Exercise $2$: Models for Data $(25$ points$)$

$y$

$x$

A model is a function that takes an input and produces a prediction $*$

Let's consider two possible models for this data:

$M_1(x)=x+5$

$M_2(x)=2x+1$

Compute the predictions of models $M_1$ and $M_2$ for the values in $x.$ These predictions should be vectors $Y$

of the same shape as $.$ Then plot the prediction lines of these two models overlayed on the "observed" data $(x,Y).$ Use plt$.$plot$()$ to draw the lines. Note: you will generate only one plot. Make sure to include axes, titles and legend.

Evaluation Metrics

How good are our models? Intuitively, the better the model, the more closely it fits the data we have. $y$

$x$

That is$,$ for each $,$ we'll compare $y,$ the true value, with $*,$ the predicted value. This comparison is often $y$

called the loss or the error. One common such comparison is squared error: $(y-{?}^{{?}^{{?}^{?}}}{)}_2.$ Averaging over all our data points, we get the mean squared error.

$1$

${}^{y_i}$

MSE $=-lonY(yi-$hat$(i{)}^i{)}_2$

Exercise $3$: Computing MSE $(25$ points$)$

Write a function for computing the MSE metric and use it to compute the MSE for the two models above, $M_1$ and $M_2.$

def MSE$($true$\_$values, predicted$\_$values$)$:

$""\text{'}"$Return the MSE between true$\_$values and predicted values.""."

# YOUR CODE HERE

print $(\text{'}$MSE for M$1$:$\text{'},$ MSE$($Y$,$ M$1))$

print $(\text{'}$MSE for M$2$:$\text{'},$ MSE$($Y$,$ M$2))$

MSE fo

r M$1$:

None

MSE fo

$r$ M$2$:

None