# Sequential Invariant Causal Prediction for an individual set S Tests whether the conditional distribution of Y given X^S is invariant across time, by assuming a linear dependence model. ```r seqICP.s(X, Y, S, test = "decoupled", par.test = list(grid = c(0, round(nrow(X)/2), nrow(X)), complements = FALSE, link = sum, alpha = 0.05, B = 100, permutation = FALSE), model = "iid", par.model = list(pknown = FALSE, p = 0, max.p = 10)) ``` ## 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: "decoupled", "combined", "trend", "variance", "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`, `B` and `permutation`. 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 (or permutations) used in the approximation of the null distribution. If the parameter `permutation` is 'TRUE' a permuatation based approach is used to approximate the null distribution, if it is 'FALSE' the scaled residuals approach is used. - `model`: string specifying the underlying model class. Either "iid" if Y consists of independent observations or "ar" if Y has a linear time dependence structure. - `par.model`: parameters specifying model. The following parameters are available: `pknown`, `p` and `max.p`. If `pknown` is 'FALSE' the number of lags will be determined by comparing all fits up to `max.p` lags using the AIC criterion. If `pknown` is 'TRUE' the procedure will fit `p` lags. ## 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. - **p**: number of lags that were used. - **model.fit**: 'lm' object of linear model fit. ## Details The function can be applied to two types of models (1) a linear model (model="iid") Y_i = a X_i^S + N_i with iid noise N_i and (2) a linear autoregressive model (model="ar") Y_t = a_0 X_t^S + ... + a_p (Y_(t-p),X_(t-p)) + N_t with iid noise N_t. For both models the hypothesis test specified by the `test` parameter is 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 <- 5*X1b+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) # apply seqICP.s to all possible sets - only the true parent set S=1 # is invariant in this example seqICP.s(Xmatrix, Y, S=numeric(), par.test=list(grid=c(0,50,100,150,200))) seqICP.s(Xmatrix, Y, S=1, par.test=list(grid=c(0,50,100,150,200))) seqICP.s(Xmatrix, Y, S=2, par.test=list(grid=c(0,50,100,150,200))) seqICP.s(Xmatrix, Y, S=c(1,2), par.test=list(grid=c(0,50,100,150,200))) ``` ## 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 `seqICP`. For non-linear models use the corresponding functions `seqICPnl` and `seqICPnl.s`. ## Author(s) Niklas Pfister and Jonas Peters

seqICP.s function