Advanced Methods for the Analysis of Semiconductor Manufacturing Process Data - Advanced Techniques in Knowledge Discovery and Data Mining - page 53

thedistancebetween x j and most distant nearest neighbor NN jk , q NN ji de-
notes the measure contribution of x j with regard to NN ji ,and ω j and ω i
denote the class a liation of x j and NN ji , respectively. The influence of
a nearest neighbor in the quality measure decays with its rank position to
n i = 0 for NN jk . The final measure q o is fine-grained and sensitive to small
changes in the feature space. Further q o is also normalized in [0,1], where 1.0
indicates no overlap in the feature space. Typically, 5 to 10 nearest neighbors
are well suited for computation of this quality measure. Simplification of the
measure is feasible, trading off fine-grained resolution in overlap computation
and, thus, sensitivity to small changes in the feature space against computa-
tional savings. An improved variant of q o takes significantly different a priori
probabilities into account

k k

q NN ji + n i

L N c

1
L

1
N c

i =1 i =1

q oi = . (2.22)

k

c =1 j =1

2 n i

i =1

Finally, compactness q c can be measured by explicitly computing the ratio
of current intra- and interclass distances. Implicitly this criterion is also used
in the computation of scatter matrices [2.12]. The compactness q c previously
introduced in [2.22] suffers from the flaw that the measure will be optimum,
if the majority of intraclass distances will be made small, i.e., class regions
with the majority of patterns will dominate the assessment and consequently
any optimization process based on the measure q c .Animprovedmeasure
q ci for different a priori probabilities and corresponding N l in the L -class
problem can be obtained by class-specific normalization during compactness
computation

L l =1 N l ( N l − 1) i =1 j = i +1 δ ( ω i ,ω j ) δ ( ω i ,l ) d X ij

1

1 2

q ci = (2.23)

N B i =1 j = i +1 (1

− δ ( ω i ,ω j )) d X ij

with

M

d X ij = ( x iq − x jq ) 2 (2.24)

q =1

and δ ( ω i ,ω j ) is the Kronecker delta, which is δ ( ω i ,ω j )=1for ω i = ω j ,i.e.,
both patterns have the same class a liation, and δ ( ω i ,ω j ) = 0 elsewhere.
Also, δ ( ω i ,l ) prescribes that only distances with ω i = ω j = l are accumulated
for the l th-class sum of intraclass distances. Further, the normalization factor
N B is given by

N N

N B =

(1 − δ ( ω i ,ω j )) . (2.25)

i =1 j = i +1