seqICPnl.s {seqICP}

Non-linear Invariant Causal Prediction for an individual set S

Description

Tests whether the conditional distribution of Y given X^S is invariant across time, by allowing for arbitrary non-linear additive dependence models.

Usage

seqICPnl.s(X, Y, S, test = "block.variance", par.test = list(grid = c(0,
  round(nrow(X)/2), nrow(X)), complements = FALSE, link = sum, alpha = 0.05, B =
  100), regression.fun = function(X, Y) fitted.values(lm.fit(X, Y)))

Arguments

X

matrix of predictor variables. Each column corresponds to one predictor variable.

Y

vector of target variable, with length(Y)=nrow(X).

S

vector containing the indicies of predictors to be tested

test

string specifying the hypothesis test used to test for invariance of a parent set S (i.e. the null hypothesis H0_S). The following tests are available: "block.mean", "block.variance", "block.decoupled", "smooth.mean", "smooth.variance", "smooth.decoupled" and "hsic".

par.test

parameters specifying hypothesis test. The following parameters are available: grid, complements, link, alpha and B. The parameter grid is an increasing vector of gridpoints used to construct enviornments for change point based tests. If the parameter complements is 'TRUE' each environment is compared against its complement if it is 'FALSE' all environments are compared pairwise. The parameter link specifies how to compare the pairwise test statistics, generally this is either max or sum. The parameter alpha is a numeric value in (0,1) indicting the significance level of the hypothesis test. The parameter B is an integer and specifies the number of Monte-Carlo samples used in the approximation of the null distribution.

regression.fun

regression function used to fit the function f. This should be a function which takes the argument (X,Y) and outputs the predicted values f(Y).

Details

The function can be applied to models of the form
Y_i = f(X_i^S) + N_i
with iid noise N_i and f is from a specific function class, which the regression procedure given by the parameter regression.fun should be able to approximate.

For both models the hypothesis test specified by the test parameter specifies the hypothesis test used to test whether the set S leads to an invariant model. For futher details see the references.

Value

list containing the following elements

test.stat

value of the test statistic.

crit.value

critical value computed using a Monte-Carlo simulation of the null distribution.

p.value

p-value.

Author(s)

Niklas Pfister and Jonas Peters

References

Pfister, N., P. Bühlmann and J. Peters (2017). Invariant Causal Prediction for Sequential Data. ArXiv e-prints (1706.08058).

Peters, J., P. Bühlmann, and N. Meinshausen (2016). Causal inference using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society, Series B (with discussion) 78 (5), 947–1012.

See Also

To estimate the set of causal parents use the function seqICPnl. For linear models use the corresponding functions seqICP and seqICP.s.

Examples

set.seed(1)

# environment 1
na <- 130
X1a <- rnorm(na,0,0.1)
Ya <- 5*X1a+rnorm(na,0,0.5)
X2a <- Ya+rnorm(na,0,0.1)

# environment 2
nb <- 70
X1b <- rnorm(nb,-1,1)
Yb <- X1b^2+rnorm(nb,0,0.5)
X2b <- rnorm(nb,0,0.1)

# combine environments
X1 <- c(X1a,X1b)
X2 <- c(X2a,X2b)
Y <- c(Ya,Yb)
Xmatrix <- cbind(X1, X2)

# use GAM as regression function
GAM <- function(X,Y){
  d <- ncol(X)
  if(d>1){
    formula <- "Y~1"
    names <- c("Y")
    for(i in 1:(d-1)){
      formula <- paste(formula,"+s(X",toString(i),")",sep="")
      names <- c(names,paste("X",toString(i),sep=""))
    }
    data <- data.frame(cbind(Y,X[,-1,drop=FALSE]))
      colnames(data) <- names
    fit <- fitted.values(mgcv::gam(as.formula(formula),data=data))
  } else{
    fit <- rep(mean(Y),nrow(X))
  }
  return(fit)
}

# apply seqICPnl.s to all possible sets using the regression
# function GAM - only the true parent set S=1 is
# invariant in this example
seqICPnl.s(Xmatrix, Y, S=numeric(), par.test=list(grid=c(0,50,100,150,200)), regression.fun=GAM)
seqICPnl.s(Xmatrix, Y, S=1, par.test=list(grid=c(0,50,100,150,200)), regression.fun=GAM)
seqICPnl.s(Xmatrix, Y, S=2, par.test=list(grid=c(0,50,100,150,200)), regression.fun=GAM)
seqICPnl.s(Xmatrix, Y, S=c(1,2), par.test=list(grid=c(0,50,100,150,200)), regression.fun=GAM)