*x)/modelMisPar)
.
Value
A list with the following elements:

X
n x p
dimensional data matrix 
environment
Indicator of the experiment or the intervention type an observation belongs to. A numeric vector of lengthn
. 
interventions
A list of lengthn
. Indicates location of interventions for each data point. 
whereInt
A list of lengthnumberInt
. Indicates location of interventions in each setting. 
noise

configs
A list with the generated adjacency matrix (trueA
) as well as all input arguments.