# Robust Cross-Validation Compute Robust Cross-Validation for selecting best model. ```r cv.gam(X, Y, init.mode = c("sLTS", "RLARS", "RANSAC"), lambda.mode = "lambda0", lmax = 1, lmin = 0.05, nlambda = 50, fold = 10, ncores = 1, gam = 0.1, gam0 = 0.5, intercept = "TRUE", alpha = 1, ini.subsamp = 0.2, ini.cand = 1000, alpha.LTS = 0.75, nlambda.LTS = 40) ``` ## Arguments - `X`: Predictor variables Matrix. - `Y`: Response variables Matrix. - `init.mode`: `"sLTS"`: a initial point is the estimate of sparse least trimmed squares. `"RLARS"`: a initial point is the estimate of Robust LARS. `"RANSAC"`: a initial point is the estimate of RANSAC algorithm. - `lambda.mode`: `"lambda0"`: Robust Cross-Validation uses grids on range [0.05lambda0,lambda0] with log scale, where lambda0 is an estimator of sparse tuning parameter which would shrink regression coefficients to zero. - `lmax`: When `lambda.mode` is not lambda0, upper bound of range of grids is lmax. - `lmin`: When `lambda.mode` is not lambda0, lower bound of range of grids is lmin. - `nlambda`: The number of grids for Robust Cross-Validation. - `fold`: the number of folds for K-fold Robust Cross-Validation. If `fold` equals to sample size, Robust Cross-Validation is leave-one-out method. - `ncores`: positive integer giving the number of processor cores to be used for parallel computing (the default is 1 for no parallelization). - `gam`: Robust tuning parameter of gamma-divergence for regression. - `gam0`: tuning parameter of Robust Cross-Validation. - `intercept`: Should intercept be fitted `TRUE` or set to zero `FALSE` - `alpha`: The elasticnet mixing parameter, with $0 \le \alpha \le 1$. `alpha=1` is the lasso penalty, and `alpha=0` the ridge penalty. - `ini.subsamp`: The fraction of subsamples in "`RANSAC`". - `ini.cand`: The number of candidates for estimating itnial points in "`RANSAC`". - `alpha.LTS`: The fraction of subsamples for trimmed squares in "`sLTS`". - `nlambda.LTS`: The number of grids for sparse tuning parameter in "`sLTS`". ## Returns - **lambda**: A numeric vector giving the values of the penalty parameter. - **fit**: All results at each lambda. - **Rocv**: The result of best model by Robust Cross-Validation. ## References Kawashima, T. and Fujisawa, H. (2017). **Robust and Sparse Regression via gamma-divergence, Entropy, 19(11).** Fujisawa, H. and Eguchi, S. (2008). **Robust parameter estimation with a small bias against heavy contamination, Journal of Multivariate Analysis, 99(9), 2053-2081.** ## Details If the "`RANSAC`" is used as the initial point, the parameter `ini.subsamp` and `ini.cand` can be determined carefully. The smaller `ini.subsamp` is, the more robust initial point is. However, less efficiency. ## Examples ```r ## generate data library(mvtnorm) n <- 30 # number of observations p <- 10 # number of expalanatory variables epsilon <- 0.1 # contamination ratio beta0 <- 0.0 # intercept beta <- c(numeric(p)) # regression coefficients beta[1] <- 1 beta[2] <- 2 beta[3] <- 3 beta[4] <- 4 Sigma <- 0.2^t(sapply(1:p, function(i, j) abs(i-j), 1:p)) X <- rmvnorm(n, sigma=Sigma) # explanatory variables e <- rnorm(n) # error terms i <- 1:ceiling(epsilon*n) # index of outliers e[i] <- e[i] + 20 # vertical outliers Y <- beta0*(numeric(n)+1) + X%*%beta res <- cv.gam(X,Y,nlambda = 5, nlambda.LTS=20 ,init.mode="sLTS") ``` ## Author(s) Takayuki Kawashima

cv_gam function