Centroid Uncertainty Bounds for Interval Type-2 Fuzzy Sets: Forward and Inverse Problems

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1 Centrod Uncertanty Bounds for Interval Type-2 Fuzzy Sets: Forward and Inverse Probles Jerry M. Mendel and Hongwe Wu Sgnal and Iage Processng Insttute Departent of Electrcal Engneerng Unversty of Southern Calforna Los Angeles, CA E-al: Abstract - Interval type-2 fuzzy sets (T2 FS play a central role n fuzzy sets as odels for words [6] and n engneerng applcatons of T2 FSs [5]. These fuzzy sets are characterzed by ther footprnts of uncertanty (FOU, whch n turn are characterzed by ther boundares upper and lower ebershp functons (MF. The centrod of an nterval T2 FS [3], whch s an nterval T FS, provdes a easure of the uncertanty n the nterval T2 FS. Intutvely, we antcpate that geoetrc propertes about the FOU, such as ts area and the center of gravtes (centrods of ts upper and lower MFs, wll be assocated wth the aount of uncertanty n an nterval T2 FS. The an purpose of ths paper s to deonstrate that our ntuton s correct and to quantfy the centrod of an nterval T2 FS wth respect to these geoetrc propertes of ts FOU. It s then possble to forulate and solve nverse probles,.e. gong fro data to paraetrc T2 FS odels. I. ITRODUCTIO Recently, Mendel [6] proposed a fuzzy set (FS odel for words that s based on collectng data fro people person ebershp functons (MFs that reflect ntra- and nterlevels of uncertantes about a word, n whch a word FS s the unon of all such person FSs. The ntra-uncertanty about a word s odeled usng nterval type-2 (T2 person FSs, and the nter-uncertanty about a word s odeled usng an equally weghted unon of each person s nterval T2 FS. Because an nterval T2 FS plays such an portant role n ths odel as well as n engneerng applcatons of T2 FSs (e.g., [5], we need to understand as uch as possble about such sets and how they odel uncertantes. Recall that an nterval T2 FS A s characterzed as [5], [8]: A = (x, u = u x ( x X u J x [,] x X u J x [,] where x, the prary varable, has doan X; u, the secondary varable, has doan J x at each x X ; J x s called the prary ebershp of x; and, the secondary grades of A all equal. Uncertanty about A s conveyed by the unon of all of the prary ebershps, whch s called the footprnt of uncertanty (FOU of A,.e. FOU( A = U J x (2 x X The upper ebershp functon ( and lower ebershp functon ( of A are two type- MFs that bound the FOU (e.g., see Fg. 5. The s assocated wth the upper bound of FOU( A and s denoted A (x, x X, and the s assocated wth the lower bound of FOU( A and s denoted A (x, x X,. e. A (x FOU( A x X (3 A (x FOU( A x X (4 The centrod of an nterval T2 FS [2], whch s an nterval T FS, provdes a easure of the uncertanty n the nterval T2 FS. Intutvely, we antcpate that geoetrc propertes about the FOU, such as ts area and the center of gravtes (centrods of ts upper and lower MFs, wll be assocated wth the aount of uncertanty n an nterval T2 FS. The an purposes of ths paper are to deonstrate that our ntuton s correct, to quantfy the centrod of an nterval T2 FS wth respect to these geoetrc propertes of ts FOU, and to then forulate and solve nverse probles,.e. gong fro data to paraetrc T2 FS odels. II. CETROID OF A ITERVAL TYPE-2 FUZZY SET Recall that the centrod, CA, of the nterval T2 FS A s an nterval set [, ] that s copletely specfed by ts left and rght end-ponts,, respectvely,.e. [3], [5] [ ] = L C A =, x θ (5 θ J x θ J x In ths equaton, prary varable x has been dscretzed for coputatonal purposes, such that x < x 2 < L < x. Unfortunately, no closed-for forulas exst to copute ; however, Karnk and Mendel [3] have developed teratve procedures for coputng these end-ponts, and recently Mendel [7] proved that gven a FOU for an nterval T2 FS, one that s syetrcal about prary varable x at x =, then the centrod of such a T2 FS s also syetrcal about x =. For such a FS t s therefore only necessary to = θ

2 copute ether or, resultng n a 5% savngs n coputaton. Before we suarze the Karnk-Mendel procedures n a for that wll be very useful to us, we ust frst ustfy the use of the length as a legtate easure of the uncertanty of A. Wu and Mendel [9] noted that accordng to Inforaton Theory uncertanty of a rando varable s easured by ts entropy [2]. Recall that a one-densonal rando varable that s unforly dstrbuted over a regon has entropy equal to the logarth of the length of the regon. Coparng the MF, C (x, of an nterval FS C, where C (x =, x [, ], (6, otherwse wth the probablty densty functon, p Y (y, of a rando [ ], where varable Y, whch s unforly dstrbuted over, p Y (y = (, y [, ], (7, otherwse we fnd that they are alost the sae except for ther apltudes. Therefore, t s reasonable to consder the extent of the uncertanty of the FS C to be the sae as (or proportonal to that of the rando varable Y. Snce the centrod of a T2 FS s an nterval set, ts length can therefore be used to easure the extent of the T2 FS s uncertanty. In the sequel, when we use sapled values of A (x and (x, naely (x A A and (x A, where =, 2,...,, we shall splfy our notaton,.e., wthout loss of generalty A (x =,..., (8a (8b A (x =,..., The Karnk-Mendel teratve procedures for coputng can be nterpreted for the purposes of ths paper as follows [9]. Defne c ( L and c ( R, for L, R, as L L c ( L x + x + = L+ (9 =L + R R c ( R x + x = R + + ( = R + The end-ponts for the centrod of an nterval T2 FS [gven by (5] are the nu of all c ( L and the axu of all c ( R, respectvely,.e. = n c ( L where and L { } = c ( L* L* = x + x = L*+ L* + = L*+ ( L* = arg n { L c( L } (2 where { c (R } = c ( R* R = ax R* = x + x = R*+ R* + = R*+ (3 R* = arg ax{ R c( R } (4 The solutons of (2 and (4, L * and R*, are obtaned usng the Karnk-Mendel teratve procedures, the detals of whch are not needed n the rest of ths paper. Because closed-for forulas do not exst for, t s possble to study how these end-ponts explctly depend upon the area of the FOU and the centrods of the upper and lower MFs of the FOU. The approach taken n the rest of ths paper s to obtan bounds for both, and to then exane the explct dependences of these bounds on the geoetrc propertes of the FOU. III. BOUDS O AD FOR A ARBITRARY FOU Theore : The end-ponts,, for the centrod of an nterval T2 FS are bounded fro below and above by (Fg. (5 (6 where = n{ c, c } (7 = ax{ c,c } (8 = = + c = x c = x ( ( (9 (2 (x (x x (x + (x x (x (x x (x + (x x Proof: Provded n the ournal verson of ths paper. (2 (22 Fg.. End-ponts (X of the centrod of A and the lower and upper bounds ( for the two end-ponts. x These theoretcal facts are establshed n [3] and [5].

3 ext, we re-express the uncertanty bounds and, that are obtaned fro (2 and (22, respectvely, n a way that provdes enorous nsghts nto these ntervals. Theore 2: Let,, A FOU, c and c denote the area under the upper MF, the area under the lower MF, the area of the FOU (note that A FOU =, the centrod of the lower MF, and the centrod of the upper MF. Then (c = (x c FOU (23 (c + (x c (c = (x c FOU (24 (c + (x c Proof: Multply the nuerator and denonators of (2 and (22 each by three Δx ters, and then take the lt as Δx. The results n (23 and (24 follow edately. Coent : Theore 2 deonstrates that the boundng ntervals (uncertanty ntervals for the end-ponts of the centrod of A are ndeed expressble n ters of geoetrc propertes of the FOU. It has not ade use of any a pror geoetrc knowledge about the FOU, e.g., the FOU s syetrc; hence ts results are ost general. Because t has not ade use of a pror geoetrc knowledge, ts results ay be proved upon by akng use of such nforaton. We explore ths further n Secton V. Theore 2 lets us obtan any new results about the uncertanty bounds. Corollary : are shft-nvarant. Proof: The proof for focuses on the two factors (c and (x c whch appear n (23. When the FOU s shfted, x x + n whch case x x +, x x +, c c + and c c +. Consequently, (23 reans unchanged when x x +. A slar arguent deonstrates that (24 reans unchanged when x x +. Coent 2: The results n Corollary ean that we obtan the sae centrod bounds for a specfc FOU regardless of where that FOU s located wth respect to ts prary varable (x. Of course, we would have hoped/ expected ths to be true, and n ths corollary our hope/ expectaton s atheatcally proved. Because of ths shftnvarance we can locate the FOU anywhere we choose to on ts x-axs. Mendel [4], [5] has collected nterval end-pont data fro people about words 2, and has observed that the uncertanty 2 A group of students were asked the queston: Below are a nuber of labels that descrbe an nterval or a range that falls soewhere ntervals about the left and rght-hand end-ponts are unequal. A non-syetrcal FOU can provde such unequal ntervals, whereas (see Corollares 4 and 5 a syetrcal FOU cannot. For a non-syetrcal FOU, c c, and t s useful to express both c and c as functons of how uch they each depart fro the centrod, (x + x 2, of a syetrcal MF. Lettng δ and δ denote the departures fro syetry for c and c, respectvely, we can express c and c as: c = ( x + x 2 + δ (25 c = ( x + x 2 + δ (26 Because x c x and x c x, t follows fro (25 and (26 that δ and δ are constraned as ( x 2 δ ( x 2 (27 ( x 2 δ ( x 2 (28 Corollary 2: An alternatve way to express and s: 2A = A 2A FOU + (x 2δ (x + 2δ 2A = A 2A FOU + (x 2δ (x + 2δ Proof: Substtute (25 and (26 nto (23 and (24. (29 (3 Corollary 3: For an nterval T2 FS s greater than, equal to, or less than f > δ δ = (3 (x 2 2 [ 4δ ] A < (x 2 2 [ 4δ ] Proof: Eq. (3 follows fro (29 and (3 and soe sple arthetc anpulatons. Exaple : Specal cases of (3 occur when: ( δ > > δ n whch case > (see Fg. 2; and, (b δ < < δ, n whch case < (see Fg. 3. It s nterestng to study (3 to establsh curves above whch the > nequalty s true and below whch the < nequalty s true. After a lot of analyss, one can show that > f: (a Δ > when δ =, or (b between to. For each label, please tell us where ths range would start and where t would end. Ths was done for two collectons of 6 and fve labels usng two dfferent groups of students. See Table 2-2 and Fg. 2- n [5] for a suary of results for the 6 labels, and Table 2-3 for a suary of results for the fve labels.

4 Δ > 2 2 ( 4Δ + ( 4Δ Δ 8Δ (32 when δ. In (32, Δ δ / (x and Δ δ / (x. For, change the nequalty n (32 fro > to. Plots of (32 for δ and fve values of / are depcted n Fg. 4. Above each curve, >, whereas below each curve <. How to use these general results to desgn or reconstruct a non-syetrcal FOU fro data are topcs that are presently under study Δ =. =.2 =.4 =.6 =.8 Fg. 4. Unversal curves of (38. ote that and A U Δ Fg. 2. on-syetrcal trangular FOU for whch δ < δ Fg. 3. on-syetrcal trangular FOU for whch δ > δ. V. BOUDS O AD FOR A SYMMETRIC FOU Interval T2 FSs wth syetrcal FOUs have been very wdely used by practtoners of T2 FSs (e.g., [4]. Splfcatons to (23 and (24 occur for such FOUs. Corollary 4: For a syetrcal FOU: (a = = and (b = Δc where [ ( ] (33 Δc = x A FOU + Proof: (a Shftng the FOU so that t s syetrcal about the orgn, t s clear that for a syetrcal FOU, c = c = ; hence, = = follows drectly fro (7 and (8. (b Substtutng c = c = nto (23 and (24, we fnd that: = A FOU x x ( + x [ ] (34 [ ( ] (35 = A FOU x x + x For the shfted FOU, syetry also x = eans ; hence, (34 and (35 reduce to the sae nuber A FOU [ x ( + ]. Because x <, we take the absolute value of to obtan the results n (33. Coent 3: For a syetrcal FOU, because = =, the results fro our boundng analyss have degenerated nto an outer-bound set that bounds the centrod [, ],.e. [, ] [, ] = [ Δc, Δc] (36 Such a set ay be too conservatve. Coent 4: Corollary 4 cannot be used as s for a syetrcal Gaussan FOU, because for such a FOU x. Ths represents yet another shortcong of tryng to use general results for syetrcal FOUs. Recall that the uncertanty bounds n Theore ade no a pror use of the syetry of a syetrcal FOU. When we do ake use of such knowledge, we obtan: Theore 3: Let A be an nterval T2 FS defned on X whose FOU s syetrcal about X. Let c H and c H denote the centrods of half of the (syetrcal lower and upper MFs, respectvely,.e. c H = x( xdx (x dx (37

5 Then, c H = x (xdx (xdx (38 = + (c H (c H + (39 = + (c H (c H 2 (4 and, by syetry, = and =. The proof of ths theore s totally dfferent fro the proof of Theore, and wll appear n the ournal verson of ths paper. It akes very heavy use of the syetry of the FOU. Coent 5: When a syetrcal nterval T2 FS A s shfted by Δ = so that A s now syetrcal about, then n (39 and (4, because and rean unchanged, and c H, c H and are all shfted by Δ, both are also shfted by Δ. Ths agan eans that and are shft-nvarant (see Corollary ; hence, n the rest of ths secton we can focus on a syetrcal nterval T2 FS that s syetrcal about the orgn. Corollary 5: For a syetrcal FOU, let Δc new = = (4 where are n (39 and (4, n whch =. Then Δc new = A FOU + (42 Proof: Ths follows drectly fro (39 and (4. Coent 6: It s nstructve to copare (42 and (33. For the rest of ths paper, we shall refer to the results n (33 as Δc old. Clearly Δc new < Δc old f < x. It s possble for > x ; so, usng our two sets of bounds, we are able to conclude that A Δc = FOU + n{ x, } = n{ Δc old, Δc new } (43 Exaple 2: Here we deterne Δc for the syetrcal trangular FOU depcted n Fg. 5. Fro the sple geoetry of ths FOU, for whch h, t follows that x = b, = b, = ha, c H = b / 3, c H = a / 3 and A FOU = = b ha so that A Δc old = x FOU = b b ha + b + ha (44 b ha Δc new = b + ha = b 2 ha 2 b ha 6ha b + ha (45 For Δc new Δc old, we requre h ( b / a2 (46 + 6b / a Fro (44, (45 and (43, t s straghtforward to study the behavor of Δc as a functon of both h and b / a. Exaple 3: Here we deterne Δc for the syetrcal Gaussan FOU depcted n Fg. 6, for whch (x = exp x 2 A ( (2σ 2 (47 (x = s exp x 2 A ( (2σ 2 (48 where s [,]. ote that Δc old =, so that Δc = Δc new. It s straghtforward to show that = 2π σ, = 2π sσ, c H = 2σ 2 and c H = 2sσ 2 ; consequently, A FOU = 2π σ( s and = 2σ ( s 2π (+ s = σ ( s 2π s Δc = σ ( s 2 2π s( + s -b -a a b h Fg. 5. Syetrcal trangular FOU. s Fg. 6. Syetrcal Gaussan FOU. (49 (5 (5 Exaple 4: Usng the FOU n Fg. 6, t s possble to solve an nterestng nverse proble. Suppose that we have collected nterval end-pont data fro a group of n people for a phrase (e.g., soe, as descrbed n footnote 2. For the purposes of ths exaple, we assue that the uncertantes about the two end-ponts of ths nterval-data are the sae. u x x

6 The case when ths s not true s currently under nvestgaton. Let x, x 2,..., x n denote the collected data for one end-pont, and x avg and Δx denote the saple average of the n ponts and the length of the ( α confdence nterval (whch s proportonal to the saple standard devaton of the n ponts. We establsh the followng two reasonable desgn equatons: x avg ( + 2 (52 Δx (53 ext, we deterne the paraeters of a FOU that satsfy (52 and (53. To that end, we assue the FOU odel of Exaple 3, fro whch t s possble to solve unquely for FOU paraeters s and σ as: s = 2x avg Δx + 3Δx 2x avg (54 σ = 2π (2x avg + Δx(2 x avg Δx (55 8Δx What ths soluton eans s: startng wth nterval data that are collected fro a group of people, we can copute the paraeters of the scaled Gaussan FOU n Fg. 6, such that the centrod of ths nterval T2 FS s guaranteed to le wthn Δc =. To the best knowledge of the authors, Exaple 4 represents the frst soluton of an nverse proble for a T2 FS. It represents a cobnng of statstcs ( x avg and Δx and uncertanty bounds for T2 FSs. In [6], Mendel coned the ter fuzzstcs for the feld of experental fuzzy sets,.e. the feld n whch data are collected fro people about MFs and related ssues are forulated and tested (e.g., []. Ths paper and especally the results n ths exaple llustrate soe aspects of type-2 fuzzstcs. VI. COCLUSIOS We have deonstrated that the centrod of an nterval T2 FS provdes a easure of the uncertanty n such a FS. The centrod s a type- FS that s copletely descrbed by ts two end-ponts. Although t s not possble to obtan closed-for forulas for these end-ponts, we have establshed closedfor forulas for upper and lower bounds of the two endponts. Most portantly, these bounds have been expressed n ters of geoetrc propertes of the FOU, naely ts area and the center of gravtes of ts upper and lower MFs. As a result, for the frst te t s possble to quantfy the uncertanty of an nterval T2 FS wth respect to these geoetrc propertes of ts FOU. Usng the results n ths paper, t s possble to exane any forward probles,.e. gven a class of FOUs (e.g., trangular, trapezodal, Gaussan we can study the bounds on the centrod as a functon of the paraeter uncertantes that defne the FOU. It s also possble to exane nverse probles,.e. gven nterval data collected fro people about a phrase, and the nherent uncertantes assocated wth that data whch can be descrbed statstcally, we can see f t s possble to establsh a paraetrc FOU such that ts uncertanty bounds are drectly connected to statstcal uncertanty bounds. Although we have provded a soluton to ths proble for one FOU, obtanng solutons for other FOUs s an open ssue that s currently under study. It s qute lkely that we wll need ore quanttatve nforaton about a FOU than ust ts centrod uncertanty bounds f we are to go fro uncertan data collected about nterval end-ponts to a unque FOU, because the centrod uncertanty bounds are over-paraeterzed for soe FOUs. Ths suggests that hgher-order oents be establshed for an nterval T2 FS, e.g., dsperson, skewness, and kurtoss. What wll be needed for these new uncertanty easures are teratve ethods for ther coputaton (analogous to the Karnk-Mendel teratve ethods for coputng the nterval end-ponts for the centrod of a T2 FS and quanttatve uncertanty bounds for the (analogous to the results presented n ths paper for the centrod of a T2 FS. Once these addtonal results have been developed, then we wll be able to establsh whether or not t s ndeed possble to go fro nterval end-pont data to a unque (non-syetrcal FOU and f so how to do ths. Connectng data and ts uncertantes to a paraetrc FOU for an nterval T2 FS s analogous to estatng paraeters n a probablty odel, and, as s well known, the latter provdes a brdge between probablty and statstcs. We hope that the ateral n ths paper wll be the start of uch research n provdng a brdge between nterval T2 FSs and type-2 fuzzstcs, soethng that we beleve s needed f coputng wth words s to becoe a realty (e.g., [4], [6]. REFERECES [] T. Blgc and I. B. Turksen, Measureent of ebershp functons: theoretcal and eprcal work, n Handbook of Fuzzy Systes, Vol. : Foundatons, (D. Dubos and H. Prade, Eds., pp , Kluwer, Boston, MA, 2. [2] T. M. Cover and J. A. Thoas, Eleents of Inforaton Theory, John Wley, ew York, 99. [3].. Karnk and J. M. Mendel, Centrod of a type-2 fuzzy set, Inforaton Scences, vol. 32, pp , 2. [4] J. M. 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