Bayes Minimum-Risk Classifier (BC).

The Bayes M inim um -risk Classifier [57] [58], m inim ises the expected cost

of m isclassified data. In this w o rk w e have u sed the sta n d a rd Bayes classifier

capable for discrim inating tw o classes. This decision rule is q u ad ratic form ula

for th e u n d e r investigation en try X an d has the form:

\

'^l

S .

V y

< r (4.33)

w h ere X : is the u n d e r investigation entry.

X ,, X2 : are the m ean vectors of the tw o classes œ ^,œ 2, respectively.

, * ^ 2 : are the covariance matrices.

The covariance m atrix can be calculated in the follow ing w ay: Let us define the

as A = ( Aj, A2, A3, A4 ). Then w e can define the covariance m atrix of A as

follows:

5” — (a,. A;)* (a,, a,.) —

_ 2 _ 2

4 1 4 4

w h ere cj = (a,. - A,. )^, by convention, (a,.-A ,.) is th e colum n vector, and

(a. - A. ) is its corresponding row vector of the covariance m atrix.

E quation 4.33, is the decision rule th at classifies the vector X into the class co ,

if true, otherw ise assigns the vector X into the class co2 . The q u an tity t , is a

th resh o ld v alu e of the classifier an d is equal to: In 12

•21

J

, w h ere p[co , )

an d p[co ^ ) are the probabilities for the vector X to be classified into the classes

(Û a n d Ù) 2/ respectively, an d p(co , ) + p(o) 2 ) = 1 • The term c.^., is the cost of

m isclassifying responses from class 7 as those of class i . It is obvious th a t if the

n u m b er of elem ents in the class co , is the sam e to th a t of the class co. , th en the

q u an tity t is equal to zero.

4.7.1. Linear Bayes Classifier.

W e can rew rite Equation 4.33, in a linear form u n d e r the assu m p tio n th at

W e can evaluate the average covariance m atrix as: S = p[co + p[co 2 ) ^ 2 • The

linear Bayes classifier can be described b y the form ula:

- X ^ y *5 -’ - X ^ y *5-1 + x ^ ) < r (4.34)

The first p a rt of the above form ula is the linear discrim inant function, the

second p a rt is th e balance p o in t half w ay b etw een the m ean vectors of each

class, 0.5* (Xj + X2). The th reshold t , in o u r case is equal to zero, since the

n u m b er of elem ents of the tw o categories is the same. E quation 4.34, can be

fu rth er sim plified if w e define the new vector =[x^ - X ^ y * S ~ \ The

E quation 4.34, can be w ritten as:

M"^*%-j*M"'*(%; - X

)

(4.35)

4.7.2. Design of the Linear Bayes Classifier.

In this section w e w ill presen t the p ro ced u re follow ed in the design of a

L inear Bayes Classifier, capable of classifying into tw o categories an u n k n o w n

e n try using tw o dim ension decision space. Let, X = [ x ^ , X2) be the u n d e r

investigation vector, A = (Aj, Aj ) be the m ean vector of th e first category, and

B =[3 ^,8 2) be the m ean vector of the second category. The first step is the

ev alu atio n of the covariance m atrices of the vector X , a n d the m ean vectors A

‘^ 1 = ( X j - A , ) ( X j - A , ) 5j — iSAjj 5!Aj2 ‘^ ■ ^ 2 1 ^^22 SBu SB,, SB,, SB,,

A fter the calculation of the tw o covariance m atrices, w e estim ate th e n ew

covariance m atrix as discussed in the p revious p arag rap h . Let, P (A )an d

P( B)be th e probabilities of the new entry to be classified into th e class A an d

B, respectively. Then w e can evaluate the n ew covariance m atrix as following:

SB,, SB,, S = P{A)*S, + P ( B y S , = P ( A ) * SA„ SA„ SA„ SA„ + p ( B y SB,, SB,, Su S,2 *^21 S22 N ex t w e evaluate the vector, by the form ula: = { A - b Y * S ^ = [ c , C2).

By su b stitu tin g the M ^ vector into the form ula of the Linear Bayes Classifier,

w e can write: C , C 2 * 1 1 I A, + B, X2 — ■“ * C C2 * 2 1 1 2 1 A2 + B2 <0: Q X i + C2X2 < 2 [C](Ai +B,)+C2 {a, +B,)^ (4.36)

The above eq u ation represents the decision ru le for the Linear Bayes Classifier,

4.8. Least Squares Minimum Distance Quadratic Classifier