[SOLVED] Regularization: ridge regression, the lasso, and elastic net math8050: homework 10

1. (90pts total, equally weighted) Comparison of ridge regression, lasso regression, and elastic net regression.
We’ll construct a matrix with n = 1000 observations (you can think about these as 1000 participants) and p
= 2000 predictors.
set.seed(123)
n <- 1000
p <- 2000
pred <- matrix(rnorm(n*p), nrow = n, ncol = p)We created an outcome variable “dv” which is calculated, first, by the sum of predictors 1 through 5. Then
by adding the sum of predictors 6 through 10 (but only influenced by these rows 80% as much as by the first
5). Then 11 through 15, time (60% as much) and so forth. Additionally, some random noise is added to each
outcome variable.dv <- (rowSums(pred[,1:5]) + .8*rowSums(pred[,6:10]) +
.6 * rowSums(pred[,11:15]) + .4*rowSums(pred[,16:20]) +
.2 * rowSums(pred[,21:25]) + rnorm(n))
We then center the predictors
pred <- scale(pred)Let’s now split our data (both the predictors and dv) into train and test sets. We’ll go with a 80-20 split.
train_rows <- sample(1:n, .8*n, replace = F)
pred.train <- pred[train_rows,]
dv.train <- dv[train_rows]
pred.test <- pred[-train_rows,]
dv.test <- dv[-train_rows]a. Perform the ridge regression using the package glmnet, in which you need to set 200 different lambda
values (the default for the function glmnet is 100). You need to generate the plot with x-axis being the
log of lambda and y-axis being the training MSE. Please also report the MSE on the testing dataset
with the model generated with a lambda equal to lambda.lse (use ?glmnet for its explanation.)b. Repeat the same steps above with Lasso Regression: fit the lasso regression with nlambda=200, plot the
training MSE against log-lambda, generate prediction on testing set using the model with lambda.lse
and compute MSE on testing set.c. Perform Elastic Net Regression: fit the elastic net regression with alpha set to be seq(0, 1, by=1/20),
plot the testing MSE against each alpha value