Fuzzification of convex bodies in Rn Текст научной статьи по специальности «Математика»

Software systems and computational methods

Правильная ссылка на статью:

Ohanyan V.K., Bardakhchyan V.G., Simonyan A.R., Ulitina E.I. — Fuzzification of convex bodies in Rn // Программные системы и вычислительные методы. - 2019. - № 2. - С. 1 - 10. DOI: 10.7256/2454-0714.2019.2.29894 URL: https://nbpublish.com'library_read_article.php?id=29894

Fuzzification of convex bodies in Rn / Фазификация выпуклых тел в □□

Оганян Виктор Кароевич

доктор физико-математических наук

профессор, зав. кафедрой теории вероятностей и математической статистики, Ереванский

государственный университет

0025, /Армения, г. Ереван, ул. Апекса Манукяна, 1 И victoohanyan@ysu.am Бардахчян Вардан Геворкович

кандидат физико-математических наук

ассистент, кафедра Актуарной Математики и финансовой Математики, Ереванский государственный

университет

0025, /Армения, г. Ереван, ул. Апекса Манукяна, 1 И vardanbardakchyan@gmail.com Симонян Арсен Рафикович кандидат физико-математических наук доцент, кафедра прикладной математики и информатики, Сочинский государственный университет 354008, Россия, Краснодарский край, г. Сочи, ул. Политехническая, 7

И oppm@mail.ru

Улитина Елена 1/Ъановна

кандидат физико-математических наук доцент, кафедра прикладной математики и информатики, Сочинский государственный университет 354008, Россия, Краснодарский край, г. Сочи, ул. Политехническая, 7

И uiitina@rambier.ru

Статья из рубрики "Математическое моделирование и вычислительный эксперимент"

Аннотация.

В исследовании особое внимание уделяется обобщению теоремы Матерона о ковариограмме на случай возможной ошибки оценки, моделируемой фаззификацией выпуклых тел. Нечеткие выпуклые тела являются естественным обобщением выпуклых тел, когда мы хотим включить погрешность измерения в нашу модель и получить все возможные выпуклые тела, которые могут привести к заданному распределению длины, зависящему от направления. Классический случай идентификации выпуклых тел не учитывает случаи, когда входная информация и измерение содержат погрешность. Это общая проблема при применении распределения отрезков линии для восстановления

ковариограммы, а затем и самого тела. В ходе исследования, мы определяем тело и запрашиваем, какой результат мы получим для распределения длины для данного нечеткого тела; широко используем нечеткую статистику и нечеткие случайные величины для расширения выпуклых тел и функций распределения по длине до нечеткого случая; используем несколько свойств нечетких чисел и методов нечеткого исчисления (в основном, интеграцию Ауманна). Было введено обобщенное нечеткое распределение, чтобы применять их в общем случае нечетких выпуклых тел. Нечеткие выпуклые тела определяются сложением с выпуклым телом и вычитанием (по Хукухаре) из его нечетких чисел в Яп. Затем выводится обобщение теоремы Матерона для нечеткого случая, основанное на методах исчисления нечеткой функции. Мы ввели нечеткую ковариограмму на основе нечетких выпуклых тел.

Ключевые слова: Теорема Матерона, Выпуклое тело, Нечеткое распределение, Нечеткая ковариограмма, Интегральная геометрия, Интегрирование Ауманна, Гауссовское поле, У-линии, Ошибка оценки, Хукухары дифференцируемость

DOI:

10.7256/2454-0714.2019.2.29894

Дата направления в редакцию:

30-05-2019

Дата рецензирования:

31-05-2019

Дата публикации:

14-06-2019

The problem of identifying of convex bodies is widely known problem in integral geometry and several tools have been introduced to deal with it (starting from support function determination, and integral representation ending with random line length distribution and more complex structures like V-lines) [1-4].

There is also a general way of identifying the convex body through its covariogram, determination of which requires other sort of measurements as well. One of the techniques suggests taking line segments' length distributions (for all directions) and recover from that covariogram of the body (for several types of convex bodies exact formula of reversion are known i-5-6!).

Methodology

Matheron's theorem provides differential tool connecting covariogram and length distribution of a parallel line of given direction. However, in most cases, measurements will contain errors. This makes precise identification of body flawed. Instead of dealing with error bounds (in covariogram) and finding upper values for it, we deal with other form of error incorporation, fuzzy sets. By doing so, we try to describe all possible convex bodies which

we could be recovered for given.

So we deal with the inverse problem in some sense. We define a body and ask what result we will get for length distribution for the given fuzzy body.

Through this procedure, we generalize Matheron's theorem for this case. We extensively use fuzzy statistics and fuzzy random variables to extend convex bodies and length distribution functions to a fuzzy case. Use several properties of fuzzy numbers and fuzzy calculus techniques (mainly Aumann integration).

First, we generalize a fuzzy distribution function for our purposes. Then define fuzzy convex bodies as a natural generalization of the convex body to the fuzzy case.

Next, we derive some useful properties of fuzzy convex bodies and fuzzy covariogram. And derive Matheron's theorem generalization for fuzzy case.

Results

1. General fuzzy distribution

Suppose we have fuzzy random variable, i.e. random variable which takes on fuzzy numbers as values.

More formally we have ^C^), with

Xa{o^)= ^(ulxjv)] (!)

a-levels. As fuzzy numbers these a-levels must obey inclusion principle, i.e. for all w-s

XaO) c XpO) for a > (3_

Generally, fuzzy densities are given by precise definition (for example in Viertl's book [7, p-55]; 1161). However, no case of discrete distribution is discussed. To define fuzzy distribution, we should analyze the idea of taking values smaller than or equal to given x.

Suppose we have two possible outcomes i.e. n = uz), and we have their probabilities

Suppose we take fetOO^OJ] and [xju^.xju^]^ Nqw we consjder some x

It is obvious that if *< min fe^i^C^)) then f0(Y) = o. It is a!so obvious that if

x > mas(xc[(u1)jxn(uz))then p^) = ^

Now let's consider the case where x is in between two fuzzy number.

So if we have ^OJ < x < Xn[>2) then Fa(x) = PC{co±})

However, if x belongs to the interval of a-levels we get the following:

For simplicity let's suppose that so that they have no intersection (or

Strictly XaOl) < XfflCoi2))_

And suppose ^(aij <s < X^CoiJ Here we either can have that outcome corresponding "i either took value less than or equal to x, or value greater than x. So in this case

So in general Fs(X) will not be fuzzy number for each x. The general case for arbitrary Q would be.

(2)

This is general formula, for fuzzy distributions.

When we are speaking about fuzzy densities we require some interval for values of cdf-s. The general framework is as follows.

(3)

with finite integral of both and And such that there is some true density function

f(X)e [^GO^GO] in the interval with highest belief.

And fuzzy distributions for continuous case will be defined

(4)

So this obviously become fuzzy function in sense that for each x it will be fuzzy number.

This structure is somewhat intuitive, while the first structure is purely constructive.

For our case we can unintendedly encounter non continuous probabilities. So we should use general formulation.

2. Fuzzy convex bodies

Now let's skip to our main problem. First of all, let's define a fuzzy case of convex body D a Rn. we will do it in the following way.

D„= D +A

« (5)

where A is some fuzzy number with support in Rn and with A<t a-cuts. Here D is understood as fuzzy number with support in Rn, whose membership function is D for any level a. Next

D„ = D — В

« (6)

where minus is understood as Hukuhara difference. So B is some fuzzy number for which it is possible, —a — Da must be satisfied.

Now the final part is

Da = {C: G is convex, and D„ Ç G Ç Da} (7)

and D is defined as a fuzzy concept having Dc as its a-levels

Proposition 1. One can note that each level Dc is a convex set, from the fact that Minkowski sum of two convex bodies is again convex. Putting it another way, ifGi'GzeDo(, then aLri + li_aJLrzfcLJo( , for a

G [0,1].

Proof. For cases a = 0, and a = 1, the proposition is obvious. Next, we give two simple arguments.

aGi and C1 _ a)G2 are convex bodies, and so aGi + C1 _ a)G2 is convex. Moreover

aü„ Ç ctG-i E aD

« (8)

and

(9)

Summing up, we get

aG1 + (1-а)Сг E Da

(10)

Proposition 2. For 1 > ß > a > 0, Dp E D,

Proof: We must show, that each element of DP is also contained in Let's first note that AP and BP ~ B<I. From which we have

D„

—P D EP is a convex set that must be summed with BP to get D. —« D B« is a convex set that must be summed with Bn to get D. As BP ~ B", —<* ~ — P . So we have

From which obviously if e , then -P ~ G ~ °P. So

and —« ~ G ~ as G is convex. So G G Da.

Remark 1. So defined D satisfy all criteria of fuzzy number except that of elements of D are convex bodies, not numbers.

3. Fuzzy covariogram and its properties.

Now we can define covariogram for D (for covariogram in classical case IA21; -U21). Though as a function of this fuzzy argument it has not developed approaches for further investigation, we try to show that it is a fuzzy number, and will try to recover its membership function.

Let s be the (n-l)-dimensional unit sphere with center at the origin.

Definitionl. We will name fuzzy covariogram of D the fuzzy concept, cf0'11), with following a-levels

where ueR° and = vn(Gn (G+ with vn(0 being n-dimensional Lebesgue measure.

Remark2. So fuzzy covariogram has a-levels, and this are collection of numbers.

Proposition 3. For 1 > P > « > 0, we have ce(°'u) E Св(Б,и)_

Proof. From Proposition 2 we have ~ so if G e than G e and

Proposition 4. For each u e Rn, C(D,U) is a fuzzy number.

Proof. By the inclusion property for D (proposition 2), we have, if l>P>°t>0

Next for completeness we should proof the continuity of membership function, which can be done in the following way. First, we will show, that the membership function satisfies the intermediate value property. And from boundedness we can imply that it is continuous.

To do it take the two layers.

. ce(D'u) and C«(M, with (3>«>o.

Let's look at the smallest

CnfDj u) C fD, u) CGÎD, u") > C ÎD, u") values —!—-- and —---. From the inclusion property we have. -- —---. Whenever

the equality is hold the intermediate value property is immediate. Suppose now

Let's take and with Gi e DP and G2 e D«. This is guaranteed by

the properties of . Now look at the convex combination of this two: For each

s g (04) there is P > TE >

Obviously, there is s* such that for each e e sGi + C1 _ e)Gs c Gz and

CisGi + (1 - s)G2) = VnftsGi + (1 - s)G2)n (sGi + (1 - s)G2+ u)) = V^fisGi + (1 - s)G2) n

(sGi + fl -s)G2 + su + (1 -s)u)) = Vn ((sG1 + (1 -s)G2) П (sfGi + u) + (l-s)(G2 + u))) = Vn ((sGi П efGi + u)) + ((1 - s)G2 П (1 - e)(Gz + u))) = eCfGi) + (1 - s)C(Gz)

(17)

This proves that intermediate value property is satisfied. And from bounded of membership function, and monotonicity of its left and right bounds of each a-levels we can say that it is continuous. The same can be done with maximal values of covariogram. And for the remaining part, the function is constant.

Remark 3. In the above proof we extensively used the property of convex bodies, for dealing with Lebesgue measure of Minkowski sum of convex bodies, and to interchange the sign of intersection and addition. In general this is forbidden, as the sum may be even non-

measurable -t111.

Next, if we define covariogram as a function of the magnitude of translation (i.e.

(GjUjt) = Vn(Gn (G +tu)) ^ then we have the f0||0wjng Here the translation is a vector (t,u), where u£S° 1 defines direction, and t determines magnitude.

Proposition 5.

defined by a-levels

Is the fuzzy function of t, i.e. the function of t e R, which take fuzzy values.

Remark 4. The later proposition is crucial, as it gives the opportunity to use Hukuhara differentiation and Aumann integral concepts.

Proposition 6

. is not t-Hu

t-Hukuhara-differentiable a.s. in

The idea behind is that c«(D'Ujt) is getting "thinner". Or in other word c«(D'u,t) is 0 starting from some t. Suppose, then ~~ t^c] is not definable in Hukuhara sense.

So we should use Aumann integration concept instead.

First, we should define distribution functions for our case. The difference from the crisp(classical) case is that the projection space is different for different elements of D. So first we should define ni(u) .

where uic

.

We know that this is homeomorphic to -t12!

nru±G = {orthogonal projection ofGto hyperplane with normal u}

And we assign a measure to each element of , bc(u) = Vn_1(nru±G)

Though we can combine them into a fuzzy concept, it is rather easier to leave it like this.

To use the definition of (1) for the fuzzy random variable, we should have the same universe of discourse, or in other words we should have the same fi. However, fi is different for different shapes. To bypass this problem, we first note that fi has not to be the same for all a-levels. So we can deal with different fi-s on different levels, but we should exactly fix fi for any a-level. So we will consider the following distribution functions.

(20)

W here w) jg j^g^ 0f intersection of a line parallel to u and intersecting nrulG or

nic(u) at some point w.

The crucial theorem here is the following

Theorem 1. ^flO'tO = iFeOU): G e and nruiG £i na}

Proof: To show this, we should literally show that any value of F(u't) G Fa,n(u't0 has corresponding G e Da with nru±G = nffi for whjch F(u,t) = Fc(u,t)

Not to embark our paper with much text, let's just give the idea of proof. For any value of

a e C0'1) and e the maximal value F^C^t) is fixed by which corresponds to

Note that this set is always non-empty. This is guaranteed by the convexity of all sets considered. So we can use the convex hull of the union of sets. Moreover, this set contains only one element. The next bound is also fixed. So any value in-between can be obtained by considering a convex combination of these two bounds. No matter which convex body is considered, while in a given direction and for given magnitude it has the desired value of distribution function.

So, when we fix the fi we can easily speak about

And next theorem which comes along as generalization of already known result

Theorem 2. u,t) = 0 ~ V»^ (compare wjth the dassica|

case in Here the integral used is Aumann integral.

Proof. This is immediate consequence of the previous theorem and Matheron theorem about covariogram. We have

where in the last line integration in Aumann sense is used, and on the previous line in Riemann-Lebesgue sense.

Discussion

Measurement error could be incorporated into model trough different error addition. An example can be error described random variable from given distribution (or distribution family). However, this approach can have several issues. First of all, this could be misleading, as the reconstruction of covariogram will require reversion. More simply suppose that you have some error field added to the given convex body. From that point, you should get rid of the error part to get the body itself. However, this will again lead to many possible alternatives of convex bodies, which you could get, as you don't know the error exactly. So the covariogram will itself represent the random field. In that situation, other approaches like support function representation can be more fruitful [13 - 15].

Next problem is concerned with possible unboundedness of error term. If we have Gaussian field the possible infinite of (very big error can occur), which is not the case of measurement error. So measurement error when dealing with convex bodies, are more likely

to be bounded [16].

For further we are going to try to get a generalization of other formulas, connecting direction dependent distribution of length, covariogram and distribution of length of line

segments [12]. Conclusion

Fuzzy convex bodies are the natural generalization of convex bodies when we want to incorporate measurement error into our model and derive all possible convex bodies that could give rise of the given direction dependent length distribution. From this point of view, fuzzy convex bodies can be seen as a collection of convex bodies (though at each level it contains not only convex bodies).

We introduced the natural generalization of fuzzy distribution function based on the ordinary distribution function. Introduced the concept of a fuzzy convex body. Discussed some of its properties making it similar up to domain with fuzzy numbers.

We introduced fuzzy covariogram based on fuzzy convex bodies.

Next, we construct a generalized fuzzy length distribution based on fuzzy distributions introduced previously. And lastly, we give equation connecting it with fuzzy covariogram, thus generalizing Matheron's theorem to the fuzzy case.

Acknowledgements

The present research was partially supported by the RA MES State Committee of Science and Russian Federation Foundation of Innovation Support in the frame of the joint research project 18RF-019 and 18-51-05010 Арм-а accordingly.

Библиография

1. Bauer Ch. Determination of Convex Bodies by Projection Functions // Geometriae Dedicata. 1998. Vol. 72. № 3, P. 309-324.

2. Ambartsoumian G. Inversion of the V-line Radon transform in a disc and its applications in imaging // Computers and Mathematics with Applications. 2012. Doi: 10.1016/j.camwa.2012.01.059

3. Cong W., Wang G. X-ray scattering tomography for biological applications // Journal of X-Ray Science and Technology. 2011. Vol. 19. P 219-227. Doi: 10.3233/XST-2011-0288

4. Brunetti S., Daurat A. Determination of Q-convex bodies by X-rays // Electronic Notes in Discrete Mathematics. 2005. Vol. 20. P. 67-81.

5. Gasparyan A.G., Ohanyan V.K. Covariogram of a parallelogram // Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences). 2014. Vol. 49 (4). P. 194-206.

6. Gasparyan A.G. and Ohanyan V. K. Recognition of triangles by covariogram // Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences). 2013. Vol. 48 (3). P. 110-122.

7. Viertl R. Statistical methods for fuzzy data. John Wiley & Sons. 2011. Doi: 10.1002/9780470974414

8. Klir G.J., Yuan B. Fuzzy Sets, Fuzzy Logic: Theory and Applications. Prentice Hall. 1995.

9. Diamond Ph., Kloeden P. Metric Spaces of Fuzzy Sets: Theory and Applications // World Scientific. 1994.

10. Matheron G. Random sets and integral geometry. Wiley. 1975.

11. Ciesielski K. Measure zero sets with non-measurable sum // Real analysis exchange. Vol. 26. № 2. P. 919-922. https://www.jstor.org/stable/44154091 .

12. Gasparyan A.G., Ohanyan V.K. Orientation-dependent distribution of the length of a random segment and covariogram // Journal of Contemporary Mathematical Analysis. 2015. Vol. 50. № 2. P. 90-97.

13. Бардахчян В.Г., Геворкян Р.А. Нечёткая оценка наименьших квадратов по выпуклым оболочкам а-уровней // Известия НАН Армении. 2017. T. 52, № 2. C. 78-84, 2017

14. Грицюк В. И. Нечеткий робастный регрессионный анализ для нечетких входных и выходных данных // Технологический аудит и резервы производства. 2015. № 6. C. 4-8.

15. Зак Ю. А. Fuzzy-регрессионные модели в условиях наличия в стохастической выборке нечисловой информации // System Research & Information technologies. 2017. №. 1. P. 88 - 96.

16. Matia F., Marichal G. N., Jimenes E. Fuzzy Modeling and Control: Theory and Applications. Atlantic Press. 2014.

17. Schneider R., Weil W. Stochastic and Integral Geometry. Springer, Berlin. 2008.