Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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C 2<br />
C 1<br />
C 1<br />
C 2<br />
C X<br />
X 2<br />
M X<br />
M X 2<br />
1<br />
C 1<br />
(a)<br />
M1 = M2 = M0 (b)<br />
C 1<br />
(c)<br />
C 2<br />
M1 = M2 = M0 Figure 4.5: Examples where LDA based on the criteria tr(S −1<br />
w Sb) fails to discriminate<br />
between classes.<br />
Let C1 and C2 be two classes, and x (i) refers to samples from Ci. The Ci-local<br />
mean <strong>of</strong> a sample x (l) from class Cl is denoted by mi(x (l) ) and is computed as:<br />
mi(x (l) ) = 1<br />
k<br />
k�<br />
j=1<br />
x (i)<br />
jNN , (4.46)<br />
where x (i)<br />
jNN is the jth NN from Ci for the sample x (l) , a sample <strong>of</strong> class Cl.<br />
Nonparametric between-class scatter matrix for two classes, denoted by Sb can<br />
be defined as:<br />
Sb = P1E{(x (1) − m2(x (1) ))(x (1) − m2(x (1) )) T |C1}<br />
+ P2E{(x (2) − m1(x (2) ))(x (2) − m1(x (2) )) T |C2} (4.47)<br />
Similarly, nonparametric within-class scatter matrix for two-class is denoted by Sw<br />
and can be defined as:<br />
Sw = P1E{(x (1) − m1(x (1) ))(x (1) − m1(x (1) )) T |C1}<br />
+ P2E{(x (2) − m2(x (2) ))(x (2) − m2(x (2) )) T |C2} (4.48)<br />
We calculate Sb and Sw for C classes using the above defined scatters for two classes<br />
(equations 4.47 and 4.48). For a set <strong>of</strong> C classes we thus get a total <strong>of</strong> C 2 scatter<br />
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