# K-fold Cross Validation This function calculates the predicted values at each point of the design and gives an estimation of criterion using K-fold cross-validation. ```r crossValidation(model, K) ``` ## Arguments - `model`: an output of the `modelFit` function. This argument is the initial model fitted with all the data. - `K`: the number of groups into which the data should be split to apply cross-validation ## Returns A list with the following components: - **Ypred**: a vector of predicted values obtained using K-fold cross-validation at the points of the design - **Q2**: a real which is the estimation of the criterion `R2` obtained by cross-validation - **folds**: a list which indicates the partitioning of the data into the folds - **RMSE_CV**: `RMSE` by K-fold cross-validation (see more details below) - **MAE_CV**: `MAE` by K-fold cross-validation (see more details below) In the case of a Kriging model, other components to test the robustess of the procedure are proposed: - **theta**: the range parameter theta estimated for each fold, - **trend**: the trend parameter estimated for each fold, - **shape**: the estimated shape parameter if the covariance structure is of type `powerexp`. The principle of cross-validation is to split the data into $K$ folds of approximately equal size $A_{1}{A1}, ..., A_{K}{AK}$. For $k=1$ to $K$, a model $Y^(-k)$ is fitted from the data $A1 U ... U AK$ and this model is validated on the fold $Ak$. Given a criterion of quality $L$ (here, $L$ could be the `RMSE` or the `MAE` criterion), the "evaluation" of the model consists in computing : $$ L_{k} = \frac{1}{n/K} \sum_{i \in A_{k}} L \left( y_{i}, Y^{(-k)} (x_{i} )\right).Lk = 1/(n/K) Sum (i in Ak) L (yi,Y^(-k)(xi). $$ The cross-validation criterion is the mean of the $K$ criterion: $L$_CV$1/K (L1+...+LK).$ The `Q2` criterion is defined as: `Q2`$=\code{R2}(\code{Y},\code{Ypred})$ with `Y` the response value and `Ypred` the value fit by cross-validation. ## See Also `R2`, `modelFit`, `MAE`, `RMSE`, `foldsComposition`, `testCrossValidation` ## Note When `K` is equal to the number of observations, **leave-one-out** cross-validation is performed. ## Author(s) D. Dupuy ## Examples ```r ## Not run: rm(list=ls()) # A 2D example Branin <- function(x1,x2) { x1 <- x1*15-5 x2 <- x2*15 (x2 - 5/(4*pi^2)*(x1^2) + 5/pi*x1 - 6)^2 + 10*(1 - 1/(8*pi))*cos(x1) + 10 } # Linear model on 50 points n <- 50 X <- matrix(runif(n*2),ncol=2,nrow=n) Y <- Branin(X[,1],X[,2]) modLm <- modelFit(X,Y,type = "Linear",formula=Y~X1+X2+X1:X2+I(X1^2)+I(X2^2)) R2(Y,modLm$model$fitted.values) crossValidation(modLm,K=10)$Q2 # kriging model : gaussian covariance structure, no trend, no nugget effect # on 16 points n <- 16 X <- data.frame(x1=runif(n),x2=runif(n)) Y <- Branin(X[,1],X[,2]) mKm <- modelFit(X,Y,type="Kriging",formula=~1, covtype="powexp") K <- 10 out <- crossValidation(mKm, K) par(mfrow=c(2,2)) plot(c(0,1:K),c(mKm$model@covariance@range.val[1],out$theta[,1]), xlab='',ylab='Theta1') plot(c(0,1:K),c(mKm$model@covariance@range.val[2],out$theta[,2]), xlab='',ylab='Theta2') plot(c(0,1:K),c(mKm$model@covariance@shape.val[1],out$shape[,1]), xlab='',ylab='p1',ylim=c(0,2)) plot(c(0,1:K),c(mKm$model@covariance@shape.val[2],out$shape[,2]), xlab='',ylab='p2',ylim=c(0,2)) par(mfrow=c(1,1)) ## End(Not run) ```

crossValidation function