Ridge OLS

Now, it is time to code ridge regression $($regularization using the $\_(2)$ norm$)$ for OLS. See equation given below.

w$=($X$^($TT$)$X$+$lambdaI$)^(-1)$X$^($TT$)$y

TODO $8(15$ points$)$: RidgeOrdinaryLeastSquares

fit$()$ TODOs

Follow the below steps to implement the fit$()$ method. Feel free to reuse code from prior homework.

Covert the ridge regression OLS equation into code to compute the optimal weights w$.$ Make sure the bias variable is not regularized, you should only regularize the weights for the features. Store the output into self.w$.$

A$.$ Hint: Use np$.$ eye to create an identity matrix I with the shape $($N$,$N$).$ Set the first element value to zero to prevent the bias from being regularized as the bias is represented by the first column in the data.

predict$()$ TODOs

Follow the below steps to implement the predict$()$ method. Feel free to reuse code from prior homework.

Compute the predictions by taking the linear combination between the weights and passed data. Return the predictions as a $2$D column vector.

class RidgeOrdinaryLeastSquares$()$:

$"""$ Perfroms ordinary least squares regression

Attributes:

lamb $($float$)$: Regularization parameter for controlling

L$2$ regularization.

w: Vector of weights

$"""$

def $\_\_$init$\_\_($self$,$ lamb: float$)$:

self.lamb $=$ lamb

self.w $=$ None

def fit$($self$,$ X: np$.$ndarray, y: np$.$ndarray$)->$ object:

$"""$ Train OLS to learn optimal weights

Args:

X: Training data given as a $2$D matrix

y: Training labels given as a $1$D vector

Returns:

The class's own object reference.

$"""$

# TODO $8\mathrm{.}1$

return self

def predict$($self$,$ X: np$.$ndarray$)->$ np$.$ndarray:

$"""$ Make predictions using learned weights

Args:

X: Testing data given as a $2$D matrix

Returns:

A $2$D column vector of predictions for each data sample in X

$"""$

# TODO $8\mathrm{.}2$

return None