Analysis of Using Z-evaluation Uncertainty in Fuzzy Inference Systems

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1 America Joural of Mathematical ad Computatioal Scieces 06; (: Aalysis of Usig Z-evaluatio Ucertaity i Fuzzy Iferece Systems Primova H. A. Niyozmatova N. A. Cetre for the Developmet of Software ad Hardwarily-Program Complex of Tashket Uiversity of Iformatio Techologies Tashket Uzbekista address xolida_primova@mail.ru (Primova H. A. _ilufar@mail.ru (Niyozmatova N. A. Keywords Discrete Z-umber Z-evaluatio Fuzzy ules Fuzzy Iferece eceived: Jue 0 06 Accepted: July 06 Published: August 8 06 Citatio Primova H. A. Niyozmatova N. A. Aalysis of Usig Z-evaluatio Ucertaity i Fuzzy Iferece Systems. America Joural of Mathematical ad Computatioal Scieces. Vol. No. 06 pp Abstract This article discusses the costructio of a model based o fuzzy iferece rules usig Z- evaluatio aimed at obtaiig coclusios with the use of vague iaccurate or icomplete iitial iformatio. A commo approach of computig Z-umbers based o the priciple of Zadeh expasio is offered. It outlies the basic arithmetic operatios o discrete Z-umbers. We cosider three types of fuzzy models for assessig the status of poorly formalized process the output of which is a liear ad o-liear depedece as well as i the form of fuzzy terms. The computatioal experimets are doe ad the results are aalyzed.. Itroductio I 0 Professor of the Uiversity Califoria (Berkeley Lotfi Zadeh the fouder of the theory of fuzzy sets fuzzy logic ad computig with words proposed the cocept of Z-umber to describe the iaccuracies (proximity of iformatio used i everyday life. Such iformatio is ot etirely accurate; i most cases people associate varyig degrees of cofidece whe expressig views describig situatios ad etc.. depedig o their experiece ituitio ad awareess The geeral approach for computig of Z-umbers is proposed i [] based o the priciple of Zadeh expasio. This approach is computatioally very complex it icludes several variatioal problems. To overcome this difficulty several simplificatios are suggested. I geeral the work [] is the begiig which opes the door to may potetial applicatios of computig Z-umbers ad claims several related importat issues. I [] the authors propose a approach for trasformig Z-umbers i fuzzy umber [] o the basis of fuzzy expectatio of a fuzzy set. Uder the proposed approach with the secod compoet B ɶ defuzzificatio operatio is performed to clear the values of alpha which is the multiplied by A ɶ. However it should be oted that the coversio of Z-umbers i the classic fuzzy umbers [ ] results i a loss of the origial iformatio. I [6] the Z-umbers are applied to the multi-criteria decisiomakig problems give i []. I this problem the weightig criteria ad the criteria for assessmet of alteratives are give i the form of Z-umbers. However i geeral evaluatig alteratives are cosidered as crisp umbers. Papers [7 8] are devoted to ew ideas approaches ad applicatios of Z-umbers i a variety of importat areas. It is proposed to cosider the Z-evaluatio ( X Bɶ i terms of the distributio capabilities of probability distributios uderlyig the correspodig Z-umber Z = ( A ɶ B ɶ. The proposed approach cosiders oly typical distributio. The author also provides a alterative formulatio of Z-data i terms of Dempster-Shafer theory [9] which

2 68 Primova H. A. ad Niyozmatova N. A.: Aalysis of Usig Z-evaluatio Ucertaity i Fuzzy Iferece Systems comprises a plurality of fuzzy type- [0]. The paper [] is devoted to the issues of calculatio of cotiuous Z-umbers ad some importat practical problems i the field of cotrol ad decisio-makig. This research is based o the use of the ormal desity fuctios for the simulatio of radom variables. Amog the importat issues raised i the paper [] we ca show IF-THEN calculatio rules described by Z- umbers. The paper [] is devoted to solve the problems i the face of ucertaity whe the relevat iformatio is described by Z-umbers. The authors formulate decisio makig problem whe the probabilities of the states of objects ad the results of the alteratives are described by Z-umbers. A decisio aalysis o the basis of the approach is proposed i []. The mai disadvatage of the proposed study is a loss of iformatio as a result of the coversio of Z-umbers ito fuzzy umbers. We ca coclude that there is o geeral ad computatioally efficiet approach to computig of Z- umbers. The developmet of a uiversal approach for the easy use for a wide rage of practical tasks such as moitorig aalysis decisio-makig ad optimizatio [ ] is oe of the problems of curret iterest. I tur this reuires the defiitio of arithmetic o Z-umbers.. Formulatio of the Problem We ca say that the theory of Z-umbers is ot sufficietly ivestigated yet (most famous publicatios relates to the period of 0-0. but scietists have already cotributed to the developmet of the theory of Z-umbers ad suggested some approaches to dealig with them which will be cosidered i this paper. Nevertheless the use of Z-umbers i the fuzzy system output still remais as a usolved problem because they are used i the represetatio of compoets of differet ature. The purpose of this article is to study the methods of usig Z-umbers i fuzzy iferece systems ad the developmet of programs based o the results of this study. To achieve this goal the followig objectives have bee idetified:. Study of existig approaches to operatios with Z- umbers.. Developmet of usig algorithm of the trasformed Z- umbers i the fuzzy output systems.. Study of the possibilities of usig Z-umbers i the output system without pre-modificatio (coversio.. Developmet of a program that implemets the proposed method.. Carryig out the experimets ad aalysis of the results. I this paper we propose a approach based o the use of the trasformed Z-umbers (the trasitio to the ormal fuzzy umbers i the fuzzy system output. The paper also aalyzes the arithmetic operatios o Z-umbers which have bee formalized at the time of writig operatio the prospects for their use i fuzzy iferece systems ad formulated the problems that arise i this case.. Discrete Z-umber The discrete Z-umber is the ordered pair Z = ( A ɶ B ɶ where A ɶ the discrete fuzzy umber plays the role of fuzzy costraits o the values that the radom variable X ca accept [ ]: X is A ɶ ad B ɶ is discrete fuzzy umber with a membership fuctio µ :{ b... b} [0]{ b... b} [0] Bɶ playig the role of a fuzzy costraits o measures of probability A ɶ : P( is Bɶ The otio of a discrete Z -umber is closely liked with the cocept of a discrete Z-umber. Give the a discrete Z- umber of Z is the umber Z of a pair cosistig of the of fuzzy umber A ɶ ad of the radom umber []: Z = ( A ɶ where A ɶ plays the same role as i discrete Z- umbers Z = ( A ɶ B ɶ ad plays a role the probability distributio p such that µ i= P( A ɶ = p P( A ɶ sup p( B ɶ. i Let the Z = ( A ɶ B ɶ ad Z = ( A ɶ B ɶ cotiuous Z- umber that describes the values of X ad X idefiite of real variables. Suppose that it is eeded for the calculatio Z = Z Z { /}. Computatio with cotiuous Z- umbers begis with the computatio of the correspodig cotiuous Z -umbers Z = ( A ɶ ad Z = ( A ɶ where i ad of the probability distributio. Thus the first eed to calculate [ 6]. i Z Z Z ( = = For simplicity cosider the case where * is the sum ad assume that X ad X are idepedet. As operads A ɶ A ɶ ad preseted i the differet types of the limitatios the values * also distiguished [6]. A ɶ A ɶ defied as µ = sup(mi{ µ µ x } I tur is probability desity p = p p x dx. Hece Z = ( A p where A ɶ = A ɶ A ɶ.

3 America Joural of Mathematical ad Computatioal Scieces 06; (: Now we realize that the "true" probability distributio p ad p are ot kow. I opposite that p ad p are i the followig costraits fuzzy µ p dx is Bɶ µ p dx is Bɶ ( ( ( x p x dx ( x p x dx µ µ Bɶ µ µ Bɶ ( (. Thus iformatio p ( x also be preseted fuzzy costrait. Problem costructio of this fuzzy restrictio is formulated which are represeted i terms of membership fuctios as o coditio { B ( A ( B A } µ ( p = sup mi µ µ p dx µ µ p dx p p p ɶ ɶ ɶ ɶ ( p = p p x dx ( x p dx x p dx p dx = ( p dx = ( x µ dx = µ dx x µ dx = µ dx (compatibility coditio for ( (compatibility coditio for (6 As you ca see ( - (6 it is a very complex oliear variatioal problem. More specifically the problem of costructig membership fuctios for a variety of package ad operads problems of probability desity which are discussed i a geeral sese. This problem may be simplified if the assumptio ca impose sufficiet to use some typical forms of distributio such as a Gaussia type distributio. However eve i this case the problem is very complex ad of aalytical ad computatioal poits of view. As it see ( - (6 it is a very complex oliear variatioal problem. More specifically the problem of costructig membership fuctios for a variety of package p ( x ad operads problems of probability p ad p desity which are discussed i a geeral sese. This problem may be simplified if the assumptio ca impose sufficiet to use some typical forms of distributio such as a Gaussia type distributio. However eve i this case the problem is very complex ad of aalytical ad computatioal poits of view. After the costructio µ p we have to go to the fial stage of of calculatig the reuired Z-umber Z = ( A ɶ B ɶ ie the defiitio B ɶ. This problem is formulated as follows: o coditio µ ( b = sup( µ ( p B ɶ p b = p µ ɶ dx A As see ewly dealig with a oliear variatioal problem. I order to overcome the complexity of operatios with cotiuous Z-umbers we offer a alterative approach. This approach cosists i cosiderig the discrete Z-umbers as discrete aalogues of cotiuous Z-umbers [ ].. The Theoretical Part A ordered triple which is iterpreted as a operator (approval U is (A B is called Z-evaluatio. A is ot a member of oly oe poit the U is a ucertai variable. I reality Z-evaluatio ca be cosidered as a limitatio o the U which is defied - by the expressio: Prob (U is A is B. This meas that we do ot kow the true probability desity for U but there is a limitatio i the form of a fuzzy subset Р of the space P of all probability desities for U. This

4 70 Primova H. A. ad Niyozmatova N. A.: Aalysis of Usig Z-evaluatio Ucertaity i Fuzzy Iferece Systems limitatio is fuzzy probability of B. Let p be a desity fuctio o U. The probability desity Prob ( p U is A (probability that U is A is determied o the basis of determiig the probability of a fuzzy subset of the proposed by Zadeh [6] as Prob p( U is A = µ A( u pu( u du. The the degree to which p satisfies Z-evaluatio Prob p( U is A is B is µ ( p = µ (Prob ( U is A = p B p = µ B( µ A( u pu ( u du. We take a followig parametric distributio is take i a form of р:. Normal distributio the desity fuctio of which is ( u m pu ( u = ormpdf ( u m σ = exp. σ π σ I this situatio for ay m σ we have ( u m Prob ( U is A = µ ( u p du = µ ( u exp du = uad( trapmf ( u[ a a a a ]* m σ A m σ A σ π σ * ormpdf ( u m σ if if The the space Р of probability distributios is the class of all ormal distributios each uiuely defied by its parameter m σ.. Uiform distributio the desity fuctio of which is 0 a t х a pu ( u = ravpdf ( u a b = /( b a a t a < x b 0 a t x > b. b b I this situatio for ay m σ we have Pr ob ( U is A = µ ( u p du = µ ( u(/( b a du = uad( trapmf ( u[ a a a a ]* ravpdf ( u a b if if a b A a b A a The the space Р of probability distributios is the class of uiform distributios each uiuely defied by its parameter a ad b. Coseuetly operatios o Z-umber give i [] will be sufficietly simplified.. Discussio of Experimetal esults Let us give a fuzzy sample of experimetal data ( X r y r r = M ; here Xr = r xr... xr - -dimesioal iput vector ad yr = ( y y... y M - output vector correspodig to it. I geeral it is reuired to build a model based o fuzzy iferece rules usig Z-estimatio ucertaity: a This approach of usig Z-umbers i the fuzzy output system gives the opportuity to take ito accout the ucertaity more effectively of whe workig with approximate iaccurate iformatio. We ca cofidetly say that this algorithm ca be developed with great success to fid wide applicatio i the solutios of both egieerig ad ecoomic problems of various kids. Let us illustrate the work of the system i a example. We cosider three types of fuzzy models for assessig the status of poorly formalized process the output of which is a liear ad o-liear depedece.. Fuzzy model the output of which is a liear depedece: x i a i b i x i i i i i i a b x a b ( = ( = (... = (... kk xi = ( ai jp bi jp y j = f x... x. p= i= the i ki ki i ki ki i ki ki = ( a b x = ( a b... x = ( a b ( ( µ a a µ a a µ ( b ( µ b µ ( a µ ( a i = i0 i i µ ( b µ ( b y c c... c i = m.

5 America Joural of Mathematical ad Computatioal Scieces 06; (: ad. Fuzzy model the output of which is represeted i the form of fuzzy terms: x =(L M ad x =(L M ad r =(H M. M the x =(L M ad M or =(H M or x =(L M ad x =(L M ad x =(L M ad x =(L M ad x =(L M ad x = (L M or x =(L M ad x =(L M ad x =(L M the r =(HM M. x =(L M ad M or =(M HM or ad x =(L M ad x =(L M ad x =(L M ad x =(L M ad x =(L M ad x =(L M ad x =(H HM the r =(M HM. x =(L HM ad =(M HM or x =(H HM ad x =(L HM ad x =(L M x =(L x =(L M ad x =(L M ad x =(L M ad x =(M x x =(M M ad x =(HM x =(HM M ad x x =(M HM x =(M HM ad x =(M HM ad x x =(M HM ad x =(H HM the r =(LM M. x =(M HM ad =(H HM or HM ad ad x =(H HM ad x =(H HM ad x =(H HM or x =(H HM ad x =(M HM ad x =(H HM ad x =(H HM ad x =(H HM the x =(M x =(H HM r =(L M. Here L-low M-medium LM- low medium HM- higher medium H-high.. Fuzzy model the output of which is represeted as a o-liear depedece: the ( ( µ a a µ a a µ ( b ( µ b µ ( a µ ( a i i0 i i µ ( b µ ( b x i a i b i x i i i i i i a b x a b y = c c c... ( ( µ a a µ a a µ ( b ( b µ µ ( a µ ( a ci c i µ ( b µ ( b ( a a ( a a µ µ µ ( b µ ( b µ ( a µ ( a c i... c i µ ( b µ ( b i = m. ( = ( = (... = (... i ki ki i ki ki i ki ki = ( a b x = ( a b... x = ( a b x

6 7 Primova H. A. ad Niyozmatova N. A.: Aalysis of Usig Z-evaluatio Ucertaity i Fuzzy Iferece Systems Software has bee developed. I this embodimet we used fuzzy iferece algorithm based o the coversio of Z- umbers i fuzzy umbers. Accordig to the results of the study we have obtaied a forecast of shortage i crop risk assessmet based o the costructio of approximatig models usig traiig ad testig of the risk of data (Figure. Charts of these relatioships are show i Figure. I the proposed model each iput variable has its ow membership fuctios of fuzzy terms (low low medium medium higher medium high which are used i the euatios. Fig.. Schedule risk assessmet for traiig ad testig data. Fig.. The surface of the "iput - output"for risk assessmet.

7 America Joural of Mathematical ad Computatioal Scieces 06; (: As a directio for further work we ca uderlie the developmet of the algorithm use of arithmetic discrete Z- umbers i systems of fuzzy output with a view to full implemetatio of Z-iformatio display mechaisms that will brig the smallest loss of iformatio cotaied i the Z- umbers. 6. Coclusio The result of this work is the developed approach to the use of Z-umbers i the system of fuzzy output by covertig Z-umbers i the classical fuzzy umbers ad the developmet of a approach to decisio-makig which summarizes the curret expected utility approach i the case of Z-iformatio. This approach i cotrast to other studies o decisio-makig withi the Z-iformatio based o a direct calculatio of the Z umbers without covertig them ito fuzzy umbers. A direct calculatio of the Z-umbers excludes loss of iformatio related to the coversio. The approach used to solve poorly-formalized process state estimatio problems. The results have showed the validity of the proposed approach. efereces [] Zadeh L. A. A ote o a Z-umber // Iformatio Scieces USA Vol 8 0 pp. 9-9 (i Eglish. [] Kag B. Wei D. Li Y. Deg Y. A Method of Covertig Z- umber to Classical Fuzzy Number //Joural of Iformatio & Computatioal Sciece USA 0 Vol 9 Issue pp (i Eglish. [] L. A. Zadeh Fuzzy sets Iformatio ad Cotrol USA 96 Vol 8 pp. 8- (i Eglish. [] Kaufma A. Gupta M. Itroductio to fuzzy arithmetic: Theory ad applicatio // Va Nostrad eihold Co New York 98 9 p. (i Eglish. [] Klir G. Yua B. Fuzzy Sets ad Fuzzy Logic Theory ad Applicatios //Pretice Hall New Jersey 99 9 p. (i Eglish. [6] Kag B. Wei D. Li Y. Deg Y. Decisio Makig Usig Z- umbers uder Ucertai Eviromet // Joural of Iformatio & Computatioal Sciece. USA 0 Vol. 8 Issue 7 pp (i Eglish. [7] Yager.. O a View of Zadeh's Z-Numbers //Advaces i Computatioal Itelligece Commuicatios i Computer ad Iformatio Sciece USA 0 Vol 99 pp (i Eglish. [8] Yager.. O Z-valuatios usig Zadeh`s Z-umbers // Iteratioal Joural of Itelliget Systems 0 Vol 7 pp (i Eglish. [9] Sevastiaov P. Numerical methods for iterval ad fuzzy umber compariso based o the probabilistic approach ad Dempster Shafer theory // Iformatio Scieces 007 Vol. 77 pp (i Eglish. [0] Zhai D. Medel J. Ucertaity measures for geeral type- fuzzy sets// Iformatio Scieces USA 0 Vol. 8 ( pp. 0-8 (i Eglish. [] Zadeh L. A. Methods ad systems for applicatios with Z- umbers// Uited States Patet Patet No.: 0 US 897 B Date of Patet: Nov. (i Eglish. [] Aliev. A. Zeialova L. M. Decisio makig uder Z- iformatio // I: P. Guo W. Pedrycz (Eds. Huma Cetric Decisio-Makig Models for Social Scieces (Studies i Computatioal Itelligece Spriger Berli Heidelberg 0 (i Eglish. [] Aliev. A. Pedrycz W. Fazlollahi B. Huseyov O. H. Alizadeh A. V. Guirimov B. G. Fuzzy logic-based geeralized decisio theory with imperfect iformatio // Iformatio Scieces 0 Vol. 89 pp. 8- (i Eglish. [] Faria M. Amato P. A fuzzy defiitio of "optimality" for may-criteria optimizatio problems // IEEE Trasactios o Systems Ma ad Cyberetics Part A: Systems ad Humas 00 Vol. ( pp. -6 (i Eglish. [] Aliev. A. Fudametals of the Fuzzy Logic-Based Geeralized Theory of Decisios //Studies i Fuzziess ad Soft Computig Spriger Berli Heidelberg 0 9 (i Eglish.