Centrod Densty of Interval Type-2 Fuzzy Sets: Comparng Stochastc and Determnstc Defuzzfcaton Ondrej Lnda, Mlos Manc Unversty of Idaho Idaho Falls, ID, US olnda@udaho.edu, msko@eee.org bstract Recently, Type-2 (T2) Fuzzy Logc Systems (FLSs) ganed ncreased attenton due to ther capablty to better descrbe, model and cope wth the ubqutous dynamc uncertantes n many engneerng applcatons. y far the most wdely used type of T2 FLSs are the Interval T2 (IT2) FLSs. Ths paper provdes a comparatve analyss of two fundamentally dfferent approaches to defuzzfcaton of IT2 Fuzzy Sets (FSs) - the determnstc Karnk-Mendel Iteratve Procedure (KMIP) and the stochastc samplng defuzzfer. s prevously demonstrated by other researchers, these defuzzfcaton algorthms do not always compute dentcal output values. In the presented work, the concept of centrod densty of an IT2 FS s ntroduced n order to eplan such dscrepances. It was demonstrated that the stochastc samplng defuzzfcaton method converges towards the center of gravty of the proposed centrod densty functon. On the other hand, the KMIP method calculates the mdpont of the nterval centrod obtaned accordng to the etenson prncple. Snce the nformaton about the centrod densty s removed va applcaton of the etenson prncple, the two methods produce nevtably dfferent results. s further demonstrated, ths dfference sgnfcantly ncreases n case of non-symmetrc IT2 FSs. Keywords-Interval Type-2 Fuzzy Sets; Defuzzfcaton; Centrod; Karnk-Mendel lgorthms; Samplng Defuzzfer I. ITRODUCTIO Type-2 Fuzzy Logc Systems (T2 FLSs), orgnally proposed by Zadeh [1], consttute powerful tool for dealng wth dynamc uncertanty. The T2 Fuzzy Sets (FSs) ntroduce addtonal desgn dmenson for modelng and descrbng the uncertanty n the specfc problem doman [2]. Unlke the Type-1 (T1) FSs wth fed membershp functons, the T2 fuzzy membershp functons are defned usng the Footprntof-Uncertanty (FOU) offerng more desgn degrees of freedom [3]. The defuzzfcaton of T2 FSs contans a typereducton phase, whch acts as a mappng between the orgnal T2 FS and the type-reduced T1 FS, also called the centrod of the T2 FS [2], [3]. Despte some new recently ntroduced representatons of general T2 FSs such as geometrc T2 fuzzy sets [4], zslces [5], - planes [6], [7] or - cuts [8], the Interval T2 (IT2) FSs are stll most commonly used [9]. IT2 FLSs have been successfully appled n a wde range of applcatons [10]-[15]. Ths paper provdes a comparatve analyss of two fundamentally dfferent approaches to defuzzfcaton of IT2 FSs [9], the Karnk-Mendel Iteratve Procedure (KMIP) [16] and the samplng defuzzfcaton method [17]. The KMIP s a determnstc approach, whch concentrates on boundng the type-reduced centrod of the IT2 FS by ts left-most and rghtmost ponts [18]. The samplng defuzzfcaton method, proposed by Greenfeld et al., uses stochastc samplng method to sample the set of all embedded fuzzy sets and provde an estmate of the true defuzzfed value [19]. The samplng defuzzfcaton method was developed as a cut-down verson of the orgnal ehaustve defuzzfcaton method, whch enumerates and defuzzfes all avalable embedded fuzzy sets [20]. Other avalable methods for defuzzfcaton of IT2 FSs are the drect defuzzfer [21], the e-tan method [22], or the collapsng method [23]. s demonstrated by other researchers, the KMIP method and the samplng defuzzfcaton method do not always compute dentcal solutons [24]. In ths paper, the concept of centrod densty of an IT2 FS s ntroduced to provde an epermental eplanaton for such dscrepances. The concept of centrod densty descrbes the hstogram of defuzzfed embedded fuzzy sets wthn the boundares of the typereduced centrod. It s shown through epermental study that the stochastc samplng defuzzfcaton method converges towards the center of gravty of the proposed centrod densty functon. On the other hand, the KMIP method computes the mdpont of the nterval centrod obtaned accordng to Zadeh s etenson prncple. ecause the nformaton about the centrod densty s removed va applcaton of the etenson prncple, the two methods produce nevtably dfferent results. Ths dfference becomes especally sgnfcant n case of non-symmetrc IT2 FSs. The rest of the paper s organzed as follows. Secton II provdes an overvew of IT2 FSs together wth a revew of the consdered defuzzfcaton technques. Secton III ntroduced the concept of centrod densty for an IT2 FS. The comparatve analyss s presented n Secton IV and the paper s concluded n Secton V.
II. ITERVL TYPE-2 FUZZY SETS Ths secton revews fundamentals about the IT2 fuzzy sets together wth a bref descrpton of the defuzzfcaton technques consdered n ths work.. Interval Type-2 Fuzzy Sets The IT2 FSs were ntroduced as a smplfcaton to the general T2 FSs, whch have been rarely appled to engneerng problems due to ther mmense computatonal complety. n IT2 FS can be descrbed as: 1/(, u) J [0,1] X uj Here, and u consttute the prmary and the secondary varables and J s the prmary membershp of varable. The secondary grades of IT2 FS are all lmted to 1. Hence, the IT2 FS can be completely descrbed by ts Footprnt of Uncertanty (FOU), whch s the area of non-zero secondary grade. The vertcal slce of an IT2 fuzzy set defnes the secondary membershp functon. It can be obtaned by nstantatng the prmary varable nto a specfc value : u J (1) (, u) ( ) 1/ u J [0,1] (2) Usng the concept of vertcal slces the FOU can be epressed as follows: FOU ( ) (3) J X ccordng the Mendel-John representaton theorem [25], the IT2 FS can be also seen as a collecton of all ts embedded fuzzy sets. n embedded fuzzy e can be descrbed as: e 1 [1/ ]/ J U [0,1] (4) Usng ths concept the IT2 FS can be also epressed as: n j e j1 (5) lternatvely, the FOU of an IT2 fuzzy set can be convenently and completely descrbed by ts upper and lower membershp functons () and () : FOU ( ) ) (6) ( ) ( ( ), X Durng the output processng stage the output IT2 FS must be frst type-reduced, whch results n the centrod of the IT2 FS C () [20]: ) 1 1 (7) J J 1 1 Ths formula was derved usng Zadeh s etenson prncple [28]. In the specal case of IT2 FSs the centrod ) s an nterval T1 FS, whch can be descrbed by ts left and rght boundary ponts y L and y R. Fnally, ths nterval centrod can be defuzzfed to obtan the fnal output value.. Ehaustve Defuzzfcaton The defnton of the generalzed centrod of the IT2 FS presented n (7), descrbes the centrod as a composton of ndvdual centrods of all of ts embedded fuzzy sets. The ehaustve defuzzfcaton method constructs ths nterval centrod by enumeratng all the embedded fuzzy sets and calculatng ther respectve centrods. Ths ehaustve algorthm for defuzzfcaton of general T2 FSs can be adapted for the case of IT2 FSs as follows [20]: Step 1: Enumerate all possble nterval type-2 embedded fuzzy sets. The are n 1 M embedded fuzzy sets, where s the resoluton of the prmary doman and M s the resoluton of the secondary doman at the th slce. Step 2: For each embedded fuzzy set fnd the mnmum secondary grade. Ths s trval as all secondary membershp grades equal to 1. Step 3: For each embedded fuzzy set calculate the doman value of the centrod of the type-2 embedded fuzzy set. Step 4: Par the computed doman value from Step 3 wth the secondary grade of 1. Step 5: For each unque doman value, the mamum secondary grade s selected. In case of IT2 FSs, ths means that only a sngle record about a defuzzfed embedded FS s kept for each unque doman value. The set of ordered pars defnes the centrod. The crsp output can then be calculated as the average of all calculated centrod coordnates. The average wll be unformly weghted, as the secondary grade of each defuzzfed embedded FS equals to 1. ote that f the prmary doman s consdered to be contnuous as opposed to the dscretzed one, the centrod becomes a contnuous nterval T1 FS. C. Karnk-Mendel Iteratve Procedure (KMIP) The centrod of the IT2 FS s an nterval T1 FS. ccordng to Karnk and Mendel, t can be completely descrbed by ts left and rght end ponts y L and y R. s derved by Karnk and Mendel, these boundary ponts can be epressed as n [20]:
y y L R L 1 R 1 ( ) L 1 1 ( ) ( ) R ( ) L1 L1 R1 R1 ( ) ( ) ( ) ( ) Ponts L and R are mportant swtchng ponts computed by the KMIP algorthm. Usng the boundary values of the typereduced centrod ) the fnal crsp defuzzfed value y can be computed as the mean of the centrod nterval: (8) (9) ( y ) L yr y (10) 2 The KMIP algorthm calculates the swtchng ponts L and R n (8) and (9) and the boundares y L and y R of the centrod of the respectve IT2 FS. The descrpton of the KMIP algorthm gven below was adopted from [20]. The algorthm conssts of two phases, whch ndependently compute the values of y L and y R. The algorthm for computng the left boundary y L can be descrbed n several steps as follows: Step 1: Intalze a vector of weghts w as follows: ( ) ( ) 1 1 w,... (11) 2 nd compute the value of y: y w 1 w 1 Step 2: Fnd swtchng pont k ( 1 k 1) such that Step 3: Set k1 k1 (12) y (13) ( ) w ( ) nd compute the value of y as: Step 4: If to Step 5. Step 5: Set y y 1 w 1 k k 1 w (14) (15) y, stop and set y L = y and L = k. Otherwse, go y y and go to Step 2. The procedure for computng the value of y R s dentcal to computng y L ecept that n Step 3 dfferent update of weghts w s used as follows: ( ) k w (16) ( ) k 1 The fnal output value s assgned to y R and R=k. In [16], the Enhanced KMIP algorthm was presented. However, the proposed enhancements only mproved the convergence of the algorthm. s both approaches produce numercally dentcal results, only the orgnal KMIP algorthm s consdered n the rest of ths paper. D. Samplng Defuzzfcaton The samplng defuzzfcaton method for T2 FS was presented and analyzed n [17], [19], [24] and [27]. Ths method follows the steps of the ehaustve defuzzfcaton method. The major dfference s that only a subset of randomly sampled embedded fuzzy set s consdered durng the calculaton. ccordng to the descrpton n [24] the samplng defuzzfcaton for IT2 FS can be descrbed n several steps as follows: Step 1: Select the requred number of embedded fuzzy sets to be sampled. Step 2: Repeat for each sample: Step 2.1: Select an embedded fuzzy set at random. Step 2.2: Fnd the doman value by defuzzfyng the sampled embedded fuzzy set. Step 2.3: Fnd the mnmum secondary grade. In case of IT2 FSs ths s trval as the secondary grade equals to 1. Step 2.4: dd the pared defuzzfed value wth ts secondary grade to the lst of pars consttutng the typereduced centrod. Step 3: Defuzzfy the type-reduced centrod. The method was epermentally tested, showng a fast convergence towards the epected defuzzfcaton value [17]. The convergence speed was dependent on the cardnalty of the sampled set. Etenson to the samplng defuzzfcaton method va usng the mportance samplng technque was proposed n [28]. It was demonstrated that ths modfcaton resulted n reduced statstcal varance of the computed defuzzfed value. III. CETROID DESITY OF IT2 FUZZY SETS The ehaustve defuzzfcaton method defuzzfes all embedded fuzzy sets and then constructs the nterval centrod
(a) Fg. 1 Interval Type-2 Gaussan fuzzy set wth uncertan mean (a) and ts correspondng dscrete centrod densty. (a) Fg. 2 IT2 Gaussan fuzzy sets wth uncertan standard devaton (a) and ts correspondng dscrete centrod densty dstrbuton. as a composton of the calculated doman values. However, n Step 5 of the ehaustve defuzzfcaton method as descrbed n Secton II. the set of defuzzfed embedded fuzzy sets s consderably reduced. Ths s acheved va applcaton of Zadeh s etenson prncple, whch maps the IT2 FS back to ts nterval centrod. Here, only a sngle centrod of the embedded fuzzy sets per each unque doman value remans n the soluton and have an mpact on the geometrcal propertes of the fnal nterval centrod. Hence, from the perspectve of the type-reducton algorthm, all embedded fuzzy sets, whch are defuzzfed to an dentcal doman value form an equvalence class. The members of each such class are here termed redundant embedded fuzzy sets. Defnton 1: Two embedded fuzzy sets 1 and 2 are redundant f they both defuzzfy nto an dentcal doman value as C ) ). ( 1 2 s an eample, consder a group of embedded fuzzy sets that are vertcally shfted n the doman of the secondary varable u. In Step 5 of the ehaustve defuzzfcaton method, those embedded fuzzy sets are treated as redundant and only a sngle representatve one remans after the type-reducton process. Ths redundancy prunng sgnfcantly smplfes the comple nner structure of the centrod. Ths nner structure s smlar to the stratfed structure of the type-reduced set presented n [19]. Ths paper further demonstrates that the dstrbuton of redundant embedded fuzzy sets provdes an nsght nto the dscrepances n the convergent behavor of the KMIP algorthm and the samplng defuzzfcaton method [1]. et, assume that apart from applyng the etenson prncple, also the number of redundant embedded fuzzy sets for each unque doman value s beng recorded durng the ehaustve defuzzfcaton process. The dstrbuton functon ( ) of ths quantty s denoted here as the centrod densty functon of the IT2 FS. Defnton 2: The centrod densty ( ) for doman value denotes the normalzed number of redundant embedded fuzzy sets that are defuzzfed to the dentcal doman value : 0 ) ( ) f ( ), f ( ) (17) 1 1 ) Here, s the number of all embedded fuzzy sets, s the embedded fuzzy set and s the normalzaton factor, whch denotes the mamum number of redundant embedded fuzzy sets for any value of the doman varable, scalng t nto the nterval between 0 and 1. Property 1: The value of the centrod densty s zero outsde the boundares of the nterval centrod: ( ) 0 [ y, y ] (18) Proof: y the defnton of the KMIP method, bound y L consttutes the left most locaton of any centrod of the avalable embedded fuzzy sets. Smlarly, bound y R s the rght most locaton of any centrod of all avalable embedded fuzzy sets. Hence, the centrods of all embedded fuzzy sets are L R
gan, the normalzaton factor s the mamum number of redundant embedded fuzzy set for any dscrete nterval n the prmary doman, scalng t to the nterval between 0 and 1. Fnally, consder the center of gravty of functon ( ), whch can be denoted as ) and computed as follows: located n ths nterval and the centrod densty s thus zero outsde ths nterval. From the computatonal perspectve, the dscrete centrod densty ( ) s more practcal, snce the prmary doman must be dscretzed for practcal mplementatons. Defnton 3: ssume that the prmary doman s dscretzed nto samples 1, 2,, wth an dstance between two consecutve samples Then the dscrete centrod densty ( ) for value denotes the normalzed number of redundant embedded fuzzy sets that are defuzzfed nto the dscretzed nterval around : 0 ) [ ] 2 ( ) f ( ), f ( ) (19) 1 (a) Fg. 3 Symmetrc IT2 Gaussan fuzzy set wth uncertan mean (a), ts centrod densty and the defuzzfed values, y center of gravty of the centrod densty dstrbuton, y samplng defuzzfer. 1 ) [ ] 2 ( ) 1 ) (20) ( ) n eample of centrod densty for two dfferent IT2 FSs s presented n Fg. 1 and Fg. 2. The fgures show symmetrc Gaussan nterval type-2 fuzzy set wth uncertan mean and uncertan standard devaton, respectvely. In both cases, varable was dscretzed nto 13 samples and each secondary membershp functon was sampled usng 3 samples, leadng to a total of 1,594,323 embedded fuzzy sets. The centrod densty functon provdes valuable nsght nto the nner structure of the nterval centrod. The only avalable method for computng the centrod densty functon s the ehaustve defuzzfcaton method that accounts for all avalable embedded fuzzy sets. However, the applcablty of ths approach s hndered by the vast amount of embedded fuzzy sets, whch makes ths method computatonally ntractable even for modest dscretzaton levels. The KMIP method offers an opposte approach. Ths algorthm only consders two specal embedded fuzzy sets and only computes the two boundary ponts y L and y R of the nterval centrod. In the fnal step of the KMIP defuzzfcaton approach, the center of gravty of ths nterval centrod s computed as the md-pont between the boundary values y L and y R. One of the greatest advantages of the KMIP approach s ts computatonal speed. The samplng defuzzfer can be seen as a mddle way between the KMIP method and the ehaustve defuzzfcaton approach. Ths method appromates the fnal result by consderng only a subset of the avalable embedded fuzzy sets. IV. 1 EXPERIMETL RESULTS In ths secton the convergent behavor of the samplng defuzzfer s epermentaly compared to the determnstc result produced wth the KMIP. It s shown that unlke the KMIP approach, the samplng algorthm converges towards the center of gravty of the respectve centrod densty dstrubuton functon. Two types of IT2 FS are consdered n the study, symmetrc and non-symmetrc.. Symmetrc IT2 fuzzy sets Consder a symmetrc IT2 fuzzy set the FOU of whch s symmetrcal about ts mean m: ( m ) ( m ) (21) ( m ) ( m ) (22)
(a) (a) Fg. 5 The convergence of the mean (a) and the standard devaton of the output value produced by the samplng defuzzfer wth ncreasng number of samples. Fg. 4 on-symmetrc Interval Type-2 Gaussan fuzzy set (a), ts centrod densty dstrbuton and the defuzzfed values ), y center of gravty of the centrod densty dstrbuton, y samplng defuzzfer. Fuzzy Set TLE I COMPRISO OF THE DEFUZZIFIED OUTPUT VLUES Centrod Densty KMIP Method Samplng Method ) y y L y R y Symmetrc 0.5 0.5 0.3940 0.3001 on- Symmetrc 0.4624 0.4311 0.6060 0.5621 0.5004 0.0005 0.4625 0.0003 symmetrcal Gaussan IT2 FS 1 wth uncertan mean s denoted n Fg. 3(a). The doman of FS 1 was agan dscretzed nto 13 samples n the doman and 3 samples for each secondary membershp functon yeldng a total of 1,594,323 avalable embedded fuzzy sets. Ths FS was frst defuzzfed usng the KMIP algorthm. et, the centrod densty functon was computed usng the ehaustve defuzzfcaton method evaluatng all 1,594,323 embedded fuzzy sets. Fnally, the IT2 FS 1 was also defuzzfed usng the samplng defuzzfer randomly selectng only 10,000 embedded fuzzy sets. The comparson of the defuzzfed values and a graphcal vew of the dscrete centrod densty functon ( ) are presented n Table I and Fg. 3. 1 It can be observed, that the centrod densty dstrbuton ( ) follows appromately a symmetrc Gaussan normal 1 dstrbuton. s demonstrated n Table I, the defuzzfed values produced by the KMIP algorthm and the center of gravty of the centrod densty functon are n agreement. The defuzzfed value produced by the samplng defuzzfer s shown to be convergng towards the correct value. In order to obtan an accurate pcture, the average of 20 defuzzfcaton cycles s reported n Table I for the samplng defuzzfer, together wth the standard devaton of the result.. on-symmetrc IT2 Fuzzy Sets et, a non-symmetrc IT2 FS 3 wth uncertan standard devaton was consdered wth ts FOU as depcted n Fg. 4(a). ll defuzzfcaton methods have been appled wth the same parameters as n the prevous eperment. The defuzzfed values and the dscrete centrod densty functon ( ) are reported n Table I and n Fg. 4. It 3 could be observed that despte the centrod densty functon
(a) (c) (d) (e) Fg. 6 The sampled centrod densty dstrbuton for the samplng defuzzfer usng 10, 100, 1.000, 10.000, 100.000 (a)-(e) sampled embedded fuzzy sets. ( ) havng the nature of a Gaussan dstrbuton, t s far 3 from beng symmetrc. Ths observaton can be attrbuted to the non-symmetrc dstrbuton of the wdth of the FOU of the orgnal IT2 FS. Ths non-symmetrc nature s also reflected n the centrod densty functon produced by the samplng defuzzfcaton method. oth functons are shfted away from the md-pont of the centrod nterval computed by the KMIP algorthm. The actual defuzzfed values can be compared n Table I. s t can be seen from the shown eamples, gven a nonsymmetrc IT2 FS, whch s a typcal case for the output fuzzy sets of IT2 FLSs, a sgnfcant dfference between the defuzzfed values produced by the KMIP algorthm and the samplng defuzzfer can be epected. The nterpretaton of these dscrepances s an open ssue. Recently, novel representatons for general T2 fuzzy sets usng -planes [7] and zslces [5] have been ntroduced. In [24] the results of defuzzfcaton of -plane based general T2 FS usng the KMIP algorthm, samplng defuzzfcaton and ehaustve defuzzfcaton have been reported. It was demonstrated that for ncreasng number of -planes the KMIP and the samplng method dd not converge to dentcal values. The authors beleve that the ntroduced concept of centrod densty can provde further nsght and eplanatons nto such dscrepances. C. Convergence of the Samplng Defuzzfcaton n addtonal set of eperments have been carred out n order to verfy the convergence of the samplng defuzzfer towards the center-of-gravty of the centrod densty functon, whch was computed usng the ehaustve defuzzfcaton technque. The samplng defuzzfer was appled to the nonsymmetrc IT2 FS 3 depcted n Fg. 4(a). The number of sampled embedded fuzzy sets ranged from 2 2 to 2 18. Each eperment was repeated 20 tmes. The mean and the standard devaton of the defuzzfed values are reported n Fg. 5. In addton, Fg. 6(a)-(e) vsually demonstrates the convergence of the sampled centrod densty functon. From the presented results t can be noted that the samplng defuzzfcaton method provdes a steady convergence towards the center of gravty of the centrod densty functon. Hence, t can be concluded that the samplng defuzzfer converges to the output value obtaned by applyng the weghted average to the centrod densty functon. On the other hand, the KMIP method computes the md-pont of the nterval centrod calculated wth accordance to the etenson prncple. The nterval centrod does not mantan nformaton about the centrod densty functon, snce t was removed durng the applcaton of Zadeh s etenson prncple. V. COCLUSIO Ths paper presented a comparatve analyss of the performance of two fundamentally dfferent defuzzfcaton technques for IT2 FSs. The stochastc samplng defuzzfcaton method and the determnstc KMIP method were consdered n ths work. The concepts of redundant embedded fuzzy sets and the noton of centrod densty functon of an IT2 FS were ntroduced. These novel concepts provded eplanaton for some of the dscrepances between the results produced by the KMIP method and the samplng defuzzfer. It was shown that the samplng defuzzfcaton approach converges towards the center of gravty of the centrod densty functon, whch can be computed usng the ehaustve defuzzfcaton method. On the other hand, the KMIP algorthm calculates the results as the md-pont of the nterval centrod computed wth accordance to the etenson prncple. Snce the applcaton of the etenson prncple durng the type-reducton removes the nformaton about the centrod densty, the KMIP and the samplng defuzzfcaton technques nevtable converge towards dfferent results. Ths dfference becomes especally sgnfcant for non-symmetrc IT2 FS. REFERECES [1] L.. Zadeh, The Concept of a Lngustc Varable and ts ppromate Reasonng - II, n Informaton Scences, o. 8, pp. 301-357, 1975.
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