CausalGraphInferMainFunc {BiCausality}

CausalGraphInferMainFunc function

Description

A framework to infer causality on binary data using techniques in frequent pattern mining and estimation statistics. Given a set of individual vectors S={x} where x(i) is a realization value of binary variable i, the framework infers empirical causal relations of binary variables i,j from S in a form of causal graph G=(V,E) where V is a set of nodes representing binary variables and there is an edge from i to j in E if the variable i causes j. The framework determines dependency among variables as well as analyzing confounding factors before deciding whether i causes j.

Note that all statistics (e.g. means) and confidence intervals as well as hypothesis testing are inferred by bootstrapping.

Usage

CausalGraphInferMainFunc(
  mat,
  alpha = 0.05,
  nboot = 100,
  IndpThs = 0.05,
  CausalThs = 0.1
)

Arguments

mat

is a matrix n by d where n is a number of transactions or samples and d is a number of dimensions.

alpha

is a significance threshold for hypothesis tests (Mann Whitney) that deploys for testing degrees of dependency, association direction, and causal direction. The default is 0.5.

nboot

is a number of bootstrap replicates for bootstrapping deployed to infer confidence intervals and distributions for hypothesis tests. The default is 100.

IndpThs

is a threshold for the degree of dependency. In the independence test, to claim that any variables are dependent, the dependency degree must greater than this value significantly. The default is 0.05.

CausalThs

is a threshold for the degree of causal direction In the causal-direction test, to claim that any variables have causal relations, the degree of causal direction must greater than this value significantly. The default is 0.1.

Value

This function returns causal inference results. #TODO: provide list of results.

depRes

The result of inferring dependencies between all pairs of variables.

ConfoundRes

The result of filtering associations without true causal directions from any confounding factor.

CausalGRes

The result of inferring causal directions between all pairs of dependent variables that have no confounding factors.

depRes$E0

An adjacency matrix of undirected graph where there is an edge between any pair of variables if they are dependent.

depRes$E0pval

A matrix of p-values from independence test of pairs of variables.

depRes$E0mean

A matrix of means of dependency degrees between variables.

depRes$E0lowbound

A matrix of lower bounds of dependency-degree confidence intervals between variables.

depRes$depInfo$'i, j'$bmean

A mean of dependency degrees between variables i and j.

depRes$depInfo$'i, j'$confInv

An alpha*100th percentile confidence interval of dependency degrees between variables i and j.

depRes$depInfo$'i, j'$testRes

A Mann-Whitney hypothesis test result for an independence test between variables i and j. The null hypothesis is that the distributions of dependency degrees of i,j differ by a location shift of IndpThs and the alternative is that distributions of dependency degrees of i,j is shifted greater than IndpThs.

depRes$depInfo$'i, j'$indices

A pair of indices of i and j in a numeric vector.

depRes$Dboot

A list of Ds (aligned list of transactions) that are generated from sampling with replacement on input samples (mat) nboot times.

ConfoundRes$E1

An adjacency matrix of undirected graph after filtering associations without true causal directions from any confounding factor.

ConfoundRes$E2

A matrix of associations that have confounding factors where E2[i,j]=0 if no confounding factor and E2[i,j]=k if k is a confounding factor of i and j.

CausalGRes$Ehat

An adjacency matrix of directed causal graph where CausalGRes$Ehat[i,j]=1 implies i causes j.

CausalGRes$EValHat

An adjacency matrix of weighted directed causal graph where edge weights are estimated means of probabilities of effect being 1 given cause being either 1 for positive association or 0 for negative association using CondProb() and bootstrapping to estimate

CausalGRes$causalInfo$'i, j'$CDirConfValInv

An alpha*100th percentile confidence interval of estimated conditional probability of effect j being 1/0 given cause i's value being either the same (positive association) or opposite (negative association).

CausalGRes$causalInfo$'i, j'$CDirConfInv

An alpha*100th percentile confidence interval of estimated causal direction degree of i cause j.

CausalGRes$causalInfo$'i, j'$CDirmean

A mean-estimated-causal-direction degree of i cause j.

CausalGRes$causalInfo$'i, j'$testRes2

A Mann-Whitney hypothesis test result for existence of causal direction. The null hypothesis is that the distributions of causal-direction degrees of i,j differ by a location shift of CausalThs and the alternative is that distributions of causal-direction degrees of i,j is shifted greater than CausalThs.

CausalGRes$causalInfo$'i, j'$testRes1

A Mann-Whitney hypothesis test result for existence of association by odd differences from oddDiffFunc(). The null hypothesis is that the distributions of absolute odd difference of i,j differ by a location shift of IndpThs and the alternative is that distributions of absolute odd difference of i,j is shifted greater than IndpThs.

CausalGRes$causalInfo$'i, j'$sign

A direction of i,j association: 1 for positive, 0 for negative, and -1 for no association.

CausalGRes$causalInfo$'i, j'$SignConfInv

An alpha*100th percentile confidence interval of i,j odd difference from bootstrapping.

CausalGRes$causalInfo$'i, j'$Signmean

A mean of i,j odd difference from bootstrapping.

Examples

resC<-CausalGraphInferMainFunc(mat = mat, nboot =50)