# Non-linear Invariant Causal Prediction Estimates the causal parents S of the target variable Y using invariant causal prediction and fits a general model of the form Y = f(X^S) + N. ```r seqICPnl(X, Y, 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)), max.parents = ncol(X), stopIfEmpty = TRUE, silent = TRUE) ``` ## Arguments - `X`: matrix of predictor variables. Each column corresponds to one predictor variable. - `Y`: vector of target variable, with length(Y)=nrow(X). - `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). - `max.parents`: integer specifying the maximum size for admissible parents. Reducing this below the number of predictor variables saves computational time but means that the confidence intervals lose their coverage property. - `stopIfEmpty`: if ‘TRUE’, the procedure will stop computing confidence intervals if the empty set has been accepted (and hence no variable can have a signicificant causal effect). Setting to‘TRUE’ will save computational time in these cases, but means that the confidence intervals lose their coverage properties for values different to 0. - `silent`: If 'FALSE', the procedure will output progress notifications consisting of the currently computed set S together with the p-value resulting from the null hypothesis H0_S ## Returns object of class 'seqICPnl' consisting of the following elements - **parent.set**: vector of the estimated causal parents. - **test.results**: matrix containing results from each individual hypothesis test H0_S as rows. The first column ind links the rows in the `test.results` matrix to the position in the list of the variable `S`. - **S**: list of all the sets that were tested. The position within the list corresponds to the index in the first column of the test.results matrix. - **p.values**: p-value for being not included in the set of true causal parents. (If a p-value is smaller than alpha, the corresponding variable is a member of parent.set.) - **stopIfEmpty**: a boolean value indicating whether computations stop as soon as intersection of accepted sets is empty. - **modelReject**: a boolean value indicating if the whole model was rejected (the p-value of the best fitting model is too low). - **alpha**: significance level at which the hypothesis tests were performed. - **n.var**: number of predictor variables. ## 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. The invariant prediction procedure is applied using the hypothesis test specified by the `test` parameter to determine whether a candidate model is invariant. For further details see the references. ## Examples ```r set.seed(2) # environment 1 na <- 120 X1a <- 0.3*rnorm(na) X3a <- X1a + 0.2*rnorm(na) Ya <- 2*X1a^2 + 0.6*sin(X3a) + 0.1*rnorm(na) X2a <- -0.5*Ya + 0.5*X3a + 0.1*rnorm(na) # environment 2 nb <- 80 X1b <- 2*rnorm(nb) X3b <- rnorm(nb) Yb <- 2*X1b^2 + 0.6*sin(X3b) + 0.1*rnorm(nb) X2b <- -0.5*Yb + 0.8*rnorm(nb) # combine environments X1 <- c(X1a,X1b) X2 <- c(X2a,X2b) X3 <- c(X3a,X3b) Y <- c(Ya,Yb) Xmatrix <- cbind(X1, X2, X3) # 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) } # Y follows the same structural assignment in both environments # a and b (cf. the lines Ya <- ... and Yb <- ...). # The direct causes of Y are X1 and X3. # A GAM model fit considers X1, X2 and X3 as significant. # All these variables are helpful for the prediction of Y. summary(mgcv::gam(Y~s(X1)+s(X2)+s(X3))) # apply seqICP to the same setting seqICPnl.result <- seqICPnl(X = Xmatrix, Y, test="block.variance", par.test = list(grid = seq(0, na + nb, (na + nb)/10), complements = FALSE, link = sum, alpha = 0.05, B =100), regression.fun = GAM, max.parents = 4, stopIfEmpty=FALSE, silent=FALSE) summary(seqICPnl.result) # seqICPnl is able to infer that X1 and X3 are causes of Y ``` ## 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 The function `seqICPnl.s` can be used to perform the individual hypothesis tests H0_S. `seqICP` and `seqICP.s` are the corresponding functions for classical sequential invariant causal prediction. ## Author(s) Niklas Pfister and Jonas Peters

seqICPnl function