We perform linear regression analysis with Lasso for the case where $Y$ is the

target variable and the variables $x_1,x_2,x_3$ and $x_4,x_5,x_6$ are highly correlated.

We generate $N=500$ groups of data, distributed as in the relations below, and

then apply linear regression analysis with Lasso to $xin{R}^{Np},yin{R}^N.$

$z_1,z_2,lon,lo{n}_1,$dots,$lo{n}_{6}N(0,1)$

$\{\begin{array}{ccc}{x}_j\text{:}=z_1+lo\frac{n_j}{5},& j=1,2,& 3\end{array}$

$x_j$:$=z_2+lo\frac{n_j}{5},j=4,5,6$

$y$:$=3z_1-1\mathrm{.}5z_2+2lon$

Fill in the blanks $(1),(2)$ below. Plot a graph showing how each coefficient

changes with $.$ n $=500$

x $=$ np$.$zeros$(($n$,6))$

z $=$ np$.$zeros$(($n$,5))$

for k in range$(2)$:

z$[$:$,$ k$]=$ np$.$random.randn$($n$)$

y $=$ ## Blank $(1)$ ##

for j in range$(3)$:

x$[$:$,$ j$]=$ z$[$:$,0]+$ np$.$random.randn$($n$)/5$

for j in range$(3,6)$:

x$[$:$,$ j$]=$ z$[$:$,1]+$ np$.$random.randn$($n$)/5$

lambda$\_$seq $=$ np$.$arange$(0\mathrm{.}1,20,0\mathrm{.}1)$

p $=6$

r $=$ len$($lambda$\_$seq$)$

coef$\_$seq $=$ np$.$zeros$(($r$,$ p$))$

cols $=["$blue$",$ "red", "green", "yellou", "purple", "orange"$]$

for i in range$($r$)$:

coef$\_$seq$[$i$,$ :$].\_=$ ## Blank$(2)$ ##

for j in range$($p$)$:

plt$.$plot$(-$np$.$log$($lambda$\_$seq$),$ coef$\_$seq$[$:$,$ j$]+0\mathrm{.}01*$ j$.$

c$=$cols$[$j$],$ label$="$X$"+$str$($j$+1))$

plt$.$ylabel$($r$"")$

plt$.$legend$($loc$=$"upper $|$left$")$

plt$.$title$("$Lasso$")$