Advanced Methods for the Analysis of Semiconductor Manufacturing Process Data - Advanced Techniques in Knowledge Discovery and Data Mining - page 50

Evaluation and assessment measures

Unsupervised Supervised

Topology Distance Signal Classifier output Feature space

Overlap
ENN

Compactness Separability

q m E NLM MSE SNR Rate q k q oi J s q ci q si

Fig. 2.12. Taxonomy of cost functions.

overlap, or compactness in the regarded feature space and ensuing systematic
dimensionality reduction. Though the classification rate or a posteriori prob-
abilities of any classifier could serve here (cf, e.g., [2.16] or [2.45]), for obvious
practical reasons, robust measures nearly free of required parameters, model
assumptions, and intricate training requirements are preferred in this work.
For instance, to measure separability, a nonparametric measure q s exploit-
ing nearest-neighbor techniques can be computed. For this class separability
assessment, the RNN-classifier [2.13] is exploited, which iteratively selects
a subset of relevant vectors as reference vectors from the training set, as
the number of these selected reference vectors T RNN is proportional to the
feature space separability. This is illustrated in Fig. 2.13, where selected ref-
erence vectors T RNN are emphasized in bold. In the case of linear separability
of class regions, one vector per class region would be required. So the quality
measure given by
q s = N − ( T RNN − L )
N

(2.14)

has 1.0 as its optimum value indicating linear separability. An improved vari-
ant of q s takes significantly different a priori probabilities in account:

L

1
L

N i − ( T RNN i − 1)

q si = . (2.15)

N i

i =1

Here N i denotes the number of patterns a liated to class ω i and T RNN i the
number of reference vectors selected for class ω i . (It is assumed here that N i
corresponds to the actual a priori probability of class ω i ). The quality mea-
sures q s and q si have O(N) complexity and thus are very fast; however, the
resolution is quite coarse, which can be detrimental for optimization schemes.
Numerous feature space configurations can be mapped on the same assess-
ment value.