# Non-linear Invariant Causal Prediction for an individual set S Tests whether the conditional distribution of Y given X^S is invariant across time, by allowing for arbitrary non-linear additive dependence models. ```r 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). ## Returns 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. ## 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. ## Examples ```r 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) ``` ## 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`. ## Author(s) Niklas Pfister and Jonas Peters

seqICPnl.s function