Vague environment based two-step fuzzy rule interpolation method

Transcription

1 Johanyák, Z. C., Kovác, Sz.: Vague Envronment-baed Two-tep Fuzzy Rule Interpolaton Method, 5th Slovakan-Hungaran Jont Sympoum on ppled Machne Intellgence and Informatc (SMI 27), January 25-26, 27 Poprad, Slovaka, ISBN , pp valable at: Vague envronment baed two-tep fuzzy rule nterpolaton method Zolt Caba Johanyák, Szlvezter Kovác Inttute of Informaton Technology, Keckemét College, GMF Faculty, Keckemét, H-6 Pf. 9, btract: The am of th paper to ntroduce a novel two-tep Fuzzy Rule Interpolaton Technque (FRIT) VEIN, baed on the concept of Vague Envronment. The trength of FRIT agant clacal fuzzy reaonng method the ablty of ganng concluon even n cae where the knowledge repreented by pare fuzzy rule bae. The FRIT VEIN ntroduced n th paper followng the tructure the Generalzed Methodology of fuzzy rule nterpolaton [], by adaptng the concept of Vague Envronment [4] for appromate decrpton of fuzzy partton [6]. Keyword: Vague Envronment, Fuzzy Rule Interpolaton, Fuzzy Set Interpolaton, SngleŰ Rule Reaonng Introducton The conventonal fuzzy rule baed ytem applyng ether the Zadeh-Mamdan- Laren [4][9][8] concept or the Takag-Sugeno [3] approach requre the dene character of the rule bae. Havng a pare rule bae they cannot fre any of the rule for ome obervaton, whch do not overlap any of the rule antecedent at leat partally. In uch cae the fuzzy ytem baed on them cannot produce an acceptable output. The problem can be olved by the applcaton of Fuzzy Rule Interpolaton Technque (FRIT). Snce 99 numerou FRIT have been propoed. They can be dvded nto two man group. The frt group produce the appromated concluon from the obervaton drectly; therefore t member are called one-tep method. The member of the econd group reach the target n two tep. In the frt tep they nterpolate a new rule whoe antecedent part overlap the obervaton at leat partally. Then n the econd reaonng tep the etmated concluon determned by a Sngle Rule Reaonng (revon) method baed on the mlarty of the obervaton and the antecedent part of the newly nterpolated rule. The man tructure of the two-tep method are ummared n the Generalzed Methodology (GM) of fuzzy rule nterpolaton ntroduced n [] by Barany et al.

2 In th paper the author propoe a novel two-tep FRIT called VEIN, baed on the concept of Vague Envronment. Th concept, orgnally ntroduced by Klawonn n [4], baed on the mlarty or ndtnguhablty of element n a unvere. Two value n a Vague Envronment are ε-dtnguhable f ther dtance greater than ε, where the dtance are weghted dtance. The weghtng factor or functon called calng functon (factor) [4]. Hence the Vague Envronment can be characterzed by t calng functon. There are way for fndng relaton between the Vague Envronment and fuzzy partton ee e.g. n [4], [6]. Therefore the concept of Vague Envronment can be alo appled for fuzzy reaonng, even for FRIT too. E.g. a one-tep FRIT method (later named a FIVE ) ntroduced n [6]. 2 Vague envronment baed two-tep fuzzy rule nterpolaton method The Vague Envronment baed two-tep fuzzy rule INterpolaton (VEIN) method eentally follow the concept lad down by the Generalzed Methodology (GM) of fuzzy rule nterpolaton []. It calculate the concluon n two tep. It. nterpolate a new rule n the poton of the obervaton, 2. determne the concluon by frng the new rule. For the facltaton and mplfcaton of the calculaton all nput and output partton are normalzed to the unt nterval. The poton of the lngutc term defned by t reference pont. We ue the centre of the core for th tak. The dtance between the et epreed by the mean of the Eucldean dtance of ther reference pont. In the frt tep the new rule determned n three tage. Thee are the followng. a. The hape of the antecedent et determned ung a et nterpolaton method called VESI, whch baed on the concept Vague Envronment and preented more detal n ecton 3. b. The poton of the conequent et calculated by a crp nterpolaton technque preented n ecton 4. c. The hape of the conequent et calculated ung the ame et nterpolaton technque a n tage a. The fnal concluon determned n the econd tep of VEIN applyng a pecal ngle rule reaonng method called REVE that alo baed on the concept Vague Envronment. Secton 5 ntroduce REVE n detal.

3 3 VESI The method VESI (Vague Envronment baed Set Interpolaton) am the determnaton of a new lngutc term n a pecfc pont, called nterpolaton pont, of a fuzzy partton. It can be appled for the determnaton of the antecedent and conequent et of the new rule n the frt tep of any FRI technque, whch follow the concept of the GM []. The method the ame regardle of t ued n cae of a rule preme or a rule conequent. It can be appled n both the cae of pare and dene partton. Smlar to the other et or rule nterpolaton/etrapolaton/appromaton technque VESI aume regularty between the lngutc term of a fuzzy partton. The frt tep of the technque the generaton of the VE of the partton, whch ha to be done only once, before tartng the fuzzy ytem baed on t. Throughout the coure of the repettve reaonng tep the orgnal VE ued n the calculaton..8 µ Fgure Spare partton wth two trapezod haped lngutc term The econd tep of VESI tart wth a gven nterpolaton pont n the current dmenon (partton), whch actually a reference pont of a fuzzy et. The bac dea that the new et conerve the properte of the VE. Let the prototypcal pont (where the memberhp value of the requeted fuzzy et equal to ) be condered to be the nterpolaton pont. Startng from t one can generate the hape of the new lngutc term ealy from the calng functon. Further on the algorthm preented by a numercal eample. For mplcty and lucdty the ample partton (fg. ) pare and t contan two trapezod haped CNF et only. The ample lngutc term are defned a follow {. /,.2 /,.3 /,.4 / }, () {.7 /,.8 /,.9 /,. / }, (2) 2

4 where the frt ubcrpt of the et denote the ordnal number of the nput dmenon and the econd ubcrpt ndcate the ordnal number of the et n the current partton (dmenon). The reference pont of the et are defned by the md pont of the core ( RP ( ). 25 and RP ( 2 ). 85 ). In th eample for the ake of mplcty the rght flank of the et and the left flank of the et 2 R,. have the ame lope n abolute value. The range of the partton [ ] The calng functon buld up from the calng factor (3)-(8) accordng to the memberhp hape of the element fuzzy et dµ d.2 L L dµ d. C C dµ d. R R dµ d. L L 2 2 dµ d. C C 2 2 dµ d. R R 2 2 5, (3), (4), (5), (6), (7), (8) Z where X the calng factor correpondng to the Z (left flank, rght flank or core) part of the hape of the fuzzy et X ( or 2 ). To be conform to the etended concept of the VE [5][7] each nterval of the calng functon delmted by prototypcal pont ncludng here alo the pare porton of the partton defned only by the neghborng flank of the et. Thu the nterval [.4,.7) characterzed by the ame calng factor a t urroundng nterval [.3,.4) and [.7,.8). In a mlar way n cae of the leadng and tralng empty (pare) porton of the partton the calng factor of the cloet et flank are ued for the generaton of the VE. Th feature enure the capablty of nterpolaton and etrapolaton for the VE. Fgure 2 preent the calng functon of the partton preented n fg.. Let we uppoe that the nterpolaton pont gven by the abca.5 and we ntend to generate a normal ( heght ( ) ) fuzzy et. The reference pont of the new lngutc term wll be dentcal wth th pont RP ( ). 5. Thu the

5 memberhp value of th pont ( RP( ) µ. The hape of the new et calculated by ntegratng the calng functon. The left and rght flank are calculated eparately. The left flank determned pont-we tartng from the reference pont and proceedng toward (the lower endpont of the range of the lngutc varable) by applyng the formula ( ) RP( ) L µ ( ) ma, d. (9) L 4 C C R L R 2 2 Fgure 2 Vague Envronment of the partton The rght flank determned n a mlar way only the endpont of the nterval and the drecton are changed { } ma, R µ d. () RP( ) µ Fgure 3 The reultng partton after the nterpolaton In th cae one proceedng toward (the upper endpont of the range of the lngutc varable). By both de the pont are calculated ung a loop contruct.

6 The termnaton condton ether the zero memberhp value of the current pont or the reachng of an endpont of the range of the lngutc varable. In our eample the reultng et trangle haped wth the charactertc pont tuated at {.4,.5,.6}. The partton contanng the nterpolated lngutc term preented on fgure 3. VESI fulfl mot of the requrement defned n [3] a General condton on FRI technque. Some relevant feature are lted below: The new et never can be an abnormal one due to the nature of the VE and to the algorthm that calculate the two flank eparately and jon them n the prototypcal pont. VESI preerve the n between feature becaue the reference pont of the new et dentcal wth the pont of the nterpolaton. The condton compatblty wth the rule bae only fulflled when the neghborng flank have the ame lope n abolute value. The fuzzne of the reult depend on the hape of the neghborng flank of the urroundng et only n the mplet cae (ee fgure and 3 ). Havng an nterpolaton pont n the cloet neghborhood of the reference pont of a lngutc term the other flank of the et alo eerce nfluence on the calculaton. Therefore the hape of the obtaned et can contan ome breakpont. a conequence of the above decrbed feature VESI not alway conerve the pece-we lnearty of the urroundng et. FRI method baed on the propoed VESI can be ealy etended to handle multdmenonal antecedent unvere a well. 4 Poton of the conequent et The poton of each conequent et of the new rule dentcal wth the reference pont of the fnal concluon n the repectve dmenon. They are calculated ndependently n the ame manner each output dmenon. Henceforth n the notaton of the equaton only the k th dmenon wll be preented. The bac aumpton of the calculaton that each rule can be vewed a a pont on a hyper-urface. The co-ordnate of th pont are the reference pont of the RP ) and the reference pont of the lngutc term of the rule antecedent ( ( ) rule conequent ( RP ( B km )). jm

7 Thu the problem of fndng the poton of the conequent et of the nterpolated rule can be reduced to a crp n a dmenonal nterpolaton of rregularly paced data, where n a the number of antecedent dmenon. Due to t eay applcablty we ue the etended veron of the Shepard nterpolaton [] for th tak [2]. It calculate the demanded abca a a weghted average of the reference pont of the conequent et that belong to the known rule RP ( B ) N m k N where ( ) ( B ) RP km wm, () B k m w m RP the reference pont of the conequent et of the nterpolated rule n the k th output dmenon, N the number of the rule, B km the conequent lngutc term of the m th rule n the k th dmenon and w m the weght attached to the m th rule. The weghtng dtance dependent. The poton of each rule antecedent and alo the poton of the obervaton n the antecedent hyper-pace can be characterzed by pont defned by the RP of the lngutc term ued a coordnate. The appled weghtng factor ue the Eucldean dtance between thee pont w m, (2) 2 n ( ( ) a d Rm, ( RP( jm ) RP( j ) j where R m the antecedent of the m th rule, the obervaton, jm the antecedent lngutc term of the m th rule n the j th nput dmenon, the obervaton n the j th dmenon. 2 j 5 REVE The concluon calculated n the econd tep of the method by frng the nterpolated rule. The antecedent part of the new rule doe not ft alway perfectly the obervaton. Therefore a revon baed ngle rule reaonng method appled, whch calculate the concluon by modfyng the conequent et of the new rule. Th modfcaton related to the dfference between the rule antecedent et and the obervaton et. REVE (Revon method baed on the Vague Envronment) a novel ngle rule reaonng method and alo baed on the concept Vague Envronment. It

8 uggeted a a complementary of the fuzzy et nterpolaton method VESI. However, t can be alo appled a ngle rule reaonng technque n cae of any two-tep fuzzy rule nterpolaton method that follow the concept of GM [] µ R L R L C C Fgure 4 One dmenonal antecedent unvere of dcoure, obervaton ( - bold lne), nterpolated antecedent et ( ) and the correpondng calng functon The man dea of REVE the conervaton of the calng functon rato n ngle rule reaonng. Havng Vague Envronment (and hence calng functon) on both the rule antecedent and conequent de, the calng functon rato between the rule obervaton and the antecedent hould be equal to the calng functon rato between the demanded concluon and the rule conequent. In other word, the mlarty of fuzzy et epreed n the form of the mlarte of the correpondng Vague Envronment, n ther calng functon rato. For multdmenonal antecedent unvere, the bac calng functon rato dea could be mply etended to mean calng functon rato a well. REVE frt calculate for each nput dmenon the rato of the calng functon, ee fg. 4 ) and the decrbng the Vague Envronment of the obervaton ( ( ) calng functon of the antecedent partton ( r j j j j, (3) where j the number of the current antecedent dmenon. Further on for a mple demontraton of the calculaton we conder a SISO (Sngle Input Sngle Output) fuzzy ytem, whch defned a follow. It antecedent unvere decrbed by () and (2) (ee fg. ). The conequent unvere gven by (4) and (5) (ee fg. 5 ). The rule bae contan two rule: B and 2 B. The obervaton 2 trangle haped and defned by (6) (ee fg. 4). The reference pont of the ) j

9 concluon ( ( ) 427 RP B. Shepard nterpolaton. {.5/,.2 /,.25/ } ) wa calculated by the etended veron of the B (4) 2 {.75/,.8 /,.85/ } B (5) {.3 /,.5/,.7/ } (6) Thu the rato of the calng functon r 5/5, 5/, 5/.5, [,.2) [.2,.3) [.8,.9) [.3,.8) [.9,] Net the harmonc mean of the antecedent rato calculated mr n a j na r j (7), (8) where n a the number of antecedent dmenon. In our eample mr r. The ret of the calculaton are done eparately for each conequent dmenon. Further on the cae of the k th output dmenon condered. In the eample k. µ.8 B B B B log() B B Fgure 5 One dmenonal conequent unvere, nterpolated conequent et ( B ), concluon ( B -bold lne), and the correpondng calng functon The bac dea of REVE the prncple of the conervaton of the mean calng functon rato. Thu the calng functon nducng the Vague Envronment of the concluon determned by conderng the ame rato between the calng functon of the concluon and the conequent partton a the mean calng functon rato calculated on the antecedent de

10 r Bk mr Bk, (9) Bk where B the calng functon of the k th conequent dmenon, and k B the calng functon decrbng the concluon n the k th dmenon (ee fg. 5 ). The calng functon of the concluon reult from the formula (9) a follow mr. (2) Bk Bk Thu the eample applcaton lead to the functon (2), a t demontrated n fgure 5. B 2,,, [,.2) [.2,.3) [.8,.9) [.3,.8) [.9,] The rght de of fg. 5 preent the calng functon of the concluon calculated for the whole range [, ]. Th done for llutraton purpoe, n practcal applcaton t ha to be calculated only for that porton of the unvere of dcoure, whch neceary for the determnaton of the hape. The memberhp functon calculated mlar to the eample preented n the ecton decrbng VESI (ee. (9) and ()). It trangle haped wth charactertc pont at {.327,.427,.527}. The method REVE ha low computatonal complety and t effectvene ncreae wth the number of conequent dmenon. It due to the fact that the calculaton on the antecedent de have to be done only once and the concluon et are determned eparately n each conequent dmenon. Thu everal tep can made parallel f the computng envronment t enable. The computatonal need can be reduced further when the method appled together wth VESI by ung the ame pregenerated antecedent and conequent Vague Envronment. The adaptaton of the harmonc mean enure that the concluon wll be crp ( ). The B ) f and only f all obervaton et are crp ( ( ) j k condton on compatblty wth the rule bae ntroduced n [3] alo fulflled (f the calng functon of the obervaton and the rule antecedent are the ame n all dmenon the concluon wll be equal to the rule conequent). Due to the rato baed conervaton prncple, the drecton of the fuzzne changng of the concluon follow the drecton of the fuzzne changng of the obervaton. Owng to thee two feature the le uncertan the obervaton the le fuzzne wll have the appromated concluon. nother advantage of the propoed REVE method the lack of verfcaton and correcton tep requred n many other method for ganng vald, conve and j k (2)

11 normal fuzzy concluon, hence from the method t traghtforward, that the concluon of REVE alway a vald conve and normal fuzzy et. The man dadvantage of REVE the lack of compatblty wth the rule bae n cae, when only an appromate calng functon can be determned n any of the antecedent or conequent partton. In addton the pece-we lnearty alo not alway conerved. Concluon FRIT offer a utable oluton for handlng fuzzy ytem where the knowledge repreented by a pare fuzzy rule bae. Th paper ntroduced a novel two-tep FRIT called VEIN, baed on the concept of Vague Envronment. an addtonal beneft of the propoed method, VEIN fulfl mot of the FRIT condton uggeted n [3]. The man advantage of the propoed method t quck nference proce, whch enure the real-tme applcablty a well. a man drawback, the lack of compatblty wth the rule bae n cae of ome memberhp functon type when appromate calng functon are requred hould be alo mentoned. Th topc ubject of further reearch work. cknowledgement Th reearch wa upported by Keckemét College GMF Faculty grant no: N76/26. Reference [] Barany, P., Kóczy, L. T. and Gedeon, T. D.: Generalzed Concept for Fuzzy Rule Interpolaton. IEEE Tran. on Fuzzy Sytem, vol. 2, No. 6, 24, pp [2] Johanyák, Z. C. and Kovác Sz.: Fuzzy Rule Interpolaton Baed on Polar Cut, Computatonal Intellgence, Theory and pplcaton, Sprnger Berln Hedelberg, 26, ISBN , pp [3] Johanyák, Z. C. and Kovác, Sz.: Survey on varou nterpolaton baed fuzzy reaonng method, Producton Sytem and Informaton Engneerng Volume 3 (26), HU ISSN , pp [4] Klawonn, F.: Fuzzy Set and Vague Envronment, Fuzzy Set and Sytem, 66 (994) [5] Kovác, Sz. and Kóczy, L. T.: pplcaton of an appromate fuzzy logc controller n an GV teerng ytem, path trackng and collon avodance trategy, Fuzzy Set Theory and pplcaton, In Tatra Mountan Mathematcal Publcaton, Mathematcal Inttute Slovak cademy of Scence, vol.6, Bratlava, Slovaka, 999, pp [6] Kovác, Sz., Kóczy, L.T.: The ue of the concept of vague envronment n appromate fuzzy reaonng, Fuzzy Set Theory and pplcaton, Tatra

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