Design of Strong Fuzzy Partitions from Cuts

Transcription

1 8th Conference of the European Socety for Fuzzy Logc Technology (EUSFLAT 03) Desgn of Strong Fuzzy Parttons from Cuts Corrado Mencar, Marco Lucarell, Cro Castello, Anna M. Fanell Dept. of Informatcs, Unversty of Bar, Italy {corrado.mencar, marco.lucarell, Trangular SFPs (TSFPs) are wdely used for modelng nterpretable fuzzy systems. They are characterzed by the use of trangular fuzzy sets to defne a fuzzy partton. Trangular fuzzy sets have a number of desrable propertes, whch are useful for nterpretablty (they are normal, convex contnuous) as well as for modelng []. However, trangular fuzzy sets have some non-dervable ponts: ths prevents ther use n modelng technques that use some gradent-based learnng technque to adapt fuzzy sets to avalable data. In such cases, usually completely dfferentable fuzzy sets are used, lke Gaussan fuzzy sets [3]; however, these fuzzy sets may not preserve some nterpretablty constrants (n partcular, the proper orderng of lngustc concepts). As a consequence, trangular fuzzy sets are preferred when the modelng process does not requre any gradent-based learnng algorthm. Abstract The adopton of trangular fuzzy sets to defne Strong Fuzzy Parttons (SFPs) s a common practce n the research communty: due to ther nherent smplcty, trangular fuzzy sets can be easly derved from data by applyng sutable clusterng algorthms. However, the choce of trangular fuzzy sets may be lmtng for the modelng process. In ths paper we focus on SFPs bult up startng from cuts (ponts of separaton between cluster projectons on data dmensons), showng that a SFP based on cuts can always be defned by trapezodal fuzzy sets. Dfferent mechansms to derve SFPs from cuts are presented compared by employng DC*, an algorthm for extractng fuzzy nformaton granules from classfed data. Keywords: Strong fuzzy parttons, α-cut, trangular fuzzy sets, trapezodal fuzzy sets The defnton of a TSFP wth n fuzzy sets s completely characterzed by n values that correspond to the prototypes of each fuzzy set: ths makes the desgn of TSFPs very smple. Usually, the prototypes are computed by some algorthm that tres to locate prototypes n order to better represent the avalable data. As an example, Herarchcal Fuzzy Parttonng (HFP) operates an teratve mergng process of trangular fuzzy sets n order to better ft avalable data, smultaneously, reduce the number of fuzzy sets n a partton [4]. The mergng process of two fuzzy sets s essentally computed by a weghted mean of the prototypes of the fuzzy sets to be merged; n ths way the number of fuzzy sets s dynamcally determned durng the desgn process. Other approaches fx the number of trangular fuzzy sets; then the locaton of prototypes s determned accordng to some optmzaton process [5] or through evolutonary algorthms [6, 7].. Introducton The key factor for the success of fuzzy logc s ts ablty of modelng perceptons rather than measurements. In many cases, perceptons can be expressed n natural language terms: ths makes knowledge expressed n fuzzy logc hghly contensve wth lngustc concepts; hence, t s easly nterpretable by users. Nevertheless, nterpretablty does not come wth fuzzy logc pso facto: t must be ensured by a number of structural semantc constrants. More specfcally, whle desgnng an nterpretable fuzzy model the data doman s represented through lngustc varables (usually one for each data feature); gven a lngustc varable, the fuzzy sets assocated to each lngustc term form a fuzzy partton of the data feature. To ensure nterpretablty, a number of constrants are mposed on the fuzzy sets of each fuzzy partton, lke dstngushablty, coverage, specal elements, so on []. The fulfllment of many nterpretablty constrants s guaranteed f Strong Fuzzy Parttons (SFPs) are adopted. Actually, SFPs are not strctly necessary for satsfyng the above mentoned nterpretablty constrants; however, they are wdely used because they smplfy the modelng process as they usually requre few parameters for ther defnton. 03. The authors - Publshed by Atlants Press In some cases, fuzzy parttons are desgned after a clusterng analyss of multdmensonal data. Ths approach enables the dscovery of multdmensonal relatonshps among data, whch can be convenently represented as fuzzy rules [8]. To ensure nterpretablty, clusters are usually projected on each nput feature, where fuzzy sets are defned so as to resemble as much as possble the projected clusters [9, 0]. Often, prototype-based clusterng s used (lke fuzzy c-means or smlar): n these cases the prototypes of multdmensonal clusters are projected on each nput feature could serve as pro44

2 Fgure : Example of fuzzy partton obtaned from cuts c, c, c 3, c 4. totypes of the fuzzy sets n a partton []. However, the smple use of multdmensonal prototypes does not gve enough nformaton about the span of clusters wthn the data doman. For such reason, an alternatve approach makes use of cuts,.e. ponts of separaton between clusters projected onto nput features []. Cuts can be convenently used to defne the bounds of the 0.5-cuts of the fuzzy sets n a fuzzy partton. More specfcally, gven a collecton of cuts, a SFP can be defned so that the 0.5-cuts of the fuzzy sets n the partton concde wth the ntervals bounded by the cuts (see fg. ). Snce the 0.5-cut of a fuzzy set s the set of elements that are most representatve for the fuzzy set, then a SFP based on cuts s a robust representaton of the projectons of multdmensonal clusters on an nput feature. In ths paper we show that a SFP based on cuts cannot be always defned by trangular fuzzy sets. The consequences of ths result mpact on the flexblty of modelng approaches based on trangular fuzzy sets: mposng the use of ths type of fuzzy sets restrcts the possbltes of representng multdmensonal relatonshps n an nterpretable way. In fact, the use of trangular fuzzy sets represents a further bas whch s not motvated by any nterpretablty requrement to be added to the structural constrants that are already taken nto account whle desgnng a fuzzy model (as known, such constrants ultmately mpose the requrement of a balance between nterpretablty accuracy). In other words, the flexblty connected to a modelng process based on the employment of SFPs may be restrcted by confnng the choce of fuzzy sets to the trangular category. As a consequence, nterest should be shfted towards a more relevant ssue concernng the possblty to defne SFPs based on cuts. In ths paper we show that ths s feasble by resortng to trapezodal fuzzy sets. Trapezodal fuzzy sets are wdely used for modelng nterpretable fuzzy systems [7, 3, 3, 4, 5]; however, n most cases trapezodal fuzzy sets requre more parameters than trangular fuzzy sets. Such parameters need to be tuned accordng to some heurstc optmzaton pro- The 0.5-cut of a fuzzy set s the (crsp) set of all elements wth membershp degree greater or equal to 0.5. cess lke genetc algorthms. The procedures we show n ths paper do not need free parameters because trapezodal fuzzy sets are defned gven a collecton of cuts only. In ths way there s no need of further optmzaton processes beyond the clusterng process that produced the cuts. In the next Secton, we provde a formal proof that trangular fuzzy sets cannot always be used to defne SFPs gven a set of cuts. Then we defne a procedure to defne SFPs based on trapezodal fuzzy sets. In Secton 3 we llustrate some examples of SFPs based on trapezodal fuzzy sets that are derved through DC* an algorthm for generatng nterpretable fuzzy parttons by usng cuts we compare them wth trangular SFPs. Some fnal notes are reported n Secton 4.. Generaton of Strong Fuzzy Parttons from cuts A SFP s a collecton of fuzzy sets A, A,..., A n+ defned on a Unverse of Dscourse X = [m, M] R such that: n+ x X : A (x) = () = A trangular fuzzy set s denoted by where: T [l, p, r] l s the leftmost bound of ts support; p s the element of ts core (also called prototype); r s the rghtmost bound of ts support. The membershp functon of a trangular fuzzy set can be convenently defned as a case-based functon: x l p l, x ]l, p] x r T [l, p, r] (x) = p r x ]p, r[ () 0, x l x r A trangular fuzzy set s well-formed f only f l p r (3) A Trangular Strong Fuzzy Partton (TSFP) s a SFP made wth trangular fuzzy sets only 3. A TSFP made of n + fuzzy sets s completely characterzed by n parameters p for =, 3,..., n. In fact, the trangular fuzzy sets of a TSFP can be defned as T [p, p, p + ] We assume that the collecton s sorted, so that t s legtmate to refer to the -th fuzzy set n a SFP. 3 Exceptonally, trapezodal fuzzy sets can be defned as leftmost rghtmost fuzzy sets. However, ths case can be safely gnored n the present argumentaton. 45

3 for =,,..., n + wth the conventon that p 0 = p = m p n+ = p n+ = M. Gven an element x X, at most two fuzzy sets have non-zero membershp n a TSFP: these fuzzy sets are sad adjacent. Furthermore, snce trangular fuzzy sets are convex, ther α-cuts are ntervals. Gven the constrant () of a SFP, t s mmedate to verfy that the 0.5-cuts of two adjacent fuzzy sets n a TSFP are also adjacent (n the sense of sharng one only one ntersecton pont). Let t, t,..., t n X a sequence of cuts, where t < t + for =,,..., n. In order to desgn a SFP based on cuts, each cut corresponds to an ntersecton pont between two adjacent fuzzy sets n a SFP; as a consequence, n cuts correspond to the ntersecton ponts of n + fuzzy sets n a SFP. (An ntersecton pont between two fuzzy sets s a pont n X where both fuzzy sets have the same non-zero membershp, see also fg..) In the followng we show that t s not always possble to buld a TSFP of n + fuzzy sets gven an arbtrary set of n cuts. We prove ths by attemptng to buld a TSFP then we hghlght the condtons that prevent the defnton of wellformed trangular fuzzy sets. The reader can refer to fg. as an llustratve example of the proof. We suppose that a trangular fuzzy set T [l, p, r ] s defned so that T [l, p, r ](t ) = 0.5 T [l, p, r ](t ) = 0.5 The membershp values on t t constran the parameters l r. In partcular, the parameter r can be obtaned by applyng the case-based defnton of a trangular fuzzy set, resultng n t r p r = 0.5 r = t p The next trangular fuzzy set T [l, p, r ] must be defned so as to satsfy the constrants () of a SFP. The parameters of the membershp functon must be therefore defned as l = p p = r = t p whle r s defned such that.e. 0.5 = t + r p r r = (t + t ) + p In order to assure well-formedness (3), the relaton p r Fgure : A sequence of cuts that prevents the generaton of a well-formed trangular fuzzy set (red dashed lne). must hold. It s easy to show that ths relaton s true f only f t + t p = r (4) Therefore, f the cuts used for parttonng do not verfy (4), t s not possble to defne well-formed trangular fuzzy sets. Ths result has a strong mpact on nterpretable fuzzy modelng. In fact, f we denote by T the collecton of all possble sets of cuts on X, by P the set of all TSFPs, then relaton (4) states that t s not possble to defne a bjectve mappng from T to P. On the other h an njectve mappng from P to T s trval: gven a TSFP, the set of cuts can be defned by selectng all the ntersecton ponts between trangular fuzzy sets. Therefore, the set T s rcher that P, thus any algorthm that carres out a collecton of cuts s potentally more flexble less based than an algorthm that produces trangular SFPs... Generaton of Trapezodal SFPs Is t possble to derve a SFP gven a set of cuts,.e. gven an element of T? The answer s affrmatve f we resort to trapezodal fuzzy sets nstead of trangular fuzzy sets. In the followng we show some procedures to derve a SFP made of trapezodal fuzzy sets gven a collecton of cuts on X. Frst, we recall the defnton of a trapezodal fuzzy set: x a b a, x ]a, b[, x [b, c] T [a, b, c, d] (x) = x d c d x ]c, d[ 0, x a x d (5) A trapezodal fuzzy set s well formed f the relatons a b c d hold. Any trangular fuzzy set s a trapezodal fuzzy set when a = l b = c = p d = r therefore t s possble to qualfy a fuzzy set as trapezodal even f ts actual shape s trangular. 46

4 A SFP made of trapezodal fuzzy sets A = T [a, b, c, d ], =,,..., n +, requres that a + = c b + = d for =,,..., n, as well as a = b = m c n+ = d n+ = M (6) In ths paper we present three approaches for desgnng trapezodal SFPs. The frst one (called Constant Slope ) defnes trapezodal fuzzy sets wth the same slope (n absolute value). Ths s the smplest approach as t does not requre addtonal knowledge for the desgn of a SFP. The second approach, called Varable Fuzzness s based on the dea that fuzzy sets wth a large support are more mprecse than fuzzy sets wth a small support. As a consequence, the slope of the trapezodal fuzzy sets s defned accordng to the dstance between two adjacent cuts. Fnally, the thrd approach extends the second one by requrng an addtonal set of Core Ponts,.e. ponts n the doman that must belong to the core of a fuzzy set. Ths approach can be used when t s a-pror known that some ponts are representatve of some concepts to be fully represented by lngustc terms.... Constant slope Gven a set of cuts t, t,..., t n X t s possble to defne a SFP made of trapezodal fuzzy sets by applyng the followng procedure. Frst, the dfferences between cuts = t + t are computed for = 0,,..., n, wth the conventon that t 0 = m t t n+ = M t n Then, the smallest dfference mn = mn { = 0,,..., n} s selected wth the correspondng ndex mn. (More than one ndex may verfy ths relaton: n such a case the frst ndex s selected.) By defnton, the nterval [t mn, t mn+] s the most specfc among all ntervals [t, t + ]. Therefore, the most specfc fuzzy set s defned, whch s trangular defned by the followng parameters: b mn = c mn = t mn+ + t mn a mn = t mn b mn = 3t mn t mn+ d mn = t mn+ c mn = 3t mn+ t mn The slopes of the oblque segments n the trangular fuzzy set have the same magntude but opposte sgns. In partcular, the ascendng segment has slope ρ + = b mn a mn = t mn+ t mn whle the descendng segment has slope ρ = c mn d mn = = ρ + t mn t mn+ We use the slopes ρ + ρ to defne the remanng fuzzy sets. By constructon, the use of these slopes assures that all trapezodal fuzzy sets are well-formed. In fact, hgher slopes (n magntude) could be also used, whle lower slopes may hamper the well-formedness of the trapezodal fuzzy sets. Gven a cut t, =,,..., n, the followng parameters are defned: a + = t ρ + b + = t + ρ + c = t + ρ = a + d = t ρ = b + Fnally, the leftmost rghtmost fuzzy sets are defned so as to be trunked at the extreme ponts of X. Therefore a = b = m c n+ = d n+ = M It s easy to verfy that a b c d for each =,,..., n +. Well-formedness of the trapezodal fuzzy sets can be thus checked by verfyng that b c for each =,,..., n +. We suppose, by contradcton, that b > c. By constructon, ths means that t + ρ + > t + ρ = t ρ + whch s equvalent to.e. ρ + < t t mn > whch s absurd by defnton of mn. 47

5 ... Varable fuzzness Ths approach s based on the dea that the fuzzness of a fuzzy set n a partton s dependent on the ampltude of the nterval between two cuts. In partcular, the smaller s such ampltude, the sharper are the related fuzzy sets. Fuzzness can be quantfed through the noton of entropy measure [6]; however t s easy to verfy that fuzzness s related to the slopes of the trapezodal fuzzy sets, so that hgh slopes lead to sharp fuzzy sets vce versa. The procedure for generatng the trapezodal fuzzy sets works as follows: for each = 0,,..., n the values + are compared the shortest s selected. (Here, t 0 s set to m t n+ s set to M to make the selecton coherent wth the dea underlyng ths approach.) If + then the descendng part of the fuzzy set A + s defned by a membershp functon that s hghest at the center of gets the value 0.5 at t +. Formally ths requres that c + = t + t + d + = t + c + As a consequence, the ascendng part of the fuzzy set A + s defned accordngly: a + = c + b + = d + By constructon, t s verfed that b + wll be smaller than the mdpont of +, thus guaranteeng well-formedness of the trapezodal fuzzy set. If > + the scheme s nverted the ascendng part of A + s frst defned by settng b + = t + + t + a + = t + b + Then, the descendng part of A + s defned accordngly: c + = a + d + = b + Fnally, the undefned parts of the leftmost rghtmost fuzzy set are set as n (6)...3. Core ponts Ths approach explots addtonal nformaton to defne the SFP. In partcular, t s assumed that n each nterval between two cuts a fnte nonempty set of ponts P [t, t + ] s avalable, wth the constrant that such ponts must belong to the core of the correspondng fuzzy set n the partton 4. The procedure for generatng the trapezodal fuzzy sets s smlar to that defned for varable fuzzness. More specfcally, for each P the mnmum maxmum elements are consdered,.e. p mn p max = mn P = max P Furthermore, the dstances between such ponts the cuts are consdered: δ left δ rght = t p max = p mn are com- for =,,..., n. For each the values of δ left pared: f δ left δ rght then otherwse 3. Example: DC* t a + = c = p max b + = d = t c d = b + = p mn c = a + = t b + δ rght One of the algorthms that uses cuts to generate fuzzy parttons from data s DC* (Double Clusterng wth A*) [7, 8]. In ths Secton we gve a bref outlne of DC* then we present some results on the use of DC* along wth the dfferent approaches for generatng SFPs. 3.. Outlne of DC* DC* (Double Clusterng wth A*) s an algorthm conceved for extractng nterpretable fuzzy nformaton granules from classfed data. Such nformaton granules are represented through nterpretable fuzzy parttons can be used to defne a set of fuzzy classfcaton rules. In essence, DC* works n three consecutve steps. In the frst step, a collecton D of classfed data n a mult-dmensonal doman X = [m, M ] [m d, M d ] s compressed through a vector quantzaton algorthm. (LVQ [9] s used n the current verson of DC*.) The resultng codebook p, p,... p c X 4 The core of a fuzzy set s the (crsp) set of all elements wth full membershp. 48

6 conssts of c mult-dmensonal classfed prototypes. (The parameter c s user-selected.) All the prototypes are projected onto each dmenson, so that for dmenson h, c classfed one-dmensonal prototypes p h, p h,... p hc [m h, M h ] are avalable. In the second step, DC* operates a clusterng process of one-dmensonal prototypes n all dmensons smultaneously. The objectve of ths clusterng process s to carry out a set of cuts SD t h, t h,... t hn [m h, M h ] for each dmenson so that any nterval [t h,, t h,+ ] ncludes one-dmensonal prototypes of the same class. Furthermore, the hyper-boxes [t,, t,+] [t,, t,+] [t d,d, t d,d +] nclude mult-dmensonal prototypes of the same class only. (The number of such prototypes can be zero.) Fnally, the number of hyper-boxes contanng prototypes s mnmal. To acheve ths complex objectve, the clusterng of the projectons addressed n the second step s defned as a combnatoral optmzaton problem, whch s faced by resortng to the A* search algorthm [0]. DC*, therefore, sts as a convenent approach to produce data clusterng makng use of cuts, represented by the mdponts between two adjacent projectons of prototypes belongng to dfferent classes. Furthermore, DC* produces a number of core-ponts, whch correspond to the onedmensonal prototypes are drectly related to the compressed representaton of data. The optmal confguraton of cuts dentfed by DC* represents the startng pont for a modelng procedure devoted to defne a SFP for each nput feature based on trapezodal fuzzy sets 5, whch corresponds the the last step of DC*. Furthermore, the nherent workng engne of DC* s orented to produce addtonal peces of nformaton, namely the prototypes dentfed by the LVQ algorthm. Therefore, DC* represents a sutable procedure to desgn SFP based on cuts core ponts (n lne wth the approach descrbed n Secton..3). SD SD3 SD4 3.. Smulaton on numercal data We recently evaluated DC* wth HFP on a number of benchmark datasets for the sake of comparson; we observed that, on the average, DC* exhbts a superor behavour n terms of accuracy/nterpretablty tradeoff []. The objectve of ths smulaton, nstead, s to evaluate the DC* behavour when dfferent strateges for generatng SFPs are adopted. Actually, the current verson of sets. 5 The orgnal verson of DC* produced Gaussan fuzzy SD5 Fgure 3: The synthetcally generated datasets adopted for the numercal smulaton. 49

7 Table : DC* classfcaton error (percentage values) when dfferent strateges are appled to generate fuzzy parttons for each of the fve datasets. SD SD SD3 SD4 SD5 CS VF CP TSFP T0.5-cuts DC* adopts the varable fuzzness approach to derve trapezodal SFPs, but t can be easly modfed to generate SFPs wth all the strateges that have been presented n the prevous secton. To ths am, we used a set of synthetcally generated datasets: one of them (SD) conssts of 00 bdmensonal examples, the other four datasets (SD, SD3, SD4, SD5) consst of 400 b-dmensonal examples. In each case, the samples belong to 3 dfferent classes. The datasets are depcted n fg. 3. DC* has been employed to process the data. The ntal clusterng has been performed consderng 4 mult-dmensonal prototypes for SD 48 multdmensonal prototypes for SD SD5. (The prototypes are proportonally dstrbuted accordng to the number of samples for each class.) The fnal fuzzy parttons have been derved by alternatvely applyng the prevously descrbed procedures: Constant Slope (CS), Varable Fuzzness (VF) Core Ponts (CP). Addtonally, two more strateges have also been tested, orented to the generaton of trangular fuzzy parttons. In the frst case (TSFP), SFPs have been obtaned by partally explotng the nformaton comng from cuts: the desgn of the trangular fuzzy sets s such that ther core ponts correspond to the mdponts of the ntervals defned by the cuts. In the second case (T0.5-cuts), the trangular fuzzy sets are shaped so that the membershp values n t,..., t n are set at 0.5. (As shown n secton, the latter mechansm leaves no guarantee to derve a SFP for sure.) Table reports the performance (n terms of percentage of classfcaton error) of DC* for each adopted strategy. It can be verfed that for each dataset the best performance s attaned by applyng the Core Ponts strategy. In general, resortng to trangular fuzzy parttons means a deteroraton n the classfcaton error values. More nterestngly, fg. 4 depcts the dfferent fuzzy parttons produced by DC* when the above mentoned strateges are appled. We show here the confguratons related to the clusterng processes performed over one of the synthetc datasets (namely, SD4); for the sake of concseness, only one nput feature s consdered n the fgures. It s mportant to hghlght how the choce for a trangular fuzzy partton forced to express a 0.5 value at the cuts ponts gves rse to a confguraton whch does not satsfy the SFP condtons. On the other h, the fuzzy parton provded through the CP approach gves a tangble dea on the fuzzness of the lngustc terms n accordance wth the core ponts provded by DC*: t s apparent that fuzzness s acceptable n the rght sde of the Unverse of Dscourse, whle crsper lngustc terms are requred to dscrmnate data n the center left sde. 4. Conclusons The defnton of fuzzy partton represents a key ssue for desgnng nterpretable fuzzy models snce fuzzy parttons are often requred to fulfll several nterpretablty constrants. In ths sense, Strong Fuzzy Parttons (SFPs) are commonly adopted as a relable tool to desgn nterpretable models, trangular SFPs are often preferred because they can be easly derved through some clusterng mechansm performed over the avalable data. In ths paper we consdered a partcular approach for defnng SFPs whch s based on cuts, that are ponts of separaton between cluster projectons on data dmensons. We dealt wth the problem of dentfyng the proper shape of fuzzy sets whle generatng SFPs from cuts, hghlghtng how the choce of trangular fuzzy sets represents an addtonal bas for the modelng process whch can be convenently removed by resortng to trapezodal fuzzy sets. Through some numercal smulatons that make use of DC*, a cut-based algorthm for generatng fuzzy parttons, we showed that the use of trapezodal fuzzy sets enables the dervaton of hghly nterpretable fuzzy parttons that are more accurate than trangular fuzzy parttons n classfcaton tasks. References [] Corrado Mencar Anna M Fanell. Interpretablty constrants for fuzzy nformaton granulaton. Informaton Scences, 78(4): , 008. [] Wtold Pedrycz. Why trangular membershp functons? Fuzzy Sets Systems, 64(): 30, May 994. [3] Shang-Mng Zhou John Q. Gan. Extractng Takag-Sugeno Fuzzy Rules wth Interpretable Submodels va Regularzaton of Lngustc Modfers. IEEE Transactons on Knowledge Data Engneerng, (8):9 04, August 009. [4] S. Gullaume B. Charnomordc. Generatng an Interpretable Famly of Fuzzy Parttons From Data. IEEE Transactons on Fuzzy Systems, (3):34 335, June 004. [5] Cheng-Lang Chen, Sheng-Nan Wang, Chung- Tyan Hseh, Feng-Yuan Chang. Theoretcal analyss of a fuzzy-logc controller wth 430

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