Problem $2$: In the class example, we apply logistic regression to build a linear classifier

to distinguish the iris species setosa $($in blue dots$)$ and versicolor $($in green dots$),$ as

shown below. We also report the coefficients and the intercept for the linear classifier.

from sklearn.linear$\_$model import LogisticRegression

model $=$ LogisticRegression$($random$\_$state $=0).f$ it $($X$\_$train.values, y$\_$train$)$

print coefficients and the intercept

print $($model$.$coef$\_)$

print$($model$.$intercept$\_)$

$4$ visualize support vectors and the boundaries

ax $=$ plt$.$gca$()$

xlim $=$ ax$.$get$\_$xlim$()$

ylim $=$ ax$.$get$\_$ylim$()$

plt$.$xlabel$(\text{'}$petal$\_$length'$)$

plt$.$ylabel$(\text{'}$petal$\_$width'$)$

$=np\mathrm{.}1$ inspace $(x\underset{?}{lim}[0],x\underset{?}{lim}[1],30)$

yy $=$ np$.$linspace $(y\underset{?}{lim}[0],y\underset{?}{lim}[1],30)$

$YY,xx=np.$meshgrid$(Yy,xx$

$xy=$ np$.$vstack $([$XX$.$ravel $(),$ YY$.$ravel$()]).$T

$z=$ model.decision$\_$function$(xy.$reshape$($XX$.$shape$)$ # get $f(x),$ i$.$e$.,$ the decision function

# plot the decision boundary $f(x)=0,$ as well as references $f(x)=-1$ and $f(x)=1.$

ax$.$contour $,$ colors$=\text{'}$k$\text{'},$ levels$=[-1,0,1],$ alpha$=0\mathrm{.}5,$ linestyles$=[\text{'}--\text{'},\text{'}-\text{'},\text{'}--\text{'}])$

plt$.$show $()$

$\left[\begin{array}{c}2\mathrm{.}31381394,0\mathrm{.}90900393\end{array}\right]$

$]$

a$)$ Write the decision boundary formula, i$.$e$.,f(x).$

b$)$ If a new species has been discovered with petal length of $2\mathrm{.}5$ and petal width of

$0\mathrm{.}75,$ what is the probability that this species belongs to setosa? $($The classes are

ordered as $[$setosa$,$ versicolor$].)$