We will now perform cross$-$validation on a simulated dataset.

$($a$)4pts$ Generate a simulated dataset as follows:

def f$($x$)$:

return x$**5-24+$x$***3$

np$.$random.seed$(1)$

x $=$ np$.$random.uniform$(0,1,$ size$=500)$

y $=$ f$($x$)+$ np$.$random.normal $(0,0\mathrm{.}01,$ size$=500)$

$($b$)4pts$ Create a scatterplot of $x$ against $y.$ Comment on what you find. $($Hint: You

may find plot$()$ helpful$)$

$($c$)[4$pts$]$ Set a random seed $123,$ and then compute the LOOCV errors that result from

fitting the polynomial functions of degree from $1$ to $7$ using the simulated data in $($a$)$:

$f_1(x)={}_0+{}_{1}x+$

$f_2(x)={}_0+{}_{1}x+{}_2{x}^2+$

vdots

$f_7(x)={}_0+{}_{1}x+{}_2{x}^2+{}_3{x}^3$cdots$+{}_7{x}^7+$

$($Hint: See Section $5\mathrm{.}3$ in ISLP for an example of how to implement cross$-$validation in

Python. You may find

from sklearn.model$\_$selection import cross$\_$validate

from ISLP.models import sklearn$\_$sm

helpful $)$

$($d$)[4$pts$]$ Repeat $($c$)$ using another random seed $12345,$ and report your results. Are your

results the same as what you got in $($c$)?$ Why?

$($e$)[4$pts$]$ Which of the models in $($c$)$ had the smallest LOOCV error? Is this what you

expected? Explain your answer.

$($f$)[4$pts$]$ Fit $f_5(x)$ using least squares. Comment on the coefficient estimates and their

statistical significance.