Federico Di Palma

Giuseppe De Nicolao

Guido Miraglia

Oliver M Donzelli

CiteSeerX

Unsupervised algorithms for the automatic classification of EWS maps: a 
comparison
Federico Di Palma, Giuseppe De Nicolao
University of Pavia
Via Ferrata 1, 27100 Pavia, Italy
{federico.dipalma-giuseppe.denicolao}@unipv.it
Oliver M. Donzelli, Guido Miraglia
STM-Microelectronics
Via Olivetti, Agrate Brianza, Italy 
{oliver.donzelli-guido.miraglia}@st.com
Abstract  –  Recently,  it  has  been  shown  that  the  
classification of Electical Wafer Sorting failure maps can  
be performed by means o funsupervised methods. In this  
work four different unsupervised methods are compared:  
SOM,  K-Means,  Neural  Gas,  and  an  Expectation  
Maximization.  The  algorithms  are  compared  using  a  
benchmark  based  on  a  probabilistic  model.  The  
performance of the classification is assessed by means of  
an new index call Index-F based on the knowledge of the  
real classification. Moreover it is studied the correlation  
between the  proposed  index  and the  following indexes:  
CH-index, D-index, I-index and average likelihood.
INTRODUCTION
If  the  devices  that  fail  at  the  Electrical  Wafer  Sorting 
(EWS)  tests  are  visualized  as  black  pixels,  the  spatial 
distribution of the failures is likely to show characteristic 
patterns.  Different  shapes  are  possible:  circular  spots, 
rings,  semi-ring,  repetitive  chessboard-like  patterns,  to 
mention the most frequent. These patterns can be used to 
trace  back  to  the  problems  that  originated  the  failures 
either  by  analyzing  their  qualitative  features  or  by 
correlating them with the lot history.  Hence, the interest 
for algorithms that perform the automatic classification of 
large  wafer  sets  on  the  basis  of  their  EWS maps.  Two 
approaches  are  possible:  the  supervised  and  the 
unsupervised ones. The supervised classifiers require the 
preliminary classification of a training set of wafers by a 
human  operator.  This  approach  is  time  consuming  and 
when the process and/or the product changes the training 
has to be performed ex-novo. Recently, it has been shown 
that  wafer  classification  can be  performed by means  of 
unsupervised methods: these algorithms create clusters of 
wafers by identifying their common features and do not 
use any training set. In [1], the Kohonen's Self Organizing 
Map (SOM) has been used successfully to classify  EWS 
wafer maps. Several other unsupervised learning methods 
may be used to classify wafer maps. 
The first  aim of this paper is  to compare four different 
unsupervised methods: SOM,  K-Means,  Neural  Gas and 
an  Expectation  Maximization  (EM)  classifier.  The 
algorithms are compared using a benchmark based on a 
probabilistic model. The performance of the classification 
is  assessed  by  means  of  an  index  (here  denoted  as  F-
index) that measures the misclassification rate.
Another  important issue is  finding indexes that measure 
the classification goodness and are appliable to real data. 
In  fact  with  real  data  the  F-index  (which  assumes  the 
knowledge of the true classification) is no longer usable. 
Hence  the  second  aim  of  this  work  is  to  study  the 
correlation between the F-index and the following indexes 
[2]: CH-index, D-index, I-index and average likelihood.
In  the  following section  the  probabilistic  model,  the  F-
index and the benchmark are presented. The third section 
is devoted to description of the clustering algorithms. The 
fourth one illustrates the results of the comparison. The 
choice  of  a  clustering  index  appliable  to  real  data  is 
discussed  in  Section  5.  The conclusion  summarizes  the 
main results.
PRELIMINARIES
To make a comparison a goodness  criterion is required. 
The proposed measure is based on the knowledge of the 
correct  classification.  Hence  the  need  for  a  benchmark 
made of simulated wafer maps whose true classification is 
known.  In  order  to  obtain  the  simulated  data  a 
probabilistic model is adopted.
Probabilistic Model
In [1] a  probabilistic model was proposed. According to 
this model the electrical test of a die has only two results: 
good (0) or failed (1). Then a binary Bernoulli variable Xd 
is used describe the outcome of the EWS test for the d-th 
die. It  was assumed that the electrical failure of a single 
device occurs  independently of the failure of the others 
with a probability P(Xd=1)=f(xd,yd) where xd and yd are the 
planar coordinates of the die d. A complete wafer  Xw can 
be created by simulating Nd Bernoulli trials. The origin of 
the planar coordinates is placed in the center of the wafer 
and the scales of the cartesian axes are such that the wafer 
has radius one. This system of coordinates allows one to 
describes  different  scales  of  integration  with  the  same 
spatial probability.
The benchmark
A  set  of  production  lots  is  often  affected  by  several 
failures with different occurrence rates. In  this work we 
simulated 8 classes each of which characterized by its own 
spatial  distribution,  reported  in  Figure  1.  The  spatial 
distributions  reflect  some  common  failure  patterns: 
standard  production (Class #2),  low yield  wafers  (Class 
#1) repetitive horizontal and vertical (Classes #5 and #6), 
1
spot and ring (Classes  #3, #4 and #8),  semi-ring (Class 
#7).  The  clustering  algorithms  were  tested  on  several 
benchmarks  with  different  number  of  devices  (Nd)  and 
wafers (Nw) as shown in Table 1
Figure 1: Spatial probability of the simulated classes
Benchmark 1 2 3 4 5 6
# Wafers 300 500 800 1200 300 500
# Dies 601 601 601 601 377 377
Benchmark 7 8 9 10 11 12
# Wafers 800 1200 300 500 800 1200
# Dies 601 601 950 950 950 950
Table 1. The twelve data set of the benchmark
Classification assessment
In a classification two kinds of error can be observed: an 
identified class includes elements coming from different 
real  classes,  or a  real  class is  splitted into two or more 
identified classes.  The first kind of misclassification can 
be disruptive for diagnosis purposes. In fact the reference 
pattern  is  explained  by  the  union  of  different  physical 
problems whose separate detection becomes problematic. 
Conversely,  if  a  real  class  is  splitted  into two or  more 
classes,  each  identified  class  is  made  of  wafer 
characterized  by  the  same  physical  problem.  This 
misclassification  is  much  less  harmful  because  in  the 
process  diagnosis  it  is  possible  to  merge  together  class 
whose reference pattern are similar. For this reason a good 
classification have to produce homogeneous clusters, each 
of which made of wafers belonging to the same real class. 
A  simply  way  to  visualize  the  number  and  type  of 
misclassified items is given by the scatter matrix [3]. In 
Table 2 the scatter matrix of a 4 cluster classification of 
100 items that belongs to 3 real classes is reported. It can 
be  observed  that  only  clusters  #2  and  #4  are 
homogeneous. Clusters #1 and #3 present respectively 4 
and 2 elements that are not homogenous. These elements 
are called "misplaced". For each identified cluster  j it is 
possible obtain the number of misplaced elements as
( ) ( ) ( )∑
=
−=
cN
i i
jiTjiTjM
~
1
,max,
where with T(i,j) denotes the scatter matrix and Ñc is the 
number  of  real  classes.  To  measure  the  degree  of 
homogeneity  of  the  classification  we  introduce  a  new 
index ,called F-index, defined as
( )∑
=
−
=
cN
icw
c jM
NN
N
F
1
~
1~
It can be shown that the F-Index ranges from 0 (optimal 
clustering) up to 1 (worst clustering).
N1 N2 N3 N4
Ñ1 16 0 14 0
Ñ2 0 45 0 0
Ñ3 4 0 2 19
Table 2. Scatter matrix of a four class clustering of 100 data 
belonging to three real classes
CLUSTERING ALGORITHMS
In this work are compared three well-known unsupervised 
clustering  algorithms:  SOM,  Neural  Gas,  K-Mean.  This 
neural  network  techniques  generate  a  cluster  for  each 
neuron. In each neuron c is stored a Nd-dimension vector 
pc that represent the centroid of the cluster c.
K-means
This algorithm is the simplest  and fastest  one. A single 
iteration is composed by two steps. In the first one each 
data is associated with the nearest  cluster center.  In  the 
second  step  the  centers  are  updated  by  computing  the 
barycenter  (centroid)  of  each cluster.  The two steps are 
repeated until the centers do not change. The parameters 
of  this algorithm are  the number  of  clusters  Nc and the 
initial values of the cluster centers  W0. It  is important to 
notice that if two cluster centers have been initialized with 
the same center, the algorithm can encounter difficulties. 
SOM
In  a  SOM network the neurons are  organized on a  r×c 
grid. At each iteration t a single wafer Xw (a binary vector) 
is  presented  to  the  network  and  winner  neuron  V is 
defined as
cw
c
pXV −= minarg
where pc is the characteristic pattern of the c-th class, that 
is a real vector whose entries belongs to [0,1]. Then all the 
neurons are updated as follows
( )( ) ( )cwcc pXVcthpp −⋅+= Γ ,,δ      (1)
2
where δ(c,V) is the distance on the grid between neurons c 
and  V and  hΓ(.,.) is  a  suitable  function  monotonic 
decreasing in both its arguments. An interesting feature of 
the SOM is that similar reference patterns are put close to 
each  other  on  the  neuron  grid.  The function  hΓ  in  (1) 
depends on a parameter set Γ which include the following 
parameters [4]: initial ηi and final ηf learning rate, initial σi 
and  final  σf  effective  width  of  the  topological 
neighbourhood and total number of iterations tf.
In [1] the authors proposed a choice of Γ for the clustering 
of  EWS maps.  It  was  observed  that  the  the  first  4 
parameters affected the speed of learning [5] while the last 
has  to  do with the overall  learning  time.  The first  four 
parameter were fixed to general purpose values [4] and a 
fine  tuning  of  the  parameter  tf.  was  performed.  The 
proposed set is reported in Table 3.
ηi ηf σi σf λi λf tf
Γ .5 .005 3 .1 - - 80 Nw
Δ .5 .005 - - 10 .1 50 Nw
Table 3.Parameter Setting for SOM and Neural Gas
Neural Gas
The Neural Gas is quite similar to the SOM algorithm. The 
main  difference  is  due  to  the  fact  that  the  neurons 
topology (e.g. the neuron grid in the SOM) is not fixed. At 
each  iteration  the  neurons  are  ordered  according  to  the 
distance  from  the  proposed  input  (wafer).  The  nearest 
neuron  has  order  0 and  the  last  has  order  Nc-1.  The 
training update of the neurons is done according to
( )( ) ( )cwcc pXcltgpp −⋅+= ∆ ,      (2)
where l(c) is the order of neuron c and gΔ(.,.)is a function 
monotonic decreasing in both its arguments. The function 
gΔ in (2) depends on a parameter set Δ [4]:
{ }ffifi t,,,, λληη=∆
where λi and λf affect the amount of information to give to 
the "far" neurons.  The consideration made for the  SOM 
method  still  hold  for  the  Neural  Gas.  In  particular  the 
same  tuning  procedure  can  be  adopted:  the  first  4 
parameter of Δ ware fixed to general purpose value and a 
fine tuning was made on  tf.  Several  classifications were 
performed on the benchmarks described in Section 2 and 
the  F-index  was  used  to  assess  the  performance.  The 
optimal parameter values are reported in Table 3. 
Expectation Maximization (EM)
Given  the  observations  X and  a  probabilistic  model 
characterized by a parameter vector θ, the likelihood L is 
given by L=P(X|θ). According to our probabilistic model 
the parameters  are the  Nc probability vectors  pc and the 
data  are  the  Nw binary  vectors Xw.  The  maximum 
likelihood estimate  θML is  the  vector  that  maximize  the 
likelihood.  The  EM  is  just  an  efficient  algorithm  to 
perform such a maximization in classification problem. A 
notable  property  of  the method is  that  it  guarantees  an 
increase of the likelihood at each iteration. 
SOM K-mean Gas EM
Random .02816 .04262 .03958 .20953
Centre .02931 .04036 .04308 .19346
Prototypes .02871 .07915 .03789 .61586
Data mean .02929 .08779 .04332 .21315
Min-Max .02943 .08182 .04310 .60589
Table 4. Value of the F-Index in 60 experiments using different 
initialization techniques
COMPARISON
All  the  algorithm  considered  have  two  types  of 
parameters:  the  number  of  clusters  Nc and  the  initial 
values of the references patterns of the classes.
The number  of  clusters  is  not  a  critical  choice  for  this 
application.  In  fact  as  already  observed,  in  the 
classification  of  EWS maps  the  main  target  is  to  find 
homogenous clusters. This is possible only if the number 
of identified classes  Nc is greater than the true one  Ñc. If 
we have some a priori knowledge on  Ñc it  makes sense 
overestimate Nc. In the benchmark, we let Nc=12.
Conversely,  the  initial  reference  pattern  may  affect  the 
performance of the classifier. In order to choose the best 
methods, different initializations were considered [6]:
Random In  this  initialization  the  patterns  are  chosen 
randomly from the valid set.
Center This initialization requires  that the valid starting 
points define a limited region; then the technique chooses 
the center of the validity set as starting point. In order to 
obtain different  values  for  each  neuron  a  small  random 
perturbation is added.
Random Prototypes Nc randomly chosen data are used as 
initialization.
Data Mean This initialization chooses as starting point the 
mean of the data set. In order to obtain different values for 
each neuron a small random perturbation is added.
Min Max This techniques use an algorithm to select  Nc 
well distanced data as starting point.
For  each  benchmark  60  different  initialization  were 
generated for each technique. For each classification the 
F-Index was evaluated. In Table 4 the average over the 12 
3
benchmarks  of  the  median  of  the  F-index  over  the  60 
experiments are reported. 
If  the  bold values  in  Table  4 are  considered,  it  can  be 
notice that SOM gives the best performances. 
MEASURING CLASSIFICATION PERFORMANCES 
ON REAL DATA 
The evaluation of the F-Index requires the knowledge of 
the true classification, that is obviously not available in a 
real problem. To assess the performance of a classification 
some other clustering index must be used. Generally these 
indexes  evaluate  the  within-class  and  the  between-class 
distances.  The  first  factor  is  an  index  of  the  wafer 
closeness  to the reference pattern while the second one 
evaluates  the  distance  between  the  clusters.  The  most 
widespread ones are [2]: Davies-Bouldin (DB), Dunn (D), 
Calinski Harabasz (CH) and I indexes. All of them regard 
a classification with compact and well separated cluster as 
a  good  one.  However,  for  our  specific  application  the 
distance  between  classes  is  not  the  main  concern, 
especially if  two or more real  classes  are close to each 
other.  A  further  index  is  the  likelihood  introduced  in 
Section 3. For computational reason we use the "average 
device likelihood" (AL) defined as:
wc NN LAL =
The  goal  of  our  study  is  to  find  the  index  that  best 
approximates the F-Index. For this purpose, from each of 
the 12 benchmarks 30 different clusterings were randomly 
created.  For  each  classification  all  the  indexes  was 
evaluated.  In  Figure 2 some significant  scatter  plots are 
reported.  Then  for  each  index  and  benchmark  the 
correlation coefficient was computed. Table 5 shows the 
average correlation coefficient with the F-Index. It can be 
noticed  that  the  AL and  CH indexes  gives  the  best 
correlation.
DB CH D I AL
.9885 -.9912 -.5649 .8939 -.9914
Table 5. Correlation Coefficient between F-Index and other 
clustering indexes.
CONCLUSION
In  this  work  a  comparison  was  made  between  some 
clustering techniques applied to EWS maps classification: 
k-means, SOM, Neural gas and EM. The last method and 
the  simulated  benchmark  are  based  on  a  simple 
probabilistic model. In order to evaluate the performances 
of  a  classification,  a  new  index,  called  F-Index,  was 
introduced.  This  index  was  created  considering  the 
possible  use  for  fault  detection  in  a  semiconductor 
manufacturing  environment.  The  classifiers  were  tested 
simulating different numbers of devices and wafers. From 
this  analysis  it  turns  out  that  the  SOM  is  most  suitable 
algorithms. The last result  presented in this work is the 
choice of a performance index to be used the classification 
of  real  data.  The  objective  was  to  find  an  index  that 
correlates well with the F-Index, which is proportional to 
the  number  of  misclassified  wafer  but  requires  the 
knowledge of the correct classification. It turns out to be 
the Average Likelihood and the CH Index.
Figure 2. Comparison of indexes measuring the classification 
goodness. The AL and the CH  index correlate well with the F-
index that is proportional to the number of misclassified items
REFERENCES
[1] F DiPalma, G DeNicolao, E Pasquinetti, G Miraglia, F 
Piccinini,  “Unsupervised  spatial  pattern  classification  of 
electrical  failures  in  semiconductor  manufacturing, 
Pattern Recognition Letters, 2005 (to appear).
[2]  U.  Maulik  and  S.  Bandyopadyay,  “Performance 
eavaluation  of  some  clustering  algorithms  and  validity 
indices,”  IEEE Trans. Pattern Anal. Machine Intell., vol. 
24, no. 12, pp. 1650–1654, 2002.
[3] A. D. Gordon, On Classification. London: Chapman & 
Hall/CRC, 1999.
[4] B. Fritzke, Some competitive learning methods, 1997, 
Ruhr-Universität  Bochum.  ftp://ftp.neuroinformatik.ruhr-
uni-bochum.de/pub/software/NN/DemoGNG/sclm.ps.gz
[5]  S.  Haykin,  Neural  Networks,  a  comprehensive  
foundation.  New  York:  MacMillan  College  Publishing 
Company, 1994.
[6]  A.  Juan,  J.  Garc´ýa-Hern´andez,  and E.  Vidal,  “Em 
initialisation for bernoulli mixture learning.” in SSPR/SPR 
workshop, 2004, pp. 635–643.
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Unsupervised algorithms for the automatic classification of EWS maps: a comparison

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Unsupervised algorithms for the automatic classification of EWS maps: a comparison

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