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Application of the Concentration Function
in Fuzzy Sets Theory
Grand PhD of Eng. Sc., professor V. A. Smagin Honored Worker of Science of Russian Federation, member of International Informatization Academy Saint Petersburg, Russia va_smagin@mail. ru
Abstract. On examples of probability theory for normal and exponential distributions the new term «concentration function» in details is studied, offered the French scientist D. Djuge. An attempt was made to disseminate this term besides probability theory on the theory of fuzzy numbers of L. Zade. Two examples for trapeze and triangular membership functions are in details considered. Recommendations on application of function of concentration in various applied areas of a science are made. The new mathematical model for function of concentration on the basis of N. M. Sedjakin&s principle and the recommendation for the further development of studied mathematical object offered.
Introduction
In the theory of random processes, according to D. Djuge, a famous French specialist in probability theory and mathematical "Theoretical and applied statistics" of statistics [1, 10] is intended to serve as a guide in the initial study of probability theory and mathematical statistics. According to this author, it is desirable to draw a distinction between applied statistics and theoretical (or probability calculation), it can be said that applied statistics, taking into account only the finite number of observations, study real observations and it are probability calculation, while theoretical statistics [8] can allow an infinite sequence of observations as the subject of the study. In our research performed in the proposed paper, we will rely only on probabilistic elements of the first chapter of the first part of the book called «Random Analysis». What exactly was our interest as an author with this book before? This studying of characteristic functions of negative random variables for correction of hyperdelta approximation of distributions of probabilities not only with the positive, but also negative initial moments and attempt of their application for the solution of the integrated equation of Wiener-Hopf by means of Fourier&s transformation [9]. As a result, some positive results were obtained. New interest in the mentioned book was caused by familiarity in it with the section «Concentration function» on page 27. Briefly, this provision can be explained as follows:
- the need to select an expert of the desired quality;
- the need to select an expert with several desirable qualities;
- the need to achieve the right quality in the expert;
- the task of selecting a group of experts;
- achieving excellence in expert groups;
- solving the necessary tasks in efficiency theory, in intelligent information system, in modern metrology, in decision theory etc.
PhD E. M. Shurygin Emperor Alexander I St. Petersburg State Transport University Saint Petersburg, Russia evgeniyok62@yandex.ru
Turning to [2], let us give an interpretation of the concept of concentration in the broad sense: «concentratio», concentration, accumulation, gathering someone, something in one place. It, like probabilistic concept, fashion, can play an important role in economic tasks. For example, Zhibra [1] makes a comment that the shoemaker makes the parts of the shoe soles in terms of the most common size determined by fashion.
However, with regard to the importance of the economic aspect of concentration, we must point out, first and foremost, the need for a deep development of concentration theory, precisely in order to meet future practical challenges. Therefore, the purpose of this article is to further study and further develop the theory related to the concept of concentration from the perspective of random processes.
P. Levy&s concentration function [1, 6, 7]. The concentration function in probability theory is one of the characteristics of a random quantity. It is used in a number of tasks in probability theory, in particular in the study of the properties of convolutions of distributions and limit sums of independent random quantities.
Definition. Let be given a random quantity £ with a distribution function F. The value concentration function £ is the function QP(x), given on the non-negative half-axis as follows:
QF(x) = max(F(t) + x + 0) - F(t)). (1)
Its properties:
The purpose of this article is to study the principle of concentration function construction on specific examples. These include the probability distributions normal and exponential. In addition, the authors of this article proposed to extend the principle of construction of concentration function [2] to the field not only of probabilistic distributions, but also of the membership functions introduced by L.A. Zade [3, 4].
Example 1. Consider a normal distribution with probability density
m = 20; a =5; t = 0,0,1 ... 1; f(t) = dnorm(t, m, a).
Figure 1 shows its distribution function F(t) = f0f(z)dz. If t = 5,10,20,25,35 and x = 3,8,13,18,23,28,33,38,43 get the following numerical data for QP (x): Magnitude t acts as a parameter for the
function QF (x). Parameter values t for the various functions Qf(x) are given to the left of the equations, and the argument values x — in the first line for all five functions.
Figures 2 and 3 show the corresponding functions y1, y2, y3, y4, y5.
Fig. 1. Normal distribution function graph
t x := (3 8 13 18 23 28 33 38 43)T
Fig. 2. Functions yl, y2, y3, y4 graphs
Example 2. Exponential distribution with probability density A = 0,05; t = 0,0,1 ...1; f(t) = Ae-Xt.
Figure 4 shows its distribution function F(t) = Jni/(z)dz.
- 3._ 8x10 r
Fig. 3. Function y5 graph
If t = 5,10,20,25,35 and x = 3,8,13,18,23,28,33,38,43 get the following numerical data for QF (x) the data shown in the table below.
t x = (3 8 13 18 23 28 33 38 43)T
Figures 5 shows the corresponding functions yl, y2, y3, y4, y5.
The more concentrated the distribution function on the axis t, the earlier the concentration function begins to grow and the
earlier it reaches the maximum value. The results of the study of concentration functions for normal and exponential distributions are shown in Examples 1 and 2.
Fig. 4. Exponential distribution function graph
Example 3. This example is devoted to the study of the concentration function not to the probability distribution function, but to the membership functions of the theory of fuzzy sets [3]. This function is a trapezoidal function of accessory with source data: a = 5, b = 10, c = 30, d = 35. Intermediate functions are entered first:
v(t) = —, k(t) = —, m(t) = 1 ,
v & b—a & d—c
which are represented first as a linear programming operator and then by a function of accessory with the argument t
Kt) ■=
b — a 1, d — t
if a <t<b; ifb < t<c; if c < t < 35.
Fig. 5. Functions yl, y2, y3, y4, y5 graphs
Figure 6 shows an image of the membership function.
Let&s find the probability density function for a given function of accessories. To do this, first find the magnitude of the area of the function ß(t), then normalization factor and auxiliary check constants
c = 25 = a04, a(t) = 25 ^ 2&5&0,04 = O&1,
And finally, we find the probability density for the membership function. Figure 7 shows this probability density.
Fig. 6. Image of function of accessory
Fig. 7. Probability density for the function of accessory graph
Intellectual Technologies on Transport. 2020. No 1
The basic distribution function F(t) by which the concentration function (t) is determined:
f (t- 5)2 250 &
°&1+(25- 5),
(30 -t)(t- 40)
i/ a < t < 6; i/ 6 < t < c; i/ c < t < 35.
Figure 8 shows the basic function F(t) = JQt a(z)dz. At values of parameter t = 5,10,20,25,35 get the following data for concentration function QF(x). Based on this data, graphs of the function are plotted on Figure 9.
x1 := (3 8 13 18 23 28)T y1 = (0,036 0,220 0,420 0,620 0,820 0,984)T
x2 := (3 8 13 18 23)T y2 = (0,120 0,320 0,520 0,720 0,884)T
x3 := (3 8 13)T y3 = (0,120 0,320 0,484)T
x4 = (3 8)T y4 = (0,120 0,284)T
x5 = (3)T y5 := (0,084)T
Fig. 8. Basic distribution function F(t) graph
The idea of constructing concentration functions also extends to the membership functions to the theory of fuzzy sets. However, for its implementation it is necessary to reduce the membership functions to functions of probability densities, and only then use the functions of probability distributions to find concentrations of accessory functions. At the same time, any breaking membership functions can be used to find the necessary concentration functions inherent in them. This is supported by Examples 3 and 4 illustrating studies first of the trapezoidal and then of the triangular membership functions.
Example 4. Consider an example of the construction of a concentration function for a triangular accessory function of the theory of fuzzy sets [3]. This function is considered a membership function with similar source data: a = 5, b = 10, d = 35. Intermediate functions are entered first
xl,x2,x3,x4,x5 Fig. 9. Concentration functions graphs
Which are represented first as a linear programming operator and then by a function of accessory with the argument x
(U(X) :=
¿/ a < x < ô; ¿/ c < x < d;
¿/ a < x < ô; ¿/ b < x < d.
Figure 10 shows the membership function, and Figure 11 shows the equivalent of its probability density. Coefficient of rationing is 1.
v(x) = , fc(x) = ^, m(x) = 1,
v J -a v J d-b v J
Fig. 10. Function of accessory graph
Below shows the distribution function and the numerical functions for drawing the graphs shown in Figure 12.
a = 5, d = 35,
&(x — 5)2
(20 — x)(x — 50) 450 ,
if a < x < b; if b < x<d.
The time points were chosen equal to: t = 10,15,20,25,20 h.
x1 ■= ■= (3 8 13 18 23)T y1 = = (0,087 0,320 0,624 0,836 0,936)T
x2 = (3 8 13 18)T y2 = (0,153 0,458 0,669 0,769)T
x3 = (3 8 13)T y3 = (0,180 0,391 0,491)T
x4 = (3 8)T y4 = (0,113 0,213)T
x5 = (3)T y5 = (0,047)T
Fig. 12. Resulting graphs y1-y5
Curves y are based on source [11]. From Figure 12 it is particularly clear that the earlier the construction of the concentration function begins (less time t), the greater the value of this function with increasing argument x and quickly reaching a concentration limit.
Fig. 11. Probability density of the function of accessory graph
Remark. From the examples discussed it follows that the concentration function for any probability distribution is a new, nested function of the random concentration value distribution QF (x) for the base distribution considered F(t). The authors were also able to consider two examples of the construction of the function of concentration of the membership functions the theory of fuzzy sets. However, the clear physical meaning of the concentration function for the author of this article remains unclear. P. Levy, in his works, cites several theorems with proposals of practical orientation, which we mentioned at the beginning of this article. They carry, in our view, theoretical value for studying the marginal behavior of the twists of random quantities, as well as in studying the behavior of sums of a significant number of random deposits. From our point of view, the study of concentration functions should be more closely linked to practical tasks. For example, with the formation of expert groups to meet the challenges of making some decisions and other challenges. Therefore, we will try to simplify the solution of the concentration problem at a simpler mathematical level. Let&s execute an additional research. Inherently, the concentration function is a function of the probability distribution given on the positive axis x e R with time parameter t. It can therefore have initial and central moments, probability density, failure intensity, and concentration lifetime:
Qp(x, t) = max(F(t + x + 0) — F(t))
qF(x,t)= ±Qf (x, t) , v^(x, t) = J™ z-qF (z, t)dz , v|(x, t) = J™z2 ■ qF(z, t)dz ,
ÄF(x, t) =
r(x, 0 = JoÄF(z, t)dz .
(10) (11) (12)
Then, the probability that the concentration value will be at least x the value will be determined as
PF (x, t)
= p-r(x,t) = e-J0 lF(z,t)dz
And since the amount of concentration resource is random, there are initial and central moments of the random amount of concentration resource. They can be defined by the formux
la (13). However, this path is difficult enough to produce calculations. Therefore, we will take a simpler and more visible way. To do this, we will perform two examples.
Example 5. Again, reference is made to Example 1 discussed above. Example. at the beginning of article. It set four points in time t on Figure 1 and nine argument values x. Each moment t has its own function of concentration distribution in Figures 2 and 4. On value of the moment t and to arguments x the probability value can be determined from their inherent values. This example relies on a normal theoretical distribution and a set of desired probability values of the concentration values. If the researcher wishes, all these initial values can be set.
Example 6. Leans on Example 4. Articles related to the construction and determination of the values of the concentration functions of the triangular membership function to the theory of fuzzy sets by L. A. Zade [3]. Its essence consists in the fact that by software first the function of concentration value distribution is built on the basis of specified initial data for the function of belonging. Note that both functions in Examples 3 and 4 are not continuous, they are characterized by discontinuity. This is illustrated in Figures 8 and 12. Then on the example of one of them we use an expression known from the theory of random processes for the probability of achieving some event (in the theory of reliability, it is the probability of no-sign operation of the system), having the form:
P(t) = e-JoA(z)dz , (14)
where A(t) — the rate of occurrence of the event (for example, failure). Expression (7) can be represented as
= p-r(t)
where r(t) = /QtA(z)dz it is commonly referred to as «reliability resource in the sense of professor N. M. Sedyakin» [5]. It can be interpreted as a «safety margin» that can be consumed during system operation. From our point of view, it can be interpreted as a value of distribution concentration. Consider the functions (Fig. 10 and 11) of the 4 examples and the resulting graphs of Figure 12. Let&s enter designation to write the concentration function by going from (10) to Q(t) = 1 -e-r(t), and replacing the variable in this formula t for x, we get the concentration value distribution function for the triangular function of accessories. And by applying the concept of resource according to N. M. Sedyakin [5] directly to the concept of resource concentration, we will get its graphical representation. The presented transition is displayed using the functions below and Figure 13. The necessary initial values of the membership function parameters can be replaced.
a := 5, b := 20, d := 35,
&(*- 5)2 450 ,
(20 — x)(x — 50)
i/ a < x < ¿; ¿/ b < x < d;
r(t) = — Zn(1 — Q(t)).
Fig. 13. Concentration value distribution function r(x)
The values of the variable t may be selected at the discretion of the researcher in accordance with (1), may also be considered to estimate the value x to construct the concentration value distribution function. For example, in our example, choose t = 5,10,15,20,30. For these values, the concentration value is estimated as follows:
r (5,01) = 2,222 • 10-7 r (5,1) = 2,222 • 10-5
r(8) = 0,020 r(10) = 0,057 r(15) = 0,251 (16)
r(18) = 0,471 r(30) = 2,890 ...
From the values of these obtained numbers it can be concluded that the value of the concentration function for the triangular function of accessory with increase of the initial basic moment of time counting monotonically increases. The scale for the resource quantity can be divided into several identical intervals and a concentration function can be constructed for each value. And then for any value of them find the amount of risk and the amount of effect (or losses) depending on the set goal of solving the problem. In this way, an answer can be found to solving some efficiency problem.
Conclusion
In the presented article authors consider the mathematical function of concentration proposed by P. Levi [6, 7] for research and solution of problems in the field of random processes. A concentration function is a function of the distribution of a plurality of random quantities depending on the magnitude of the concentration interval located on the positive real axis of the original, basic function of the distribution of random quantities. The concentration interval is defined as the distance between two fixed moments of time, its magnitude can vary from zero to an infinite value. The power of the concentration function (probability) depends on the length of this interval and the random values enclosed inside it, provided that the interval can be located anywhere in the axis of the basic distribution function. Thus, for any basic distribution, the built concentration function is a monotonically increasing function from zero to a unit value.
The built concentration function based on the first basic function can itself be regarded as a new basic function for constructing a new concentration function, etc. This conclusion is drawn by us, the authors of this article, on the basis of the study of the first concentration function. For the construction of functions (13) and (14) the principle of linear prediction from the book [11] was used.
The principle of the construction of the concentration function by the authors of this article extends to the membership function of the theory of fuzzy sets, provided that the membership function is normalized. This allows you to work not only with continuous functions, but also with functions that have breaks. Two examples are given to confirm this fact.
In addition, the authors have shown that the value of concentration can be determined on the basis of the value of the «reliability resource» proposed by Professor N. M. Sedyakin [5] in 1965. The use of concentration function can be used in solving many problems of a variety of nature, for example, in efficiency theory, in intelligent information system, in modern metrology and in decision theory.
References
Применение функции концентрации в теории
нечетких множеств
д.т.н., профессор В. А. Смагин заслуженный деятель науки РФ, действит. член Международной академии информатизации Санкт-Петербург, Россия va_smagin@mail. ru
Аннотация. На примерах теории вероятностей для нормального и экспоненциального распределений детально изучается новый термин «функция концентрации», предложенный французским ученым Д. Дюге. Предпринята попытка распространить это нововведение помимо теории вероятностей на теорию нечетких чисел Л. Заде. Детально рассмотрены два примера для трапецеидальной и треугольной функций принадлежностей. Даются рекомендации на применение функции концентрации в различных прикладных областях науки. Предложена новая математическая модель для функции концентрации на основе принципа Н. М. Седя-кина и рекомендация для дальнейшего развития изучаемого математического объекта.
к.т.н. Е. М. Шурыгин Петербургский государственный университет путей сообщения Императора Александра I Санкт-Петербург, Россия evgeniyok62@yandex.ru
Интеллектуальные технологии на транспорте. 2020. № 1
