The use of the concept of vague environment in approximate fuzzy reasoning

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1 Kovác, Sz., Kóczy,.T.: The ue of the concept of vague envronment n approxmate fuzzy reaonng, Fuzzy Set Theory and pplcaton, Tatra Mountan Mathematcal Publcaton, Mathematcal Inttute Slovak cademy of Scence, vol.., pp.69-8, Bratlava, Slovaka, (997). Draft veron. The ue of the concept of vague envronment n approxmate fuzzy reaonng Szlvezter Kovác Computer Centre, Unverty of Mkolc Mkolc-Egyetemváro, Mkolc, H-355, Hungary e-mal: zkzlv@gold.un-mkolc.hu and ázló T. Kóczy Department of Telecommuncaton an Telematc, Techncal Unverty of Budapet Sztoczek u., Budapet, H-, Hungary btract Ung the concept of vague envronment decrbed by calng functon [] ntead of the lngutc term of the fuzzy partton gve a mple way for fuzzy approxmate reaonng. Wdely ued way of fuzzy approxmate reaonng baed on mlarty meaure decrbed by dtance meaure of fuzzy et []. The approxmate fuzzy reaonng need a lot of computatonal effort, becaue the dffcult way of calculatng the dtance of the fuzzy et [3]. In mot of the practcal applcaton, the unvere of the fuzzy partton (ued a prmary et of the fuzzy rulebae) can be decrbed by vague envronment (baed on the mlarty or ndtnguhablty of the element []). The mlarty relaton n a vague envronment can be defned by a calng functon. Comparng a decrpton of a unvere gven by a fuzzy partton to the way of ung the concept of vague envronment we can ay, that the lngutc term of the fuzzy partton are pont n the vague envronment, whle the hape of the fuzzy et decrbed by the calng functon. So the mlarty meaure of fuzzy et needed for approxmate reaonng can be calculated a vague dtance of pont. Th cae the prmary fuzzy et of the antecedent and the conequent part of the fuzzy rule are pont n ther vague envronment, o the fuzzy rule themelve are pont n ther vague envronment too (n the vague envronment of the fuzzy rulebae). It mean, that the queton of approxmate fuzzy reaonng can be reduced to the problem of nterpolaton of the rule pont n the vague envronment of the fuzzy rulebae relaton [5,6].. Connecton between mlarty of fuzzy et and vague dtance of pont n a vague envronment The concept of vague envronment baed on the mlarty or ndtnguhablty of the element. Two value n the vague envronment are ε-dtnguhable f ther dtance grater then ε. The dtance n vague envronment are weghted dtance. The weghtng factor or functon called calng functon (factor) []. Two value n the vague envronment X are ε-dtnguhable f x ( x, x ) ( x) dx where δ ( x x ) ε > δ x, the vague dtance of the value x, x and (x) the calng functon on X

2 For fndng connecton between fuzzy et and a vague envronment we can ntroduce the memberhp functon x ( x) a a level of mlarty x to x, a the degree to whch x ndtnguhable to x []. The α-cut of the fuzzy et x ( x) the et whch contan the element that are ( α)-ndtnguhable from x (ee Fg..): δ ( ab, ) α b x( x) mn { δ ( ab, ), } mn ( x) dx, a Fg.. It very eay to reale (ee Fg..), that th cae the vague dtance of pont a and b (δ ( a, b )) bacally the Dcontency Meaure (S D ) of the fuzzy et and B (where B a ngleton): S up x (, ) δ ( ab, ), D x X B( ) δ a b f [ ] where B the mn t-norm. It mean, that we can calculate the dcontency meaure between member fuzzy et of a fuzzy partton and a ngleton, a vague dtance of pont n the vague envronment of the fuzzy partton. The dfference between the dcontency meaure and the vague dtance, that the vague dtance are not lmted to [,]. That why they are ueful n approxmate fuzzy reaonng. So f t poble to decrbe all the fuzzy partton of the prmary fuzzy et (the antecedent and conequent unvere) of our fuzzy rulebae, and the obervaton a ngleton, we can calculate the dcontency meaure of the antecedent prmary fuzzy et of the rulebae and the obervaton, and the dcontency meaure of the conequent prmary fuzzy et and the conequence (we are lookng for) a vague dtance of pont n the antecedent and conequent vague unvere.. Generatng vague envronment from the fuzzy partton of the lngutc term of the fuzzy rule The vague envronment decrbed by t calng functon. For generatng a vague envronment we have to fnd an approprate calng functon, whch decrbe the hape of all the term n the fuzzy partton. The method propoed by Klawonn [], for choong the calng functon, gve an exact decrpton of the fuzzy term after ther recontructon from the calng functon (e.g.fg..): d x ( ) '( x) dx

3 Fg.. fuzzy et and t calng functon So we alway fnd a calng functon, f we have only one fuzzy et n the fuzzy partton. Uually the fuzzy partton contan more than one fuzzy et, o th method requre ome retrcton for the memberhp functon of the term [] (e.g.fg.3.): ext ff { } d x ( ) '( x) mn ( x), j( x) > ' ( x) ' j ( x), j I dx Z PS PM P X There are a calng functon ext, whch There are no calng functon decrbe decrbe all the fuzzy et both the fuzzy et Fg.3. Generally the above condton not fulfllng, o the queton how to decrbe all fuzzy et of the fuzzy partton wth one unveral calng functon. For th reaon we propoe to ue the approxmatve calng functon. 3. The approxmatve calng functon The approxmatve calng functon an approxmaton of the calng functon decrbe the term of the fuzzy partton eparately. The mplet way of generatng the approxmatve calng functon the lnear nterpolaton of the calng functon between the neghbourng term. Suppong that the fuzzy term are trangle, each fuzzy term can be charactered by two contant calng functon, the calng factor of the left and the rght lope of the trangle. So a trangle haped fuzzy term can be charactered by three value (by a trple), by the value of the left and the rght calng factor and the value of t core pont (e.g.fg..).

4 Fg.. Two trangle haped fuzzy et charactered by two trple, by the left and the rght calng factor and the value of the core pont Ung the method of lnear nterpolaton, we are calculatng the approxmatve calng functon a a pecewe lnear functon, whch a lnear nterpolaton of the rght de calng factor of the left neghbourng term and the left de calng factor of the rght neghbourng term (e.g.fg.5.): where (x) x x + ( x) ( x x) + x [ x, x +), [, n ] x, n + the approxmatve calng functon the core of the th term of the approxmated fuzzy partton are the left and rght de calng factor of the th trangle haped term of the approxmated fuzzy partton the number of the term n the approxmated fuzzy partton Fg.5. pproxmatve calng functon generated by lnear nterpolaton of two trangle haped fuzzy term (Fg..), and the approxmated fuzzy partton The man problem of the lnearly nterpolated calng functon, that they cant handle the bg dfference n neghbourng calng factor or crp fuzzy et correctly. If there are bg dfference n the neghbourng calng factor, the bgger calng factor domnatng the maller factor (ee e.g.fg.7,8). For example f one of the neghbourng fuzzy et crp (t calng factor nfnte), the lope of the lnearly nterpolated calng functon nfnte too, o both the fuzzy et decrbed by th calng functon wll be crp. To olve th dlemma, we ugget to adopt the followng non-lnear functon for nterpolatng the neghbourng calng factor: ( x) w ( d +) w ( d +) k w k w k w ( d +) k ( x x +) k w ( d +) ( x x +) + + +, x [ x, x +), [, n ], +, + < w + k w

5 where w +, [, n ] d x+-x, [, n ] (x) the approxmatve calng functon x the core of the th term of the approxmated fuzzy partton, are the left and rght de calng factor of the th trangle haped term of the approxmated fuzzy partton k contant factor of entvty for neghbourng calng factor dfference n the number of the term n the approxmated fuzzy partton The above functon ha the followng ueful properte: If the neghbourng calng factor are equal, (x) lnear [ ) ( x), x x, x + + If one of the neghbourng calng factor nfnte e.g. (the rght de of the th term crp) then x x and + fnte, x [ x, x+) ( x) otherw e mlarly x x + and + fnte, x [ x, x+) ( x) otherw e Fg.6. and Fg.9. how ome example for the applcaton of the propoed nonlnear functon. + Fg.6. Example for the propoed non-lnear functon (x, x,, k)

6 Fg.7. Fuzzy et wth bg dfference n neghbourng calng factor value ( << B) Fg.8. nearly nterpolated calng functon of fuzzy et hown on Fg.7., and thee et a the approxmatve calng functon decrbe them (,B ) Fg.9. pproxmatve calng functon generated by the propoed non-lnear functon (k), and the orgnal fuzzy partton (,B) a th calng functon decrbe t (,B ). Calculatng the concluon by approxmatng the vague pont of the rulebae If the vague envronment of a fuzzy partton (the calng functon or the approxmatve calng functon) ext, the member et of the fuzzy partton can be charactered by pont n the vague envronment. (In our cae the pont are characterng the core of the term, whle the hape of the memberhp functon are decrbed by the calng functon.) If all the vague envronment of the antecedent and conequent unvere of the fuzzy rulebae are ext, all the prmary fuzzy et (lngutc term) ued n the fuzzy rulebae can be charactered by pont n ther vague envronment. So the fuzzy rule (buld on the prmary fuzzy et) can be charactered by pont n the vague envronment of the fuzzy rulebae too. Th cae the approxmatve fuzzy reaonng can be handled a a clacal nterpolaton tak. pplyng the concept of vague envronment (the dtance of pont are weghted dtance), we can ue the clacal nterpolaton method for approxmatve fuzzy reaonng. For example we can adopt the method of lnear rule nterpolaton of two fuzzy rule for vague envronment. Th nterpolaton method deal only wth two rule

7 from rule bae, whoe antecedent are the cloet flankng antecedent to the obervaton. The Fundamental Equaton of the near Interpolaton of Two Fuzzy ule : dt(,x) : dt(x,) dt(b,y) : dt(y,b) where x and B B the two fuzzy rule flank the obervaton x B [,] n ene of orderng (The retrcton for the two choen rule are: the flankng of the obervaton on the antecedent de and the extence of the orderng on the conequent de. The dt(f,g) denote the fuzzy dtance. See more detaled n [3].) Wthout hurtng generalty we uppoe, that the conequence unvere of the fuzzy rule one dmenonal (multdmenonal cae can be decompoed to one dmenonal one) and antecedent unvere multdmenonal. Subttutng the formula of vague dtance to the equaton of the lnear nterpolaton of two fuzzy rule we get (the multdmenonal dtance are n Eucldean ene): x m y B m x ( x ) dx : x ( x ) dx ( y) dy ( y) dy Y : Y x B y where x the th calng functon of the m dmenonal antecedent unvere y the calng functon of the conequence unvere For ung of th method n one dmenonal antecedent unvere cae ee example on Fg.. Fg.. near nterpolaton of two fuzzy rule n approxmated vague envronment, the approxmatve calng functon are generated by lnear nterpolaton of the calng value pont,, B, B, the fuzzy rule are : B, : B The next example for nterpolaton n vague envronment the followng method. It an extenon of the method of lnear nterpolaton of two fuzzy rule to all the rule n the fuzzy rulebae (propoed n []). Th method calculate the concluon a a weghted um of the conequent part of the rule. The weghtng factor are nverely proportonal to the dtance between the

8 obervaton and the correpondng rule antecedent. Subttutng the formula of vague dtance to the equaton we get: j m δ x ( x, ) x ( x) dx j, x δ y ( y, y) n (, ) w δ y y w y y n y w j j y ( δ ( x, ) ) x ( y) dy y Where the rule are B, the number of the rule n, x the obervaton, y the concluon, δ x ( f, g) and δ y ( f, g) the vague dtance of f and g pont on the antecedent and conequent de, p the factor determne the entvene of the method for dtant rule. See for example Fg.. p Orgnal fuzzy partton:.5 X : X : S M.5 S X pproxmatve calng functon obtaned from the orgnal fuzzy partton:.5 S M.5 S X X Fuzzy partton decrbed by the approxmatve calng functon:.5 S M.5 S X Y: S M S M S M X Y Y Y ulebae: f X S and X S then YM f X M and X S then Y f X and X S then Y f X S and X then YS f X M and X then YM f X and X then Y X

9 Y Y 3 3 X 3 3 X 3 X 3 X Fuzzy nterpolatve reaonng of x fuzzy rule (p3) Max-mn compoton wth centre of gravty defuzzfcaton Fg.. Interpolaton of x fuzzy rule n approxmated vague envronment, compared to the max-mn compoton wth centre of gravty defuzzfcaton 5. Generatng the fuzzy concluon decrbed by the pont of the vague concluon, ung the vague envronment of the conequence unvere The vague concluon calculated by rule nterpolaton bacally one pont. For tranformng th pont to a fuzzy concluon, we have to examne the conequence unvere. Suppong that the term n the fuzzy partton of the conequence unvere decrbe all the man properte of the conequence unvere and the calng functon approxmated from th term proper, we can calculate the memberhp functon of the fuzzy concluon a a level of mlarty to the vague concluon n the vague envronment of the conequence unvere (e.g. Fg.): yv y( y) mn { δy( y, yv), } mn y( y) dy,, y where y the calng functon of the conequence unvere the vague concluon y v 6. Extendng the method of vague envronment baed approxmatve fuzzy reaonng from ngleton to fuzzy obervaton Ung the dea of equal Dcontency Meaure we can extend the concept of vague envronment from mlarty relaton between the member et of a fuzzy partton and ngleton to mlarty relaton between the member of two fuzzy partton: S up x (, ) δ ( ab, ), D x X B( ) δ a b f [ ] We would lke to fnd a reultant calng functon ( x), whch unfe the two calng functon of the two ndependent fuzzy partton, generatng a fuzzy et, whch ha the ame dcontency meaure to a ngleton, a the two fuzzy et decrbed by the orgnal calng functon had (ee Fg..): x δ ( ab, ) B ( x) dx ( x) dx ( ) a b x b a x dx

10 Fg.. The reultant calng functon ( x), ha the ame dcontency meaure to a ngleton, a the two fuzzy et decrbed by the orgnal calng functon have Th queton very eay to olve f the calng functon are contant (Fg..): C CB m C ( x a) CB ( b x ) ( b a) C ( b a) C + CB C CB ( x) B( x) C ( x) x X C + C x + x B ( ) ( ) It very eay to proof, that generally the reultant calng functon not ext, but the reult of the contant calng functon cae, a a knd of approxmaton, eem to be ueful n practcal applcaton. Ung the propoed formula ( ( x) above) we can unfy two ndependent fuzzy partton (two ndependent vague envronment) nto one, mplfyng the queton of calculatng the dcontency meaure of two arbtrary fuzzy et to calculatng the dcontency meaure of one fuzzy et and a ngleton - to calculate the vague dtance of two pont n a vague envronment (e.g.fg.3.). In practcal ene t mean, that f t poble to decrbe the unvere of obervaton by vague envronment, we can unfy thee vague envronment wth the approprate vague envronment of the antecedent unvere of the fuzzy rulebae. Th way we can tranform the method of vague envronment baed approxmatve fuzzy reaonng from ngleton to fuzzy obervaton. B Fg.3. Unfyng the calng functon, B - the calng functon of fuzzy partton, B to - the calng functon of fuzzy partton (th cae B a ngleton) Concluon Ung the concept of vague envronment n mot of the practcal cae we can bult approxmate fuzzy reaonng method mple enough to be a good alternatve of the clacal Compotonal ule of Inference method n practcal applcaton.

11 The advantage of the method propoed n th paper (compared to CI) are the followng: - the computatonal effort needed for the concluon can be reduced by reducng the number of the fuzzy rule (the unmportant fllng rule can be elmnated), - the propoed method gve concluon n cae of nuffcent evdence (pare fuzzy rulebae) too, - ung the propoed approxmate fuzzy reaonng method, f crp concluon needed, t can be fetched drectly from the vague concluon (there no addtonal defuzzfcaton tep needed). eference [] Turken, I.B., Zhong, Z.: n pproxmate nalogcal eaonng Schema Baed on Smlarty Meaure and Interval-valued Fuzzy Set, Fuzzy Set and Sytem, vol.3, pp 33-36, (99). [] Klawonn, F.: Fuzzy Set and Vague Envronment, Fuzzy Set and Sytem, 66, pp7-, (99). [3] Kóczy,. T., Hrota, K.: Interpolatve reaonng wth nuffcent evdence n pare fuzzy rule bae, Informaton Scence 7, pp 69-, (99). [] Va, Gy., Kalmár,., Kóczy,.T.: Extenon of Fuzzy ule Interpolaton Method, Int. Conference on Fuzzy Set Theory and pplcaton, ptovky Mkula, Czecho-Slovaka, (99). [5] Kovác, Sz., Kóczy,.T.: Fuzzy ule Interpolaton n Vague Envronment, Proceedng of the 3rd. European Congre on Intellgent Technque and Soft Computng, pp.95-98, achen, Germany, (995). [6] Kovác, Sz.: New pect of Interpolatve eaonng, Proceedng of the 6th. Internatonal Conference on Informaton Proceng and Management of Uncertanty n Knowledge-Baed Sytem, pp.77-8, Granada, Span, (996).

Distance based similarity measures of fuzzy sets

Distance based similarity measures of fuzzy sets Johanyák, Z. C., Dr. Kovác Sz.: Dtance baed mlarty meaure of fuzzy et, SAMI 2005, 3rd Slovakan-Hungaran Jont Sympoum on Appled Machne Intellgence, Herl'any, Slovaka, January 2-22 2005, ISBN 963 75 35 3,