Advanced Methods for the Analysis of Semiconductor Manufacturing Process Data - Advanced Techniques in Knowledge Discovery and Data Mining - page 47

Dimensionality reduction methods

Linear methods Non linear methods

Signal preserving Discrimination

Signal
preserving

Distance
preserving

Topology
preserving

PCA
(

FA Scatter

matrices

FS & FW M-PCA CCA NLM

(

LSB Visor TOPAS Koontz &

Fukunaga

ICA ) MDS )

BP (auto-
associative)

BP (net
pruning)

DIPOL-
SOM

SOM
ViSOM

BP (discr.
analysis)

Fig. 2.11. Taxonomy of dimensionality reduction methods.

d Xij , and, thus, implicitly the data structure, shall be preserved in the NLM
according to the cost function E ( m ):

N j

( d Xij − d Yij ( m )) 2
d Xij

E ( m )= 1
c

. (2.7)

j =1 i =1

Here

d

− y jq ( m )) 2 d Yij ( m )= ( y iq ( m ) (2.8)

q =1

denotes the distance of the respective data points in the visualization plane
and

M

d Xij = ( x iq − x jq ) 2 (2.9)

q =1

in the original data space and

j

N

c = d Xij . (2.10)

j =1 i =1

Based on a gradient descent approach, the new coordinates of the N pivot
vectors in the visualization plane y i are determined by:
y iq ( m +1)= y iq ( m ) − MF ∗ ∆y iq ( m ) (2.11)

with