In the case of two groups, we might consider using linear regression as a classification method. Specifically, given data (91, x1), (9n, xn) (where 91, 9n each take one of two possible values 0 or 1), we choose Bo, to minimize n (9i - Box) 2 We can then use the value of Bo + xTB to classify an observation ; for example, we might classify x to g = 1 if BoxTB is greater than some threshold. (a) Define Show that (b) Now define n n Xi 3 = ((x x) (x x) - *7)*((x-2) - i=1 i=1 x1 n n (1 9i) xi (1 - 9) i=1 n n i=1 to be the means of observations within the two groups and S == 1 n 2 ( x n x gr ) ( x i x gr ) T i=1 to be an estimate of a common covariance matrix for the two groups. Show that B = kS(x1 x0) for some constant k. (Hint: Note that n (xi - x) (xi - x)T - i=1 n n ((1 9i) (xi x) (xi x) i=1 n i=1 ==== i=1 N - n Ii (xi 1) (xi x1) + [(1 9i) (xi xo) (xi xo) and x = X + (1 ) where = (91 + + In)/n.)