Artificial cross-sectional data.

Description

A data set of 500 intances, generated from a network consisting of six continuous variables.

Usage

crossdata6V

Format

A data frame with six continuous variables: x1,...,x6.

Reshape longitudinal data

Description

Reshape longitudinal data with t time slices into a longitudinal data with two time slices.

Usage

dataReshape(theData = NULL, numTime = NULL)

Arguments

Value

A data frame representing longitudinal data with two time slices, such that the first n data points contain the relations that occur in the first two time slices t_0 and t_1. The next n data points contain the relations that occur in time slices t_1 and t_2. The i-th subset of n data points contain the relations in time slices t_i-1 and t_i. The reshaped data can be used as data input for function stableSpec when computing longitudinal data.

Author(s)

Ridho Rahmadi r.rahmadi@cs.ru.nl

Examples

the_data <- longiData4V3T
num_time <- 3
reshaped_the_data <- dataReshape(the_data, num_time)

Scoring the given SEM models.

Description

Compute the model chi-square and model complexity of the given SEM models.

Usage

getModelFitness(theData = NULL, allModelString = NULL, numTime = NULL,
  longitudinal = NULL, co = NULL, mixture = NULL)

Arguments

Value

a matrix of models including their fitness': chi-square and model complexity.

Author(s)

Ridho Rahmadi r.rahmadi@cs.ru.nl

Examples

the_data <- crossdata6V
#assummed that variable 5 does not cause variables 1, 2, and 3
models <- modelPop(nPop=5, numVar=6, longitudinal=FALSE,
consMatrix = matrix(c(5, 1, 5, 2, 5, 3), 3, 2, byrow=TRUE))

model_fitness <- getModelFitness(theData=the_data,
allModelString=models, numTime=1, longitudinal=FALSE,
co="covariance", mixture = FALSE)

Artificial longitudinal data.

Description

A data set of 400 intances, that is generated from a network consisting of four continuous variables and three time slices t_0,...,t_2.

Usage

longiData4V3T

Format

A data frame with twelve continuous variables: x1,...,x4 are for time slice t_0, x5,...,x8 are for time slice t_1, and x9,...,x12 are for time slice t_2

Random SEM models.

Description

Generating recursive (acyclic) SEM models represented by binary vectors.

Usage

modelPop(nPop = NULL, numVar = NULL, longitudinal = NULL,
  consMatrix = NULL)

Arguments

Details

This function generates nPop random SEM models which are represented by binary vectors; 1 means there is a causal path from, e.g., variable A to B and 0 otherwise. In addition, the generated models have passed the cyclic test to ensure they are all acyclic. The function also includes minPop models which representing models from each model complexity, i.e., minPop = numVar(numVar-1)/2+1, if longitudinal = FALSE, or minPop = (numVar(numVar-1)/2+1)+numVar^2, otherwise. If nPop <= minPop then this function will generate minPop models.

Value

nPop or minPop by m matrix, where m is the length of the binary vector depending of the given number of variables and also whether longitudinal or cross-sectional model.

Author(s)

Ridho Rahmadi r.rahmadi@cs.ru.nl

Examples

#assumming a prior knowledge that variable 1 does not cause variable 2
models <- modelPop(nPop=25, numVar=6,
longitudinal=FALSE, consMatrix = matrix(c(1, 2), 1, 2))
models

Plot of edge and causal path stability.

Description

Plot each of the stability of causal path and edge including the threshold of stability and model complexity.

Usage

plotStability(listOfFronts = NULL, threshold = NULL, stableCausal = NULL,
  stableCausal_l1 = NULL, stableEdge = NULL, longitudinal = NULL)

Arguments

Value

Plot of causal path and edge stability for every pair of variables, including plots of all edge stabilites and all cauasl path stabilities.

Author(s)

Ridho Rahmadi r.rahmadi@cs.ru.nl

Examples

the_data <- crossdata6V
numSubset <- 1
num_iteration <- 5
num_pop <- 10
mut_rate <- 0.075
cross_rate <- 0.85
longi <- FALSE
num_time <- 1
the_co <- "covariance"
#assummed that variable 5 does not cause variables 1, 2, and 3
cons_matrix <- matrix(c(5, 1, 5, 2, 5, 3), 3, 2, byrow=TRUE)
th <- 0.1
to_plot <- FALSE

result <- stableSpec(theData=the_data, nSubset=numSubset,
iteration=num_iteration,
nPop=num_pop, mutRate=mut_rate, crossRate=cross_rate,
longitudinal=longi, numTime=num_time,
co=the_co, consMatrix=cons_matrix, threshold=th, toPlot=to_plot)

plotStability(listOfFronts=result$listOfFronts, threshold=th,
stableCausal=result$causalStab,
stableCausal_l1=result$causalStab_l1,
stableEdge=result$edgeStab,
longitudinal=longi)

Repairing a SEM model that is cyclic.

Description

Repairing a SEM model that is cyclic.

Usage

repairCyclicModel(stringModel = NULL, numVar = NULL, longitudinal = NULL)

Arguments

Details

The main idea of this function is to seek cyclic(s) with any possible length from a given model, and then to cut the cyclic, so as to make the model acyclic. Moreover, this function is used in stableSpec to ensure no cyclic model in the computation.

Value

a binary vector with the same length of input, representing a repaired model (acyclic).

Author(s)

Ridho Rahmadi r.rahmadi@cs.ru.nl

Examples

num_vars <- 6
longi_a <- FALSE
longi_b <- TRUE

# Assume that the generated model below is cyclic
# a cross-sectional model
model_a <- round(runif(num_vars * num_vars))
# a longitudinal model
model_b <- c(round(runif(num_vars * num_vars)),
round(runif(num_vars * (num_vars-1))))

repaired_model_a <- repairCyclicModel(stringModel=model_a, numVar=num_vars,
longitudinal=longi_a)

repaired_model_b <- repairCyclicModel(stringModel=model_b, numVar=num_vars,
longitudinal=longi_b)

repaired_model_a
repaired_model_b

Stable specifications of constrained structural equation models.

Description

Search stable specifications (structures) of constrained structural equation models.

Usage

stableSpec(theData = NULL, nSubset = NULL, iteration = NULL,
  nPop = NULL, mutRate = NULL, crossRate = NULL, longitudinal = NULL,
  numTime = NULL, seed = NULL, co = NULL, consMatrix = NULL,
  threshold = NULL, toPlot = NULL, mixture = NULL, log = NULL)

Arguments

Details

This function performs exploratory search over recursive (acyclic) SEM models. Models are scored along two objectives: the model fit and the model complexity. Since both objectives are often conflicting we use NSGA-II to search for Pareto optimal models. To handle the instability of small finite data samples, we repeatedly subsample the data and select those substructures that are both stable and parsimonious which are then used to infer a causal model.

Value

a list of the following elements:

• listofFronts is a list of optimal models for the whole range of model complexity of all subsets.

• causalStab is a list of causal path stability for the whole range of model complexity

• causalStab_l1 is a list of causal path stability of length 1 for the whole range of model complexity

• edgeStab is a list of edge stability for the whole range of mdoel complexity

• relCausalPath is n by n matrix of relevant causal path, where n is the number of variables. Each positive element i,j represents the stability of causal path from i to j.

• relCausalPath_l1 is n by n matrix of relevant causal path with length 1, where n is the number of variables. Each positive element i,j represents the stability of causal path from i to j with length 1.

• relEdge is n by n matrix of relevant edge, where n is the number of variables. Each positive element i,j represents the stability of edge between i to j.

• If argument toPlot = TRUE, then a visualization of relevant model structures is generated. Otherwise an object of graph is returned. An arc represents a causal path, and an (undirected) edge represents strong association where the direction is undecidable. The graph is annotated with reliability scores, which are the highest selection probability in the top-left region of the edge stability graph.

• allSeed is an integer vector representing seeds that are used in subsampling data. This can be used to replicate the result in next computation.

Author(s)

Ridho Rahmadi r.rahmadi@cs.ru.nl, Perry Groot, Tom Heskes. Christoph Stich is the contributor for parallel support.

References

Rahmadi, R., Groot, P., Heins, M., Knoop, H., and Heskes, T. (2016) Causality on cross-sectional data: Stable specification search in constrained structural equation modeling. Applied Soft Computing, ISSN 1568-4946, http://www.sciencedirect.com/science/article/pii/S1568494616305130.

Rahmadi, R., Groot, P., Heins, M., Knoop, H., & Heskes, T. (2015). Causality on Longitudinal Data: Stable Specification Search in Constrained Structural Equation Modeling. Proceedings of AALTD 2015, 101.

Fox, J., Nie, Z., and Byrnes, J. (2015). sem: Structural Equation Models. R package version 3.1-6. https://CRAN.R-project.org/package=sem

Ching-Shih Tsou (2013). nsga2R: Elitist Non-dominated Sorting Genetic Algorithm based on R. R package version 1.0. https://CRAN.R-project.org/package=nsga2R

Kalisch, M., Machler, M., Colombo, D., Maathuis, M. H., and Buehlmann, P. (2012). Causal inference using graphical models with the R package pcalg. Journal of Statistical Software, 47(11), 1-26.

Meinshausen, N., and Buehlmann, P. (2010). Stability selection. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(4), 417-473.

Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002), A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6(2), 182-197.

Chickering, D. M. (2002). Learning equivalence classes of Bayesian-network structures. The Journal of Machine Learning Research, 2, 445-498.

Examples

# Cross-sectional data example,
# with an artificial data set of six continuous variables.
# Detail about the data set can be found in the documentation.
# As an example, we only run one subset.
# Note that stableSpec() uses foreach to support
# parallel computation, which could issue a warning
# when running sequentially as the following example. However
# the warning can be just ignored.

the_data <- crossdata6V
numSubset <- 1
num_iteration <- 5
num_pop <- 10
mut_rate <- 0.075
cross_rate <- 0.85
longi <- FALSE
num_time <- 1
the_seed <- NULL
the_co <- "covariance"
#assummed that variable 5 does not cause variables 1, 2, and 3
cons_matrix <- matrix(c(5, 1, 5, 2, 5, 3), 3, 2, byrow=TRUE)
th <- 0.1
to_plot <- FALSE
mix <- FALSE

result <- stableSpec(theData=the_data, nSubset=numSubset,
iteration=num_iteration,
nPop=num_pop, mutRate=mut_rate, crossRate=cross_rate,
longitudinal=longi, numTime=num_time, seed=the_seed,
co=the_co, consMatrix=cons_matrix, threshold=th,
toPlot=to_plot, mixture = mix)

##########################################################
## Parallel computation is possible by
## registering parallel backend, e.g., package doParallel.
## For example, add the following lines on top of
## the example above.
#
# library(parallel)
# library(doParallel)
# cl <- makeCluster(detectCores())
# registerDoParallel(cl)
#
## Then call stableSpec() as normal.
##
## Note that makeCluster() and detectCores() are
## from package parallel, and registerDoParallel()
## is from package doParallel. For more detail
## check the aforementioned packages' documentations.
###########################################################