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# function to fold data into k folds. this returns a list of matrices where
# the 1st column in each is the response and all other columns are predictors
fold <- function(y, X, k){
n <- length(y)
lapply(0:(k-1)*n/k + 1, function(i){
cbind(y, X)[seq(from=i, length.out=n/k),]
# function to compute MSE for datasets with different numbers of folds
sim <- function(n=50, p=3){
# create the full dataset. predictors are uniform in [0,1],
# uncorrelated, and with slopes evenly spaced from 0 to 2
X <- cbind(1, matrix(runif(p*n), ncol=p))
y <- X %*% cbind(c(0, seq(0,2,length.out=p))) + rnorm(n)
# fold the same data 1 of 4 different ways and get MSE for each fold
sapply(c(2,5,10,n), function(k){
dat <- fold(y, X, k)
# do the computations below for the ith fold
# then average the resulting MSEs over all k folds
mean(sapply(seq(k), function(i){
# create the training set by deleting the ith fold
train <- Reduce(rbind, dat[seq(k)[-i]])
# fit model to training set. parameters are computed by hand
# (using the normal equations) for speed reasons
beta <- solve(t(train[,-1]) %*% train[,-1]) %*% t(train[,-1]) %*% train[,1]
# compute mean squared error for test set
yh <- matrix(dat[[i]], ncol=p+2)[,-1] %*% beta
e2 <- (matrix(dat[[i]], ncol=p+2)[,1] - yh)^2
# compute the mean squared error across all n/k test observations
# estimate sim time: linear extrapolation from 100 iterations
system.time(replicate(100, sim(n=50, p=3)))
# 1.901 seconds
# estimated time: 3.16 minutes
# run full simulation (10000 iterations)
results <- replicate(10000, sim(n=50, p=3))
rownames(results) <- paste("k =",c(2,5,10,50))
# print means and variances of MSEs for each k
variance=apply(results, 1, var)), 3)
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