Polynomial classifier with one feature. Generate $200$ points $x^{(1)},$dots,$x^{(200)},$ uniformly

spaced in the interval $-1,1,$ and take

$y^{(i)}=\{\begin{array}{ccc}+1,& -0\mathrm{.}5x^{(i)}<mn\; id="EditorElement\_851646">0</mn><mn\; id="EditorElement\_851647">.</mn><mn\; id="EditorElement\_851648">1</mn><mtext\; id="EditorElement\_851650"></mtext><mi\; id="EditorElement\_851651">o</mi><mi\; id="EditorElement\_851653">r</mi><mtext\; id="EditorElement\_851655"></mtext>& 0\mathrm{.}5x^{(i)}\\ -1& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

for $i=1,$dots,$200.$ Fit polynomial least squares classifiers of degrees $0,$dots,$8$ to this

training data set.

$($a$)$ Evaluate the error rate on the training data set. Does the error rate decrease when

you increase the degree?

$($b$)$ For each degree, plot the polynomial tilde$(f)(x)$ and the classifier hat$(f)(x)=$sign$(tilde(f)(x)).$

$($c$)$ It is possible to classify this data set perfectly using a classifier hat$(f)(x)=$sign$(tilde(f)(x))$

and a cubic polynomial

tilde$(f)(x)=c(x+0\mathrm{.}5)(x-0\mathrm{.}1)(x-0\mathrm{.}5),$

for any positive $c.$ Compare this classifier with the least squares classifier of degree

$3$ that you found and explain why there is a difference.