In this exercise we will generate simulated data and use it to fit a lasso model. Use

the code from class, ISLR text, or a LLM to help with completing each task. $[10$

marks$]$

$($a$)$ Use the rnorm$()$ function to generate a predictor X of length n $=100,$ as well

as a $$noise$$ vector $$ of length n $=100($Hint: make the standard deviation

much smaller for generating the noise vector $).$ Just paste your code, not all

the data values in your answer.

$($b$)$ Generate a response vector Y of length n $=100$ according to the model

Y $=\backslash$beta $0+\backslash$beta $1$X $+\backslash$beta $2$X$2+\backslash$beta $3$X$3+,$

where $\backslash$beta $0,\backslash$beta $1,\backslash$beta $2$ and $\backslash$beta $3$ are constants of your choice. Just paste your code in

your response, not all the generated data.

$($c$)$ Create a data frame with the response variable Y and the predictors

X$,$ X$2,$ X$3,...,$ X$10.$ Run the following code, replacing Y and data with your

response variable and data frame name and paste the output as your answer

to this part.

library$($glmnet$)$

x $<-$ model.matrix$($Y ~ $.,$ data $=$ data$)[,-1]$

lasso.model $<-$ cv$.$glmnet$($x$,$ Y$,$ alpha $=1)$

plot$($lasso$.$model$)$

best$\_$lambda $<-$ lasso.model$lambda.min

lasso.coef $<-$ predict$($lasso$.$model, s $=$ best$\_$lambda, type $=$ "coefficients"$)$

# Print the best lambda and coefficients

print$($paste$("$Best lambda:", best$\_$lambda$))$

print$("$Lasso Coefficients:"$)$

print$($lasso$.$coef$).$

$($d$)$ Explain what the code in part $($c$)$ is doing. You will need to lookup what

the cv$.$glmnet$()$ function does and research $$cross$-$validation$.$ This can be

completed with a few sentences. How close are the coefficients to the actual

coefficients you chose in $($b$)?$