Research Article A Proposal to Speed up the Computation of the Centroid of an Interval Type-2 Fuzzy Set

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1 Advances n Fuzzy Systems Volume 23, Artcle ID 58969, 7 pages Research Artcle A Proposal to Speed up the Computaton of the Centrod of an Interval Type-2 Fuzzy Set Carlos E. Celemn and Mguel A. Melgarejo Laboratoro de Automátca e Intelgenca Computaconal, Unversdad Dstrtal Francsco Jose de Caldas, Carrera 8 o. 4-62, Pso 7, Bogota, Colomba Correspondence should be addressed to Mguel A. Melgarejo; mgbet@gmal.com Receved 28 August 22; Accepted 2 October 22 Academc Edtor: M. Onder Efe Copyrght 23 C. E. Celemn and M. A. Melgarejo. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Ths paper presents two new algorthms that speed up the centrod computaton of an nterval type-2 fuzzy set. The algorthms nclude precomputaton of the man operatons and ntalzaton based on the concept of uncertanty bounds. Smulatons over dfferent knds of footprnts of uncertanty reveal that the new algorthms acheve computaton tme reductons wth respect to the Enhanced-Karnk algorthm, rangng from 4 to 7%. The results suggest that the ntalzaton used n the new algorthms effectvely reduces the number of teratons to compute the extreme ponts of the nterval centrod whle precomputaton reduces the computatonal cost of each teraton.. Introducton Type-2 fuzzy logc systems (T2-FLS) theory and ts applcatons have grown n recent years [ 3]. One of the man problems related to the mplementaton of these systems s type reducton, whch computes the generalzed centrod of a type-2 fuzzy set (T2-FS) [4 6]. Ths operaton becomes computatonally smpler when performed over a partcular class of T2-FS, namely, nterval type-2 fuzzy set (IT2-FS) [5, 7]. Bascally, the centrod of an IT2-FS s an nterval [3, 8]. Therefore, computng ths centrod can be consdered as an optmzaton problem that fnds the extreme ponts that defne the nterval [6]. FastcomputatonoftypereductonforIT2-FSssan attractve problem, whch s crtcal snce type-reducton procedures for more general type-2 fuzzy sets (T2-FSs) make use of nterval type-2 fuzzy computatons [4, 9 ]. Up-todate, several teratve approaches to computng type reducton for IT2-FSs have been proposed [5, 2 8]. The Karnk- Mendel (KM) algorthms are the most popular procedures used n nterval type-2 fuzzy logc systems (IT2-FLS) [7]. It has been demonstrated that the KM algorthms converge monotoncally and superexponentally fast. These propertes are hghly desrable n teratve algorthms [3]. However, KM procedures converge n several teratons and demand a consderable amount of arthmetc and memory lookup operatons [7, 8, 2, 9], for real IT2-FLS mplementatons. In the case of IT2 fuzzy controllers, the computatonal cost of type reducton s an mportant subject [2, 2]. The overall complexty of the controller largely depends on the type-reducton and defuzzfcaton stages. Thus developng strateges for reducng the computatonal burden and necessary resources for mplementng these two stages s hghly convenent. In addton, there are other applcatons of IT2- FLS n whch the complexty of the hardware and software platforms should be reduced n order to guarantee the fastest possble executon of fuzzy nferences [22, 23]. onteratve type reducton of IT2-FS can be acheved by means of uncertanty bounds [7]; however, ths method s approxmate and nvolves relatvely complex computatons. Other methods for computng the centrod of an IT2-FS have been proposed, for example, geometrc type-reducton [4], genetcally optmzed type-reducers [8], nterval-analyssbased type-reducton [24], and samplng defuzzfcaton and collapsng defuzzfcaton [2]. Although these alternatve methods exhbt nterestng propertes, they are not as popular as the KM algorthm.

2 2 Advances n Fuzzy Systems An enhanced verson of the KM algorthm () was presented n [6]. Ths verson ncludes a better ntalzaton and some computatonal mprovements. Expermental evdence shows that the algorthm saves about two teratons, whch means a reducton n computaton tme of more than 39% wth respect to the orgnal algorthm. By the tme the algorthm was presented, and the teratve algorthm wth stop condton () was also ntroduced by Melgarejo et al. [4, 5, 9]. Ths algorthm deals wth the problem of computng type-reducton for an IT2-FS by usng some propertes of the centrod functon [3, 9]. Expermental evdence showed that s faster than the algorthm n some cases. Recently, has been enhanced by Wu and e [8] leadng to a new verson called, whch proved to be the fastest algorthm wth respect to the and algorthms for several type-2 fuzzy logc applcatons. The mprovement ntroduced n manly deals wth modfyng the teratve procedure for computng the rght pont of the nterval centrod. Although and outperform the algorthm, the ntalzaton of the -type algorthms s somewhat trval and does not provde an approprate startng pont, whch can lmt the convergence speed of these algorthms when calculatng the ponts of the type-reduced set. An extensve and nterestng comparson of several type-reducton methods s provded by Wu n [25]. Ths study confrmed the convenence of over the A for reducng the computatonal cost of an IT2-FLS. The experments that were lmted to nterpreted language mplementatons also showed that and the enhanced opposte drecton searchng algorthm (EODS) exhbted smlar performance over dfferent applcatons; however, the EODS outperformed n low-resoluton cases [26]. Regardng the aforementoned problem, ths work consders a dfferent ntalzaton perspectve for teratve typereducton algorthms based on the concept of nner-bound set for a type-reduced set [3]. Ths knd of ntalzaton has not been tred n type-reducton computng untl now. As t wll be analyzed later, the nner-bound ntalzaton reduces the dstance that the algorthms need to cover n order to fnd the optmal ponts of the nterval centrod. In addton, bearng n mnd the computatonal burden of teratve algorthms, ths paper also focuses on proposng an alternatve strategy to speed up ther computaton based on the concept of precomputng. Precomputaton n algorthmc desgn s a powerful concept that reduces the necessary arthmetc operatons. Current lterature does not report the useofprecomputatonnthetype-reductonstagenit2- FLSs. Usng nner bound sets ntalzaton and precomputaton, Both the and KM algorthms have been restated, leadng to faster algorthms hereafter refered to and. Our experments consderng two dfferent mplementaton perspectves (.e., nterpreted language and compled language) show that and outperform A and. In fact, tmng results suggest that may be the best opton for mplementng IT2-FLSs n dedcated hardware, whereas A may support the computaton of large-scale smulatons of IT2-FLSs over general purpose processors. The paper s organzed as follows. Secton 2 provdes some background about the centrod of an nterval type-2 fuzzy set. and are presented n Secton 3. Secton4 s devoted to a computatonal comparatve study of teratve type-reducton methods whch consders two types of mplementaton: compled-language-based and nterpretedlanguage-based. Fnally, we draw conclusons n Secton Background Thepurposeofthssectonstoprovdesomebascnformaton about the centrod of nterval type-2 fuzzy sets. The reader who s not famlar wth ths theory s nvted to consult [3, 5, 3]amongothers. 2.. Type-2 Fuzzy Sets. Accordng to Mendel [5] atype-2 fuzzy set, denoted A, s characterzed by a type-2 membershp functon μ A (x, u) where x Xand μ A (x, u) : A ={((x, u),μ A (x, u)) u J x [, ]}. () 2.2. Interval Type-2 Fuzzy Sets. An IT2-FS s a partcular case of a T2-FS whose values n μ A (x, u) are equal to one [27]. An IT2-FS s fully descrbed by ts footprnt of uncertanty (FOU) [3]. ote that an IT2-FS set can be understood as a collecton of type- fuzzy sets (.e., tradtonal fuzzy sets). The membershp functons of these sets are contaned wthn the FOU; thus, they are called embedded fuzzy sets [5, 3] Centrod of an IT2-FS. Let A be an IT2-FS, and ts centrod c( A) s an nterval gven by [5]: c (Ã) = [y l ( A), y r ( A)] = [y l,y r ], (2) where y L and y R are the solutons to the followng optmzaton problems: y L = y R = mn f [f, f ] max f [f, f ] = x f = f, = x f = f, where the unverse of dscourse X as well as the upper and lower membershp functons have been dscretzed n ponts. A smpler way to understand the centrod of an IT2-FS s to thnk about t as a set of centrods that are computed from all the embedded fuzzy sets [3, 5]. Clearly, n order to characterze an nterval set, t s only necessary to fnd ts mnmum and maxmum. The Karnk-Mendel teratve algorthms [5, 6, 7] arethemostpopularproceduresfor computng these two ponts. (3)

3 Advances n Fuzzy Systems Computng the Centrod of an IT2-FS. It has been demonstrated that y L and y R canbecomputedntermsofthe upperandlowermembershpfunctonsofthefootprntof uncertanty (FOU) of A [3, 5] as follows: y L = mn = x f f [f, f ] = f = = x f + =L+ x f = f + =L+ f, µ(x) y R = max = x f f [f, f ] = f = R = x f + =R+ x f R = f + =R+ f, (4) y L y l yr y R x where f and f are the values of the lower and upper membershp functons, respectvely, consderng that the unverse of dscourse X has been dscretzed nto ponts x. In addton L and R are two swtch ponts that satsfy the followng: x L y L x L+, x R y R x R Uncertanty Bounds of the Type-Reduced Set. The typereduced set of an nterval type-2 fuzzy gven n (2) canbe approxmated by means of two nterval sets called nner- and outer-bound sets [7]. The end y L and y R ponts of the typereduced set of an nterval type-2 fuzzy set are bounded from below and above by (5) y l y L y l, (6) y r y R y r, (7) where [y l,y r ]and[y l, y r ]arethenner-andouter-boundsets, respectvely. For the purposes of ths paper only the nner bound set s consdered, so t s computed as follows: y A = = x f = f, y B = = x f = f, y l = mn (y A,y B ), y r = max (y A,y B ) Propertes of the Centrod Functon. The defnton and correspondng study of the contnuous centrod functon are provded by Mendel and Lu n [3]. Theoretcal studes derved from (6) can be legtmated for the dscrete centrod functon [2]. Regardng ths fact, Mendel and Lu demonstrated several nterestng propertes of the contnuous centrod functon, whch are applcable also to the dscrete one. Some of these propertes can be summarzed sayng that the centrod functon decreases or has flat spots to the left of ts mnmum, and t ncreases or has flat spots to the rght of (8) (9) Fgure : Unform random FOU: f (red lne) and f (blue lne). The arrows ndcates the computaton of mnmum and maxmum. Orange arrows correspond to, green to and blue to. ths value. Addtonally, ths functon has a global mnmum when k = L. The rght-centrod functon y r has smlar propertes snce ths functon has a global maxmum when k = R. Thus, t ncreases to the left of the maxmum and decreases to the rght. 3. Modfed Iteratve Algorthms for Computng the Centrod of an Interval Type-2 Fuzzy Set 3... The -2 algorthm s presented n Table.The algorthm conssts of four stages: sortng, precomputaton, ntalzaton, and recurson. Sortng and precomputaton are the same for computng y L and y R. Precomputaton stores the partal results of computng (8); only two dvsons are requred. Those results wll be used along the other stages of the algorthm. The ntalzaton of IACS2 s characterzed by ts dependence from the nner-bound set. Ths feature ntroduces a bg dfference wth prevous algorthms snce they use fxed ntalzaton ponts (e.g., A,, and ); however, regardng ths partcularty, s smlar to the KMA, because both algorthms nclude a FOU-dependent ntalzaton. The works wth the swtch ponts startng nsde the centrod, teratng smlarly as type algorthms but from the y mn toward left sde whle the mnmum y s reached,tsntheoppostesensetoand.the procedure for computng the maxmum y R begns n y max andgoestotherghtsdeuntltsfound,andthesearch drecton s the same of but opposte to the proposed n. The advantage of ths ntalzaton for computng the centrod s that always the swtch ponts start close to the y L and y R ; therefore, less teratons are used. Fgure depcts a unform random FOU wth ts y L,y R,y mn,ory l and the y max or y r,andthearrowsshow the drecton of the search of each procedure from the ntal swtch pont to the mnmum or maxmum. Also, n ths example, t s possble to see the space scanned by every

4 4 Advances n Fuzzy Systems Table : algorthm flow. The mnmum extreme pont y L of the centrod of an nterval type-2 fuzzy set over x wth upper membershp functon f and lower membershp functon f can be computed usng the followng procedure: The maxmum extreme pont y R of the centrod of an nterval type-2 fuzzy set over x wth upper membershp functon f and lower membershp functon f can be computed by the followng procedure: () Sort x (=,2,...,) n ascendng order and call the sorted x by the same name, but now x x 2 x.matchtheweghtsf wth ther respectve x and renumber them so that ther ndex corresponds to the renumbered x. (2) Precomputaton: V [n] = n f (5) = H [n] = n f (6) = T [n] = n x f (7) = Z [n] = n x f (8) = wth n =, 2,.... T [] y A = V [] (9) Z [] y B = H [] (2) (3) Intalzaton: Intalzaton: y mn = mn(y A,y B ) (2) y max = max(y A,y B ) (32) Fnd ( )sothat Fnd R ( R )so that x L y mn <x L + x R y max <x R + D l ( )=T[ ]+(Z[] Z[ ]) (22) E R (R )=Z[R ]+(T[] T[R ]) (33) P l ( )=V[ ] + (Y [] Y[ ]) (23) Q R (R )=Y[R ] + (X [] X[R ]) (34) y mn = D (24) y P max = E R(R ) L Q R (R ) (35) D=D l ( ) (25) E=E R (36) P=P l ( ) (26) Q=Q R (37) k=. (27) k=r +. (38) (4) Recurson: Recurson: α k = f k f k (28) α k = f k f k (39) D=D x k α k (29) E=E x k α k (4) P=P α k (3) Q=Q α k (4) y l = D P. (3) y r = E Q. (42) (5) If y l y mn,theny mn =y l, k=k and go to step 4 else y L =y mn and stop. If y R y max,theny max =c R and k=k+go to step 4 else y R =y max and stop. algorthm, suggestng that ths space s proportonal to the amount of teratons requred to converge. ote that has the shortest arrows, whch mean few teratons. In Table (2), y mn corresponds to the mnmum of the nner-bound set; thus accordng to (6) y mn y L. Therefore Table (2) provdes an ntalzaton pont that s always greater than or equal to the mnmum extreme pont y L ;In addton, t s possble to fnd an nteger so that x L y mn < x L +. ItseasytoshowthatTable (22) (24) s computng the left centrod functon wth k=,whchfxes the startng pont of recurson from the values obtaned n the precomputaton stage. Recurson Table (28) (3) computes the left centrod functon wth ntal pont n k =, and each decrement n k leads to a decrement n y l (k). Bydongsomealgebrac operatons,tseasytoshowthaty l (k) = ( k = x f + =k+ x f )/( k = f + =k+ f ) = (D L =k+ x (f f ))/(P L =k+ (f f )). If sums are computed recursvely, t leads to y l (k) = (D k x k (f k f k ))/(P k (f k f k )). Snce L gven that y l y mn n Table (2), teratons should decrease k from. Thus, a decrement of k nduces adecrementnf k from f k to f k.in[5], the followng was demonstrated. Condton. If x k <y(f f ),theny(f f ) ncreases when f k decreases.

5 Advances n Fuzzy Systems 5 The KM algorthm for fndng the mnmum extreme pont y L of the centrod of an nterval type-2 fuzzy set over x wth upper membershp functon f and lower membershp functon f can be restated as follows: Table2:algorthmflow. The KM algorthm for fndng the maxmum extreme pont y R of the centrod of an nterval type-2 fuzzy set over x wth upper membershp functon f and lower membershp functon f can be reexpressed as follows: () Sort x (=,2,...,)n ascendng order and call the sorted x by the same name, but now x x 2 x.matchtheweghtsf wth ther respectve x and renumber them so that ther ndex corresponds to the renumbered x. (2) Precomputaton: V [n] = n f (43) = H [n] = n f (44) = T [n] = n x f (45) = Z [n] = n x f (46) = wth n =, 2,.... T [] y A = V [] (47) Z [] y B = H []. (48) (3) Set Set y mn = mn(y A,y B ) (49) y max = max(y A,y B ) (56) Fnd k( k )sothat Fnd k( k ) so that x k y mn <x k+ x k y max <x k+ D l (k) = T [k] + (Z [] Z[k]) (5) E R (k) =Z[k] + (T [] T[k]) (57) P l (k) = V [k] + (Y [] Y[k]) (5) Q R (k) =Y[k] + (X [] X[k]) (58) y= D l(k) P l (k). (52) y= E R(k) Q R (k). (59) (4) Fnd k [, ] such that Fnd k [, ]such that x k y<x k +. x k y<x k +. (5) If k =k,stopandsety L =y,elsecontnue. Ifk =k,stopandsety R =y,elsecontnue. (6) Set Set D l (k )=T[k ]+(Z[] Z[k ]) (53) E R (k )=Z[k ]+(T[] T[k ]) (6) P l (k )=V[k ]+(Y[] Y[k ]) (54) Q R (k )=Y[k ]+(X[] X[k ]) (6) y = D l(k ) P l (k ). (55) y = E R(k ) Q R (k ). (62) (7) Set y=y, k=k andgotostep4. Sety=y, k=k andgotostep4. Condton 2. If x k >y(f f ),theny(f f ) ncreases when f k ncreases. ote that k s greater than L whch mples that x k >y L ; thus, t s guaranteed that x k > y l (k) satsfyng Condton. Therefore, t can be concluded that a decrement n f k wll nduce a decrement n y l (k). Fnally, the stop condton s stated from the propertes of the centrodfuncton [3]. They ndcate that the left-centrod functon has a global mnmum n k=l,sotsfoundwhen the functons start to ncrease. The procedure to compute y s only analyzed n ths secton. The full demonstraton of ths procedure s provded n the appendx for the readers who are nterested n followng t. The computaton of y R obeys smlar conceptual prncples, and t can be understood by usng the deas suggested above or those provded n the appendx The KM2 algorthm s presented n Table 2.Ths algorthm uses the same conceptual prncples of KMA for fndng y L and y R [5]. The algorthm ncludes precomputaton, ntalzaton, and searchng loop. Precomputaton and ntalzaton of are the same for, havng always the ntal swtch ponts near to the extreme ponts y L and y R. The searchng loop corresponds to steps 4 7. The core of the loop s Table 2 (53) (55) and (A.2) (A.4), whch can be easly demonstrated by makng equal to k or k.theseequatons calculate y l (k) or y r (k) wthfewsumsandsubtractonstakng advantage of pre-computed values. Fnally, the stop condton s the same for the A algorthm, whch was stated n [6].

6 6 Advances n Fuzzy Systems µ(x) µ(x) x x µ(x) µ(x) (c) x (d) x Fgure 2: FOUs cases used n the experments. Unform random FOU. Centered unform random FOU. (c) Rght-sded unform random FOU. (d) Left-sded unform random FOU. 4. Computatonal Comparson wth Exstng Iteratve Algorthms: Implementaton Based on Compled Language 4.. Smulaton Setup. The purpose of these experments s to measure the average computng tme that a typereducton algorthm takes to compute the nterval centrod of a partcular knd of FOU when the algorthm s mplemented asaprogramnacompledlanguagelkec.inaddton, complementary varables lke standard devaton of the computng tme and the number of teratons requred by each algorthm to converge are also regstered. The algorthms consdered n these experments are [4], [8], and A [6], whch are confronted aganst and. For the sake of comparson, the algorthm s regarded as a reference for all the experments. Four FOU cases were selected to evaluate the performance of the algorthms. All cases corresponded to random FOUs modfed by dfferent envelopes. The frst one consderedaconstantenvelopethatsmulatedtheoutputofafuzzy nferenceengnenwhchmostoftherulesarefredata smlar level. In the second one, a centered envelope was used to smulate that rules wth consequents n the mddle regon of the unverse of dscourse are fred. The last two cases were proposed to smulate frng rules whose consequents are n the outer regons of the unverse of dscourse. For the sake of clarty, several examples of the FOUs consdered n ths work aredepctednfgure2. The followng expermental procedure was appled to characterze the performance of the algorthms over each FOU case descrbed above: we ncreased from to wth step sze of, and then was ncreased from to wth step sze of. For each, 2 Monte Carlo smulatons were used to compute y L and y R. In order to overcome the problem of measurng very small tmes,. smulatons were computed n each Monte Carlo tral; thus, the tme of a type-reducton computaton corresponded to the measured tme dvded by.. The algorthms were mplemented as C programs and compled as SCILAB functons. The experments were carred out over SCILAB 5.3. numerc software runnng over a platform equpped wth an AMD Athlon(tm) 64 2Dual Core Processor GHz, 2 GB RAM, and Wndows XP operatng system Results: Computaton Tme. The results of computaton tme for unform random FOUs are presented n Fgure 3. exhbted the best performance of all algorthms for between and. For greater than, was thefastestalgorthmfollowedby.thepercentage

7 Advances n Fuzzy Systems 7 Average computaton tme (s).4e 5.2E 5 E 5 8E 6 6E 6 4E 6 2E 6 Rato E+ 6E 7 STD of the computaton tme (s) 5E 7 4E 7 3E 7 2E 7 E 7 E+ (c) Fgure 3: Unform random FOU results (compled language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of the computaton tme. of computaton tme reducton (PCTR) wth respect to the A (.e., PCTR = (t o t ekm )/t ekm where t o and t e are the average computaton tmes of a partcular algorthm and A, resp.) of was about 3% for =,whle n the case of, the largest PCTR was about 4% for =. ote that and exhbted poorer performance wth respect to A when grows beyond ponts; however was the fastest algorthm for = ponts. The standard devaton of the computaton tme for and was always smaller than that of the A. Fgure 4 presents the results for centered random FOUs. exhbted the largest PCTR for smaller than ponts, about 4%. When was greater than, was the fastest algorthm. Ths algorthm provded a maxmum PCTR of about 5% for =. Thestandarddevaton of the computaton tme for all algorthms was often smaller than that of A. In the case of left-sded random FOUs (Fgure 5), was the fastest algorthm for smaller than ponts. Ths algorthm provded a PCTR that was between 7% and 5%. Here agan was the fastest algorthm for greater than ponts wth a PCTR that was about 45% for ths regon. In ths case, exhbted the worst performance and, n general, the standard devaton of the computaton tme was smlar for all the algorthms. The results concernng rght-sded random FOUs are presented n Fgure 6. was the algorthm that provded the largest PCTR for smaller than ponts. Ths percentage was between 55% and 6%. accounted for the largest PCTR for greater than ponts. The reducton acheved by ths algorthm was between 5% and 6%. exhbted the worst performance n ths case. The standard devaton of the computaton tme was smlar n all algorthms for smaller than ; however, a bg dfference wth respect to A was observed for greater than ponts Results: Iteratons. The amount of teratons requred by each algorthm to fnd y R was recorded n all the experments. In Fgures 7, resultsforthemeanofthe

8 8 Advances n Fuzzy Systems.4E 5.4 Average computaton tme (s).2e 5 E 5 8E 6 6E 6 4E 6 2E 6 Rato E+ 3.5E 7 STD of the computaton tme (s) 3E 7 2.5E 7 2E 7.5E 7 E 7 5E 8 E+ (c) Fgure 4: Centered random FOU results (compled language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of the computaton tme. number of teratons regardng each FOU case are presented. A comparson of the -type algorthms and also of the KM-type algorthms are provded for dscusson. In the case of a unform random FOU (Fgure 7), reduced consderably the number of teratons when compared to the and algorthms. The percentage reducton was about 75% for = wth respect to. On the other hand, no mprovement was observed for over the A n ths case; however, the computaton tme wassmallerforaspresentednprevoussubsecton. In the case of a centered random FOU, the provded a sgnfcant reducton n the number of teratons compared to the and algorthms, as shown n Fgure 8. The percentage reducton was about 88% for = wth respect to. Regardng the KM-type algorthms, the reduced the number of teratons wth respect to the A for smaller than 5 ponts. However, theperformanceofsqutesmlartothatofa for greater than ponts. Regardng the results from the left-sded random FOU case (Fgure 9), an nterestng reducton n the number of teratons was acheved by compared to the and algorthms. The reducton percentage s around 9% for = wth respect to. dsplayed a reducton percentage rangng from 3% for = to 83% for =wth respect to A. The algorthm outperformed the other two type algorthms for a rght-sded random FOU (Fgure ), as observed n the prevous cases. Consderng the KMtype algorthms, provded the largest reducton n the number of teratons wth respect to A for = ponts. The reducton percentage was about 8% n ths case. also outperformed A for greater resolutons Dscusson. We beleve that the four experments reported n ths secton provde enough data to clam that and are faster than exstng teratve algorthms lke A,, and as long as the algorthms are

9 Advances n Fuzzy Systems 9 Average computaton tme (s).4e 5.2E 5 E 5 8E 6 6E 6 4E 6 2E 6 E+ 3E 7 Rato STD of the computaton tme (s) 2.5E 7 2E 7.5E 7 E 7 5E 8 E+ (c) Fgure 5: Left-sded random FOU results (compled language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of the computaton tme. coded as compled language programs. The experments are not only lmted to one type of FOU, nstead, typcal FOU cases that are expected to be obtaned at the aggregated output of an IT2-FLS were used. may be regarded as the best soluton for computng type-reducton n an IT2-FLS when the dscretzaton of the output unverse of dscourse s smaller than ponts. On the other hand, mght be the fastest soluton when dscretzaton s bgger than ponts.thuswebelevethatshouldbeusedn real-tme applcatons whle should be reserved for ntensve applcatons. Intalzatonbasedontheuncertantyboundsprovedto be more effectve than prevous ntalzaton schemes. Ths s evdent from the reducton acheved n the number of teratons when fndng y R (smlar results were obtaned n the case of y L ). We argue that the nner-bound set s an ntal pont that adapts tself to the shape of the FOU. Ths sclearlydfferentfromthefxedntalpontsthatpertan to and A, where the uncertanty nvolved n the IT2-FS s not consdered. In addton, a smple analyss of the uncertanty bounds n [7] shows that the nner-bound set largely contrbutes to the computaton of the outer-bound set, whch s used to estmate the nterval centrod. 5. Computatonal Comparson wth Exstng Iteratve Algorthms: Implementaton Based on Interpreted Language 5.. Smulaton Setup. The same four FOU cases of the prevous secton were used to characterze the algorthms mplemented as nterpreted programs. Snce an nterpreted mplementaton of an algorthm s slower than a compled mplementaton, the expermental setup was modfed. In ths case, 2 Monte Carlo trals were performed for each FOU case; however only smulatons were computed for each tral, so the tme for executng a type-reducton s the measured tme dvded by.

10 Advances n Fuzzy Systems Average computaton tme (s).8e 5.6E 5.4E 5.2E 5 E 5 8E 6 6E 6 4E 6 2E 6 E+ STD of the computaton tme (s) E 6 9E 7 8E 7 7E 7 6E 7 5E 7 4E 7 3E 7 2E 7 E 7 E+ (c) Rato Fgure 6: Rght-sded random FOU results (compled language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of the computaton tme. The algorthms were mplemented as SCILAB programs. The experments were carred out over SCILAB 5.3. numerc software runnng over a platform equpped wth an AMD Athlon(tm) 64 2 Dual Core Processor GHz, 2 GB RAM, and Wndows XP operatng system Results: Executon Tme. The results of computaton tme for unform random FOUs are presented n Fgure. acheved the best performance among all algorthms wth a PCTR of around 6%, and the PCTR was close to 5%. The standard devaton of the computaton tme for the and was always smaller than that of A. Thus, consderng an mplementaton usng an nterpreted language, should be the most approprated soluton for type-reducton n an IT2-FLS. Fgure 2 s smlar to Fgure 3. These fgures present the results for centered random FOUs and left-sded random FOUs, respectvely. was the fastest algorthm, followed by,,, and A as the slowest. The PCTR was almost smlar to the PCTR for greater than 2, whch was about 6%. When =, the PCTR was about 46%. The standard devaton of computaton tme was smlar for all algorthms for smaller than 3 although t was sgnfcantly dfferent wth respect to the A for greater than 3 ponts. The results wth rght-sded random FOUs n Fgure 4 show that was the fastest algorthm n all the cases, followed by wth a smlar performance. The PCTR for greater than 3 was about 76% wth, whle, wth, t was about 74% over the A computaton tme. For the other cases the reducton was around 7% and 64%, respectvely. The standard devaton of the computaton tme for all algorthms was often smaller than that of A. 6. Conclusons Two new algorthms for computng the centrod of an IT2- FS have been presented, namely, and. follows the conceptual lne of algorthms lke [4]

11 Advances n Fuzzy Systems 7 6 Mean of the number of teratons Mean of the number of teratons Mean of the number of teratons Mean of the number of teratons Fgure 7: Unform random FOU, teratons, -type algorthms and KM-type algorthms. Fgure 8: Centered random FOU, teratons: -type algorthms and KM-type algorthms. and [8]. s nspred n the KM and algorthms [6]. The new algorthms nclude an ntalzaton thatsbasedontheconceptofnner-boundset[7], whch reduces the number of teratons to fnd the centrod of an IT2-FS. Moreover, precomputaton s consdered n order to reduce the computatonal burden of each teraton. These features have led to motvatng results over dfferent knds of FOUs. The new algorthms acheved nterestng computaton tme reductons wth respect to the algorthm. In the case of a compled-language-based mplementaton, exhbted the best reducton for resolutons smaller than ponts between 4% and 7%. On the other hand, exhbted the best performance for resolutons greater than ponts, resultng n a reducton rangng from 45% to 6%. In the case of an nterpretedlanguage-based mplementaton, was the fastest algorthm, exhbtng a computaton tme reducton of around 6%. The mplementaton of IT2-FLSs can take advantage of the algorthms presented n ths work. Snce reported the best results for low resolutons, we consder that ths algorthm should be appled n real-tme applcatons of IT2- FLSs. Conversely, can be consdered for ntensve applcatons that demand hgh resolutons. The followng step n ths paper wll be focused to compare the new algorthms and wth the EODS algorthm, snce t demonstrated to outperform n low-resoluton applcatons [25].

12 2 Advances n Fuzzy Systems Mean of the number of teratons Mean of the number of teratons Mean of the number of teratons Mean of the number of teratons Fgure 9: Left-sded random FOU, teratons, -type algorthms and KM-type algorthms. Fgure : Rght-sded random FOU, teratons, -type algorthms and KM-type algorthms. Appendx Theproofofthealgorthm-2forcomputngy s dvded n four parts. It wll be stated that Table (2) fxes a vald ntalzaton pont. It wll be demonstrated that Table (22) (24) computes the left centrod functon wth k=. (c) It wll be demonstrated that recurson of step 4 computes the left centrod functon wth ntal pont n k =, and each decrement n k leads to a decrement n y l (k). (d) The valdty of the stop condton wll be stated. Proof. From Table (5) (2), t follows that T [] y A = V [] = = x f = f, (A.) Z [] y B = H [] = = x f = f, (A.2) y mn = mn ( = x f = f y mn = y l., = x f = f ), (A.3) (A.4)

13 Advances n Fuzzy Systems 3 Average computaton tme (s) Average computaton tme (s) Rato.6.4 Rato STD of the computaton tme (s) STD of the computaton tme (s) (c) (c) Fgure : Unform random FOU results (nterpreted language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of computaton tme. Fgure 2: Centered random FOU results (nterpreted language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of computaton tme.

14 4 Advances n Fuzzy Systems 4 25 Average computaton tme (s) Average computaton tme (s) Rato.6 Rato STD of the computaton tme (s) STD of the computaton tme (s) (c) (c) Fgure 3: Left-sded random FOU results (nterpreted language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of computaton tme. Fgure 4: Rght-sded random FOU results (nterpreted language mplementaton of the algorthms). Average computaton tme, rato wth respect to the A, and (c) standard devaton of computaton tme.

15 Advances n Fuzzy Systems 5 It s clear that y mn n Table (2) corresponds to the mnmum of the nner-bound set [7], so accordng to (6) y mn y L.ThusTable (2) provdes an ntalzaton pont thatsalwaysgreaterthanorequaltothemnmumextreme pont y L.Inaddton,Itspossbletofndannteger so that x L y mn <x L +. From Table (22), t follows that Expandng t wthout alterng the proporton y l (k) = ( k = + x f + =k+ =k+ x f x f + =k+ =k+ x f ) x f =k+ x f D l ( )=T[ ]+(Z[] Z[ ]), D L =D l ( )= = D L =D l ( )= x f +( = = x f + From Table (23), t follows that x f = + = x f. x f ), (A.5) ( k = f + =k+ =k+ f ) f +, =k+ f =k+ f + f =k+ y l (k) = = x f + = + x f + =k+ x f =k+ x f = f + = + f + =k+ f =k+ f, y l (k) = = x f + = + x f =k+ x (f f ) = f + = + f =k+ (f, f ) P l ( )=V[ ]+(Y[] Y[ ]), P L =P l ( )= = P L =P l ( )= Therefore from Table (24) y mn = D P L f +( = = f + f = f. = + f ), = = x f +( = x f = x f ) = f +( = f = f ), (A.6) y l (k) = D =k+ x (f f ) P L =k+ (f f ), y l (k) = D =k+ x (f f ) P L =k+ (f f ). (A.9) (A.) Itcanbeobservedthat(A.) s a proporton composed by ntal values D L and P L and two terms that can be computed by recurson D(k) and P(k),so y l (k) = D D(k) P L P(k), D (k) = x (f f ), =k+ (A.) y mn = = x f + = x f + = + x f = x f = f + = f + = + f = f, y mn = = x f + = + x f = f + = + f. (A.7) P (k) = (f f ). =k+ Sums are computed by recurson as D k =D k x k (f k f k ), P k =P k (f k f k ), (A.2) (A.3) (c) Frst we show that recurson computes the left centrod functon wth ntal pont and decreasng k. Consderng y l (k) = k = x f + =k+ x f k = f + =k+ f. (A.8) y l (k ) = D k P k = D k x k (f k f k ) P k (f k f k ). (A.4) Snce L gven that y l y mn n Table (2), teratons decrease k from accordng to (A.). It has been demonstrated that (A.4) s the same(a.8) computed recursvely from ntal pont.so,byperformng

16 6 Advances n Fuzzy Systems adecrementofk n step 5, a decrement s ntroduced n f ; that s, f = (f,...,f )=(f,...f k, f k,f k+,...f ), f=(f,...,f )=(f,...f k,f k,f k+,...f ). (A.5) In [5], t s demonstrated that f x k <y(f,...,f ),then y(f,...,f ) ncreases when f k decreases. If x k >y(f,...,f ),then y(f,...,f ) ncreases when f k ncreases. (A.6) (A.7) ote that k s greater than L whch mples that x k >y L ;thus from the centrod functon propertes, t s guaranteed that x k >y l (k), satsfyng(a.7). Therefore, t can be concluded that a decrement n f k as (A.5) wll nduce a decrement n y l (k). (d) The stop condton s stated from the propertes of the centrod functon. Snce the centrod functon has a global mnmum n L, once recurson has reached k=l, y l (k) wll ncrease n the next decrement of k because (A.7)snolonger vald. So by detectng the ncrement n y l (k),tcanbesadthat the mnmum has been found n the prevous teraton. 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