FUZZY LOGIC FUNDAMENTALS

Transcription

1 3.fm Page 6 Monday, March 26, 200 0:8 AM C H A P T E R 3 FUZZY LOGIC FUNDAMENTALS 3. INTRODUCTION The past few years have wtnessed a rapd growth n the number and varety of applcatons of fuzzy logc (FL). FL technques have been used n mage-understandng applcatons such as detecton of edges, feature extracton, classfcaton, and clusterng. Fuzzy logc poses the ablty to mmc the human mnd to effectvely employ modes of reasonng that are approxmate rather than exact. In tradtonal hard computng, decsons or actons are based on precson, certanty, and vgor. Precson and certanty carry a cost. In soft computng, tolerance and mpresson are explored n decson makng. The exploraton of the tolerance for mprecson and uncertanty underles the remarkable human ablty to understand dstorted speech, decpher sloppy handwrtng, comprehend nuances of natural language, summarze text, and recognze and classfy mages. Wth FL, we can specfy mappng rules n terms of words rather than numbers. Computng wth the words explores mprecson and tolerance. Another basc concept n FL s the fuzzy f then rule. Although rule-based systems have a long hstory of use n artfcal ntellgence, what s mssng n such systems s machnery for dealng wth fuzzy consequents or fuzzy antecedents. In most applcatons, an FL soluton s a translaton of a human soluton. Thrdly, FL can model nonlnear functons of arbtrary complexty to a desred degree of accuracy. FL s a convenent way to map an nput space to an output space. FL s one of the tools used to model a multnput, multoutput system. Soft computng ncludes fuzzy logc, neural networks, probablstc reasonng, and genetc algorthms. Today, technques or a combnaton of technques from all these areas are used to desgn an ntellgence system. Neural networks provde algorthms for learnng, classfcaton, and optmzaton, whereas fuzzy logc deals wth ssues such as formng mpressons and reasonng on a semantc or lngustc level. Probablstc reasonng deals wth uncertanty. Although there are substantal areas of overlap between neural networks, FL, and probablstc reasonng, 6

2 3.fm Page 62 Monday, March 26, 200 0:8 AM 62 Chapter 3 FUZZY LOGIC FUNDAMENTALS n general they are complementary rather than compettve. Recently, many ntellgent systems called neuro fuzzy systems have been used. There are many ways to combne neural networks and FL technques. Before dong so, however, t s necessary to understand basc deas n the desgn of FL technques. In ths chapter, we wll ntroduce FL concepts such as fuzzy sets and ther propertes, FL operators, hedges, fuzzy proposton and rule-based systems, fuzzy maps and nference engne, defuzzfcaton methods, and the desgn of an FL decson system. 3.2 FUZZY SETS AND MEMBERSHIP FUNCTIONS Zadeh ntroduced the term fuzzy logc n hs semnal work Fuzzy sets, whch descrbed the mathematcs of fuzzy set theory (965). Plato lad the foundaton for what would become fuzzy logc, ndcatng that there was a thrd regon beyond True and False. It was Lukasewcz who frst proposed a systematc alternatve to the bvalued logc of Arstotle. The thrd value Lukasewcz proposed can be best translated as possble, and he assgned t a numerc value between True and False. Later he explored four-valued logc and fve-valued logc, and then he declared that, n prncple, there was nothng to prevent the dervaton of nfnte-valued logc. FL provdes the opportunty for modelng condtons that are nherently mprecsely defned. Fuzzy technques n the form of approxmate reasonng provde decson support and expert systems wth powerful reasonng capabltes. The permssveness of fuzzness n the human thought process suggests that much of the logc behnd thought processng s not tradtonal twovalued logc or even multvalued logc, but logc wth fuzzy truths, fuzzy connectveness, and fuzzy rules of nference. A fuzzy set s an extenson of a crsp set. Crsp sets allow only full membershp or no membershp at all, whereas fuzzy sets allow partal membershp. In a crsp set, membershp or nonmembershp of element x n set A s descrbed by a characterstc functon µ ( x ), where µ ( x) = f x A and µ ( x) = 0 f x A. Fuzzy set theory extends ths concept by defnng partal membershp. A fuzzy set A on a unverse of dscourse U s characterzed A A A by a membershp functon µ ( x that takes values n the nterval. Fuzzy sets represent A ) [ 0,] commonsense lngustc labels lke slow, fast, small, large, heavy, low, medum, hgh, tall, etc. A gven element can be a member of more than one fuzzy set at a tme. A fuzzy set A n U may be represented as a set of ordered pars. Each par conssts of a generc element x and ts grade of membershp functon; that s, A= {( x, µ n, x s called a support value f. A ( x) ) x U} µ ( x ) > A 0 2 k A lngustc varable x n the unverse of dscourse U s characterzed by T( x) = { Tx, T x,..., Tx } 2 k and µ ( x ) = { µ x, µ x,.., µ x }, where T( x) s the term set of x that s, the set of names of lngustc values of x, wth each T x beng a fuzzy number wth membershp functon µ x defned on U. For example, f x ndcates heght, then T( x) may refer to sets such as short, medum, or tall. A membershp functon s essentally a curve that defnes how each pont n the nput space s mapped to a membershp value (or degree of membershp) between 0 and. As an example, consder a fuzzy set tall. Let the unverse of dscourse be heghts from 40 nches to 90 nches. Wth a crsp set, all people wth heght 72 or more nches are consdered tall, and all people wth heght of less than 72 nches are consdered not tall. The crsp set membershp functon for set tall s shown n Fgure 3.. The correspondng fuzzy set wth a smooth membershp functon s shown n Fgure 3.2. The curve defnes the transton from not tall and shows the degree of mem-

3 3.fm Page 63 Monday, March 26, 200 0:8 AM FUZZY SETS AND MEMBERSHIP FUNCTIONS 63 tall 0.8 µ(x) nput Fgure 3. Crsp membershp functon. tall 0.8 µ(x) nput Fgure 3.2 An example of a fuzzy membershp functon.

4 3.fm Page 64 Monday, March 26, 200 0:8 AM 64 Chapter 3 FUZZY LOGIC FUNDAMENTALS bershp for a gven heght. We can extend ths concept to multple sets. If we consder a unverse of dscourse from 40 nches to 90 nches, then, to descrbe heght, we can use three term values such as short, average, and tall. In practce, the terms short, medum, and tall are not used n the strct sense. Instead, they mply a smooth transton. Fuzzy membershp functons representng these sets are shown n Fgure 3.3. The Fgure shows that a person wth heght 65 nches wll have membershp value for set medum, whereas a person wth heght 60 nches may be a member of the set short and also a member of the set medum; only the degree of membershp vares wth these sets. Varous types of membershp functons are used, ncludng trangular, trapezodal, generalzed bell shaped, Gaussan curves, polynomal curves, and sgmod functons. Fgure 3.3 shows trapezodal membershp functons. Trangular curves depend on three parameters a, b, and c and are gven by 0 for x < a x a for a x < b b a f ( x; a, b, c) = c x for b x c c b 0 for x > c (3.) short medum tall µ(x) heght Fgure 3.3 Trapezodal membershp functons.

5 3.fm Page 65 Monday, March 26, 200 0:8 AM FUZZY SETS AND MEMBERSHIP FUNCTIONS 65 Trapezodal curves depend on four parameters and are gven by 0 for x < a x a for a x < b b a f( xabcd ;,,, ) = for b x < c d x for c x < d d c 0 for d x (3.2) The π-shaped membershp functons are gven by (Garratano and Rley, 993) S x; c b, c b 2, c for x c f( x; b, c) = S x; c, c + b 2, c + b for x > c (3.3) where S( x; a, b, c) represents a membershp functon defned as 0 for x < a 2 2 ( x a) for a x < b 2 ( c a) Sxabc ( ;,, ) = 2 2 ( x c) for b x c 2 ( c a) for x > c (3.4) In Equaton (3.4), a, b, and c are the parameters that are adjusted to ft the desred membershp data. The parameter b? s the half wdth of the curve at the crossover pont. The Gaussan and π- shaped membershp functons are shown n Fgures 3.4 and 3.5, respectvely. Gaussan curves depend on two parameters σ and c and are represented by 2 ( x c) f(; x σ,) c = exp 2 2σ (3.5) In desgnng a fuzzy nference system, membershp functons are assocated wth term sets that appear n the antecedent or consequent of rules.

6 3.fm Page 66 Monday, March 26, 200 0:8 AM 66 Chapter 3 FUZZY LOGIC FUNDAMENTALS cold cool normal warm hot 0.8 µ(x) temperature Fgure 3.4 Gaussan membershp functons. cold cool normal warm hot 0.8 µ(x) temperature Fgure 3.5 π-shaped membershp functons.

7 3.fm Page 67 Monday, March 26, 200 0:8 AM LOGICAL OPERATIONS AND IF THEN RULES LOGICAL OPERATIONS AND IF THEN RULES Fuzzy set operatons are analogous to crsp set operatons. The mportant thng n defnng fuzzy set logcal operators s that f we keep fuzzy values to the extremes (True) or 0 (False), the standard logcal operatons should hold. In order to defne fuzzy set logcal operators, let us frst consder crsp set operators. The most elementary crsp set operatons are unon, ntersecton, and complement, whch essentally correspond to OR, AND, and NOT operators, respectvely. Let A and B be two subsets of U. The unon of A and B, denoted A B, contans all elements n ether A or B; that s, µ ( x ) f x A or x B A B =. The ntersecton of A and B, denoted A B, contans all the elements that are smultaneously n A and B; that s, µ ( x ) f x A and x B. The complement of A s denoted by, and t contans all elements that are not n A; that s µ ( x ) = f x A, and µ ( x) = 0 f x A. The truth tables for A B = A A A these operators are shown n Fgure 3.6. In FL, the truth of any statement s a matter of degree. In order to defne FL operators, we have to fnd the correspondng operators that preserve the results of usng AND, OR, and NOT operators. The answer s mn, max, and complement operatons. These operators are defned, respectvely, as ( x) max ( x), ( x) ( x) mn ( x), ( x) ( x) ( x) µ = µ µ A B A B µ = µ µ A B A B µ = µ Α Α (3.6) The formulas for AND, OR, and NOT operators n Equaton (3.6) are useful for provng other mathematcal propertes about sets; however, mn and max are not the only ways to descrbe the ntersecton and unon of two sets. Zadeh (965) defned fuzzy unon and fuzzy ntersecton as ( x) ( x) ( x) ( x) ( x) ( x) ( x) ( x) µ =µ +µ µ µ A B A B A B µ =µ µ A B A B (3.7) Fgure 3.6 AND OR NOT A B A B A B A B A A Truth tables for AND, OR, and NOT operators.

8 3.fm Page 68 Monday, March 26, 200 0:8 AM 68 Chapter 3 FUZZY LOGIC FUNDAMENTALS In more general terms, the ntersecton of two fuzzy sets A and B s specfed by a bnary mappng T that aggregates two membershp functons as ( x) T( ( x), ( x) ) µ µ µ A B = A B (3.8) For example, the bnary operator T may represent the multplcaton of µ. A x, µ B x These fuzzy ntersecton operators are referred to as T-norm (trangular norm) operators, and they meet the followng basc requrements: = T( b a) ( ) = ( ) boundary: T 0, 0 = 0, T a, = T, a = a monotoncty: T a, b T c, d f a c and b d commutatvty: T a, b, assocatvty: T a, T b, c T T a, b, c (3.9) The frst requrement ensures the correct generalzaton of crsp sets. The second requrement mples that a decrease n the membershp values n A and B cannot produce an ncrease n the membershp value of the ntersecton of sets A and B. The thrd requrement specfes that the operaton s nsenstve to the order n whch fuzzy sets are combned, and the fourth requrement enables us to take the ntersecton of any number of fuzzy sets and any order of parwse groupngs. Smlar to fuzzy ntersecton, the fuzzy unon operator s specfed by the followng bnary mappng S: (, ) µ S µ x µ x = A B A B (3.0) These fuzzy unon operators are known as T-conorm or S-norm operators, and they satsfy the followng requrements: = Sba ( ) = ( ) boundary: S, =, S a, 0 = S 0, a = a monotoncty: Sab, Scd, f a cand b d commutatvty: Sab,, assocatvty: SaSbc,, SSab,, c (3.)

9 3.fm Page 69 Monday, March 26, 200 0:8 AM LOGICAL OPERATIONS AND IF THEN RULES 69 Several T-norms and S-norms have been suggested n the lterature (Yager, 980; Dubos and Prade, 980; Schwezer and Sklar, 963, Sugeno, 977). One example of a par of S-norm and T-norm operators s the bounded sum and bounded product: [ ] [ ] x y= mn, x+ y x y= max 0, x+ y (3.2) Most applcatons use mn for fuzzy ntersecton, max for fuzzy unon, and µ x for complementaton. We have to remember that operators used n FL, such as unon, ntersecton, A and complement, reduce to ther crsp logc counterparts when the membershp functons are restrcted to 0 or. Fuzzy nference systems consst of f then rules that specfy a relatonshp between the nput and output fuzzy sets. Fuzzy relatons present a degree of presence or absence of assocaton or nteracton between the elements of two or more sets. Let U and V be two unverses of dscourse. A fuzzy relaton RUV, s a set n the product space U V and s characterzed by the membershp functon µ ( xy, ), where x U and y V, and µ ( xy, ) [ ]. Fuzzy relatons R R 0, play an mportant role n fuzzy nference systems. FL uses notons from crsp logc. Concepts n crsp logc can be extended to FL by replacng 0 or values wth fuzzy membershp values. A sngleton fuzzy rule assumes the form f x s A, then y s B, where x U and y V, and has a membershp functon, µ ( xy, ), where µ. The f part of the rule, x s A, s A B ( xy A B, ) [ 0,] called the antecedent or premse, whle the then part of the rule, y s B, s called the consequent or concluson. Interpretng an f then rule nvolves two dstnct steps. The frst step s to evaluate the antecedent, whch nvolves fuzzfyng the nput and applyng any necessary fuzzy operators. The second step s mplcaton, or applyng the result of the antecedent to the consequent, whch essentally evaluates the membershp functon µ xy. It can be seen that n crsp logc a rule A B, s fred f the premse s exactly the same as the antecedent of the rule, and the result of such rule frng s the rule s actual consequent. In fuzzy logc, a rule s fred so long as there s a nonzero degree of smlarty between the premse and the antecedent of the rule. For most applcatons, the fuzzy membershp functon µ xy for a gven relaton s obtaned wth the mnmum or A B, product mplcaton, gven, respectvely, as follows: ( x) ( x) ( x) ( x) mn ( x), ( x) µ =µ µ A B A B µ = µ µ A B A B (3.3) (3.4)

10 3.fm Page 70 Monday, March 26, 200 0:8 AM 70 Chapter 3 FUZZY LOGIC FUNDAMENTALS It was Mamdan (977) who frst proposed the mnmum mplcaton, and later Larsen (980) proposed the product mplcaton. The mnmum and product nferences have nothng to do wth tradtonal prepostonal logc; hence, they are collectvely referred to as engneerng mplcatons. Detals of mplcaton methods can be found n the classc tutoral paper by Mendel (995). 3.4 FUZZY INFERENCE SYSTEM A fuzzy nference system (FIS) essentally defnes a nonlnear mappng of the nput data vector nto a scalar output, usng fuzzy rules. The mappng process nvolves nput/output membershp functons, FL operators, fuzzy f then rules, aggregaton of output sets, and defuzzfcaton. An FIS wth multple outputs can be consdered as a collecton of ndependent multnput, sngle-output systems. A general model of a fuzzy nference system (FIS) s shown n Fgure 3.7. The FLS maps crsp nputs nto crsp outputs. It can be seen from the fgure that the FIS contans four components: the fuzzfer, nference engne, rule base, and defuzzfer. The rule base contans lngustc rules that are provded by experts. It s also possble to extract rules from numerc data. Once the rules have been establshed, the FIS can be vewed as a system that maps an nput vector to an output vector. The fuzzfer maps nput numbers nto correspondng fuzzy membershps. Ths s requred n order to actvate rules that are n terms of lngustc varables. The fuzzfer takes nput values and determnes the degree to whch they belong to each of the fuzzy sets va membershp functons. The nference engne defnes mappng from nput fuzzy sets nto output fuzzy sets. It determnes the degree to whch the antecedent s satsfed for each rule. If the antecedent of a gven rule has more than one clause, fuzzy operators are appled to obtan one number that represents the result of the antecedent for that rule. It s possble that one or more rules may fre at the same tme. Outputs for all rules are then aggregated. Durng aggregaton, fuzzy sets that represent the output of each rule are combned nto a sngle fuzzy set. Fuzzy rules are fred n parallel, whch s one of the mportant aspects of an FIS. In an FIS, the order n whch rules are fred does not affect the output. The defuzzfer maps output fuzzy sets nto a crsp number. Gven a fuzzy set that encompasses a range of output values, the defuzzfer nput x fuzzfer nference engne defuzzfer output y rule base Fgure 3.7 Block dagram of a fuzzy nference system.

11 3.fm Page 7 Monday, March 26, 200 0:8 AM FUZZY INFERENCE SYSTEM 7 returns one number, thereby movng from a fuzzy set to a crsp number. Several methods for defuzzfcaton are used n practce, ncludng the centrod, maxmum, mean of maxma, heght, and modfed heght defuzzfer. The most popular defuzzfcaton method s the centrod, whch calculates and returns the center of gravty of the aggregated fuzzy set. FISs employ rules. However, unlke rules n conventonal expert systems, a fuzzy rule localzes a regon of space along the functon surface nstead of solatng a pont on the surface. For a gven nput, more than one rule may fre. Also, n an FIS, multple regons are combned n the output space to produce a composte regon. A general schematc of an FIS s shown n Fgure 3.8. Consder a multnput, multoutput system. Let x = x, x2,..., x n be the nput vector and T y = ( y be the output vector. The lngustc varable n the unverse of dscourse U, y2,..., y m ) x 2 k 2 k s characterzed by T( x ) = { Tx, Tx,..., Tx } and µ ( x ) = { µ x, µ x,..., µ x } where T ( x) s a term set of x; that s, t s the set of names of lngustc values of x, wth each T x beng a fuzzy member and the membershp functon µ x defned onu. As an llustraton, we consder a fuzzy nference system wth two nputs ( n = 2) and one output ( m = ). Let the two nputs represent the number of years of educaton and the number of years of experence, and let the output of the system be salary. Let x ndcate the number of years of educaton, T( x ) represent ts term set {low, medum, hgh}, and the unverse of dscourse be [ 0 5]. Let x 2 ndcate the number of years of experence, the unverse of dscourse be [ 0 30], and the correspondng term set be {low, medum, hgh}. Smlarly, lngustc varable y n the unverse of dscourse V s characterzed by T( y ) = { T, where s a term set of y; that s, T s the set of names of ln- 2 l y, Ty,..., Ty } T ( y) gustc values of y, wth each beng a fuzzy membershp functon defned on V. If the T y Producton rules T µ y Fuzzfer f x n T x th y n T y Aggregaton Defuzzfcaton x x 2 x 3 f x 2 n T x 2 th y n T y y x n f x 3 n T x 3 th y n T y f x n n T x n th y n T y Fgure 3.8 Schematc dagram of a fuzzy nference system.

12 3.fm Page 72 Monday, March 26, 200 0:8 AM 72 Chapter 3 FUZZY LOGIC FUNDAMENTALS varable y represents salary, then T y represents a term set {very low, low, medum, hgh, very hgh}, and the unverse of dscourse s [20 200], whch represents the mnmum and maxmum n thousands of dollars that s, 20,000, and 200,000, respectvely. In order to map nput varables x and x 2 to output y, t s necessary that we frst defne the correspondng fuzzy sets. The membershp functons for the nput and output varables are shown n Fgure 3.9. The frst step n evaluatng the output of a FIS s to apply the nputs and determne the degree to whch they belong to each of the fuzzy sets. The fuzzfer block performs the mappng from the nput feature space to fuzzy sets n a certan unverse of dscourse. A specfc value x s then mapped to 2 2 the fuzzy set T x wth degree µ x, to fuzzy set T x wth degree µ x, and so on. In order to perform ths mappng, we can use fuzzy sets of any shape, such as trangular, Gaussan, π-shaped, etc. A fuzzy rule base contans a set of fuzzy rules R. A sngle f then rule assumes the form f x s T x then y s T y. An example of a rule mght be f educaton s hgh and experence s hgh, then salary s very hgh. For a multnput, multoutput system, R= R, R 2,..., Rn (3.5) low medum hgh Degree of membershp Fgure 3.9a educaton Fuzzy membershp functons for nput.

13 3.fm Page 73 Monday, March 26, 200 0:8 AM FUZZY INFERENCE SYSTEM 73 low medum hgh degree of membershp experence Fgure 3.9b Fuzzy membershp functon for nput2. very_low low medum hgh very_hgh degree of membershp Fgure 3.9c salary Fuzzy membershp functon for output.

14 3.fm Page 74 Monday, March 26, 200 0:8 AM 74 Chapter 3 FUZZY LOGIC FUNDAMENTALS where the th fuzzy rule s ( ) ( ) p q R = f x s T, and..., x s T then y n T, and..., Y n T x p x y q y (3.6) The p precondtons of R form a fuzzy set Tx T, and the consequent s the x T 2 xp unon of q ndependent outputs. If we consder a multnput, sngle-output system, then the consequent reduces to ( y s. For the gven example, the rules are stated as T ) R : f educaton s low and experence s low, then salary s very low R 2 : f educaton s low and experence s medum, then salary s low R 3 : f educaton s low and experence s hgh, then salary s medum R 4 : f educaton s medum and experence s low, then salary s low R 5 : f educaton s medum and experence s medum, then salary s medum R 6 : f educaton s medum and experence s hgh, then salary s hgh R 7 : f educaton s hgh and experence s low, then salary s medum R 8 : f educaton s hgh and experence s medum, then salary s hgh R 9 : f educaton s hgh and experence s hgh, then salary s very hgh Interpretng an f then rule s a three part process: (a) Resolve all fuzzy statements n the antecedent to a degree of membershp between 0 and ; (b) f there are multple parts to the antecedent, apply fuzzy logc operators and resolve the antecedent to a sngle number between 0 and, s the result beng the degree of support for the rule; and (c) apply the mplcaton method, usng the degree of support for the entre rule to shape the output fuzzy set. If the rule has more than one antecedent, the fuzzy operator s appled to obtan one number that represents the result of applyng that rule. For example, consder an th rule : f s x and 2 s 2 then s y R x T x T y T (3.7) Then the frng strength or membershp of the rule can be defned as ( ( x ) ) x 2 2 α = mn µ, µ x x or (3.8) α ( x ) ( x ) =µ x µ x2 2 Equaton (3.8) represents fuzzy ntersecton wth the mnmum or product operators. Each fuzzy rule yelds a sngle number that represents the frng strength of that rule. The frng strength s then used to shape the output fuzzy set that represents the consequent part of the rule. The mplcaton method s defned as the shapng of the consequent (the output fuzzy set), based

15 3.fm Page 75 Monday, March 26, 200 0:8 AM FUZZY INFERENCE SYSTEM 75 on the antecedent. The nput for the mplcaton process s a sngle number gven by the antecedent, and the output s a fuzzy set. Two methods are commonly used: the mnmum and the product methods, represented, respectvely, by Equatons (3.9) and (3.20), respectvely. ( w) ' mn, ( w) y y µ = α µ ( w) ' ( w) y y µ =αµ (3.9) (3.20) where w s the varable that represents the support value of the membershp functon. For the gven example, f we assume that educaton equals 0 years and experence equals 8.6 years, then t can be seen n Fgure 3.0 that rules R 5 and R 6 fre. After we obtan frng strengths of the rules, we need to combne the correspondng output fuzzy sets nto one composte fuzzy set. The process of combnng output fuzzy sets nto a sngle set s called aggregaton, a processthat unfes the outputs of all the rules. Essentally, aggregaton takes all fuzzy sets that represent the output for each rule and combnes them nto a sngle fuzzy set that s used as the nput to the defuzzfcaton process. Aggregaton occurs only once for each output varable. The nputs to the aggregaton process are truncated or modfed output fuzzy sets obtaned as the output of the mplcaton process. The output of the aggregaton process s a sngle fuzzy set that represents the output varable. Snce the aggregaton method s commutatve, the order n whch the rules are executed s not mportant. The commonly used aggregaton method s the max method. If we have two rules wth output fuzzy sets represented by two fuzzy sets µ y (w) and µ y2 (w), then, combnng the two sets, we obtan the output decson 2 ( w) max ( w) ( w) µ = µ µ y y y (3.2) Notce that the last result s a membershp curve. The output of aggregaton of fuzzy sets n our example s shown n Fgure 3.0. In order to get a crsp value for output y, we need a defuzzfcaton process. The nput to the defuzzfcaton process s a fuzzy set (the aggregate output fuzzy set), and the output of the defuzzfcaton process s a sngle crsp number. The most commonly used defuzzfcaton method s the centrod calculaton. Methods of defuzzfcaton are dscussed n the next secton. The fuzzy nference process defnes the mappng surface y= f ( x, whch s llustrated, x2) n Fgure 3.. The nference process can be descrbed completely n the fve steps shown n Fgure 3.2. Step : Fuzzy Inputs The frst step s to take nputs and determne the degree to whch they belong to each of the approprate fuzzy sets va membershp functons.

16 3.fm Page 76 Monday, March 26, 200 0:8 AM Fgure 3.0 Fuzzy rules. 76

17 3.fm Page 77 Monday, March 26, 200 0:8 AM FUZZY INFERENCE SYSTEM 77 Fgure 3. Mappng surface. Step 2: Apply Fuzzy Operators Once the nputs have been fuzzfed, we know the degree to whch each part of the antecedent has been satsfed for each rule. If a gven rule has more than one part, the fuzzy logcal operators are appled to evaluate the composte frng strength of the rule. Step 3: Apply the Implcaton Method The mplcaton method s defned as the shapng of the output membershp functons on the bass of the frng strength of the rule. The nput for the mplcaton process s a sngle number gven by the antecedent, and the output s a fuzzy set. Two commonly used methods of mplcaton are the mnmum and the product. Step 4: Aggregate all Outputs Aggregaton s a process whereby the outputs of each rule are unfed. Aggregaton occurs only once for each output varable. The nput to the aggregaton process s the truncated output fuzzy sets returned by the mplcaton process for each rule. The output of the aggregaton process s the combned output fuzzy set. Step 5: Defuzzfy The nput for the defuzzfcaton process s a fuzzy set (the aggregated output fuzzy set), and the output of the defuzzfcaton process s a crsp value obtaned by usng some defuzzfcaton method such as the centrod, heght, or maxmum. As an example, we consder a system that determnes dnner n a restaurant on the bass of the servce receved. We consder nput membershp functons wth dfferent degrees of overlap. Here, the nput x denotes the qualty of the

18 3.fm Page 78 Monday, March 26, 200 0:8 AM 78 Chapter 3 FUZZY LOGIC FUNDAMENTALS Fuzzfy nputs Apply fuzzy operators Apply mplcaton method Aggregate all output fuzzy sets Defuzzfy Fgure 3.2 Fuzzy nference process. servce, whch s represented by a number between 0 and 20, where 20 desgnates very good and 0 very poor. The nput x s represented by the term set {very poor, poor, average, good, very good}. The output y represents the tp, whch vares between 5 and 30 percent, and s gven by the term set {very cheap, cheap, average, generous, very generous}. The nput output fuzzy sets for ths example are shown n Fgures 3.3 and 3.4, respectvely. We have assumed the followng fve rules that defne mappng y = f ( x) : R : f servce s very poor, then tp s very cheap R 2 : f servce s poor, then tp s cheap R 3 : f servce s average, then tp s average R 4 : f servce s good, then tp s generous R 5 : f servce s very good, then tp s very generous

19 3.fm Page 79 Monday, March 26, 200 0:8 AM FUZZY INFERENCE SYSTEM 79 very-poor poor average good very-good µ(x) servce Fgure 3.3 Input fuzzy sets. v. cheap cheap average generous v-generous µ(x) tp Fgure 3.4 Output fuzzy sets.

20 3.fm Page 80 Monday, March 26, 200 0:8 AM 80 Chapter 3 FUZZY LOGIC FUNDAMENTALS tp(%) servce Fgure 3.5 Mappng surface. We have mplemented the FIS wth these rules. The mappng functon for the system s shown n Fgure 3.5. In order to see the effect of overlap between nput fuzzy sets on the mappng functons, we used another set of fuzzy nput output membershp functons, llustrated n Fgures 3.6 and 3.7, respectvely. The correspondng mappng functon s shown n Fgure 3.8. The mappng functon y = f ( x) depends on the number of nput fuzzy sets and ther shapes. It can be seen from the precedng example that, as we ncrease the overlap between the nput fuzzy sets, the mappng functon becomes smoother. 3.5 DEFUZZIFICATION A fuzzy nference system maps an nput vector to a crsp output value. In order to obtan a crsp output, we need a defuzzfcaton process. The nput to the defuzzfcaton process s a fuzzy set (the aggregated output fuzzy set), and the output of the defuzzfcaton process s a sngle number. Many defuzzfcaton technques have been proposed n the lterature. The most commonly used method s the centrod. Other methods nclude the maxmum, the means of maxma, heght, and modfed heght method. The fve methods may be descrbed as follows: (a) Centrod defuzzfcaton method: In ths method, the defuzzfer determnes the center of gravty (centrod) y ' of B and uses that value as the output of the FLS. For a contnuous aggregated fuzzy set, the centrod s gven by y' = s yµ s µ B B y dy ydy (3.22)

21 3.fm Page 8 Monday, March 26, 200 0:8 AM DEFUZZIFICATION 8 very-poor poor average good very-good µ(x) servce Fgure 3.6 Input fuzzy sets. v.-cheap cheap average generous v. generous 0.8 µ(x) tp Fgure 3.7 Output fuzzy sets.

22 3.fm Page 82 Monday, March 26, 200 0:8 AM 82 Chapter 3 FUZZY LOGIC FUNDAMENTALS tp(%) Fgure servce Mappng surface. where S denotes the support of µ y. Often, dscretzed varables are used so that y B can be approxmated as shown n Equaton (3.23), whch uses summatons nstead of ntegraton. y' = n = yµ ( y ) B = n µ B ( y ) (3.23) The centrod defuzzfcaton method fnds the balance pont of the soluton fuzzy regon by calculatng the weghted mean of the output fuzzy regon. It s the most wdely used technque because, when t s used, the defuzzfed values tend to move smoothly around the output fuzzy regon. The technque s unque, however, and not easy to mplement computatonally. The method of centrod defuzzfcaton s depcted n Fgure 3.9. (b) Maxmum-decomposton method: In ths method, the defuzzfer examnes the aggregated fuzzy set and chooses that output y for whch µ ( y s the maxmum, as B ) shown n Fgure Unlke the centrod method, the maxmum-decomposton

23 3.fm Page 83 Monday, March 26, 200 0:8 AM DEFUZZIFICATION 83 Center of gravty µ(y) Fgure 3.9 x Centrod defuzzfcaton method. method has some propertes that are applcable to a narrower class of problems. The output value for ths method s senstve to a sngle rule that domnates the fuzzy rule set. Also, the output value tends to jump from one frame to the next as the shape of the fuzzy regon changes. (c) Center of maxma: In a multmode fuzzy regon, the center-of-maxma technque fnds the hghest plateau and then the next hghest plateau. The mdpont between the centers of these plateaus s selected as shown n Fgure 3.2. µ(y) Fgure 3.20 Defuzzfcaton (maxmum-decomposton method). x

24 3.fm Page 84 Monday, March 26, 200 0:8 AM 84 Chapter 3 FUZZY LOGIC FUNDAMENTALS µ(y) Fgure 3.2 Defuzzfcaton (average of maxmums). y (d) Heght defuzzfcaton: In ths method, the defuzzfer frst evaluates µ B y at y' and then computes the output of the FLS, where denotes the center of gravty of fuzzy sets B. Output y h n ths case s gven by y h = m y ' µ ( y ) B = m = µ B ( y ) (3.24) where m represents the number of output fuzzy sets obtaned after mplcaton and y represents the centrod of fuzzy regon. Ths technque s easy to use because the centers of gravty of commonly used membershp functons are known ahead of tme. Regardless of whether mnmum or product nference s used, the fuzzy nference process essentally defnes the mappng of the gven vector of crsp values to an output crsp value usng fuzzy rules stored n the knowledge base. The fuzzy nference process just dscussed s known as Mamdan s fuzzy nference method. Sugeno (977) suggested a fuzzy nference method that s smlar to Mamdan s. In Sugeno s method, the frst two parts, namely, mappng nputs to fuzzy membershp functons and applyng fuzzy operators, are the same as n Mamdan s method. The man dfference between the two s n the evaluaton of the output membershp functons. In Sugeno s method, the output membershp functon s a constant or a lnear functon. A fuzzy rule for the zero-order Sugeno method s of the form f x s A and y s B then C = K, where A and B are fuzzy sets n the antecedent and K s a constant. The frst-order Sugeno model has rules of the form f x s A and y s B then C = px+ qy+ r, where A and B are fuzzy sets n the antecedent and p, q, and r are constants.

25 3.fm Page 85 Monday, March 26, 200 0:8 AM FUZZY SET REPRESENTATION WITH A CUBE FUZZY SET REPRESENTATION WITH A CUBE Kosko (997) provded a geometrc representaton of fuzzy sets. The geometry of fuzzy sets nvolves both the doman x = { x, x 2,..., xn}, and the range of mappng µ. The A : x [ 0,] geometry of fuzzy sets ads us n descrbng fuzzness and defnng fuzzy concepts. Kosko represents the fuzzy power set F ( 2 x ), the set of all fuzzy subsets of x, by a cube. A pont n the cube n represents a fuzzy set. The set of all fuzzy subsets equals the unt hypercube I = [ 0,] n. Vertces n of the cube, I, defne crsp sets. A two-dmensonal hypercube s shown n Fgure We may vew the fuzzy subset P( 0.6, 0.3) as a pont n the unt hypercube. Wth ths formulaton, we defne fuzzy set ntersecton by the parwse mnmum, unon by the parwse maxmum, and complement by a reversal of order, as follows: µ = mn µ, µ A B A B µ = max µ, µ A B A B C A µ = µ A (3.25) A = Consder two fuzzy sets, and B = n a fourdmensonal cube. The followng are varous ntersectons, unons, and complements of these sets: A A A B= A B= A C C = A C = A = ( ) ( ) ( ) ( ) ( ) (3.26) (0,) (,) P(0.6, 0.4) (0,0) (,0) Fgure 3.22 Unt cube representng a fuzzy set.

26 3.fm Page 86 Monday, March 26, 200 0:8 AM 86 Chapter 3 FUZZY LOGIC FUNDAMENTALS C The overlap-ft vector A A n ths example s not equal to all zeros, and the overlap-ft C vector A A s not equal to the vector of all ones. A dstance between two fuzzy sets s often requred to calculate fuzzness or entropy. The dstance of order l between two ponts A and B can be defned as n l d A B x x ( ; ) = µ A µ B = l l (3.27) where n represents the dmenson of the unt cube, l s the order of the dstance measure, x represents th varable, µ A( x) represents membershp values for fuzzy set A, and µ B( x) denotes membershp values for fuzzy set B. For l =, Equaton (3.27) represents the Hammng dstance, whereas for l = 2, t represents the Eucldean dstance. Ths leads us to a smple defnton of the sze S( A) of set A as the sum of ts components or the fuzzy Hammng dstance between the orgn and the pont n the hypercube representng the set: S A n = A = (3.28) The poston of the fuzzy set n the unt hypercube determnes the set s fuzzness. Snce entropy s a measure of uncertanty, the term fuzzy entropy s defned to quantfy fuzzness. The fuzzy entropy of fuzzy set A, E( A), vares from 0 to on the unt hypercube. The vertces of the cube have zero entropy, whle the mdpont has the maxmum, or unty, entropy. Kosko (997) defned entropy as the rato of the dstance between the pont defnng the fuzzy subset and the nearest vertex to the dstance between the pont and the farthest vertex: l d E( A) = l d ( A, Anear ) ( A, A ) far (3.29) The fuzzy entropy theorem states that the entropy can be wrtten as S A E( A) = S A C ( A ) C ( A ) (3.30) Kosko (986, 987) provded a geometrc proof of the fuzzy entropy theorem. Wth ths representaton, the fuzzy nference process can be represented as the mappng of a fuzzy set from one cube to another.

27 3.fm Page 87 Monday, March 26, 200 0:8 AM HEDGES HEDGES A lngustc hedge or modfer s an operaton that modfes the meanng of a term or a fuzzy set. For example, f hot s a fuzzy set, then very hot, more or less hot, and extremely hot are examples of hedges that are appled to that fuzzy set. Hedges can be vewed as operators that act upon a fuzzy set s membershp functon to modfy t. Hedges play the same role n fuzzy producton rules that adjectves and adverbs play n Englsh sentences. There are hedges that ntensfy the characterstcs of a fuzzy set (very, extremely), that dlute the membershp curve (somewhat, rather, qute), that form the complement (not), and that approxmate a scalar to a fuzzy set (about, close to, approxmately). The mechancs underlyng the hedge operaton s.3 generally heurstc n nature. For example, µ A ( x) s used frequently to mplement the hedge slghtly. Zadeh s orgnal defnton of the hedge very ntensfes the fuzzy regon by squarng the 2 membershp functon at each pont n the set µ A ( x). On the other hand, the hedge somewhat dlutes the fuzzy regon by takng the square root of the membershp functon at each pont along 0.5 the set µ A x. A generalzaton of the concentrator hedge s con( A ) n ( x) ( x) µ = µ A (3.3) where n. Ths hedge smply replaces the exponent of the ntensfcaton functon wth a real postve number greater than unty. Fgure 3.23 shows the hedge that uses the general concentrator form wth n = 3. The complement of very s a hedge group represented by somewhat, rather, and qute. These hedges bascally dlute the force of a fuzzy set membershp functon. A gener- hot µ(x) very hot temperature Fgure 3.23 Concentrator hedge.

28 3.fm Page 88 Monday, March 26, 200 0:8 AM 88 Chapter 3 FUZZY LOGIC FUNDAMENTALS alzaton of the dlator hedge smply replaces the exponent of the ntensfcaton functon wth a real postve number less than unty, expressed as a fracton ( /n). The generalzed dlator edge s defned as dl( A) ( x) n ( x) µ =µ A (3.32) where n. Generalzed dlator hedges are shown n Fgure The contrast hedges change the nature of fuzzy regons by makng the regon ether less fuzzy (ntensfcaton) or more fuzzy (dffuson). Hedges such as postvely, absolutely, and defntely are contrast hedges, changng a fuzzy set by rasng the truth values above ( 0.5 ) and decreasng all the truth values below ( 0.5), thus reducng the overall fuzzness of the regon (Fgure 3.25). These hedges are represented by 2 ( A ( A) ) A( A) n ( A) 2 µ f µ 0.5 µ nf ( A) = 2( µ A ) otherwse (3.33) µ(x) 0.4 rather hot hot temperature Fgure 3.24 Dlator hedge.

29 3.fm Page 89 Monday, March 26, 200 0:8 AM HEDGES µ(x) 0.4 postvely tall 0.2 tall heght (nches) Fgure 3.25 Intensfcaton hedge. Smlarly, a hedge such as generally changes the fuzzy surface by reducng all truth values above ( 0.5 ) and ncreasng all truth values below ( 0.5), as shown n Fgure These hedges are represented by 2 µ A ( A) f µ A( A) µ def ( A) = µ 2 A ( A) otherwse 2 (3.34) Snce a hedge s lngustc n nature, multple hedges can be appled to a sngle fuzzy regon. The approxmaton hedges are an mportant class of transformers. They not only broaden or restrct exstng bell-shaped fuzzy regons, but also convert scalar values nto bell-shaped fuzzy regons. The most often used approxmate hedge s the about hedge, whch creates a space that s proportonal to the heght and wdth of the generated fuzzy space.

30 3.fm Page 90 Monday, March 26, 200 0:8 AM 90 Chapter 3 FUZZY LOGIC FUNDAMENTALS µ(x) generally tall tall heght (nches) Fgure 3.26 Dffuson hedge. 3.8 FUZZY SYSTEMS AS FUNCTION APPROXIMATORS A fuzzy system can be used to approxmate a functon. Kosko (997) descrbed a class of addtve fuzzy systems. An addtve fuzzy system approxmates a functoon by coverng ts graph wth fuzzy patches. The approxmaton mproves as the fuzzy patches grow n number. Addtve fuzzy systems have a feed-forward archtecture that resembles the feed-forward multlayer neural systems used to approxmate functons. Addtve fuzzy systems are dfferent from conventonal fuzzy nference systems. Addtve fuzzy systems add the then parts of fred f-then rules, whereas conventonal fuzzy nference systems combne the then part wth parwse maxma. The fuzzy mappng functon for the adaptve fuzzy system F: X Y that approxmates a functon f : x y s shown n Fgures 3.27 and Fuzzy patches approxmate the functon. The approxmaton mproves wth the number of patches. However, the computatonal cost also ncreases wth the number of patches. A schematc dagram for an addtve fuzzy system s presented n Fgure A Cartesan product space for the system s shown n Fgure Rules for the addtve system are of the form If X s A, then Y s B. Addtve fuzzy systems fre all rules n parallel and average the scaled then part sets. The class of addtve fuzzy systems represents a large number of addtve systems. The most commonly used model for an addtve fuzzy

31 3.fm Page 9 Monday, March 26, 200 0:8 AM FUZZY SYSTEMS AS FUNCTION APPROXIMATORS 9 y Fgure 3.27 x Functon approxmaton wth four fuzzy patches. y Fgure 3.28 x Functon approxmaton wth fve fuzzy patches. If A then B x A If A 2 then B 2 defuzzfer y If A n then B n Fgure 3.29 Addtve fuzzy nterference system.

32 3.fm Page 92 Monday, March 26, 200 0:8 AM 92 Chapter 3 FUZZY LOGIC FUNDAMENTALS LP SP Y ZE If X = ZE, then Y = SN SN LN Fgure 3.30 LN SN ZE SP LP X Fuzzy rule as a state patch. system s the standard addtve model (SAM), whch defnes a functon gven by Kosko (992), vz., y= F x = = m j= m j= yb y dy bydy wva j j j yj wva j j j xc ( x) (3.35) where V j s the volume of the jth then-part set B j and w j s the weght of the jth rule (often, w j = ). The term c y s the centrod of the jth output set. Ft valu a escales the then-part set B j, and m ( x) s the number of rules. The SAM equaton departs from the lngustc context of earler fuzzy models. The complexty of a SAM system depends on the complexty of the f-part fuzzy sets A j and the dmensonalty of the problem. Smple sets such as trapezods and bell curves lead to effcent approxmaton.

33 3.fm Page 93 Monday, March 26, 200 0:8 AM EXTRACTION OF RULES FROM SAMPLE DATA POINTS EXTRACTION OF RULES FROM SAMPLE DATA POINTS Earler, we saw that n order to desgn a FIS, we need a rule base that contans fuzzy rules. Usually, these rules are obtaned from expert knowledge. However, on many occasons, we may not know rules, but we may have sample data ponts or tranng samples n the nput/output spaces. In stuatons lke ths, t s possble to generate fuzzy rules that defne the mappng surface. The generated rules then can be used to desgn an FIS that performs the desred mappng. Wang and Mendel (99) suggested a systematc method for extractng fuzzy rules from sample data ponts. The method conssts of fve steps. They have also shown that the mappng surface can approxmate any real contnuous functon on a compact set to a desred degree of accuracy. In ths method, t s possble to combne nformaton of two knds: numerc and lngustc. Consder a functon y= f x, x2. We can desgn an FIS wth two nputs and one output to approxmate the functon, usng the followng steps: Step. Dvde the nput/output space nto fuzzy regons: Dvde each doman nterval nto 2N + regons. Let the regons be denoted as S N (small N),..., S (small ), CE (central), B (bg ),..., B N (bg N). The number of regons can be dfferent for each varable. Assgn each regon a fuzzy membershp functon. Fuzzy membershp functons for x, x 2, and y are shown n Fgure 3.3a c. S 2 S CE B B µ(x) x Fgure 3.3a Fuzzy membershp functon.

34 3.fm Page 94 Monday, March 26, 200 0:8 AM 94 Chapter 3 FUZZY LOGIC FUNDAMENTALS S 2 S CE B B µ(x ) x 2 Fgure 3.3b Fuzzy membershp functon. S CE B µ(y) y Fgure 3.3c Fuzzy membershp functon.

35 3.fm Page 95 Monday, March 26, 200 0:8 AM EXTRACTION OF RULES FROM SAMPLE DATA POINTS 95 Step 2. Generate fuzzy rules from gven data ponts: Frst determne the degree of the gven data pont x(), x2(), and y() for each membershp functon, and assgn the pont to the regon wth the maxmum degree of membershp. Then obtan a rule from the gven data pont. The rule may look lke the followng: R : f x s B and x 2 s CE, then y s B Step 3. Assgn a degree to each rule: There are many data ponts, each generatng a rule. Therefore, some rules may conflct. The degree of a rule can be defned as follows: For rule R, vz., f x s A and x 2 s B, then y s C, the degree of the rule s defned as =µ µ µ DR x x y A B 2 c (3.36) where µ A x, µ B x2, and µ C y represent membershp values n fuzzy sets A, B, and C, respectvely. In practce, we may have a pror nformaton about data ponts. We therefore assgn a degree to each data pont that represents expert belef n the rule. The degree of the rule then can be wrtten as DR =µ x µ x µ yµ A B 2 c (3.37) Step 4. Create a combned FAM bank, shown n Fgure Cells n the FAM bank are to be flled wth fuzzy rules, by assgnment. If there s more than one rule n any cell, then the rule wth the maxmum degree s used. If there s a rule wth an OR operator, t flls all the cells n that row or column. Step 5. Determne a mappng based on the combned FAM bank. We used the followng defuzzfcaton to determne output y for gven nputs ( x : The output membershp regon o, x2) s gven by µ =µ µ ( x ) ( x ) o 2 2 (3.38) The centrod defuzzfcaton method can be used to obtan the crsp output y. We can vew the precedng fve-step procedure as a block. Input to the block conssts of examples and expert rules, and the output of the block s a mappng from nput to output space. Wang and Mendel (99) used the fve-step process for a chaotc tme-seres predcton problem. A chaotc tme seres whch s suffcently complcated that t appears to be random can be

36 3.fm Page 96 Monday, March 26, 200 0:8 AM 96 Chapter 3 FUZZY LOGIC FUNDAMENTALS B 2 B X 2 CE S S 2 S 2 S CE B B 2 Fgure 3.32 x The form of a FAM bank. generated from a determnstc nonlnear system. The tme seres Wang and Mendel used s generated from the dfferental equaton () 0.2x ( t τ) 0 + ( ) dx t = 0.xt dt x t τ () (3.39) where t > 7. Those two researchers generated and plotted the tme seres wth,000 ponts, usng the frst 700 ponts to tran the system and the last 300 ponts to test the system. They approxmated the tme seres usng both an FLS and a neural network and compared the results. They showed that the approxmaton could be greatly mproved by dvdng the doman nto fner ntervals. An FLS can approxmate any arbtrary contnuous functon to any desred degree of accuracy. The ablty to approxmate an arbtrary functon s an mportant property that can be used n many mage-processng applcatons. Gven dscrete data ponts, we can generate a functon passng through those ponts. The functon can then be used n applcatons such mage nterpolaton or predctve mage codng. 3.0 FUZZY BASIS FUNCTIONS From the prevous secton, we know that a FIS s a nonlnear system that maps a crsp nput vector x to a scalar y. The mappng can be represented as y= f ( x). Dscussons n the prevous sectons offer a geometrc nterpretaton of a FIS. Km and Mendel (995) provded the

37 3.fm Page 97 Monday, March 26, 200 0:8 AM FUZZY BASIS FUNCTIONS 97 mathematcal formula that defnes the mappng functon for a FIS. Ther expanson for the FIS mappng functon s y= f x = M l= M y' = p l= = p µ ' µ ' ( x ) ( x ) (3.40) where M denotes the number of rules, y ' represents the center of gravty of the output fuzzy set that s assocated wth the rule R, p s the dmenson of vector x, and µ '( x) represents nput membershp functons. Ths expanson s vald only when we choose sngleton fuzzfcaton functons, product nference, maxmum-product composton, and heght defuzzfcaton (Mendel, 995). Equaton (3.40) can be rewrtten as M ' y = f x = y φ x l= (3.4) were the φ x are called fuzzy bass functons (FBFs) and are gven by ( x) φ = p l= = µ ' = m p ( x ) µ ' ( x ) (3.42) Equaton (3.42) s vald for sngleton fuzzfcaton. The representaton n Equaton (3.40) s referred to as the fuzzy bass functon expanson. Relatonshps between the fuzzy functon and other bass functons (such as trgonometrc functons) were studed extensvely by Km and Mendel (993). It can be seen that the FBF expanson s essentally a sum over M rules, each of whch generates an FBF. The rules n the FBF expanson can be obtaned from numerc data as well as from expert knowledge. It s therefore convenent to decom fposeas x y= f x = f x + f x N L (3.43) Equaton (3.43) can be rewrtten as Mn l ' N ' N L k y= f x = y φ x + yφ x M = k= (3.44)

38 3.fm Page 98 Monday, March 26, 200 0:8 AM 98 Chapter 3 FUZZY LOGIC FUNDAMENTALS where M = MN + ML and MN FBFs are assocated wth numerc data, and M L fuzzy bass functons are assocated wth lngustc nformaton. These FBFs are gven by s s= ( x N ) MN p φ = p j= s= s s= L ( x) k ML p φ = p µ µ j= s= ( x ) ' µ ( x ) ' ( x ) ' µ s s ( x ) ' s s s s (3.45) (3.46) It can be seen from Equatons (3.45) and (3.46) that each FBF s normalzed by nformaton that s assocated wth both numerc and lngustc nformaton. Because the decomposton of an FBF depends on M, where M = M, a mappng N + ML functon assocated wth an FLS can be expressed as a summaton over FBFs. Unlke other classcal bass functons (e.g., trgonometrc functons), whch are nherently orthogonal, FBFs are not orthogonal. They are, however, mportant and unque, because of the fact that they are the only bass functonsthat can nclude numerc, as well as lngustc, nformaton. 3. DESIGN AND IMPLEMENTATION OF A FUZZY INFERENCE SYSTEM FISs map vectors from nput space to output space. A number of other methods, ncludng neural networks, mathematcal functons, and conventonal control systems, can perform a smlar mappng. For any gven problem, f a smpler soluton s already avalable, we must try that frst. Alternatve methods may be used nstead of an FLS. However, among the many advantages FISs have are that they are flexble, they have the ablty to model any nonlnear functon to any arbtrary degree of accuracy, they are based on rules that can be specfed wth a natural language, they can be bult from expert knowledge, and they are tolerant to mprecse data. Fuzzy logc technques can be used to complement other technques, such as neural networks or genetc algorthms. In order to desgn an FIS, we must frst decde nputs, outputs, each of ther domans, and fuzzy nference rules. Mappng rules can be obtaned from numerc data or from expert knowledge. We also need to decde nput and output membershp functons, overlap between these functons, mplcaton and aggregaton methods, and the defuzzfcaton method. Wth the choce of these parameters, we can desgn a varety of FISs. In order to mplement an FIS, we can select any method and algorthm and any programmng envronment. Today, many software and hardware tools are avalable for desgnng and mplementng fuzzy logc systems. Most software tools provde extensve debuggng and optmzaton features, as well as a graphcal user nterface (GUI) envronment that makes fuzzy logc system desgn smple and easy. One