=Paper=
{{Paper
|id=Vol-2098/paper29
|storemode=property
|title=Ensembles of Clustering Algorithms for Problem
of Detection of Homogeneous Production
Batches of Semiconductor Devices
|pdfUrl=https://ceur-ws.org/Vol-2098/paper29.pdf
|volume=Vol-2098
|authors=Ivan Rozhnov,Victor Orlov,Lev Kazakovtsev
}}
==Ensembles of Clustering Algorithms for Problem
of Detection of Homogeneous Production
Batches of Semiconductor Devices==
https://ceur-ws.org/Vol-2098/paper29.pdf
Ensembles of Clustering Algorithms for Problem
of Detection of Homogeneous Production
Batches of Semiconductor Devices
Ivan Rozhnov1 , Victor Orlov1 , and Lev Kazakovtsev1,2
1
Reshetnev Siberian State University of Science and Technology,
prosp. Krasnoyarskiy Rabochiy 31, 660031, Krasnoyarsk, Russia
2
Siberian Federal University,
prosp. Svobodny 79, 660041, Krasnoyarsk, Russia
levk@bk.ru
Abstract. To complete the on-board equipment of space systems with
a highly reliable electronic component base (ECB), specialized test cen-
ters perform hundreds of tests to analyze each semiconductor device.
One of the requirements is that the shipped lot of products must be
made from a single batch of raw materials (wafers) which is not guar-
anteed if the devices are not manufactured for use in the space industry
only. To solve the problem of detecting homogeneous production batches,
various clustering algorithms are implemented on multidimensional data
of test results. In practice, it is impossible to predict in advance which
of the algorithms in each particular case will show the most adequate
results and the use of the ensemble approach is promising. Most of the
clustering algorithms for the problem of dividing the ECB mixed lot into
two homogeneous production batches show rather high accuracy. With
an increase in the number of homogeneous production batches in the
mixed lot, the accuracy decreases.
Authors propose an approach to constructing an ensemble of clustering
algorithms based on co-occurrence matrices with weight coefficients. Re-
sults of computational experiments on specially mixed lots of the ECB
show that for the such large-scale problems, the use of the ensemble
approach allows to achieve a higher adequacy of the results. Individual
algorithms can show results that exceed the ensemble’s accuracy, but
the accuracy of the ensemble is still higher than the averaged accuracy
of individual algorithms.
Keywords: Clustering algorithms · Electronic component base · Semi-
conductor devices · Ensembles of algorithms
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org
Ensembles of Clustering Algorithms 339
1 Introduction
Intensive use of big data in various areas leads to increased interest of researchers
in methods and tools for processing and analysing datasets of huge volumes and
diversity. One of the promising directions of big data analysis is the cluster anal-
ysis, which allows solving such problems as reducing the size of the initial data
set, identifying patterns, etc [14]. The goal of the automatic grouping (cluster-
ing) is detection of a natural grouping of a number of samples, points or objects.
The solution of the clustering problem is reduced to the development of an al-
gorithm or an automated system capable of detecting these natural groupings
in unmarked data.
Clustering [13] is segmentation through the allocation of certain associations
of homogeneous elements which are considered as independent objects with cer-
tain properties [3]. As a result, the clustering procedure forms ”clusters”, i.e.
groups of very similar objects [25].
A criterion for the clustering quality is some functional which depends on the
scatter of objects within the group (cluster) and the distances between them [4].
Modern methods of cluster analysis offer a wide variety of methods for revealing
heterogeneous groups of parameters. The most common of these methods is the
k-means procedure [21, 2]. Algorithms implementing this method are local op-
timization algorithms which depend on a choice of initial parameters (centroids
of clusters). At the same time, for many problems, the preferred methods of
identifying groups in data must produce reproducible results.
The on-board units of spacecrafts must be equipped with a highly reliable
electronic component base (ECB). First of all, it is necessary to prevent counter-
feit products that do not meet the reliability requirements, ensure the purchase
of ECB from authorized suppliers and passing through the 100 % input con-
trol, additional rejection tests and destructive physical analysis (DPA) of ECB.
Individual rejection tests of components are essential [24]. The shipped ECB
lots (batches) may be inhomogeneous, collected from several production batches
[23]. Therefore, the test results of the DPA of several ECB samples cannot be
extended to the entire lot (batch) of components unless we are sure that all
components of this lot are manufactured as a single production batch from a
single batch of wafers. Relatively small fluctuations in the manufacturing pro-
cess can radically affect the sensitivity to radiation and other characteristics of
the semiconductor devices.
ECB clustering is important in terms of ensuring reliability and, even more,
radiation resistance. Ionizing radiation as a physical factor of the space environ-
ment determines the period of active existence of space systems.
At present, there is a tendency to use collective methods in cluster analysis
[10, 27]. The algorithms of cluster analysis are not universal: each algorithm has
its own special field of application. In case of different types of data sets, to select
clusters, a researcher needs to apply a set of various algorithms to select the best
one. The ensemble (collective) approach allows to reduce the dependence of the
final solution on the parameters of the original algorithms and obtain a stable
solution, even in case of noise and emissions in the dataset [4].
340 I. Rozhnov et al.
2 Results of Various Standalone Clustering Algorithms
As datasets for our experiments, we used the results of non-destructive tests of
mixed production batches of the ECB performed in a specialized test center.
The composition of the mixed production batches was known in advance. These
mixed batches were completed from several obviously homogeneous batches of
the ECB:
- 140UD25AVK: 2 production batches (ECB clusters) and comparatively
small data volume (56 data vectors of dimensionality 18);
- 3OT122A: 2 batches (767 data vectors of dimensionality 10);
- 1526LE5: 6 batches (963 data vectors of dimensionality 41).
Our problem was to divide the mixed batch into homogeneous components
and analyze the quality of this division.
We used 5 common clustering algorithms [6]: k-Means [21, 22, 7, 1], k-Means-
fast [11], k-Means-kernel [9], k-Medoids [12], EM algorithm (Expectation Max-
imization) [8].
In addition to the actual form of the clustering algorithm, the result is signif-
icantly influenced by the parameters of the algorithms which can be optimized
by their values. By optimization, we mean the selection of such values of some
optimized parameter at which the maximum clustering accuracy is ensured, that
is, the best match of the result of clustering to the true partition of the mixed
batch into homogeneous batches of ECB is achieved. As an optimized parameter
in the k-Means, k-Means (fast) and k-Medoids algorithms, we used the type of
distance measure. For the k-Means (kernel) algorithm, tried to use various types
of the kernel (dot / radial kernel). For the EM algorithm, we tried to find the
optimal number of optimization steps in each iteration.
At the output of this process, we evaluate our results by the accuracy. By
accuracy, we mean the proportion of data objects assigned to the ”right” cluster.
This ”correctness” can be assessed by having a sample of marked data, for which
it is known in advance that they are assigned to a particular cluster. In this case,
our samples are combined from data from separate homogeneous batches of ECB.
The results are summarized in Table 1.
As we can see from this table, the clustering algorithms with relatively small
data volumes and small number of production batches (clusters) show rather
high accuracy. With the increase in data volumes and the number of clusters,
clustering accuracy decreases.
For clustering models, the most important parameter affecting the result is
the distance measure used. The use of special measures sometimes allows us
to adapt simple models like k-means to rather complex clustering problems. In
case of using some complex and non-standard distance measures, a sufficient
condition for the applicability of the measure of distance is the existence of
an algorithm for solving the corresponding Weber problem, i.e. the problem of
finding the center of the cluster [15, 26].
For comparative analysis, in addition to the problems of ECB batches clus-
tering, we analyzed the features of clustering algorithms and their ensembles on
the most common data sets from the UCI Machine Learning Repository [20]
Ensembles of Clustering Algorithms 341
with comparable data volumes and dimensionalities :
- Cryotherapy [18, 19] - 2 clusters (90 data vectors of dimensionality 6);
- pima-indians-diabete - 2 clusters (768 data vectors of dimensionality 8);
- ionosphere - 2 clusters (351 data vectors of dimensionality 34);
- Iris - 3 clusters (150 data vectors of dimensionality 4);
- Zoo - 7 clusters (101 data vectors of dimensionality 16).
Results of standalone algorithms were summarized in Table 2.
Due to problems concerning the behavior of the EM algorithm with com-
paratively small datasets in the multidimensional space (all objects of a small
cluster in a multidimensional space belong to the same hyperplane and the corre-
sponding probability distribution represented by its covariation matrix collapses
into the hyperplane), our realizatio of the EM algorithm did not allow to obtain
results for some particular cases (”no result” in Table 2). In particular cases,
analogous problem arose in the procedure which optimized parameters of the
other algorithms.
3 Ensembles of Clustering Algorithms
The ensemble approach is one of the most promising directions in cluster analysis
[14]. The following basic techniques for constructing an ensemble of algorithms
are commonly used [5]:
1. Finding a consensus partition, i.e. consistent partitioning with several
available solutions, optimal for some criterion;
2. Calculation of a consistent matrix of similarity/differences (co-occurrence
matrix).
When forming the final solution, an ensemble uses the results obtained by
various algorithms.
Let us consider an example of an ensemble of algorithms [14]. It is a com-
bination of separate algorithms, each of which offers its own partition, and a
hierarchical agglomerative algorithm that combines the resulting solutions with
a special mechanism.
In the first step, each algorithm splits the data into clusters using its objective
function, based on the distance metric or on the likelihood function. Then, the
accuracy and weight of the view of the algorithm in the ensemble are calculated
by the following equation:
Acci
Wi = PL (1)
i=1 Acci
where Acci is the accuracy of the ith algorithm, i.e. the ratio of the number
of correctly clustered objects to the volume of the entire sample, and L is the
number of the algorithms in our ensemble.
For each partition obtained, our algorithm compiles a preliminary binary
matrix of differences of size n×n (where n is the number of objects) to determine
whether the objects of the partition are included in the same clusters. After that,
our algorithm calculates a matched matrix of differences, each element of which
342 I. Rozhnov et al.
Table 1. Results of computational experiments with standalone clustering algorithms
on ECB production batches
Accuracy / optimized parameter value
Algorith
140UD25AVK 3OT122A 1526LE5
2 batches 2 batches 6 batches
k-Means 100.00 76.53 50.57
(Euclidean (Euclidean (Euclidean
Distance) 1 Distance) Distance)
k-Means(fast) 100.00 67.67 50.57
(Euclidean (Euclidean (Euclidean
Distance) Distance) Distance)
k-Means(kernel) 100.00 59.19 47.14
(radial kernel) (radial kernel) (radial kernel)
k-Medoids 100.00 60.63 48.60
(Euclidean (Euclidean (Euclidean
Distance) Distance) Distance)
EM 96.43 90.09 No result
(100 optimization (100 optimization
steps in each steps in each
iteration) iteration)
k-Means Optim. 100.00 76.53 63.03
(Euclidean (Euclidean (Overlap
Distance) Distance) Similarity)
k-Means(fast) Optim. 100.00 76.53 50.99
(Euclidean (Euclidean (KernelEuclidean
Distance) Distance) Distance)
k-Means(kernel) Optim. 53.57 67.67 30.22
(dot kernel) (dot kernel) (dot kernel)
k-Medoids Optim. 100.00 91.79 55.97
(Euclidean (Euclidean (Manhattan
Distance) Distance) Distance)
EM Optim. 96.43 95.44 No result
(40 optimization (95 optimization
steps) steps)
1
optimization parameter value
Ensembles of Clustering Algorithms 343
Table 2. Results of computational experiments with standalone clustering algorithms
on datasets from the UCI repository
Accuracy / optimized parameter value
Algorith
Cryotherapy Pima-indians- Ionosphere Iris Zoo
diabetes
2 batches 2 batches 2 batches 3 batches 7 batches
k-Means 56.67 66.02 71.23 89.33 75.25
(Euclidean (Euclidean (Euclidean (Euclidean (Euclidean
Distance) 1 Distance) Distance) Distance) Distance)
k-Means(fast) 56.67 66.02 71.23 89.33 75.25
(Euclidean (Euclidean (Euclidean (Euclidean (Euclidean
Distance) Distance) Distance) Distance) Distance)
k-Means(kernel) 55.56 51.17 55.56 93.33 54.46
(radial (radial (radial (radial (radial
kernel) kernel) kernel) kernel) kernel)
k-Medoids 57.78 54.43 68.09 76.67 79.21
(Euclidean (Euclidean (Euclidean (Euclidean (Euclidean
Distance) Distance) Distance) Distance) Distance)
EM 56.67 65.62 No result 96.67 No result
(100 steps) (100 steps) (100 steps)
k-Means Optim. 75.56 66.28 No result 96.67 83.17
(Camberra (Manhattan (Cosine (Manhattan
Distance) Distance) Similarity Distance)
Distance)
k-Means(fast) 75.56 66.28 No result 96.67 83.17
Optim. (Camberra (Manhattan (Cosine (Manhattan
Distance) Distance) Similarity) Distance)
k-Means(kernel) 53.33 65.10 64.10 33.33 40.59
Optim. (dot (dot (dot (dot (dot
kernel) kernel) kernel) kernel) kernel)
k-Medoids Optim. 73.33 66.02 72.36 97.33 80.20
(Camberra (Dynamic (Jaccard (Cosine (Cosine
Distance) Time Similarity Similarity Similatity
Warping Distance) Distance) Distance)
Distance)
EM Optim. 56.67 66.28 No result 96.67 No result
(1 optimi- (1 optimi- (101 optimi-
zation zation zation
step) step) steps)
1
optimization parameter value
344 I. Rozhnov et al.
is a weighted sum (using the weight of equation (1)) of the elements of the
preliminary matrices. The obtained matrix is used as input for the algorithm of
hierarchical agglomerative clustering. Then, using common techniques, such as
determining the jump in the agglomeration distance, we can choose the most
suitable cluster solution.
As mentioned above, to obtain the best partitioning into clusters, a binary
matrix of similarity/differences for each partition in the ensemble is constructed:
Hi = hhi (i, j)i
where h(i, j) = 0 if both ith and jth elements belong to the same cluster, and
1, otherwise.
The next step in composing an ensemble of clustering algorithms is to compile
a matched matrix of binary partitions.
L
X
H ∗ = hh∗ (i, j)i, h∗ (i, j) = wi hi (i, j)
i=1
where wi is the weight of the ith algorithm.
The most popular clustering algorithms often fail for certain datasets that do
not match well with the modeling assumptions [10]. Ensembles which include
approaches such as k-means that are better suited to low-dimensional spaces in
combination with other approaches designed for high-dimensional sparse spaces
(spherical k-means, Jaccard-based clustering, EM-clustering with spherical Gaus-
sian distributions [24] etc.) perform well across a wide range of data dimen-
sionality [27]. At the same time, in high-dimensional cases, the choice of the
best clustering models in not evident: sometimes, algorithms designed for high-
dimensional data fail to improve the results of the simplest models such as k-
means [24].
For constructing an ensemble (Table 3), we take three or five best algorithms
showing the highest accuracy for each specific dataset (Table 1).
For the 140UD25AVK dataset, we used k-Means, k-Means(kernel) and k-
Medoids to construct an ensemble of three best algorithms; for 3OT122A dataset,
we used EM-Optim., k-Medoids-Optim. and EM; for 1526LE5, we used k-Means-
Optim., k-Medoids-Optim. and k-Means(fast)-Optim.
Analogous results for various datasets from the UCI Repository are shown
in Table 4 and Table 5.
A fragment of calculation of ensemble results for dataset 3OT122A is given in
Table 6. In most rows, some of standalone algorithms demonstrate wrong result
and the ensemble improves this situation.
4 Conclusions
Our computational experiments show that any clustering algorithms for the
problem of dividing a batch of ECB batch into two homogeneous batches can be
used with rather high accuracy. With increase in the number of homogeneous
Ensembles of Clustering Algorithms 345
Table 3. Results of computational experiments with ensembles on ECB production
batches (accuracy)
ECB mixed production 140UD25AVK 3OT122A 1526LE5
batch / algorithm 2 batches 2 batches 6 batches
The best standalone 100 95.44 63.03
algorithm
Averaged accuracy of 3 100 92.44 56.66
best algorithms
Averaged accuracy of 5 100 86.08 54.23
best algorithms
Ensemble of 3 algorithms 100 95.04 57.01
Ensemble of 5 algorithms 100 95.44 52.54
Table 4. Algorithms for each dataset sorted by their accuracy
Dataset Cryotherapy Pima-indians- ionosphere Iris Zoo
/range 2 clusters diabetes 2 clusters 3 clusters 7 clusters
2 clusters
1 k-Means k-Means k-Medoids k-Medoids k-Means
-Optim -Optim -Optim -Optim -Optim
2 k-Means(fast) k-Means(fast) k-Means EM k-Means(fast)
-Optim -Optim -Optim
3 k-Medoids EM k-Means(fast) k-Means-Optim k-Medoids
-Optim -Optim
4 k-Medoids k-Means k-Medoids k-Means(fast) k-Medoids
-Optim
5 EM k-Means(fast) k-Means EM k-Means
(kernel)-Optim
Table 5. Results of computational experiments with ensembles on datasets from the
repository (accuracy)
Dataset/algorithm Cryotherapy pima-indians- ionosphere Iris Zoo
2 clusters diabetes 2 clusters 3 clusters 7 clusters
2 clusters
The best standalone 75.56 66.28 72.36 97.33 83.17
algorithm
Averaged accuracy of 3 74.82 66.28 71.61 96.89 82.18
best algorithms
Averaged accuracy of 5 67.78 66.18 69.40 96.80 80.20
best algorithms
Ensemble of 3 algorithms 75.56 66.28 71.23 96.71 83.17
Ensemble of 5 algorithms 75.56 65.89 68.66 96.67 81.15
346 I. Rozhnov et al.
Table 6. Comparison of results of an ensemble of 5 algorithms and standalone algo-
rithms (incorrect results of the ensemble are marked by ”*”)
No. of semicon- True EM-Optim k-Medoids EM k-Means k-Means Ensemble
ductor device batch -Optim (fast)
in the mixed batch number -Optim
1 1 1 2 1 2 1 1
2 1 1 2 1 1 1 1
3 1 1 1 1 1 1 1
4 1 1 1 1 1 1 1
5 1 1 2 1 2 2 2*
6 1 1 2 1 2 1 1
7 1 1 2 1 1 1 1
8 1 1 2 1 1 1 1
9 1 1 2 1 1 1 1
10 1 1 1 1 1 1 1
... ... ... ... ... ... ... ...
71 2 2 2 2 1 1 2
72 2 2 2 2 2 2 2
73 2 2 2 2 2 2 2
74 2 2 2 2 1 1 2
... ... ... ... ... ... ... ...
production batches in the mixed batch, the accuracy decreases. For different
data sets, the best results are demonstrated by different algorithms.
Using ensemble approach allows achieving higher accuracy in comparison
with standalone clustering algorithms. In this case, individual algorithms are able
to show results that exceed the ensemble’s accuracy, however, the accuracy of the
ensemble is still higher than the averaged accuracy of the individual algorithms.
It is also necessary for a particular problem to take into account the number of
algorithms used in the ensemble, in connection with the fact that the accuracy
of the ensemble of clustering algorithms for various data depends on the number
of algorithms in the ensemble.
In practice, the accuracy of clustering cannot be determined due to the lack
of information on the actual classes in the sample and it is impossible to predict
a priori which of the algorithms in the particular case shows the most adequate
results. Thus, usage of an ensemble approach to our problem is a promising
research direction. In particular, the application of the ensemble approach in
combination with the clustering algorithms that provide the best result within
the framework of the given clustering model [17, 16] will make it possible to
obtain results which are both more adequate and reproducible under repeated
runs of the algorithm and hence verifiable.
The last table shows that for three of five data sets, the ensembles of algo-
rithms show results that are worse than the averaged value of the individual
algorithms from which they are composed. This is typical for ensembles of both
three and five best algorithms.
Ensembles of Clustering Algorithms 347
Though for our problem of mixed production batch separation, the ensemble
approach does not show an advantage over individual algorithms in all cases, in
general, the ensemble approach allows reducing the dependence of the obtained
results on the features of using separate algorithms to a specific data set. Taking
into account that the best results for different data sets are achieved by different
algorithms, selection of some set of the best algorithms that show good results
for many problems of such class increases the reliability of the process of homo-
geneous ECB production batch separation.
Acknowledgement. Results were obtained in the framework of the state task
No. 2.5527.2017/8.9 of the Ministry of Education and Science of the Russian
Federation.
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