Fuzzy Logic : Introduction

1 Fuzzy Logic : IntroductionDebasis SamantaIIT Samanta(IIT Kharagpur)Soft Computing / 69 What is Fuzzy Logic ? Fuzzy Logic is a mathematical languageto express means it has grammar, syntax, semantic like a language are some other mathematical languages also known Relational algebra(operations on sets) Boolean algebra(operations on Boolean variables) Predicate Logic (operations on well formed formulae (wff), alsocalled predicate propositions) Fuzzy Logic deals with Fuzzy Samanta(IIT Kharagpur)Soft Computing / 69A brief history of Fuzzy LogicFirst time introduced by Lotfi Abdelli Zadeh (1965), University ofCalifornia, Berkley, USA (1965).He is fondly nick-named asLAZD ebasis Samanta(IIT Kharagpur)Soft Computing / 69A brief history of Fuzzy logic1 Dictionary meaning offuzzyis not clear, noisy : Is the picture on this slide is Fuzzy ?

2 2 Antonym of Fuzzy iscrispExample: Are the chips crisp?Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Example : Fuzzy Logic vs. Crisp logicMilkWaterCocaSpite Crisp answer Yes or No True or False Crisp Is the liquid colorless? Yes No A liquidDebasis Samanta(IIT Kharagpur)Soft Computing / 69 Example : Fuzzy Logic vs. Crisp logicFuzzy answer May be May not be Absolutely Partially etc Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Example : Fuzzy Logic vs. Crisp Logic Fuzzy Is the person honest? Extremely honest Very honest Honest at times Extremely dishonest 99 75 55 35 Ankit Rajesh Santosh Kabita SalmonScoreDebasis Samanta(IIT Kharagpur)Soft Computing / 69 World is Fuzzy ! Our world is better described with fuzzily! Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Concept of Fuzzy systemFuzzy element(s) Fuzzy set(s) Fuzzy rule(s) Fuzzy implication(s)(Inferences) Fuzzy systemOU T P U T I N P U T Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Concept of Fuzzy setTo understand the concept offuzzy setit is better, if we first clear ouridea ofcrisp = The entire population of = All Hindu population ={h1,h2,h3.}

3 ,hL}M = All Muslim population ={m1,m2,m3, .. ,mN}HMXU niverse of discourseHere, All are the sets of finite numbers of a set is called crisp Samanta(IIT Kharagpur)Soft Computing / 69 Example of Fuzzy setLet us discuss about Fuzzy = All students in = AllGood ={(s, g)|s X}and g(s) is a measurement of goodness of :S ={(Rajat, ), (Kabita, ), (Salman, ), (Ankit, )} Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy set vs. Crisp setCrisp SetFuzzy Set1. S ={s|s X}1. F = (s, )|s X and (s) is the degree of It is a collection of It is collection of or-dered Inclusion of an el-ement s X into S iscrisp, that is, has Inclusion of an el-ement s X into F isfuzzy, that is, if present,then with a degree Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy set vs. Crisp setNote:A crisp set is a Fuzzy set, but, a Fuzzy set is not necessarily acrisp :H ={(h1, 1), (h2, 1).

4 , (hL, 1)}Person ={(p1, 1), (p2, 0), .. , (pN, 1)}In case of a crisp set, the elements are with extreme values of degreeof membership namely either 1 or to decide the degree of memberships of elements in a Fuzzy set? the cities of comfort can be judged?Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Example: Course evaluation in a crisp way1EX = Marks 902A = 80 Marks<903B = 70 Marks<804C = 60 Marks<705D = 50 Marks<606P = 35 Marks<507F = Marks<35 Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Example: Course evaluation in a crisp way10 FPDCBAEX355060708090100 Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Example: Course evaluation in a Fuzzy way10 FPBAEX355060708090100 DCDebasis Samanta(IIT Kharagpur)Soft Computing / 69 Few examples of Fuzzy setHigh TemperatureLow PressureColor of AppleSweetness of OrangeWeight of MangoNote: Degree of membership values lie in the range [ ].

5 Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Some basic terminologies and notationsDefinition 1: Membership function (and Fuzzy set)IfXis a universe of discourse and x X, then a Fuzzy setAinXisdefined as a set of ordered pairs, that isA={(x, A(x))|x X}where A(x) is called the membership functionfor the Fuzzy : A(x) map each element ofXonto a membership grade (ormembership value) between 0 and 1 (both inclusive).Question:How (and who) decides A(x) for a Fuzzy setAinX?Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Some basic terminologies and notationsExample:X = All cities in IndiaA = City of comfortA={(New Delhi, ), (Bangalore, ), (Chennai, ), (Hyderabad, ), (Kolkata, ), (Kharagpur, 0)}Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Membership function with discrete membershipvaluesThe membership values may be of discrete Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Membership function with discrete membershipvaluesEither elements or their membership values (or both) also may be ofdiscrete Number of children (X)A ={(0, ),(1, ),(2, ).}

6 (10, )}Note : X = discrete valueHow you measure happiness ??A = Happy family Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Membership function with continuousmembership (X)B = Middle aged 4150110()Bxx BNote : x = real value= R+Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: SupportSupport: The support of a Fuzzy setAis the set of all pointsx Xsuch that A(x)>0 ADebasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: CoreCore: The core of a Fuzzy setAis the set of all pointsxinXsuch that A(x) =1 core (A) = {x | A(x) = 1} Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: NormalityNormality: A Fuzzy setAis a normal if its core is non-empty. In otherwords, we can always find a pointx Xsuch that A(x) = Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: Crossover pointsCrossover point: A crossover point of a Fuzzy setAis a pointx Xat which A(x) = That isCrossover(A) ={x| A(x) = }.

7 Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: Fuzzy SingletonFuzzy Singleton: A Fuzzy set whose support is a single point inXwith A(x) =1 is called a Fuzzy singleton. That is|A|=|{x| A(x) =1}|= 1. Following Fuzzy set is not a Fuzzy Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: -cut and strong -cut -cut and strong -cut:The -cut of a Fuzzy setAis a crisp set defined byA ={x| A(x) }Strong -cut is defined similarly :A ={x| A(x)> }Note: Support(A) =A0 and Core(A) = Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: ConvexityConvexity: A Fuzzy setAis convex if and only if for anyx1andx2 Xand any [0,1] A( x1+ (1 - )x2) min( A(x1), A(x2))Note : A is convex if all its - level sets are convex. Convexity (A )= A is composed of a single line segment function is Membership Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: BandwidthBandwidth:For a normal and convex Fuzzy set, the bandwidth (or width) is definedas the distance the two unique crossover points:Bandwidth(A) =|x1-x2|where A(x1) = A(x2) = Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: SymmetrySymmetry:A Fuzzy setAis symmetric if its membership function around a certainpointx=c, namely A(x + c) = A(x - c) for all x Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy terminologies: Open and ClosedA Fuzzy setAisOpen leftIf limx A(x) = 1 and limx + A(x) = 0 Open right:If limx A(x) = 0 and limx + A(x) = 1 ClosedIf.

8 Limx A(x) = limx + A(x) = 0 Open leftOpen rightClosedDebasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy vs. ProbabilityFuzzy: When we say about certainty of a thingExample: A patient come to the doctor and he has to diagnose so thatmedicine can be prescribed a medicine with certainty 60% that the patient issuffering from flue. So, the disease will be cured with certainty of 60%and uncertainty 40%. Here, in stead of flue, other diseases with someother certainties may : When we say about the chance of an event to occurExample: India will win the T20 tournament with a chance 60% meansthat out of 100 matches, India own 60 Samanta(IIT Kharagpur)Soft Computing / 69 Prediction vs. ForecastingThe Fuzzy vs. Probability is analogical to Prediction vs. ForecastingPrediction: When you start guessing about : When you take the information from the past job andapply it to new main difference:Predictionis based on the best guess from based on data you have actually recorded and packedfrom previous Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy MembershipFunctionsDebasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy membership functionsA Fuzzy set is completely characterized by its membership function(sometimes abbreviated asMFand denoted as ).

9 So, it would beimportant to learn how a membership function can be expressed(mathematically or otherwise).Note:A membership function can be on(a) a discrete universe of discourse and(b) a continuous universe of ANumber of children (X)A = Fuzzy set of Happy family BAge (X)B = Young age 10204050 Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy membership functionsSo, membership function on a discrete universe of course is , a membership function on a continuous universe ofdiscourse needs a special figures shows a typical examples of membership functions. xxx< triangular > < trapezoidal > < curve >x< non-uniform >x< non-uniform > Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy MFs : Formulation and parameterizationIn the following, we try to parameterize the different MFs on acontinuous universe of MFs :A triangular MF is specified by three parameters{a,b,c}and can be formulated as (x;a,b,c) = 0ifx ax ab aifa x bc xc bifb x c0ifc x(1) Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy MFs: TrapezoidalA trapezoidal MF is specified by four parameters{a,b,c,d}and canbe defined as follows:trapeziod(x;a,b,c,d) = 0ifx ax ab aifa x b1ifb x cd xd cifc x d0ifd x(2) Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy MFs: GaussianA Gaussian MF is specified by two parameters{c, }and can bedefined as below:gaussian(x.)

10 C, ) =e 12(x c ) Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy MFs: Generalized bellIt is also called Cauchy MF. A generalized bell MF is specified by threeparameters{a,b,c}and is defined as:bell(x; a, b, c)=11+|x ca|2bcc+ac-a2ba 2babSlope at x =xySlope at y =Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Example: Generalized bell MFsExample: (x)=11+x2;a=b=1 andc=0; Samanta(IIT Kharagpur)Soft Computing / 69 Generalized bell MFs: Different shapesChanging aChanging bChanging aChanging a and bDebasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy MFs: Sigmoidal MFsParameters:{a,c}; wherec= crossover point anda= slope atc;Sigmoid(x;a,c)=11+e [ax c] = aDebasis Samanta(IIT Kharagpur)Soft Computing / 69 Fuzzy MFs : ExampleExample : Consider the following grading system for a = Marks 90 Very good = 75 Marks 90 Good = 60 Marks 75 Average = 50 Marks 60 Poor = 35 Marks 50 Bad= Marks 35 Debasis Samanta(IIT Kharagpur)Soft Computing / 69 Grading SystemA Fuzzy implementation will look like the can decide a standard Fuzzy MF for each of the Fuzzy Samanta(IIT Kharagpur)Soft Computing / 69 Operations on Fuzzy SetsDebasis Samanta(IIT Kharagpur)Soft Computing / 69 Basic Fuzzy set operations: UnionUnion (A B): A B(x)= max{ A(x), B(x)}Example:A={(x1, ), (x2, ), (x3, )}andB={(x1, ), (x2, ), (x3, )}.