Page 34 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN CLASSIFICATION 23

Listing 2.2
PRTools code for estimating decision boundaries taking account of the
cost.

load nutsbolts;
cost ¼ [ 0.20 0.07 0.07 0.07 ; . . .
0.07 0.15 0.07 0.07 ; . . .
0.07 0.07 0.05 0.07 ; . . .
0.03 0.03 0.03 0.03];
w1 ¼ qdc(z); % Estimate a single Gaussian per class
% Change output according to cost
w2 ¼ w1*classc*costm([],cost);
scatterd(z);
plotc(w1); % Plot without using cost
plotc(w2); % Plot using cost

2.1.1 Uniform cost function and minimum error rate

A uniform cost function is obtained if a unit cost is assumed when an
object is misclassified, and zero cost when the classification is correct.
This can be written as:

1 if i ¼ k
!
Cð^ ! i j! k Þ¼ 1 ði; kÞ with: ði; kÞ¼ ð2:9Þ
0 elsewhere
(i,k) is the Kronecker delta function. With this cost function the condi-
tional risk given in (2.4) simplifies to:

K
X
!
!
Rð^ ! i jzÞ¼ Pð! k jzÞ¼ 1 Pð^ ! i jzÞ ð2:10Þ
k¼1;k6¼i
Minimization of this risk is equivalent to maximization of the posterior
probability P(^ ! i jz). Therefore, with a uniform cost function, the Bayes
!
decision function (2.8) becomes the maximum a posteriori probability
classifier (MAP classifier):

^ ! ! MAP ðzÞ¼ argmaxfPð!jzÞg ð2:11Þ
!2O

Application of Bayes’ theorem for conditional probabilities and cancel-
lation of irrelevant terms yield a classification, equivalent to a MAP