Stochastic behavioral models. Classification
Stochastic behavioral models are specified by a difference evolutionary equation for the probabilities of binary alternatives. The classification of stochastic behavioral models is analyzed by the limit comportment of alternatives probabilities. The main property of classification is characterized b...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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irk-123456789-1420582018-09-25T01:22:54Z Stochastic behavioral models. Classification Koroliouk, D.V. Bertotti, M.L. Koroliuk V.S. Системный анализ Stochastic behavioral models are specified by a difference evolutionary equation for the probabilities of binary alternatives. The classification of stochastic behavioral models is analyzed by the limit comportment of alternatives probabilities. The main property of classification is characterized by three types of equilibrium: attractive, repulsive, and dominant. The stochastic behavioral models are classified by using the stochastic approximation. Стохастичні моделі поведінки визначаються різницевими еволюційними рівняннями для ймовірностей бінарних альтернатив. Класифікація стохастичних моделей поведінки здійснюється по граничній поведінці альтернатив ймовірностей. Основна властивість класифікації характеризується трьома типами рівноваги: притягувальним, відштовхувальним і домінантним. Класифікація стохастичних моделей поведінки здійснюється з використанням дифузійної апроксимації. Стохастические модели поведения определяются разностными эволюционными уравнениями для вероятностей бинарных альтернатив. Классификация стохастических моделей поведения осуществляется по предельному поведению альтернатив вероятностей. Основное свойство классификации характеризуется тремя типами равновесия: притягивающим, отталкивающим и доминирующим. Классификация стохастических моделей поведения осуществляется с использованием диффузионной аппроксимации. 2016 Article Stochastic behavioral models. Classification / D.V. Koroliouk, M.L. Bertotti, V.S. Koroliuk // Кибернетика и системный анализ. — 2016. — Т. 52, № 6. — С. 60-72. — Бібліогр.: 11 назв. — англ. 0023-1274 http://dspace.nbuv.gov.ua/handle/123456789/142058 519.21 en Кибернетика и системный анализ Інститут кібернетики ім. В.М. Глушкова НАН України |
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Системный анализ Системный анализ Koroliouk, D.V. Bertotti, M.L. Koroliuk V.S. Stochastic behavioral models. Classification Кибернетика и системный анализ |
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Stochastic behavioral models are specified by a difference evolutionary equation for the probabilities of binary alternatives. The classification of stochastic behavioral models is analyzed by the limit comportment of alternatives probabilities. The main property of classification is characterized by three types of equilibrium: attractive, repulsive, and dominant. The stochastic behavioral models are classified by using the stochastic approximation. |
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Article |
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Koroliouk, D.V. Bertotti, M.L. Koroliuk V.S. |
| author_facet |
Koroliouk, D.V. Bertotti, M.L. Koroliuk V.S. |
| author_sort |
Koroliouk, D.V. |
| title |
Stochastic behavioral models. Classification |
| title_short |
Stochastic behavioral models. Classification |
| title_full |
Stochastic behavioral models. Classification |
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Stochastic behavioral models. Classification |
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Stochastic behavioral models. Classification |
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stochastic behavioral models. classification |
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Інститут кібернетики ім. В.М. Глушкова НАН України |
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2016 |
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Системный анализ |
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http://dspace.nbuv.gov.ua/handle/123456789/142058 |
| citation_txt |
Stochastic behavioral models. Classification / D.V. Koroliouk, M.L. Bertotti, V.S. Koroliuk // Кибернетика и системный анализ. — 2016. — Т. 52, № 6. — С. 60-72. — Бібліогр.: 11 назв. — англ. |
| series |
Кибернетика и системный анализ |
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AT korolioukdv stochasticbehavioralmodelsclassification AT bertottiml stochasticbehavioralmodelsclassification AT koroliukvs stochasticbehavioralmodelsclassification |
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2025-07-10T14:03:44Z |
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2025-07-10T14:03:44Z |
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1837268970927816704 |
| fulltext |
UDC 519.21
D.V. KOROLIOUK, M.L. BERTOTTI, V.S. KOROLIUK
STOCHASTIC BEHAVIORAL MODELS. CLASSIFICATION
1
Abstract. Stochastic behavioral models are specified by a difference evolutionary
equation for the probabilities of binary alternatives. The classification of stochastic
behavioral models is analyzed by the limit comportment of alternatives
probabilities. The main property of classification is characterized by three types of
equilibrium: attractive, repulsive, and dominant. The stochastic behavioral models
are classified by using the stochastic approximation.
Keywords: evolutionary behavioral process, stochastic behavioral process,
equilibrium, attractive model, repulsive model, dominant models, generator of
discrete Markov process.
INTRODUCTION
The purpose of this paper is classification and analysis of the dynamics of
discrete-time behavioral processes. The discrete time instants are also called stages.
The stochastic behavioral processes are described by normalized sums of sample
random variables with a finite number of values, which correspond to a set of possible
decisional inferences.
The dynamics of the behavioral process is determined by the regression functions
of increments. The evolutionary behavioral processes are given by a solution of
difference evolutionary equations.
A simple but very important case is the stochastic behavioral process, defined by
normalized sums of sample random variables with two possible values, for example �1.
The fundamental property of the regression function of increments, in this case, means
that the probability of sampling values are determined by two directing parameters,
one of which promotes (increases) the probability of a certain behavioral choice, and
the other inhibits (reduces) the probability of the alternatives. Such general principle of
“stimulus and deterrence” [1] is present in a wide variety of natural stochastic
processes that describe, for example, the interaction of molecules in chemical
reactions [2] or the procedure of buying and selling in the economy [3].
The important analysis task, related to the stochastic behavioral processes, is the
description of possible behavioral scenarios under unrestricted growth of the stages. In
this context, an important task is to classify the behavioral processes by asymptotic
properties of behavioral frequencies (probabilities) for increasing the number of stages.
The classification of the evolutionary behavioral process is realized by using
specific properties of regression function of increments, based on the idea of
Wright–Fisher [2, 4], where the regression function of increments has three
equilibrium points, which play an essential role in the models classification. The
classifiers, corresponding to every equilibrium, are introduced for classification of the
of stochastic behavioral processes by using stochastic approximation approach.
It should be noted that the stochastic behavioral processes considered here, in the
starting positions are similar to the stochastic learning models, studied in details in
numerous articles and monographs [5, 6]). However, the stochastic learning models are
focused primarily on the study of the asymptotic properties by an unrestricted growth
of the sample volume. In the meanwhile, the stochastic behavioral processes under
consideration are investigated under unrestricted growth of the stages by a fixed
sample volume, assuming–of course–that the sample size is large enough.
60 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6
1
One of the authors, D.V. Koroliouk, was supported by the Free University of Bolzano/Bozen Visiting
Professorship Program in 2015 which provided the basis of research on the subject of this publication.
� D.V. Koroliouk, M.L. Bertotti, V.S. Koroliuk, 2016
Stochastic behavioral models serve as a model of intelligence, driven by
“stimulation–deterrence” factors. Similarly, the property stochastic behavioral models
“gain–loss” are used in the analysis of processes in economics. The dynamics of
population genetics processes are also subject to the stochastic behavioral models
properties listed above [4, 7]. In particular, in the book [8] linear stochastic behavioral
models are also applied.
The paper contains three sections.
In section 1 there is the introduction of the evolutionary behavioral processes [9],
determined by solutions of difference evolutionary equation (proposition 2). The
increments of these processes are given by time-homogeneous regression functions.
Section 2 contains three variants of behavioral models classification [10], their
justification and interpretation by the limit of evolutionary behavioral processes. The
generic model classification is realized by using classifier functions.
In section 3 for stochastic behavioral models, the classification is established by
using a stochastic approximation approach [11] and using the classifier functions
introduced in section 2.
1. EVOLUTIONARY BEHAVIORAL PROCESS
The binary stochastic behavioral process is given by the averaged sum of sample
values
S k
N
kN n
n
N
( ) : ( )�
�
�
1
1
� , k � 0,
(1)
of independent and identically distributed by n random variables �n k( ), 1 � �n N ,
k � 0, which take binary values �1.
The binary stochastic behavioral process (1) can be represented by difference of
the frequency processes:
S k S k S k kN N N( ) ( ) ( ),� � � � 0. (2)
The frequency stochastic behavioral processes S k
N
� ( ), k � 0, are defined as
follows:
S k
N
k k
N n
n
N
� �
�
� ��( ) : ( ),
1
0
1
� , (3)
where
� �n nk I k n N k� � � � � � �( ) : ( ) , ,{ }1 1 0.
The frequency behavioral processes S k
N
� ( ), k � 0, describe the relative part of
positive and negative values of the sample terms �n k( ), 1 � �n N , k � 0.
It is obvious that the identity
S k S k k
N N
�
� �( ) ( ) 1 0
takes place.
Hence, the frequency stochastic behavioral processes (3) can be represented as
follows:
S k S k k
N N
� � � �( ) [ ( )] ,
1
2
1 0. (4)
The properties of the binary stochastic behavioral process (1) imply the
corresponding properties of the frequency stochastic behavioral processes (3), which
can present a particular interest for applications. In what follows, all the three
stochastic behavioral processes (1) and (3) will be analyzed in a parallel way.
The dynamics, by k � 0, of evolutionary behavioral processes (1) will be studied
in terms of the following conditional probabilities:
C k E k S k C kn N( ) : [ ( )| ( ) ( )] � �1 1� , 1 0� � �n N k, , (5)
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and
P k E k S k P k n N kn N�
� �
� � � � � �( ) : [ ( )| ( ) ( )], ,1 1 1 0� . (6)
It is easy to see that the conditional probabilities (5), (6) have the following
representations:
P k E S k S k P k k
N N�
� �
� � � �( ) : [ ( )| ( ) ( )] ,1 1 0, (7)
C k E S k S k C k kN N( ) : [ ( )| ( ) ( )] , � � �1 1 0. (8)
It is obvious that the conditional expectations (7), (8) satisfy the following
relation (see (2)):
C k P k P k k( ) ( ) ( ),� � � � 0.
The relations (7), (8) imply that the conditional expectations (5), (6) do not
depend on the sample size N .
The dynamics of the evolutionary behavioral processes will be studied in terms of
increments:
� �P k P k P k C k C k C k k� � � � � � � �( ) : ( ) ( ), ( ) : ( ) ( ),1 1 1 1 0. (9)
Proposition 1. The probabilities increments (9) are given by the difference
evolutionary equations:
�P k V P k P k k� � � �( ) ( ( ), ( )),1 0� , (10)
and their difference is given by the following difference evolutionary equation:
�C k V C k k( ) ( ( )), � � �1 0, (11)
where the regression functions of increments are:
V p p p p V p V p( , ) : ( ) � � � �� � , (12)
and also
V c V p p( ): ( , )� �2 . (13)
Here, by definition (see (4)):
p c� � �
1
2
1[ ] .
The linear component of the regression function of increments (11) is defined by the
real-valued directing parametersV� , which satisfy the following condition: �
�1 1V .
Remark 1. The regression function of increments (11) contains the linear term
( )V p V p � �� , which reflects the main principle of interactions “stimulation–deterrence”
in behavior processes. The additional multiplier p p � corresponds to the property of
absorption in the extreme points ( , )0 1 . Similar multipliers are present in Wright–Fisher
genetic model [4, Ch.10; 7, Ch. 12].
The linear component in (12) has equilibrium points:
V ( , )� � � �0 0 0,
defined as follows:
� �
�
� �0 00 1, � ,
and the equilibrium values ( , )� � � , determined by the following equations:
V V � � �� � �� � � �0 1, .
Hence, the equilibrium values ( , )� � � are expressed as follows:
� � �� � V V V V V� / , : .
The difference
� � �:� � � (14)
is the equilibrium of the regression function of increments (13): V ( )� � 0.
The equilibrium points ( , )� �� allow to express the linear component of the
regression function of increments (11) as follows:
V p p Vp p p( , ) ( ) � � � �� � � � . (15)
62 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6
Similarly, the regression function of increments in (12), using the relations (11)
and (14), acquires the following form:
V c V c c( ) ( )( )� � �
1
4
1 2 � . (16)
The regression function of increments representations (15), (16) allow to
formulate the difference evolutionary equations (10), (11) in the following form.
Proposition 2. The evolutional behavioral processes P k� ( ), C k( ), k � 0, are
determined by solutions of the following difference evolutionary equations:
�P k VP k P k P k� � � � � �( ) ( ) ( )( ( ) )1 � � , (17)
�C k V C k C k( ) ( ( ))( ( ) ) � � � �1
1
4
1 2 � .
(18)
Remark 2. The evolutional behavioral processes, defined by the evolutionary
equations (10), are characterized by two parameters V� , whereas the evolutionary
behavioral processes, defined by the evolutionary equations (17), (18), are
characterized by two parameters:V , � � in (17) andV , � in (18), correspondingly.
Now, it is possible to classify the evolutionary behavior processes C k( ) and
P k� ( ), k � 0, varying the parameters V , V� , � or � � .
2. CLASSIFICATION OF THE MODELS
2.1. Classification of evolutionary behavioral models. The characterization of
evolutional behavioral processes, given by the difference evolutionary equations
(17) or (18), is carried out by the properties of the regression functions of
increments (15) or (16).
Consider, for definiteness, the frequency evolutional behavioral processes P k ( ),
k � 0. The models classification can be realized in different ways.
Firstly, it is natural to characterize the different models by mean of the directing
parameters V� and V V V� � . It is possible to use the equations (10), (11).
Behavioral model classification-I
The directing parametersV ,V� ,V V V� � characterize the following behavioral
models:
�A : V V� ��0 0, attractive,
�R : V V
��0 0, repulsive,
�A : V V V� � � �0 0, attractive dominant, ,
�R : V V V
� � �0 0, repulsive dominant, ,
�A �: V V V� � �� 0 0, attractive dominant, ,�
�R �: V V V
� �� 0 0, repulsive dominant, .�
Behavioral model classification-II
The directing parameters V , � � , � � � �1 characterize the following
behavioral models:
�A : V �
�0 0 1, � attractive ,
�R : V
�0 0 1, � repulsive ,
�A : V � � �� 0 0, � � attractive dominant, ,
�R : V
� � �0 0, � � repulsive, dominant ,
�A �: V � � � �0 0, � � attractive dominant, ,�
�R �: V
� �� 0 0, � � repulsive dominant, .�
Behavioral model classification-III
The directing parametersV , � characterize the following behavioral models:
�A : V �
0 1, | |� attractive ,
�R : V
0 1, | |� repulsive ,
�A : V � �0 1, � attractive dominant, ,
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6 63
�R : V
� �0 1, � repulsive dominant, ,
�A �: V � � �0 1, � attractive dominant, ,�
�R �: V
�0 1, � repulsive dominant, .�
The classification of the evolutionary behavioral models is based on the analysis
of asymptotic comportment of evolutionary behavioral processes (7) and (8), given by
the solutions of difference evolutionary equations (17) and (18).
2.2. Justification of the classification-I and the classification-II. Without
generality limitation, only frequency evolutionary behavioral process P k ( ), k � 0, is
considered for definiteness.
The dynamics of evolutionary behavioral processes, given by the solution of
evolutionary equation (17) is determined by the regression functions of increments in
the following form:
�P k VP k P k P k � � � �( ) ( ) ( )( ( ) )1 � . (19)
The corresponding regression function of increments
V p Vp p p � � � �( ) : ( )( )1 �
has zeros at the points 0, 1, � , in which its first derivative assumes the following
values:
� � � � � � � � � � � � V V V V V V V V( ) , ( ) , ( )0 1� � � � � .
To justify the Classification-II, we start considering.
� The model � A, where the equilibrium value � of the frequency behavioral
process satisfies the inequality:
0 1
� . (20)
Suppose for definiteness an initial value of the frequency behavioral probability
P �( )0 � , with the consequence that P � �( )0 0� . Then, by (19) it is clear (by
induction) that �P k
( ) 0, k � 0. Hence, P k ( ), k � 0, is decreasing and lower
bounded random sequence (Fig. 1).
This implies that:
� � �
��
lim ( ) *
k
P k P � . (21)
So that, by the convergence (21) and by (19), the following limit equation takes place:
0 � � � � Vp p p* * *( )� . (22)
Therefore,
p �* � . (23)
In a similar way, supposing P
( )0 � , it is obtainable
lim ( ) *
k
P k p
��
� � �
and the equality (23) remains
valid.
So, the model �A has an
attractive equilibrium point 0 1
�
and lim ( )
k
P k
��
� � .
� The model �R, where the
frequency behavioral probabilities
with equilibrium � satisfy the
inequality (20).
In the case P �( )0 � , the
increments satisfy the relations:
�P k �( ) 0, k � 0. Hence, P k ( ),
k � 0, is an increasing and upper
64 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6
Fig. 1. Function V p ( ) in the model �A
Current value, p
R
eg
re
ss
io
n
fu
n
ct
io
n
,
C
V �
0 0 1; �
�
bounded random sequence. This
implies that lim ( ) *
k
P k p
��
� . Now,
the limit equation (22) ensures that
p �* 1 .
Similarly, in the case P
( )0 � ,
the convergence to zero of the random
sequence P k ( ), k � 0, is obtainable.
Obviously (Fig. 2), there are the
following relations:
P k P � � �( ) ( )0 0 0for � ,
P k P � � �( ) ( )0 0 1for � .
Thus, the frequency behavioral
probability P k ( ) increases in the
interval ( , )� 1 and decreases in the
interval ( , )0 � .
Hence, the equilibrium point �
is repulsive, i.e. there takes place the
convergence:
lim ( ) *
k
P k p
��
� �
�
�
�
�
0 0 0
1 0 1
, ( ) ;
, ( ) .
if
if
P
P
�
�
� The model � A : V � 0,
V V �� �0 . In this model, the
equilibrium point satisfies the
inequalities (Fig. 3):
� �� � �0 1, .
Accordingly, the regression func-
tion of increments satisfies V p �( ) 0
for all p �[ , ]0 1 . Consequently, the
probability of a positive alternative in-
creases and we have the following
limit (see (22)):
lim ( )
k
P k
��
�1. (24)
� The model �R : V
0,
V V �� �0 . In this model, the
equilibrium point satisfies the
inequalities: � � �� �0 1, .
The regression function of
increments satisfies the inequality
(Fig. 4):V p �( ) 0 for all p �[ , ]0 1 .
Hence, the behavior of a positive
alternative probability is described by
the limit relation (24).
� The model �A � : V � 0, V V� � �0 . As above, there is a limit:
lim ( )
k
P k
��
� 0. (25)
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6 65
Fig. 2. Function V p ( ) in the model �R
Current value, p
R
eg
re
ss
io
n
fu
n
ct
io
n
,
C
V
0 0 1; �
�
Fig. 3. Function V p ( ) in the model �A
Current value, p
R
eg
re
ss
io
n
fu
n
ct
io
n
,
C
V � � 0 1; �
�
Fig. 4. Function V p ( ) in the model �R
R
eg
re
ss
io
n
fu
n
ct
io
n
,
C
V
0 0; �
�
Current value, p
� The model �R � :V
0,V V� � �0 . Again, in a similar way as above, there is
the limit (25).
2.3. Evolutionary behavioral models interpretation. The classification of the
behavioral models presented in Section 2.2 refers to the limit behavior analysis (as the
number of stages k � �) of the increments �P k� ( ), which are solutions of the
difference equations (17).
� In the model � A (Fig. 5), the probabilities of the alternatives P k� ( ), k � 0, tend
to equilibrium values � � for any initial value P� ( )0 . The equilibriums � � are attractive.
Namely, we have
lim ( )
k
P k
��
� �� � .
� In the model �R (Fig. 6), the behavior of the alternative probabilities P k� ( ) as
k � � depends on the initial conditions. The equilibriums � � are repulsive.
Specifically, we have
lim ( ) , ( ) ;
k
P k P
�� � � ��
0 0if � lim ( ) , ( ) .
k
P k P
�� � � �� �1 0if �
� In the models �A and �R (dominant +), for any initial value P� ( )0 , the
advantage of the alternative + is verified: P k �( ) 1 as k � �, while P k� �( ) 0 as
k � �. In this case, there are two different situations.
� If V � 0 (Fig. 7), the attraction of the positive alternative probability to
equilibrium � � 1 occurs.
More specifically,
lim ( ) , lim ( )
k k
P k P k
��
��
�� �1 0. (26)
� If V
0 (Fig. 8), the repulsion of the positive alternative probability from the
equilibrium � � 0 occurs.
Again, the limits in (26) hold true.
� In the models �A � and �R � (dominant �), for any initial value P� ( )0 , the
advantage of the alternative is verified: P k� �( ) 1 as k � �, while P k �( ) 0 as
k � �. Also, in this case, there are two different situations.
� If V � 0 (Fig. 9), the attraction of the positive alternative probability to the
equilibrium � � 0 occurs.
Accordingly,
lim ( )
k
P k
��
� 0, lim ( )
k
P k
��
� �1 . (27)
66 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6
Fig. 5. Behavior of P k� ( ) in the model �A with increasing k
�A V: ;�
�0 0 1�
Fig. 7. Behavior of P k� ( ) in the model �A with increasing k
�A V � � �� : ;0 0� �
Fig. 6. Behavior of P k� ( ) in the model �R with increasing k
�R V: ;
�0 0 1�
� If V
0 (Fig. 10), the repulsion of the positive alternative probability from the
equilibrium � � 1 occurs.
Again, the relations (27) hold true.
2.4. Discussion of the behavioral models classification. The foregoing
classification of behavioral models can be used to explain phenomena of collective
behavior and to predict trends in collective decisions.
1. For parameters of the model � A, the preferred choice of alternatives varies at
each stage, and there is an attracting equilibrium point � � . The behavioral
probabilities P k� ( ) converge, as k � �, to the equilibrium values, � � .
This means that, in the behavioral process, in the long run, the stationary
frequencies � � of choice of alternatives are observable: N� subjects select the
alternative +, and N� � subjects select the alternative �.
2. For parameters of the model �R, there is a repulsive equilibrium point, which
distinguishes the initial values. For initial values P� �
( )0 � , the alternatives
probabilities � converge to 0 as k � �; for P� ��( )0 � , the alternatives probabilities
converge to 1. Accordingly, there are two absorbing equilibrium points: 0 or 1.
This law seems to deserve the greatest attention of specialists. The following
interpretation of this law can be given, for example, in the education system.
Each team class has a certain average intelligence �, 0 1
� , and the result of
a long behavioral process strongly depends on the initial conditions. If the proportion
of scholars taking the right decision exceeds �, then eventually the whole class will do
the same. If that proportion is less than �, then in the behavioral process the class
average intelligence will reduce to zero.
This law describes the effects of the intelligence of individual subjects on the
collective intelligence.
3. For parameters of the model �A �, �R �, the advantage of one of the
alternatives is preserved at any stage: for all initial conditions there exists a unique
equilibrium absorbing point as k � �: P k �( ) 1, P k� �( ) 0 or P k �( ) 0,
P k� �( ) 1.
In this case, one of the alternatives dominates the selection.
The regularities of the behavioral process formulated above require experimental
verification.
2.5. Classifiers. The evolutionary behavioral process is given by a solution of
difference evolutionary equation (18) with regression function of increments (16).
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Fig. 8. Behavior of P k� ( ) in the model �R with increasing k
�R V
� � �: ;0 0� �
Fig. 9. Behavior of P k� ( ) in the model �A � with increasing k
� A V� � � � �: ;0 0� �
Fig. 10. Behavior of P k� ( ) in the model �R � with increasing k
�R V�
� �� : ;0 0� �
Now, we introduce the classifier functions which discriminate the behavioral
models of the classification-III.
The classifier functions are constructed using truncated regression function of
increments by the elimination of the corresponding equilibrium multipliers. The
classifier functions are defined by the following expressions:
V c V c� ( ) : ( )� � �1 2 , | |c
1 , (28)
for equilibrium point �, and
V c V c c� � � � �1 1( ) : ( )( )� , | |c
1 , (29)
for extremal points �1 .
The classifier functions (28) and (29) have an essential property: they assume
negative values for all c � � ( , )1 1 in the corresponding classification-III models.
The attractive model � A : V � 0, | |�
1 , is discriminated, by the classifier (28),
which takes negative value: V c V c� ( ) ( )� � �
1
4
1 02 for all c � � ( , )1 1 .
The regression function of increments (16) in the difference evolutionary
equation (18), has the following representation:
V c V c c( ) ( )( )� �
1
4
� � .
Hence, the evolutionary behavioral process, determined by solution of the
difference equation (18), is increasing for c � �( , )1 � and decreasing for c � ( , )� 1 .
The repulsive model �R :V
0, | |�
1 is discriminated, by the classifiers (29), as
follows:
V c V c c � �
1 1 0( ) : ( )( )� for c � ( , )� 1 , (30)
and
V c V c c� � � � �
1 1 0( ) : ( )( )� for c � �( , )1 � . (31)
The regression function of increments (16) in the difference evolutionary
equation (18), has the following representation:
V c
V c c c
V c c c
( )
( )( ), ( , );
( )( ),
�
� �
�
�
1
4
1 1
1
4
1
1
1
for
for
�
( , ).�
�
�
�
�
� 1 �
The inequalities (30), (31) mean that the evolutionary behavioral processes
determined by the difference evolutionary equation (18) is increasing for c � ( , )� 1 up
to c1 1� , and decreasing for c � �( , )1 � up to c� � �1 1. Such a comportment of the
evolutionary behavioral process is classified as repulsive property of the extremal
points � � � { }1 1, .
The model �A : V � 0, � � 1 (attractive dominant +) and the model �R :
V
0, � � �1(repulsive dominant +) are discriminated by the classifier (29) as follows:
V c V c c � �
1 1 0( ) : ( )( )� for all c � � ( , )1 1 .
For these models, the regression function of increments (16) in the difference
evolutionary equation (18), is used in the following representation:
V c V c c( ) ( )( )� �
1
4
11 for all c � � ( , )1 1 .
Hence, the evolutionary behavioral processes are increasing for all c � � ( , )1 1 ,
up to the value 1 for both models �A and �R .
Similarly, the limit behavior of the models � A � and �R � are obtainable.
68 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6
3. STOCHASTIC BEHAVIORAL MODEL
3.1. Basic definitions. Let us focus our attention on the stochastic behavioral
models described by averaged sums:
S k
N
kN n
n
N
( ) : ( )�
�
�
1
1
� , k � 0, (32)
of sample values �n k( ), 1 � �n N , k � 0, which take two values �1 with the
conditional expectation of increments (5), that satisfy the difference evolutionary
equation (18).
The stochastic behavioral processes, based on the averaged sums (32), are
determined by two components:
— evolutionary component V S kN( ( )), described by the evolutionary equation (18);
— stochastic component, described by the martingale-difference:
� � �� N N N Nk S k E S k S k( ) ( ) [ ( )| ( )] � � 1 1 1 , k � 0. (33)
So, the stochastic behavioral process is determined by a solution of the stochastic
difference equation
� �S k V S k kN N N( ) ( ( )) ( ) � � 1 1� , k � 0. (34)
The stochastic component (33) is characterized by its two first moments:
E k kN� � ( ) ,
�0 0,
E k S k c c N kN N[( ( )) | ( ) ] ( ) / ,� � �2 2 0� � � , � 2 21
4
1( ) : ( )c c� � .
3.2. Stochastic behavioral models in a stochastic approximation scheme.
The convergence problem of stochastic behavioral processes, described by the
difference stochastic equation (34), are more complicated that the analogous problems
of convergence of the evolutionary behavioral processes described by the difference
evolutionary equation (18).
In order to get an effective result, the stochastic approximation approach can be
used.
At the beginning, consider a stochastic behavioral process, normalized by the
sequence a t a tk k( ) : /� , t k Nk � / , satisfying the stochastic approximation
procedure [11]:
a tk
k
( )
�
�
� � �
1
, a t k
k
k
2
1
0
�
�
�
� �( ) , . (35)
The normalized stochastic behavioral process in t k Nk � / , k � 1, is considered in
discrete-continuous time scale with the step � : /�1 N :
� �� N Nt a t S k( ) ( ) ( )� 1 , t t tk k�
1, k � 0.
The next step is the normalization of the directing parameter V and of the
stochastic component increments, as described hereunder.
Definition 1. The stochastic behavioral process, under the stochastic
approximation conditions (35), is determined by a solution of the difference stochastic
equation
� � �� � � � �N N N Nt a t V t t t( ) ( )[ ( ( )) ( ( )) ( )]� � 1 , (36)
where V c( ) is defined in (16) and the stochastic component satisfies the following
conditions:
E tN� �� ( )
0, E t t NN N[( ( )) | ( )] /� � �� � � �2 1 .
Remark 3. The stochastic behavioral process � N kt( ), t kk � �, k � 1 , is
a discrete Markov process, characterized by its two conditional first moments:
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E t t a t V tN N N[ ( )| ( )] ( ) ( ( ))� �� � �� � , (37)
E t a t V c t c a t cN N[( ( ) ( ) ( ) ) | ( ) ] ( ) ( )� � �� � � � �2 2 2 . (38)
The second moment of the stochastic behavioral process increments, defined by
the difference stochastic equation (36), is calculated using (37), (38):
E t t c a t B c
N N N[( ( )| ( ) ] ( ) ( )� �� �2 2� � , B c c V cN ( ) : ( ) ( )� � 2 2 �. (39)
3.3. Classification-III for stochastic behavioral processes. The classification of
stochastic behavioral processes, according to the behavioral model classification-III,
described above in section 2.3, is based on the limit theorems for a normalized
stochastic behavioral process in the stochastic approximation scheme.
The behavioral model classification is characterized by an almost surely
convergence of the normalized behavioral process � N t( ).
Theorem 1. The behavioral model classification-III give the following limit
results:
�A V: � 0, | |�
1
P t t
t N
1 lim ( ) ;
��
� � �� � as (40)
�R V:
0, | |�
1
P t
t N
N
N
1
1 0
1 0
lim ( )
, ( ) ,
, ( ) ,��
�
�
�
�
�
�
�
� �
� �
if
if
as t � � ; (41)
�A V �: ,0 � � 1; �R V
: 0, � � �1
P t t
t N
1 1lim ( )
��
� � �� as �
� �N N( ) : | ( )|0 0 1 ; (42)
�A V� �: 0, � � �1; �R V�
: 0, � � 1
P t t
t N
1 1lim ( )
��
� � � �� as �
� �N N( ) : | ( )|0 0 1 . (43)
Proof. According to the approach proposed in the monograph of Nevelson–Hasminskii
[11, Ch. 2, Sec. 7], the generator of the discrete Markov process (37)–(39) is used as
follows:
L c NE c t c t cN N N� � � �( ) [ ( ( )) ( )| ( ) ]� � �� � , (44)
on the test-functions
�( ) : ( ) , | | , , ,c c c c c� � � � � 0
2
01 1 1{ }� . (45)
Lemma 1. On the test-functions (45), the generator (44) has the following
representations:
L c c a t V c c c a t B cN c N( ) ( ) ( )( ) ( ) ( )� � � � 0
2
0
2 22
0
, (46)
where
B c c V c NN ( ) : ( ) ( ) /� � 2 2 , (47)
and by definition, the classifiers are as follows:
V c
V c c
V c c
c0
0
1 0 1
( ) :
( ), ,
( ), .
�
�
� �
�
�
� �
� �for
for
(48)
The Theorems 2.7.1 and 2.7.2 from Nevelson–Hasminskii monograph [11],
adapted to the stochastic behavioral processes (36), are represented here below.
Theorem 2 [11, Ch. 2, Sec. 7]. Let there exist a non-negative functionV c( ), | |c � 1,
with zero point c0 : V c( )0 0� , satisfying the inequality
sup
| |
[ ( )( )]
c c h
V c c c
� �
�
0
0 0 for a fixed h � 0, (49)
70 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6
and the additional positive component, defined by (47), is bounded:
| ( )| | ( ) ( ) / |B c c V c N KN � �� 2 2 .
Then the stochastic behavioral process � N t( ), t � 0, converges with probability 1
to the equilibrium point c0 :
� N
P
t c( )
1
0� �� , as t � �.
The main condition (49) of Theorem 2, using Lemma 1, is transformed as follows:
V c c c V c c cc( )( ) ( )( )� � �0 0
21
4 0
, c � � { }1 1, , � .
Hence, by Theorem 2.7.2 [11], the stochastic behavioral process � N t( ), t � 0,
characterized by the generator (46)–(48), converges with probability 1, as t � �, to
the equilibrium point c0 1 1� � { }, , � . The limits (40)–(43) of Theorem 1 are proved.
CONCLUSIONS
The statistical data analysis in the stochastic behavioral process consists primarily
of the the original parameters V� evaluation. In a model of stochastic behavioral
process with a number of alternatives bigger than two, the behavior regularities
become naturally more complex. However, in essence, the main features are always
the same: after a sufficiently large number of stages, the behavioral process of
a large population of subjects is characterized by three types of equilibria:
attracting, repulsive and absorbing. In processes with many alternatives, the
behavior equilibrium state (in the parameter space) can be a point, a line, a plane,
etc., up to a R-dimensional subspace.
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// Advanced Lectures on Machine Learning. — Berlin; Heidelberg: Springer, 2004. — P. 169–207.
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with applications in science and engineering. — New York: Springer-Verlag, 2002. — 501 p.
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— 365 p.
9. K o r o l i o u k D . Binary statistical experiments with persistent non-linear regression // Theor.
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10.1007/s10559-015-9755-4.
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Iss. 47. — 244 p.
Íàä³éøëà äî ðåäàêö³¿ 04.03.2016
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6 71
Ä.Â. Êîðîëþê, Ì.Ë. Áåðòîòò³, Â.Ñ. Êîðîëþê
ÑÒÎÕÀÑÒÈ×Ͳ ÌÎÄÅ˲ ÏÎÂÅIJÍÊÈ. ÊËÀÑÈÔ²ÊÀÖ²ß
Àíîòàö³ÿ. Ñòîõàñòè÷í³ ìîäåë³ ïîâåä³íêè âèçíà÷àþòüñÿ ð³çíèöåâèìè åâî-
ëþö³éíèìè ð³âíÿííÿìè äëÿ éìîâ³ðíîñòåé á³íàðíèõ àëüòåðíàòèâ. Êëà-
ñèô³êàö³ÿ ñòîõàñòè÷íèõ ìîäåëåé ïîâåä³íêè çä³éñíþºòüñÿ ïî ãðàíè÷í³é ïî-
âåä³íö³ àëüòåðíàòèâ éìîâ³ðíîñòåé. Îñíîâíà âëàñòèâ³ñòü êëàñèô³êàö³¿ õàðàê-
òåðèçóºòüñÿ òðüîìà òèïàìè ð³âíîâàãè: ïðèòÿãóâàëüíèì, â³äøòîâõóâàëüíèì
³ äîì³íàíòíèì. Êëàñèô³êàö³ÿ ñòîõàñòè÷íèõ ìîäåëåé ïîâåä³íêè çä³éñíþºòüñÿ
ç âèêîðèñòàííÿì äèôóç³éíî¿ àïðîêñèìàö³¿.
Êëþ÷îâ³ ñëîâà: åâîëþö³éíèé ïîâåä³íêîâèé ïðîöåñ, ñòîõàñòè÷íèé ïðîöåñ
ïîâåä³íêè, ð³âíîâàãà, ïðèòÿãóâàëüíà ìîäåëü, â³äøòîâõóâàëüíà ìîäåëü,
äîì³íàíòí³ ìîäåë³, ãåíåðàòîð äèñêðåòíîãî ìàðêîâñüêîãî ïðîöåñó.
Ä.Â. Êîðîëþê, Ì.Ë. Áåðòîòòè, Â.Ñ. Êîðîëþê
ÑÒÎÕÀÑÒÈ×ÅÑÊÈÅ ÌÎÄÅËÈ ÏÎÂÅÄÅÍÈß. ÊËÀÑÑÈÔÈÊÀÖÈß
Àííîòàöèÿ. Ñòîõàñòè÷åñêèå ìîäåëè ïîâåäåíèÿ îïðåäåëÿþòñÿ ðàçíîñòíûìè
ýâîëþöèîííûìè óðàâíåíèÿìè äëÿ âåðîÿòíîñòåé áèíàðíûõ àëüòåðíàòèâ.
Êëàññèôèêàöèÿ ñòîõàñòè÷åñêèõ ìîäåëåé ïîâåäåíèÿ îñóùåñòâëÿåòñÿ ïî ïðå-
äåëüíîìó ïîâåäåíèþ àëüòåðíàòèâ âåðîÿòíîñòåé. Îñíîâíîå ñâîéñòâî êëàññè-
ôèêàöèè õàðàêòåðèçóåòñÿ òðåìÿ òèïàìè ðàâíîâåñèÿ: ïðèòÿãèâàþùèì, îòòàë-
êèâàþùèì è äîìèíèðóþùèì. Êëàññèôèêàöèÿ ñòîõàñòè÷åñêèõ ìîäåëåé ïîâå-
äåíèÿ îñóùåñòâëÿåòñÿ ñ èñïîëüçîâàíèåì äèôôóçèîííîé àïïðîêñèìàöèè.
Êëþ÷åâûå ñëîâà: ýâîëþöèîííûé ïîâåäåí÷åñêèé ïðîöåññ, ñòîõàñòè÷åñêèé
ïðîöåññ ïîâåäåíèÿ, ðàâíîâåñèå, ïðèòÿãèâàþùàÿ ìîäåëü, îòòàëêèâàþùàÿ ìî-
äåëü, äîìèíèðóþùèå ìîäåëè, ãåíåðàòîð äèñêðåòíîãî ìàðêîâñêîãî ïðîöåññà.
Êîðîëþê Äìèòðèé Âëàäèìèðîâè÷,
êàíäèäàò ôèç.-ìàò. íàóê, ñòàðøèé íàó÷íûé ñîòðóäíèê Èíñòèòóòà òåëåêîììóíèêàöèé è ãëîáàëüíîãî
èíôîðìàöèîííîãî ïðîñòðàíñòâà ÍÀÍ Óêðàèíû, Êèåâ, e-mail: dimitri.koroliouk@ukr.net.
Áåðòîòòè Ìàðèÿ Ëåòèöèÿ,
ïðîôåññîð ìàòåìàòèêè Ñâîáîäíîãî Óíèâåðñèòåòà Áîëüöàíî/Áîçåíà, Áîëüöàíî, Èòàëèÿ,
e-mail: marialetizia.bertotti@unibz.it.
Êîðîëþê Âëàäèìèð Ñåìåíîâè÷,
àêàäåìèê ÍÀÍ Óêðàèíû, ñîâåòíèê äèðåêöèè Èíñòèòóòà ìàòåìàòèêè ÍÀÍ Óêðàèíû, Êèåâ,
e-mail: vskorol@yahoo.com.
72 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2016, òîì 52, ¹ 6
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