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Problem $1$A $-$ Fisher LDA $[35$pts$]$; $\{(5)/(7)\}$ Given the following training data set:

Class $1$: Consists of $5$ data points $(1,2),(2,3),(3,3),(4,5),(5,5)$

Class $2$: Consists of $6$ data points $(1,0),(2,1),(3,1),(3,2),(5,3),(6,5)$

Find Fisher's normal vector w and intercept b $($separation point$)$ that define the classification plane

f:bf orthogonal to w that intercepts the latter at a point bw $(5,4).$

Show the following computations:

$2$pts: Compute $\backslash$mu $\_(1)$ and $\backslash$mu $\_(2),$ and S$\_($B$),$ the between$-$class scatter matrix

$2$pts: Compute SC$\_(1)$ and SC$\_(2),$ and S$\_($W$),$ the within$-$class scatter matrix

$2$pts: Compute S$\_($W$)^(-1)$S$\_($B$)$ matrix

$2$pts: Find the best vector $()/($bar$)($w$)$ that defines discriminating hyperplane between the classes

$2$ pts : Having found the $()/($bar$)($w$),$ find the separation point.

$2$ pts : Classify the new vector $(5,4).$