=Paper=
{{Paper
|id=Vol-3018/Paper_4
|storemode=property
|title=Models and Evolutionary Methods for Objects and Systems Clustering
|pdfUrl=https://ceur-ws.org/Vol-3018/Paper_4.pdf
|volume=Vol-3018
|authors=Maryna Antonevych,Nataliia Tmienova,Vitaliy Snytyuk
|dblpUrl=https://dblp.org/rec/conf/intsol/AntonevychTS21
}}
==Models and Evolutionary Methods for Objects and Systems Clustering==
https://ceur-ws.org/Vol-3018/Paper_4.pdf
Models and Evolutionary Methods for Objects and Systems
Clustering
Maryna Antonevych, Nataliia Tmienova and Vitaliy Snytyuk
Taras Shevchenko National University of Kyiv, B. Havrylyshyna Str., 24, Kyiv, 04116, Ukraine
Abstract
As we know, clustering refers to the problems of Data Mining. The clustering problem is
usually solved without a supervisor with using the features of the object or system. It is
necessary to carry out clustering in various fields: agriculture, medicine, biology, etc. In this
paper propose to carry out clustering using two objective functions and constructed
evolutionary population methods based on genetic algorithms, evolutionary strategies and the
method of deformed stars.
New elements of such methods for calculation of objective function are developed,
advantages of the population approach before known classical methods of clustering are
shown. A comparative analysis of the proposed clustering methods effectiveness using a
known sample of iris data and a sample of randomly generated data is fulfilled. The
peculiarities of each proposed method application, its advantages and disadvantages are
determined.
Keywords 1
Objects, systems, clustering, models, evolutionary methods
1. Introduction
The diversity of the modern world, the need and desire to know it are the most important reasons
for the systematization, clustering and classification of objects and systems. The dynamics of the
amount of information and the limitations of human perception require the creation of data filtering
systems, their organization, determining the importance and so on. Obviously, these factors
emphasize the importance of developing and using modern clustering methods based on new data
technologies.
The globalization of the processes of the modern world, the various challenges facing humanity,
make us think about its future. The famous writer John Galsworthy wrote: "If you do not think about
the future, you may not have it." However, thinking about the future is not enough, we need to predict
it and use intelligent decision-making methods to reduce future risks.
It is known that effective prediction is based on the results of solving the problem of
identification, both structural and parametric. Today, most often, the basis of prediction methods are
time series, or the dependence of unknown characteristics on known factors, one of which may be
time. Prediction using time series is based on the analysis of retrospective data and the construction of
autoregressive models, usually linear. Otherwise, there are multifactorial dependencies, the values of
the factors of which are in a certain area, taking into account the possible values.
II International Scientific Symposium «Intelligent Solutions» IntSol-2021, September 28–30, 2021, Kyiv-Uzhhorod, Ukraine
EMAIL: marina.antonevich@gmail.com (A. 1); tmyenovox@gmail.com (A. 2); snytyuk@knu.ua (A. 3)
ORCID: 0000-0003-3640-7630 (A. 1); 0000-0003-1088-9547 (A. 2); 0000-0002-9954-8767 (A. 3)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
37
It is important to note that the economy of the modern world is unstable, the processes in it are
heterogeneous and non-stationary. The use of data from the whole possible spectrum for their
identification is incorrect, as their logical-dynamic nature can lead to averaging and displacement of
results. That is why the output data should be divided into clusters that correspond to homogeneous
processes and allow adequate identification and high-precision prediction.
2. Clustering Problem and Analysis of Methods its Solving
The urgency of solving the clustering problem has led to the development of a large number of
clustering methods. Since this problem has an optimization character, the No Free Lunch Theorem
(NFL) is applicable to it [1]. The meaning of this theorem is that there is no single method that can
best find a solution to any optimization problem. That is why both the development of clustering
methods and the study of their effectiveness continue.
Historically, hierarchical methods were developed first. They were divided into agglomerative
and divisive methods of clustering [2,3]. For the first methods, clusters were formed by merging
smaller groups into larger ones, for the second, on the contrary, clusters were formed by dividing
larger groups into smaller ones. The class of similar methods includes dendrograms, as well as
decision trees. Typical representatives of the agglomerative paradigm are Ward's method, single
linkage, pair-group method using arithmetic mean and complete linkage, and the pair-group method
using centroid average.
The next large group of clustering methods includes algorithms of the Forel family of the
Novosibirsk school of data analysis [4], the method of K-means and K-medians [5], EM-algorithm
[6], and others. Their feature is the iterative nature of the search and refinement of cluster centers.
In recent decades, clustering methods based on new paradigms, usually inspired by the structure and
functioning of biological and social systems have been developed. In this case, the affiliation of
objects to clusters could be ambiguous, with a certain degree of correspondence, the object could
belong to a particular cluster. This is the method of C-means. The method of C-means is a method of
fuzzy clustering, which is based on the idea of calculating subjective conclusions [7]. Much of the
clustering methods were developed based on the use of neural networks. In the foreground here is the
Kohonen neural network with the implementation of the principles of WTA (winner take all) and
unambiguous assignment of the object to a particular cluster and SoftMax with the assignment of the
object to several clusters with certain probabilities [8].
Another approach to developing clustering methods is to use technologies that imitate elements
of natural evolution. Such technologies are genetic algorithms, evolutionary strategies, swarm
algorithms, imitation of metal annealing, and others. Like the above methods, they belong to the Soft
Computing paradigm [9,10]. Evolutionary methods have a certain feature in comparison with other
methods: under certain conditions with a certain probability, they guarantee finding the global
optimum of the objective function. That is why the development and study of the effectiveness of
such methods is an urgent task.
Note that in the article we did not aim to study methods based on other hypotheses about the
shape of clusters, their functional nature, etc. [11-13]. The developed methods are designed to solve
the classical problem of clustering, where objects are points in space of a certain dimension, because
the solutions of such a problem are most often used by decision makers in solving practical problems
of today [14].
When solving the clustering problem, the proposed metric which determines the distance
between object plays an important role, s. It is necessary to consider what kind of similarity is being
considered. In addition to distance, it can be an existing analytical relationship between objects or a
form of their condensation. The problems of studying possible metrics were studied in [15-18].
3. Clustering Method on the Base of Evolutionary Strategy
Suppose that objects or systems of research are characterized by a certain range of properties.
Then
38
each object can be represented as a point in the n -dimensional space of the following properties, that
is:
Xi ( xi1 , xi 2 ,..., xin ), i 1, m,
where n is the number of properties, m is the number of objects.
Data about objects are in the matrix
x11 x12 ... x1n
x21 x22 ... x2 n
X
... ... ... ...
xm1 xm 2 ... xmn .
It is necessary, based on the data of the matrix X , to divide the set of studied objects into
clusters. As a rule, the number of clusters is known a priori, but there are problems where the optimal
number of clusters also needs to be determined.
Consider the first problem and assume that the number of clusters is K , and K m . At the first
stage we normalize the elements of the matrix X and get the matrix X ' {xij' }, i 1, m, j 1, n , its
elements are
xij min xij
x
' i
, xij [0,1].
max xij min xij
ij
i i
As a result of normalization, the points from the matrix X will lie in a single hypercube
[0,1]n . Remind that the distance between two points in n -dimensional space is equal to
n
dij d ( X i , X j ) ( ( xik x jk ) 2 )1/ 2 , i, j 1, m .
k 1
We write the objective function for the problem of clustering the objects described above:
K
F (C ) d (Ci , X j ) , (1)
i 1 j
where C is the set of point-centers of the clusters, Ci is the point-center of the i -th cluster, j is the
index that indicates the affiliation of X j to the i -th cluster. There are problems in which the essential
requirement is that the centers of the clusters must be as far apart as possible. Therefore, in this case,
function (1) is modified to the form:
K K K
b
F (C ) ( d (Ci , X j )) / ( d (Ci , C j )) (2)
i 1 j i 1 j i
or
K K K
F (C ) ( d (Ci , X j )) ( d (Ci , C j )). (3)
i 1 j i 1 j i
Let us solve the problem of finding cluster centers
C * arg min F (c) . (4)
c[0,1]n
To do this, we use an evolutionary strategy ( ) with an elite selection of elements of the new
population, where is the number of potential parental solutions, is the number of potential
descendant solutions. The authors of the evolutionary strategy consider that the necessary condition to
ensure the diversity of the population of solutions is 7 .
Step 1. Initialization of evolutionary strategy parameters. t 0 .
Step 2. Generation of a population of potential solutions of problem (1-4),
C t {Cijtl }, i 1, k , j 1, n, l 1, . Thus, the number of potential solutions is equal (in most cases
10-30).
Step 3. Build matrices of distances from the centers of clusters to objects:
39
d11tl d12tl ... d1tlm
tl tl
... d 2tlm
Ae 21
t d d 22
,
... ... ... ...
tl tl
c d k1 d ktl2 ... d km
where l 1, , dij d (Ci , X j ) , Ci is the center of the i -th cluster in the l -th potential solution.
tl tl tl
Step 4. Find the value of the analog of the objective function (1) for each of the potential solutions,
m
F ( Aet ) min dijtl , l 1, . (5)
i
j 1
Thus, the value of the objective function is matched to each potential solution.
Step 5. Generate a population of potential descendant solutions as follows. For every l -th potential
solution:
Perform times.
For each i -th center of the cluster:
Generate a random number r {1,2,..., n} and modify the r -th coordinate:
2 2
Cirtl Cirtl ( N (0, )) , where ( N (0, )) is a normally distributed random number
9 9
2
with zero mathematical expectation and variance .
9
If Cir 0 , then Cir 1 Cir .
tl tl tl
If Cir 1 , then Cir Cir 1.
tl tl tl
This means that the descendant solution will differ from the parent solution by one
coordinate. Write the descendant solution in the intermediate population C P .
Step 6. Calculate function (4) for all descendant solutions from C P .
Step 7. Combine C t C p and arrange potential solutions in descending order of the objective
function (4).
Step 8. t t 1 . Build a population C t from the best population solutions C t 1 C p .
Step 9. If the stop condition is not carry out, go to step 3.
Step 10. Output of the result (the centers of clusters).
As is known, evolutionary strategy is a method of global optimization, implemented, as a rule, by
a population algorithm. Unlike other evolutionary modeling algorithms, the evolutionary strategy does
not use binary coding or other types of coding. At the beginning of the algorithm, a population of
potential solutions (cluster centers) is generated.
If there are three clusters, then we generate, for example, 20 triplets of cluster centers. For each
such triple, we find the points that belong to the corresponding cluster and calculate the values of the
objective function. Then we store these triplets and the values of the function in the intermediate
population. Next, for each triple, we generate triplets-offsprings using the normal distribution on the
principle that the point of the initial triangle is subject to modification, and the modified points form
new triplets.
According to the authors of the evolutionary strategy [10], to ensure population diversity and the
optimal number of computational procedures, the number of solutions-offsprings must refer to the
number of solutions-parents at least 7 to 1.
Thus, for example, for 20 initial triplets should be at least than 140 modified triplets-offsprings.
For these 140 triplets, the values of the objective function are also calculated and the corresponding
data are also entered into the intermediate population. Thus, in the intermediate population there are
160 triplets and values of the objective function. Arrange these 160 triplets according to the values of
the objective function. The first 20 triplets will be written in another intermediate population and will
be considered as 60 separate points.
Next, we randomly form 20 triplets and the computational process is iteratively repeated until the
stop criterion is met. Note that in the proposed method we use the well-known 3-sigma rule, which
40
determines the value of the standard deviation of the points-offsprings from the parent points, which
significantly affects the depth of the study neighborhood the parent solution and the rate of
convergence of the algorithm.
4. Clustering on the Base of Genetic Algorithm
Genetic algorithm as well as evolutionary strategy under certain conditions is a method of global
optimization. That is why its elements can be used to solve the problem of clustering. Consider the
appropriate method. Assume that the points to be broken lie in a rectangular hyperparallelepiped of n
-dimensional space. Find the rationing, all points will be in a single hypercube [0,1]n . The
coordinates of the points form a matrix X (see above), their number is m . Assume that the number
of clusters is K . The solution of problem (1) - (4) will be the vector of cluster centers
C* (C1* , C2* ,..., CK* ) . The search for the vector C * is organized according to the following algorithm.
Step 1. t 0 (iteration number). Initialization of method parameters n , m , K .
Step 2. Generate uniform distributed in [0,1]n points (potential centers of clusters)
C (C1 , C2 ,..., Cl ), K l m ,
C j (C1 j , C2 j ,..., Cn j ), j 1, l .
Step 3. From the elements of the matrix C form matrix
Ci11 Ci12 Ci1k
2
Ci Ci22 Ci2k
Pt 1 ,
Ciq Ciq2 q
Cik
1
where q is the number of K -gons, and none of the K -gons coincides with the other, that is
i j ! v, w : Civj Ciwj .
Step 4. For each K -gon find the distance from it to each point to be assigned to the cluster, that is
calculate the distance
d (Civj , xl ) d ((Civj1 , Civj2 , , Civjn ),( xl1 , xl2 , xln )) j 1, k , l 1, m .
Fq 0 .
Крок 5. For each object to be clustered, we find the center of the cluster, the distance to which is the
smallest:
Сivj min d (Civj , xl ) ,
*
j
Fq Fq d (Civj , xl ) .
*
Step 6. Arrange K -gons by the value of the function Fq in ascending order. The vertices of the best
K -gons are entered in the intermediate population Gt .
Step 7. Next, we perform the usual crossover operations on them by panmixing, inbreeding or
outbreeding.
Step 8. Perform a mutation operation.
Step 9. If the stop condition is not met, then t t 1 and go to step 3. Otherwise, display the centers
of the clusters and calculate to which cluster the objects belong.
Stop criteria can be one of the following conditions:
1. Achieving a certain predetermined number of iterations.
2. The increment of the objective function on adjacent iterations does not exceed a small
predetermined number.
41
When applying the elements of the genetic algorithm, it is necessary to take into account the fact
that its convergence to the global optimum is not guaranteed, not least because, in the general case,
the condition of continuity in the transition from integers to binary numbers is not met. In particular,
for example, the numbers 15 and 16 differ by one, and their binary counterparts 01111 and 10000
differ as much as possible. In order to ensure the continuity of calculations, it is proposed to use the
Gray code. Its advantage is the ease of obtaining from the usual binary representation and ensuring the
continuity of the computational process.
5. Clustering on the Base of Method of Deformed Stars. Type 1
Another evolutionary method that can be used to solve the clustering problem is the method of
deformed stars, which was previously proposed by the author in [19, 20]. In this paper, the simplest
case was considered when the solution space was one or two-dimensional, and the stars were in the
form of a segment or a triangle. Further, the method was improved [19]; in the two-dimensional case,
a generalization was obtained for quadrangles and pentagons. And, finally, in [20], a universal method
of deformed stars was obtained for solving the global optimization problem in the n-dimensional case.
The main ideas of the method are that the analysis of the set of potential solutions provides more
information about finding the global optimum of a certain functional dependence, and various
deformations of polygons that form potential solutions allow exploring the area of probable global
optimum. Similar ideas can be used to solve the clustering problem, because it has an objective
function, the minimum of which must be found, and objects that are points in n -dimensional space.
The illustration of the method of deformed stars for the clustering problem is given in Fig.1 and
Fig.2. The first figure shows two-point stars that can rotate, stretch, or compress in two-dimensional
space, thus approaching the real centers of the two clusters. If it is assumed that there will be three
clusters, then the stars will be triangles. The transformations over them will be shown below.
Figure 1: Two-point stars Figure 2: Three-point stars
Remind that the objects to be clustered are the points in the hypercube [0,1]n in the n -
dimensional space. Then the steps of the method will be as follows:
Step 1. Determine the size of the sample population of potential solutions (cluster centers) m .
Usually, m{20,21,...50} . t 0 .
1 2 m
Step 2. Generate a population of potential solutions Pt {x , x ..., x }, X i {xi1 , xi 2 ,..., xin }, i 1, m .
Step 3. Randomly form k triangles whose vertices are potential solutions. Note that one vertex can
belong to several triangles, k m .
Step 4. For each j -triangle:
42
Step 4.1. Find the value of the objective function Fj .
Step 4.2. For each l -th vertex ( l 1,3 ) we rotate the l -th vertex, form new points:
xjl xl cos l xl sin l ,
xjl xl sin l xl cos l ,
xjl xl , ,
Step 4.3. For a triangle composed of new points, we find the value of the objective function G j .
Step 4.4. For each l -th vertex of the initial triangle we play a uniformly distributed number
(0,1) . Also, let us play a number {1,2,..., n} and a random number {1;1} . The new
vertex will be as follows:
l
x ( xl1 , xl2 ,..., xl , xl 1 ,..., xln ) .
Step 4.5. For a triangle formed from new vertices, we find the value of the objective function H j .
Step 5. t t 1 . The points of the triangle to which corresponds max{Fj , G j , H j } are entered in
the population Pt . If j n , then j j 1 , and go to step 4.
Step 6. If the stop condition is not carry out, go to step 3.
Step 7. The end of the algorithm.
The peculiarity of this algorithm is that it is not necessary to look for the centers of polygons in
n-dimensional spaces, in contrast to the classical algorithm MODS [16]. For simplification, the
rotations of the points are made relative to the coordinates center in a randomly selected plane. The
operations of stretching and compressing the vertices of the polygon to simplify the algorithm are also
performed on one randomly determined spatial coordinate. The proposed algorithm is a global
optimization algorithm, because in this case there is a probabilistic convergence with the number of
iterations, which goes to infinity. A necessary condition for such convergence is the elite selection of
elements of the population of potential solutions-offsprings.
6. Clustering on the Base of Method of Deformed Stars. Type 2
The method of deformed stars can be a basic algorithm for solving a wide range of optimization
problems, one of which is clustering. In addition, the method of deformed stars is a parametric method
that allows wide variational possibilities for creating new elements of the method and their
optimization. In particular, in the proposed clustering method, triangles whose vertices are the centers
of the clusters are randomly generated. For the n -dimensional case, n -triangles are considered. The
operation of mutation is to create descendant triangles, the vertices of which are the midpoints of the
segments that connect the nearest points of the three triangles. These points will form an intermediate
population. The population of descendants is formed from the points of the initial population and the
points of the intermediate population, but those that correspond to the triangles with the best values of
the objective function.
Let us all preconditions of the method application are fulfilled. Steps 1, 2, 3 remain the same.
Step 4. For each triangle we find the values of the objective function: F1 , F2 ,..., Fk .
Step 5. Perform k 1 times:
Step 5.1. According to the principle of tournament selection (in proportion to the values
F1 , F2 ,..., Fk ) to determine randomly i, j {1,2,..., k}, i j .
Step 5.2. Mark the vertices of j -th triangle Aj , B j , C j .
Step 5.3. Find the vertex closest to Ai , that is, the vertex that corresponds
min{d ( Ai , Aj ), d ( Ai , B j ), d ( Ai , C j )} . Let us mark it as D j Aj B j C j .
Step 5.4. Connect by the segments points Ai and D j , and find its middle:
ai1 d j1 ai2 d j2 ain d jn
Al ( , ,..., ) , where Ai (ai , ai ,..., ai ) and D j (d j , d j ,..., d j ) .
1 2 n 1 2 n
2 2 2
43
Step 5.5. Find the vertex closest to Bi , that is, the vertex that corresponds
min{d ( Bi , Aj ), d ( Bi , B j ), d ( Bi , C j )} , provided that this vertex is not a vertex D j . Let it be
E j Aj B j C j , E j D j . Similarly, we find the middle of the segment that connects the
points Bi and E j , let it be a point Bl .
Step 5.6. Two points left: Ci and G j . The middle of the corresponding segment will be Cl .
Step 5.7. Points Al , Bl , Cl are recorded in the population of the new generation.
Step 6. Perform the generation of random three points ( Ak , Bk , Ck ) in [0,1]n . Mutation.
Step 7. t t 1 . Pt ( A1 , B1 , C1 ,..., Ak , Bk , Ck ) .
Step 8. If the stop condition is not carry out, go to step 3.
Step 9. The end of the algorithm.
The difference between the second method of clustering and the previous methods is that only
one modification of the method of deformed stars is used. Such a modification is a special type of
mutation operation. Mutation allows to simultaneously avoid hitting the local optimum and more
deeply explore the environment of finding a potential optimum. This algorithm is easier to implement,
its execution time is relatively short.
7. Analysis of experimental results
The proposed methods belong to the class of clustering methods based on the evolutionary
paradigm. Its main idea is to gradual approximation the desired solution based on the development
and movement of the population of potential solutions. Population development under certain
conditions provides a variety of search and depth of research in a particular area. This search is slower
than in other methods, but allows you to work with different, not necessarily differentiable objective
functions, which will allow you to generalize them later to determine clusters of objects based on
metrics other than Euclidean.
We will conduct experiments and test the effectiveness of the proposed methods. To do this,
consider a well-known sample of data on irises, which contains fields such as Petal Length and Widht,
Sepal Length and Width, Classes Types. To illustrate the operation of clustering methods without
general restrictions, we use only the data of the fields Petal Length and Widht. Another sample was
randomly generated based on a uniform distribution. The number of elements in both samples is 150
points. It is assumed that these data represent three clusters.
The experiments were performed using the following clustering methods: K-means, Forel, CGA
(clustering with use of the genetic algorithms elements), CES (clustering with use of the evolutionary
strategy elements), and CMODS (clustering with use of the method of deformed stars elements). The
algorithm stop criterion is a fixed number of iterations. The results of the calculations are recorded in
Table 1 and Table 2. Data of Table 1 correspond to the objective function (1), Data of Table 2
correspond to the objective function (3).
Table 1
Test results for the objective function of type (1)
Random points Irises
300 iter 1000 iter 2000 iter 1000 iter 2000 iter
K-means 34.77 32.83 34.2 29.44 29.43
Forel 46.02 47.17 44.6 33.81 33.33
CGA 35.92 35.75 34.72 30.85 31.4
CES 34.83 33.34 34.24 35.66 34.23
CMODS 34.49 32.52 33.92 29.36 29.32
The first experiment concerned randomly generated points (Fig. 3). The Forel method showed
consistently the worst results. The best results are shown by CMODS, and these results are on average
better by 0.86% than in the method of K-means, the results of which are in the second place. The
number of iterations has a negligible effect on the accuracy of clustering, except for the CGA method,
44
where the value of the objective function decreases monotonically with the increasing number of
iterations. For the Petal Length and Widht iris clustering experiment, there is a reverse trend for CGA.
The results of the CMODS method remain the best.
In another experiment (objective function (3)) it was necessary to minimize intracluster distances
and maximize distances between cluster centers. For random points, as the number of iterations
increases, the value of the objective function increases. The CMODS method demonstrates the best
accuracy again. Methods based on other population algorithms show good results, but they are worse
than the results of the K-mean method. It is worth noting that for problems with higher data
dimensions, the results of evolutionary methods will show better results.
Table 2
Test results for the objective function of type (2)
Random points Irises
1000 iter 2000 iter 1000 iter 2000 iter
K-means 32.11 33.34 27.11 27.11
Forel 38.71 43.12 36.39 36.39
CGA 32.49 35.07 28.89 28.98
CES 31.26 33.64 32.36 31.15
CMODS 30.92 33.07 27.06 26.79
Figure 3: Random points distribution
Figure 4: Irises data
45
Figure 5: Fitness-function for random points
Figure 6: Fitness-function for irises
8. Conclusions and prospects
The obtained results testify to the prospects of development, research, and progress of
evolutionary optimization technologies in general and their application to solve the clustering
problem. Such methods are of particular value when the objective clustering function is complex,
possibly with a polyextreme dependence, and the characteristic space of objects or systems is
multidimensional.
It is worth paying attention to the method of deformed stars, which has demonstrated its
advantages in most experiments. Such results can be explained by the fact that when searching for the
optimal location of clusters, information from a certain number of points at the same time is taken into
account. In methods based on elements of a genetic algorithm or evolutionary strategy, such
information is insufficient. In particular, in a genetic algorithm, each potential solution contains
information about two parental solutions, and in an evolutionary strategy, each solution contains
information about only one parental solution. Note that in MODS the amount of information in the
system is determined by the shape of the star, and the more vertices it has, the more information there
is. At the same time, computation time increases. That is why the qualification of the researcher plays
46
a decisive role in solving the problems of clustering using the ideas of evolutionary population
algorithms. The proposed methods require modification and can be used in clustering problems, when
the degree of reliability of objects or systems is not only the distance between them but also, for
example, functional dependence.
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