# Cross-validation for Ridge Redundancy Analysis This function performs cross-validation to evaluate the performance of Ridge Redundancy Analysis (RDA) models. It calculates the mean squared error (MSE) for different ranks and ridge penalty values through cross-validation folds. The function also supports centering and scaling of the input matrices. The range of lambda for the cross-validation is automatically calculated following the method of "glmnet" (Friedman et al., 2010). When we have a matrix of response variables (Y; n times q matrix) and a matrix of explanatory variables (X; n times p matrix), the largest lambda for the validation is obtained as follows [REMOVE_ME]$$ \lambda_{\text{max}} = \frac{\max_{j \in \{1, 2, \dots, p\}} \sqrt{\sum_{k=1}^{q} \left( \sum_{i=1}^{n} (x_{ij}\cdot y_{ik}) \right)^2}}{N \times 10^{-3}} [REMOVE_ME_2]$$ Then, we define $\lambda_{min}=10^{-4}\lambda_{max}$, and the sequence $\lambda$ is generated based on the range. Also, to reduce the computation, the variable sampling is performed for the large matrix of X and Y (by default, when the number of the variables is over 1000). Alternatively, the range of lambda can be specified manually. ```r rrda.cv( Y, X, maxrank = NULL, lambda = NULL, num.lambda = 50, nfold = 5, folds = NULL, sample.X = 1000, sample.Y = 1000, scale.X = FALSE, scale.Y = FALSE, center.X = TRUE, center.Y = TRUE, verbose = TRUE ) ``` ## Arguments - `Y`: A numeric matrix of response variables. - `X`: A numeric matrix of explanatory variables. - `maxrank`: A numeric vector specifying the maximum rank of the coefficient Bhat. Default is `NULL`, which sets it to `(min(15, min(dim(X), dim(Y))))`. - `lambda`: A numeric vector of ridge penalty values. Default is `NULL`, where the lambda values are automatically chosen. - `num.lambda`: A number of lambda generated (only when the lambda is not given by user). Default is 50. - `nfold`: The number of folds for cross-validation. Default is 5. - `folds`: A vector specifying the folds. Default is `NULL`, which randomly assigns folds. - `sample.X`: A number of variables sampled from X for the lamdba range estimate. Default is 1000. - `sample.Y`: A number of variables sampled from Y for the lamdba range estimate. Default is 1000. - `scale.X`: Logical indicating if `X` should be scaled. If `TRUE`, scales `X`. Default is `FALSE`. - `scale.Y`: Logical indicating if `Y` should be scaled. If `TRUE`, scales `Y`. Default is `FALSE`. - `center.X`: Logical indicating if `X` should be centered. If `TRUE`, scales `X`. Default is `TRUE`. - `center.Y`: Logical indicating if `Y` should be centered. If `TRUE`, scales `Y`. Default is `TRUE`. - `verbose`: Logical indicating. If `TRUE`, the function displays information about the function call. Default is `TRUE`. ## Returns A list containing the cross-validated MSE matrix, lambda values, rank values, and the optimal lambda and rank. ## Description This function performs cross-validation to evaluate the performance of Ridge Redundancy Analysis (RDA) models. It calculates the mean squared error (MSE) for different ranks and ridge penalty values through cross-validation folds. The function also supports centering and scaling of the input matrices. The range of lambda for the cross-validation is automatically calculated following the method of "glmnet" (Friedman et al., 2010). When we have a matrix of response variables (Y; n times q matrix) and a matrix of explanatory variables (X; n times p matrix), the largest lambda for the validation is obtained as follows $$ \lambda_{\text{max}} = \frac{\max_{j \in \{1, 2, \dots, p\}} \sqrt{\sum_{k=1}^{q} \left( \sum_{i=1}^{n} (x_{ij}\cdot y_{ik}) \right)^2}}{N \times 10^{-3}} $$ Then, we define $\lambda_{min}=10^{-4}\lambda_{max}$, and the sequence $\lambda$ is generated based on the range. Also, to reduce the computation, the variable sampling is performed for the large matrix of X and Y (by default, when the number of the variables is over 1000). Alternatively, the range of lambda can be specified manually. ## Examples ```r ## Not run: set.seed(10) simdata<-rdasim1(n = 100,p = 200,q = 200,k = 3) X <- simdata$X Y <- simdata$Y cv_result<- rrda.cv(Y = Y, X = X, maxrank = 5, nfold = 5) rrda.summary(cv_result = cv_result) ##Complete Example## # library(future) # <- if you want to compute in parallel # plan(multisession) # <- if you want to compute in parallel # cv_result<- rrda.cv(Y = Y, X = X, maxrank = 5, nfold = 5) # cv # plan(multisession) # <- To come back to sequential computing # rrda.summary(cv_result = cv_result) # cv result p <- rrda.plot(cv_result) # cv result plot print(p) h <- rrda.heatmap(cv_result) # cv result heatmao print(h) estimated_lambda<-cv_result$opt_min$lambda # selected parameter estimated_rank<-cv_result$opt_min$rank # selected parameter Bhat <- rrda.fit(Y = Y, X = X, nrank = estimated_rank,lambda = estimated_lambda) # fitting Bhat_mat<-rrda.coef(Bhat) Yhat_mat <- rrda.predict(Bhat = Bhat, X = X) # prediction Yhat<-Yhat_mat[[1]][[1]][[1]] # predicted values cor_Y_Yhat<-diag(cor(Y,Yhat)) # correlation summary(cor_Y_Yhat) ## End(Not run) ```

rrda.cv function