Problem $2($Coding$)12$ pts:

For the following tasks use the following California House Price Data:

$[1]1$ from sklearn.datasets import fetch$\_$california$\_$housing

$2$ housing $=$ fetch$\_$california$\_$housing$()$

$[20]1$ data$\_x=$ pd$.$DataFrame$($housing$[\text{'}$data$\text{'}],$ columns $=$ housing.feature$\_$names$)$

$2$ data$\_$X$.$head$(2)$

$1$ data$\_$y $=$ housing.target

$2$ data$\_$y$|$

array$([4\mathrm{.}526,3\mathrm{.}585,3\mathrm{.}521,$dots,$0\mathrm{.}923,0\mathrm{.}847,0\mathrm{.}894])$

$1$ data$\_$y $=$ housing.target

$2$ data$\_$y$|$

array$([4\mathrm{.}526,3\mathrm{.}585,3\mathrm{.}521,$dots,$0\mathrm{.}923,0\mathrm{.}847,0\mathrm{.}894])$

Split the dataset into training and test sets of $80$:$20$ ratio $($use random$\_$seed $=2024)$ and test$\_$size $=$

$0\mathrm{.}20.$ You must train the linear regression model using the training data and compute MSE using the test

dataset.

Apply Multiple Linear Regression $($MLR$)$ using normal $($least square solution$).$ You must not use any direct

or in$-$built package for MLR$.$

a$.(2\mathrm{.}5$ pts$)$ Check the five assumptions $($mentioned in the classroom$)$ of MLR $($use $-$ training dataset$)$

and proper interpretation $-$ why the assumptions are met or not

b$.(2$ Pts$)$ Derive the normal equation for linear regression.

c$.(2$ Pts$)$ Apply the standardization technique to all features to ensure that all features have a

consistent scale. Utilize 'fit$\_$transform' for the training data and 'transform' for the test data to

prevent data leakage.

d$.(3$ pts$)$ Find optimal values of intercept and coefficients using the normal equation of the linear

regression $(b_{\text{o}\text{p}\text{t}}=(x^{T}x{)}^{-1}{x}^{T}y)$ using the training data. To avoid inverse matrix error, you may

use pseudo inverse $($np$.$ling.pinv$)$

e$.(1$ pts$)$ Find $($ hat$(y))($predict for each datapoints of $x_{-}$test$)-$ show in dataframe $-$ making two columns:

y$\_$actual & hat$(y)\_$predict

f$.(1+0\mathrm{.}5=1\mathrm{.}5pts)$ Finally, for the test dataset:

a$.$ Calculate coefficient of determination $(R^2)$ and interpret the result

b$.$ Find MSE $($mean of sum of squares of error $($residual$)$