Estimate connectivity matrix of a directed graph with linear effects and hidden variables.

Description

This function estimates the connectivity matrix of a directed (possibly cyclic) graph with hidden variables. The underlying system is required to be linear and we assume that observations under different shift interventions are available. More precisely, the function takes as an input an (nxp) data matrix, where n is the sample size and p the number of variables. In each environment j (j in {1, \ldots, J}) we have observed n_j samples generated from

X_j= X_j * A + c_j + e_j

(in case of cycles this should be understood as an equilibrium distribution). The c_j is a p-dimensional random vector that is assumed to have a diagonal covariance matrix. The noise vector e_j is assumed to have the same distribution in all environments j but is allowed to have an arbitrary covariance matrix. The different intervention settings are provided to the method with the help of the vector ExpInd of length n = (n_1 + ... + n_j + ... + n_J). The goal is to estimate the connectivity matrix A.

Usage

backShift(X, ExpInd, covariance=TRUE, ev=0, threshold =0.75, nsim=100,
          sampleSettings=1/sqrt(2), sampleObservations=1/sqrt(2),
          nodewise=TRUE, tolerance = 10^(-4), baseSettingEnv = 1,
          verbose = FALSE)

Arguments

Value

A list with elements

Author(s)

Christina Heinze-Deml <heinzedeml@stat.math.ethz.ch>

References

Dominik Rothenhaeusler, Christina Heinze, Jonas Peters, Nicolai Meinshausen: backShift: Learning causal cyclic graphs from unknown shift interventions. Advances in Neural Information Processing Systems (NIPS) 28, 2015. arXiv: http://arxiv.org/abs/1506.02494

See Also

ICP and hiddenICP for reconstructing the parents of a variable under interventions on all other variables. getParents and getParentsStable from the package CompareCausalNetworks to estimate the connectivity matrix of a directed causal graph, using various possible methods (including backShift).

Examples

## Simulate data with connectivity matrix A

seed <- 1
# sample size n
n <- 10000
# 3 predictor variables
p  <- 3
A <- diag(p)*0
A[1,2] <- 0.8
A[2,3] <- -0.8
A[3,1] <- 0.8

# divide data into 10 different environments
G <- 10

# simulate
simulation.res <- simulateInterventions(
                    n, p, A, G, intervMultiplier = 2,
                    noiseMult = 1, nonGauss = FALSE,
                    fracVarInt = 0.5, hidden = TRUE,
                    knownInterventions = FALSE,
                    simulateObs = TRUE, seed)

environment <- simulation.res$environment
X <- simulation.res$X

## Compute feedback estimator with stability selection

network <- backShift(X, environment, ev = 1)

## Print point estimates and stable edges

# true connectivity matrix
print(A)
# point estimate
print(network$Ahat)
# shows empirical selection probability for stable edges
print(network$AhatAdjacency)

Computes a simple model-based bootstrap confidence interval for success of joint diagonalization procedure. The model-based bootstrap approach assumes normally distributed error terms; the parameters of the noise distribution are estimated with maximum likelihood.

Description

Computes a simple model-based bootstrap confidence interval for success of joint diagonalization procedure. The model-based bootstrap approach assumes normally distributed error terms; the parameters of the noise distribution are estimated with maximum likelihood.

Usage

bootstrapBackShift(
  Ahat,
  X,
  ExpInd,
  nrep,
  alpha = 0.05,
  covariance = TRUE,
  baseInd = 1,
  tolerance = 0.001,
  verbose = FALSE
)

Arguments

Value

A list with the following elements:

• bootsSumOffDiags Vector of length nrep with sum of off-diagonal elements after joint diagnolization procedure in each of the bootstrap samples.

• sumOffDiagsBackShift Sum of off-diagonal elements after joint diagnolization procedure in original estimation.

• jointDiagSuccess TRUE if sumOffDiagsBackShift lies within bootstrap confidence interval.

• lower Lower bound of bootstrap confidence interval.

• upper Upper bound of bootstrap confidence interval.

• lowerBasic alpha/2 quantile of empirical bootstrap distribution.

• upperBasic 1 - alpha/2 quantile of empirical bootstrap distribution.

Computes the matrix \Delta \Sigma_{c,j} resulting from the joint diagonalization for a given environment (cf. Eq.(7) in the paper). If the joint diagonalization was successful the matrix should be diagonal for all environments $j$.

Description

Computes the matrix \Delta \Sigma_{c,j} resulting from the joint diagonalization for a given environment (cf. Eq.(7) in the paper). If the joint diagonalization was successful the matrix should be diagonal for all environments $j$.

Usage

computeDiagonalization(estConnectivity, X, env, whichEnv, main = NULL)

Arguments

Example adjacency matrix

Description

An example for an adjacency matrix A to be used as input to simulateInterventions. The entry A_{ij} contains the edge from node i to node j.

Usage

data("exampleAdjacencyMatrix")

Format

A matrix with 10 rows and 10 columns.

References

Used in simulations in:

Dominik Rothenhaeusler, Christina Heinze, Jonas Peters, Nicolai Meinshausen (2015): backShift: Learning causal cyclic graphs from unknown shift interventions arXiv preprint: http://arxiv.org/abs/1506.02494

Examples

data("exampleAdjacencyMatrix")
plotGraphEdgeAttr(estimate = exampleAdjacencyMatrix, plotStabSelec = FALSE, 
                  labels = colnames(exampleAdjacencyMatrix),
                  thres.point = 0, thres.stab = NULL, main = "True graph")

Generates a connectivity matrix A.

Description

Generates a connectivity matrix A with cycle product smaller than 1.

Usage

generateA(p, expNumNeigh, minCoef, maxCoef, cyclic, verbose = FALSE)

Arguments

Details

If expNumNeigh and maxCoef are large, function may fail to find a connectivity matrix with cycle product smaller one. In this case, try to lower these parameters.

Value

A list with two elements

• A Connectivity matrix

• sizeCycle Size of the cycle, if cyclic was set to TRUE.

Performance metrics for estimate of connectiviy matrix A.

Description

Computes various performance metrics for estimate of connectiviy matrix A.

Usage

metricsThreshold(trueA, est, thres = seq(0.01, 1, by = 0.01))

Arguments

Value

A data frame with the following columns:

• Threshold Value at which point estimate est was thresholded.

• SHD Structural Hamming distance between trueA and est.

• TPR.Recall True positive rate / recall value

• FPR False positive rate

• Precision Precision value

Examples

# true A
p  <- 3
A <- diag(p)*0
A[1,2] <- 0.8
A[2,3] <- -0.8
A[3,1] <- 0.8

# say an estimated connectivity matrix is given by:
A.est <- matrix(rnorm(p*p, 1e-3, 1e-3), ncol = p)
diag(A.est) <- 0
A.est[1,2] <- 0.76
A.est[2,3] <- -0.68
A.est[3,1] <- 0.83
 
# compute metrics with threshold 0.25
metricsThreshold(A, A.est, thres = 0.25)

Plots the joint diagonalization. I.e. if it was successful the matrices should all be diagonal.

Description

Plots the joint diagonalization. I.e. if it was successful the matrices should all be diagonal.

Usage

plotDiagonalization(estConnectivity, X, env, whichEnv, main = NULL)

Arguments

Plotting function to visualize directed graphs

Description

Given a point estimate of the connectivety matrix or the adjacency matrix, this function visualizes the directed graph using plot.igraph from the package igraph. If a point estimate is plotted, the edges' intensity reflects the magnitude of the coefficients. If the result is an adjacency matrix estimated by stability selection then the edges' width reflects how often an edge was selected and the intensity reflects the magnitude of the coefficients (if this information is also provided).

Usage

plotGraphEdgeAttr(
  estimate,
  plotStabSelec,
  labels,
  thres.point,
  edgeWeights = NULL,
  thres.stab = 0.75,
  main = "",
  edge.color = "blue",
  ...
)

Arguments

Details

Currently not all options of igraph are used; additional arguments are ignored.

Plots the estimated intervention variances.

Description

Plots the estimated intervention variances.

Usage

plotInterventionVars(estIntVars, trueIntVars = NULL, scales_facet = "free")

Arguments

Simulate data of a causal cyclic model under shift interventions.

Description

Simulate data of a causal cyclic model under shift interventions.

Usage

simulateInterventions(
  n,
  p,
  A,
  G,
  intervMultiplier,
  noiseMult,
  nonGauss,
  hiddenVars,
  knownInterventions,
  fracVarInt,
  simulateObs,
  seed = 1
)

Arguments

Value

A list with the following elements:

• X (nxp)-dimensional data matrix

• environment Indicator of the experiment or the intervention type an observation belongs to. A numeric vector of length n.

• interventionVar (Gxp)-dimensional matrix with intervention variances.

• interventions Location of interventions if knownInterventions was set to TRUE.

• configs A list with the following elements:

  • trueA True connectivity matrix used to generate the data.

  • G Number of environments.

  • indexObservationalData Index of observational data

  • intervMultiplier Multiplier steering the intervention strength

  • noiseMult Multiplier steering the noise level

  • fracVarInt If knownInterventions was set to TRUE, fraction of variables that were intervened on in each environment.

  • hiddenVars If TRUE, hidden variables exist.

  • knownInterventions If TRUE, location of interventions is known.

  • simulateObs If TRUE, environment 1 contains observational data.

References

Dominik Rothenhaeusler, Christina Heinze, Jonas Peters, Nicolai Meinshausen (2015): backShift: Learning causal cyclic graphs from unknown shift interventions. arXiv preprint: http://arxiv.org/abs/1506.02494