$3$ LDA $[45$ pts$]$

The binary LDA classification of $x^{22}$ was solved by finding the direction of the projection hyperplane

$(wprop{S}_w^{-1}(m_2-m_1))$ in the lecture. In this problem, the binary LDA classification model will be

trained and the train model will generalize the test data. Please display the results for the full credit

consideration.

Generate the data set and split it into train and test sets.

from sklearn import datasets

import numpy as np

X$,$ y $=$ datasets.make$\_$blobs$($n$\_$samples $=100,$ n$\_$features $=4,$ centers $=2,$

cluster$\_$std$=1\mathrm{.}5,$ random$\_$state$=123)$

y $=$np$.$array $([-1$ if i$=0$ else $1$ for i in y $])$

from sklearn.model$\_$selection import train$\_$test$\_$split

X$\_$train, X$\_$test, y$\_$train, y$\_$test $=$ train$\_$test$\_$split$($X$,$ y$,$

test$\_$size$=0\mathrm{.}2,$ random$\_$state$=42)$

a$.10pts$ Write a method that computes the between$-$class scatter matrix $S_B.$ Compute $S_B$ using a

train data.

b$.[10$ pts$]$ Write a method that computes and returns the within$-$class scatter of each class, the total

within$-$class scatter $(S_w),$ and its inverse $(S_w^{-1}).$ Calculate $S_w$ and $S_w^{-1}$ using the train data.

c$.10pts$ NumPy.linalg.eig$(*$