Shadowed Type-2 Fuzzy Logic Systems

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1 Shadowed Type-2 Fuzzy Logc Systems Dumdu Wjayaseara, ndrej Lnda, Mlos Manc Unversty of daho daho Falls, D, US bstract General Type-2 Fuzzy Logc Systems (GT2 FLSs) are an extenson to Type-1 (T1) FLS where at least one Fuzzy Set (FS) s a GT2 FS. However, due to the hgh computatonal complexty of operatons on GT2 FSs, GT2 FLSs have been rarely used n practcal applcatons. nstead, nterval Type-2 (T2) FLSs whch employ constraned T2 FSs, have been wdely used. Despte ther superor computatonal complexty, T2 FLSs lac the expressve power of GT2 FSs when descrbng varous sources of uncertanty. Further, t s unclear how to derve an T2 FLS from a specfc GT2 FLS. To allevate these ssues, ths paper outlnes a novel concept of Shadowed Type-2 Fuzzy Logc Systems (ST2 FLS). The ST2 FLS conssts of prevously proposed ST2 FSs, whch are T2 FSs wth secondary membershp functons represented as Shadowed Sets (SSs). Because ST2 FSs are drectly nduced by GT2 FSs, the entre desgn of the ST2 FLS can be automatcally derved from a specfc GT2 FLS. Furthermore, the proposed ST2 FLS was shown to approxmate GT2 FLS more accurately compared to T2 FLS, whle mantanng the computatonal effcency of T2 FLS. ndex Terms General Type-2 Fuzzy Logc Systems, nterval Type-2 Fuzzy Logc Systems, Shadowed Sets, Shadowed Type-2 Fuzzy Sets, Uncertanty Modelng G. NTRDUCTN ENERL Type-2 Fuzzy Logc Systems (GT2 FLSs) were orgnally desgned as an extenson to T1 FLSs [1]. Whle the archtecture of GT2 FLS s very smlar to Type-1 (T1) FLS, they dffer n the nature of ndvdual Fuzzy Sets (FSs), where GT2 FLSs use GT2 FSs to model the fuzzy rule antecedents and consequents. The concept of GT2 FSs was orgnally proposed by Lotf Zadeh [2] to address the problem of over-specfcaton of the real-valued membershp degrees of T1 FSs. GT2 FSs use membershp degrees that are themselves FSs. Despte the powerful uncertanty modelng capablty of GT2 FSs, the hgh computatonal complexty of computng wth GT2 FSs sgnfcantly hndered ther practcal use. s a consequence, GT2 FLSs have been rarely appled n practce [3]. Recently, there has been a renewed nterest n the area of GT2 FSs and GT2 FLSs due to the recently ntroduced representatons of geometrc T2 FSs [4], [5] or the -planes [6], [7], [8] and the zslces [9], [10] representatons. The hgh computatonal complexty of GT2 FLSs led to a wde spread of applcatons of ther constraned verson the nterval T2 (T2) FLSs [11], [12]. The T2 FLSs consst of T2 FSs whch restrct the form of the secondary membershp functons to ntervals [13]. Ths smplfcaton allows the development of effcent algorthms for fuzzy nference wth T2 FSs [14]. However, the restrcted nterval secondary membershp functons can be seen as a sgnfcant lmtaton n stuatons where more complex representaton of secondary uncertanty s requred [15], [16]. Hence, on one sde there are GT2 FLSs wth rch uncertanty modelng capablty but wth unfavorable computatonal complexty. n the other sde there are T2 FLSs whch provde effcent computatonal framewor but at the prce of sgnfcantly restrctng the optons for modelng varous sources of uncertanty [9]. Ths paper proposes a new class of FLSs, the Shadowed Type-2 Fuzzy Logc Systems (ST2 FLSs), whch consttute a compromse of both methods. The proposed class of ST2 FLS s based on the prevously proposed concept of ST2 FSs [17]. n ST2 FS s a GT2 FS wth all secondary membershp functons represented as Shadowed Sets (SSs) [18], [19], [20], [21], [22]. The computatonal complexty of processng ST2 FSs s sgnfcantly reduced because t s able to tae advantage of the effcent fuzzy operatons on T2 FSs. However, at the same tme, the ST2 FSs offer mproved descrpton of uncertanty, whch s captured usng the SSs rather than smple nterval values for the secondary fuzzy membershp functons. Smlar representaton to ST2 FSs, named Shadowed Fuzzy Sets was recently outlned n [23], [24]. Ths paper outlnes the desgn of ST2 FLSs, whch employs ST2 FSs to model the fuzzy rule antecedents and consequents. The ST2 FLS desgn s automatcally derved from an orgnal GT2 FLS so as to preserve the uncertanty modeled by the orgnal GT2 FSs. ST2 FLSs can thus offer mproved modelng of uncertanty when compared to T2 FLS whle also provdng effcent computatonal framewor snce the secondary membershp grades can only attan three values of 0, 1, or completely uncertan (shadowed) grade of [0, 1]. Smlarly, as demonstrated n the expermental results secton, ST2 FLSs are also computatonally effcent snce they apply the effcent fuzzy nference mechansms of T2 FLSs. The rest of the paper s organzed as follows. Secton revews the prevously publshed concept of ST2 FSs. The novel archtecture and the desgn of ST2 FLS are outlned n Secton. Several examples of ST2 FLSs are presented n Secton V and the paper s concluded n Secton V.

2 (a) Fg. 1 Secondary membershp functon of GT2 FS and ts segmentaton usng two selected -planes and the optmzaton functon V ( ).. SHDWED TYPE-2 FUZZY SETS Ths secton provdes an overvew of the concept of ST2 FSs whch was prevously proposed n [17]. The concept of Shadowed Sets (SSs) was orgnally developed to mprove the observablty and nterpretablty of T1 FSs and to allevate the ssues of excessve precson n descrbng mprecse concepts usng T1 fuzzy membershp functons [18]-[21]. n SS s drectly nduced by a T1 FS. Based on the T1 fuzzy membershp grades, the SS can be dvded nto three regons: excluson, core and shadow [18]-[21]. n ST2 FS s drectly nduced by a GT2 FS by transformng all the T1 fuzzy secondary membershp functons nto ther SS forms [17]. n ths paper all secondary membershp functon of the respectve GT2 FSs are assumed to be convex fuzzy sets. Hence, the secondary membershp functons f x ( can be descrbed as: s s s g x ( u L ( x 0), sl ( x 1) g x ( [0,1] 1 u L ( x 1), sr ( x 1) f x ( (1) hx ( u R ( x 1), sr ( x 0) hx ( [0,1] 0 therwse where g x ( and h x ( are monotoncally non-decreasng and monotoncally non-ncreasng functons n ther respectve domans.. Representaton of ST2 FSs n ST2 FS s nduced by a GT2 FS. The process of constructng constrants all the secondary membershp functons of to be SSs. The ST2 FS can be seen as functonal mappng:,, : X [0,1] {0,1,[0, 1]} (2) Here, the secondary membershp of 1 corresponds to the core of the ST2 FSs, secondary membershp of 0 corresponds to the excluson regon and the absolutely uncertan grade of [0, 1] corresponds to the shadow. The ST2 FS membershp functon can be expressed as follows: {( x,, ( x, x X, u [0,1], ( x, {0,1,[0, 1]}} (3) The process of constructng an ST2 FS based on a GT2 FS ncludes elevaton, reducton and balancng of the membershp grades. The ST2 FS s constructed usng a sutable threshold. The core core( ) of ST2 FS can be descrbed as a footprnt of where all secondary membershp degrees are greater than 1. core( ) { x, u ( x,, x X, u[0,1], ( x, (1 )} (4) The excluson regon excl() of ST2 FS can be defned as a footprnt of where all secondary membershps are less than threshold : excl( ) { x, u ( x,, x X, u[0,1], ( x, } (5) Fnally, The shadow regon sh() of ST2 FS can be constructed as a footprnt of where all secondary membershps are between thresholds values and 1 : sh( ) { x, u ( x,, x X, u[0,1], ( x, (1 )} (6) The process of locatng the optmal value of threshold conssts of fndng a par of -planes at levels and 1, whch optmze a ftness functon V ( ). The objectve functon s composed of three components, whch express the R amount of uncertanty n regons that were reduced ( V ( ) ), E B elevated ( V ( ) ) or balanced ( V ( ) ). Usng the notaton depcted n Fg. 1(a) the ndvdual components can be expressed as:

(a) n ST2 FS can be completely descrbed usng ts nner and outer boundares and.

The outer boundary mars the boundary between the excluson and the shadow regon. Smlarly, the nner boundary mars the transton from the shadow to the core regon. Ths s depcted n Fg.

Ths smplfed vew offers a convenent way to fully descrbe the ST2 FS as: {, } (12) V R s L ( x ) ( ) ( x, dudx ( x, dudx (7) V x X 0 B V E x X 1 x X s R ( x 1 ) s L ( x 1 ) s R ( x ) ( ) ( x, dudx (8)

cases where the GT2 FS s represented n the -plane framewor wth a fnte number of -planes, the soluton can be obtaned as: arg mn V ( ) (11) opt Fg. 2 GT2 FS (a) and ts nduced ST2 FS.

3 (a) n ST2 FS can be completely descrbed usng ts nner and outer boundares and. Each boundary s composed of two T1 fuzzy membershp functons, the lower ( ( x), ( x) ) and the upper ( ( x), ( x) ) membershp functons. The outer boundary mars the boundary between the excluson and the shadow regon. Smlarly, the nner boundary mars the transton from the shadow to the core regon. Ths s depcted n Fg. 3, whch shows a GT2 FS and ts derved ST2 FS. Ths smplfed vew offers a convenent way to fully descrbe the ST2 FS as: {, } (12) V R s L ( x ) ( ) ( x, dudx ( x, dudx (7) V x X 0 B V E x X 1 x X s R ( x 1 ) s L ( x 1 ) s R ( x ) ( ) ( x, dudx (8) ( ) dudx (9) s L ( x 1 ) s R ( x ) dudx x X ( x x X ) ( x 1 ) s L By combnng all three components the optmzaton functon V ( ) can be constructed as: R E B V ( ) V ( ) V ( ) V ( ) (10) n practcal cases where the GT2 FS s represented n the -plane framewor wth a fnte number of -planes, the soluton can be obtaned as: arg mn V ( ) (11) opt Fg. 2 GT2 FS (a) and ts nduced ST2 FS. s R n example of the optmzaton functon V ( ) s depcted n Fg. 1. Here, both and are T2 FSs. B. Set theoretc operatons wth ST2 FSs Here, the three elementary operatons of ntersecton, unon and complement on ST2 FSs are revewed. The ntersecton (meet) of two ST2 FSs and B can be defned as follows: B {, } { B, B } { B, B } (13) The method for ntersecton of two T2 FSs descrbed n [15] can be used to calculate ndvdual components. The unon (jon) of two ST2 FSs and B can be defned as follows: B {, } { B, B } { B, B } (14) The method for unon of two T2 FSs can be used to calculate ndvdual components n (14). Fnally, the complement of a ST2 FSs can then be obtaned as follows: {, } {, } x X (15) The method for computng the complement an T2 FS can be used to calculate ndvdual components n (15). n example demonstratng the applcaton of ST2 FS operatons s depcted n Fg. 3. (a) (c) (d) Fg. 3 Two ST2 FSs and B (a), the results of the set theoretc operatons of meet B, jon B (c) and complement (d).

C. Type-reducton of ST2 FSs Smlar to the basc set theoretc operatons, the typereducton of ST2 FSs also taes advantage of the wellestablshed and computatonally effcent algorthms of T2 FSs.

4 C. Type-reducton of ST2 FSs Smlar to the basc set theoretc operatons, the typereducton of ST2 FSs also taes advantage of the wellestablshed and computatonally effcent algorthms of T2 FSs. The centrod of an ST2 FS denoted as C can be descrbed usng two nterval T1 FSs descrbng the nner and the outer centrods C and C : C { C, C } (16) The nner and the outer centrod can be computed by ndependently type-reducng the nner and the outer boundary sets Fg. 4 Centrod of the ST2 FS from Fg. 3 and. Hence: C { C, C } { c, c, c, c } (17) The outer centrod C mars the boundary between the excluson regon and the shadowed regon of the centrod. Smlarly, the nner centrod l r l r C creates a boundary between the shadowed boundary and the core regon. n example of the centrod of ST2 FS s depcted n Fg. 4. D. Defuzzfcaton of ST2 FSs Prevously, three methods for defuzzfcaton of the centrod of ST2 FSs were proposed, namely the optmstc, pessmstc and weghted defuzzfcaton methods. The output values y, y P and y W of each method can be expressed as follows: y cl cr (18) 2. SHDWED TYPE-2 FUZZY LGC SYSTEMS Ths Secton descrbes frst the archtecture and nference of the proposed ST2 FLSs. Next, the desgn of the ST2 FLS based on a specfc GT2 FLS s outlned.. ST2 FLS rchtecture and nference n FLS s a rule based system wth ndvdual rule antecedents and consequents represented as FSs. Whle many dfferent types of FLS can be found n lterature (e.g. Mamdan or Taag-Sugeno), ths paper focuses on the Mamdan type of FLS. The archtecture of the Mamdan FLS can be decomposed nto four major parts: nput fuzzfcaton, fuzzy nference engne, fuzzy rule base and output defuzzfcaton or processng, as shown n Fg. 5. For the case of the proposed ST2 FLS the fuzzy rule base s populated wth lngustc mplcatve fuzzy rules n the followng form: Rule R : F x 1 s 1 ND ND xn s n THEN y s B (21) Each nput x s frst fuzzfed by computng the membershp to the respectve antecedent ST2 FS. The membershp grade of nput x wth respect to ST2 FSs membershp functon at of denoted as ( x ). Fg. 5 rchtecture of a Mamdan FLS s equal to the secondary at coordnate x, whch can be The output of rule R can be expressed as a ST2 fuzzy membershp functon ( x, y), whch can be computed by R applyng the fuzzy meet operatons to the rule antecedents and the consequents: y P cl cr (19) 2 y W N 1 N w( x ) x 1 w( x ) (20) Here, w(x ) s a specfc weghtng functon, e.g. a trapezodal weghtng functon (see Fg. 4). Fg. 6 Parallel processng nner and outer boundares of the ST2 FLS

5 (a) Fg. 7 GT2 FS for the nputs (a) and the output (a) Fg. 8 ST2 FS for the nputs (a) and the output (nduced by the GT2 FS n Fg. 7) ( x, y) ( x )... R 1 1 n ( x n ) B ( y) (22) The ST2 FSs meet operaton can be computed accordng to the descrpton provded n (13). For completeness sae, (22) can be smplfed nto: n x, y) [ 1 ( x )] B ( y) R ( (23) ssumng that there are K dstnct fuzzy rules, the output ST2 FSs B ( y ) can be computed by aggregatng the ndvdual rule outputs va the jon operaton: K B( y) ( x, ) (24) 1 R y The ST2 FSs meet operaton can be computed accordng to the descrpton provded n (14). Fnally, the output ST2 FS B ( y ) s frst type-reduced n the frst phase of output processng and subsequently defuzzfed nto a termnal real-valued output n the second phase. s prevously dscussed, by applyng the T2 FSs type-reducton algorthms such as the Enhanced Karn Mendel algorthm [25], [26] ndvdually to the nner and the outer boundares of the output ST2 FSs B ( y ) ts centrod C can be obtaned. B Ths centrod can then be defuzzfed usng any of the prevously dscussed defuzzfcaton technques for ST2 FSs from Secton. D. By followng the descrbed operatons of meet, jon and type-reducton on ST2 FSs, the entre fuzzy nference process wth ST2 FLS can be thought of as a parallel processng of two T2 FLSs, one for the nner and one for the outer boundary T2 FSs of the ST2 FSs. The two results are then merged durng the defuzzfcaton stage. Ths nterpretaton s depcted n Fg. 6.

nput 1 Fg. 9 utput surface of the ST2 FLS Fg. 10 utput surface of the GT2 FLS (a) Fg. 11 T2 FS for the nputs (a) and the output (usng the FU of the GT2 FS n Fg. 7) B. ST2 FLS Desgn TBLE.

6 nput 1 Fg. 9 utput surface of the ST2 FLS Fg. 10 utput surface of the GT2 FLS (a) Fg. 11 T2 FS for the nputs (a) and the output (usng the FU of the GT2 FS n Fg. 7) B. ST2 FLS Desgn TBLE. FUZZY RULE BSE nput 2 Low Medum Hgh Low Low Low Medum Medum Low Medum Hgh Hgh Medum Hgh Hgh When compared to T2 FLS, one of the major advantages of the proposed ST2 FLS s that ts desgn s drectly nduced from a GT2 FLS. s a consequence of the desgn process the ST2 FLS preserves the uncertanty modeled by the orgnal GT2 FLS. The desgn process of obtanng an ST2 FLS based on a GT2 FLS can be descrbed n several steps as follows: Step 1: For all antecedent and consequent GT2 FSs fnd the optmal splttng -planes by mnmzng functon V ( ) accordng to (10). TBLE. CMPRSN RESULTS BETWEEN T2 FLS ND ST2 FLS Method MSE verage Runtme GT T ST2 (ptmstc) ST2 (Pessmstc) ST2 (Weghted) Step 2: Convert all antecedent and consequent GT2 FSs nto ST2 FSs usng the dentfed splttng -planes. Step 3: Preserve the fuzzy rule base of the GT2 FLS. Step 4: Modfy the fuzzy nference process to mplement the fuzzy nference operaton on ST2 FSs as descrbed n (22) and (24).

EXPERMENTL RESULTS Ths secton demonstrates the fuzzy nference process wth the proposed ST2 FLS on a smple use case.

7 (a) Fg. 12 Squared errors of the T2 FLS (a) and the ST2 FLS. (Note the dfferent scales used n both fgures). Step 5: Modfy the output processng to mplement the type-reducton and defuzzfcaton operaton on ST2 FSs as descrbed n Sectons.C and.d. V. EXPERMENTL RESULTS Ths secton demonstrates the fuzzy nference process wth the proposed ST2 FLS on a smple use case. The constructed FLS s composed of two nputs and a sngle output, all parttoned usng three antecedent and consequent FSs, respectvely. Frst, an ntal GT2 FLS was created. The antecedent and consequent GT2 FSs were constructed usng trangular prmary membershp functon wth symmetrcally postoned Gaussan secondary membershp functon as shown n Fg. 7(a) and Fg. 7. For the nference process the GT2- FSs shown were decomposed nto 200 -planes. The fuzzy rule base used s depcted n Table. Second, the GT2 FLS was transformed nto a ST2 FLS usng the method outlned n Secton.B. The optmal values of values were calculated and used to dentfy splttng - planes of each GT2 FS (Step 2). The ST2 FSs that were drectly nduced by the GT2 FSs are shown Fg. 8(a) and Fg. 8, respectvely. Usng the nference and defuzzfcaton process detaled n Secton., the output surface for the ST2 FLS can be constructed. Ths ST2 FLS output surface computed usng the weghted defuzzfcaton method s shown n Fg. 9. For comparson, the output surface for the orgnal GT2 FLS s depcted n Fg. 10. t can be observed, that both control surfaces are very smlar. To further nvestgate the modelng capablty of the ST2 FLS t was compared to an T2 FLS. The respectve T2 FLSs was constructed based on the orgnal GT2 FLS, where ndvdual T2 FSs were mplemented as the Footprnt-f- Uncertanty of the GT2 FSs. The T2 FSs are shown n Fg. 11. The T2 FLS output surface and ST2 FLS output surface were then compared to the orgnal GT2 FLS output surface. Ths comparson was performed by calculatng the squared error between the output of the T2 FLS and ST2 FLS and the output of the GT2 FLS. n addton, the average computatonal tme of the fuzzy nference process for a sngle nput-output par acheved by each method was also measured. Table shows the mean squared error (MSE), and the computaton tme of each method. Fg. 12(a) and Fg. 12 show the squared error for each nput value for T2 FLS and ST2 FLS (weghted defuzzfcaton), respectvely (note the dfferent scales used n Fg. 12). The results show that the ST2 FLS s capable of modelng the GT2 FLS better than T2 FLS whle mantanng the computatonal effcency of an T2 FLS. V. CNCLUSN Ths paper presented a novel concept of Shadowed Type-2 Fuzzy Logc Systems. The ST2 FLS s an FLS usng ST2 FSs for ts antecedents and consequents. Snce the ST2 FSs are drectly nduced by GT2 FSs, the entre desgn of the ST2 FLS can be automatcally derved from a specfc GT2 FLS. The ST2 FLSs can thus offer mproved modelng of uncertanty when compared to T2 FLS whle also provdng effcent computatonal framewor. The expermental results demonstrated the feasblty of the proposed concept of ST2 FLS and ts comparson to both T2 FLS and GT2 FLS n terms of both modelng accuracy and computatonal complexty. s future wor the advantages of ST2 FLS compared to T2 FLS wll be further nvestgated by applyng the presented methodology to real world problems. REFERENCES [1] N. Karn, J. M. Mendel, Q. Lang, Type-2 Fuzzy Logc Systems, n Trans. n Fuzzy Systems, vol. 7, no. 6, pp , Dec [2] L.. Zadeh, The Concept of a Lngustc Varable and ts pproxmate Reasonng -, n nformaton Scences, No. 8, pp , 1975.

8 [3] H. Tahayor,. G. B. Tettamanz, G. D. nton, pproxmated Type-2 Fuzzy Set peratons, n Proc. EEE nternatonal Conference n Fuzzy Systems, Canada, pp , [4] S. Coupland, R. John, Geometrc Type-1 and Type-2 Fuzzy Logc Systems, n EEE Trans. on Fuzzy Systems, vol. 15, no. 1, pp. 3-15, February [5] S. Coupland, R. John, Fast Geometrc Method for Defuzzfcaton of Type-2 Fuzzy Sets, n EEE Trans. on Fuzzy Systems, vol. 16, no. 4, pp , [6] F. Lu, n effcent centrod type-reducton strategy for general type-2 fuzzy logc system, n nformaton Scences, vol. 178, pp , [7] J. M. Mendel, F. Lu, D. Zha, -Plane Representaton for Type-2 Fuzzy Sets: Theory and pplcatons, n EEE Transacton on Fuzzy Systems, vol. 17, no. 5, pp , ct [8] J. M. Mendel, Comments on -Plane Representaton for Type-2 Fuzzy Sets: Theory and pplcatons, n EEE Trans. on Fuzzy Systems, vol. 18, no. 1, pp , Feb [9] C. Wagner, H. Hagras, Toward General Type-2 Fuzzy Logc Systems Based on zslces, n EEE Transacton of Fuzzy Systems, vol. 18, no. 4, pp , ug [10] C. Wagner, H. Hagras, n approach for the Generaton and daptaton of zslces based General Type-2 Fuzzy Sets from nterval Type-2 Fuzzy Sets to Model greement wth pplcaton to ntellgent Envronments, n Proc. EEE World Congress on Computatonal ntellgence, Barcelona, Span, pp , July 18-23, [11] H. Hagras, Type-2 FLCs: New Generaton of Fuzzy Controllers, n EEE Computatonal ntellgence Magazne, pp , Feb [12] T. Derel,. Bayasoglu, K. ltun,. Durmusoglu, B. Tursen, ndustral applcatons of type-2 fuzzy sets and systems: concse revew, n Computers n ndustry, vol. 62, ssue: 2, pp , [13]. Lnda, M. Manc, Uncertanty-Robust Desgn of nterval Type-2 Fuzzy Logc Controller for Delta Parallel Robot, n EEE Trans. on ndustral nformatcs, vol. 7, no. 4, pp , Nov [14]. Lnda, M. Manc, nterval Type-2 Fuzzy Voter Desgn for Fault Tolerant Systems, n nformaton Scences, vol. 181, ssue: 14, pp , July [15] J. M. Mendel, Uncertan Rule-Based Fuzzy Logc Systems: ntroducton and New Drectons, Prentce-Hall, Upper Saddle Rver, NJ, [16] J. M. Mendel, R. John, F. Lu, nterval Type-2 Fuzzy Logc Systems Made Smple, n EEE Trans. on Fuzzy Systems, vol. 14, no. 6, pp , [17]. Lnda, M. Manc, Shadowed Type-2 Fuzzy Sets Type-2 Fuzzy Sets wth Shadowed Secondary Membershp Functons, n Proc. of FUZZ- EEE, 2012, wthn, EEE World Congress n Computatonal ntellgence, June 10-25, [18] W. Pedrycz, Shadowed Sets: Representng and Processng Fuzzy Sets, n EEE Trans. on Systems, Man, and Cybernetcs Part B: Cybernetcs, vol. 28, no. 1, pp , Feb [19] W. Pedrycz, nterpretaton of clusters n the framewor of shadowed sets, n Pattern Recognton Letters, vol. 26, ssue: 15, pp , Nov [20] S. Mtra, W. Pedrycz, B. Barman, Shadowed c-means: ntegratng fuzzy and rough clusterng, n Pattern Recognton, vol. 43, ssue: 4, pp , prl [21] W. Pedrycz, From fuzzy sets to shadowed sets: nterpretaton and computng, n nternatonal Journal of ntellgent Systems, vol. 24, ssue: 1, pp , Jan [22] J. Zhou, W. Pedrycz, D. Mao, Shadowed sets n the characterzaton of rough-fuzzy clusterng, n Pattern Recognton, vol.: 44, ssues: 8, pp , ug [23] H. Tahayor,. Sadeghan, Handlng uncertantes of membershp functons wth Shadowed Fuzzy Sets, n Proc. of World utomaton Congress (WC), 2012, pp. 1-5, June 24-28, [24] H. Tahayor,. Sadeghan, Shadowed Fuzzy Sets: Framewor wth More Freedom Degrees for Handlng Uncertantes Than nterval Type-2 Fuzzy Sets and Lower Computatonal Complexty Than General Type-2 Fuzzy Sets, n New Concepts and pplcatons n Soft Computng Studes n Computatonal ntellgence, vol. 417, pp , [25] D. Wu, J. M. Mendel, Enhanced Karn-Mendel lgorthms, n EEE Trans. on Fuzzy Systems, vol. 17, No. 4, pp , ug [26] N. Karn, J. M. Mendel, Centrod of a type-2 fuzzy set, n nformaton Scences, vol. 132, pp , [27] C. Wagner, H. Hagras, Novel Methods for the Desgn on General Type-2 Fuzzy Sets based on Devce Characterstcs and Lngustc Labels Surveys, n Proc FS World Congress, Eusflat World Conference, July, 2009.

Type-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data

Type-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data Malaysan Journal of Mathematcal Scences 11(S) Aprl : 35 46 (2017) Specal Issue: The 2nd Internatonal Conference and Workshop on Mathematcal Analyss (ICWOMA 2016) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES