Fuzzy Sets and Fuzzy Logic. KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur,

Transcription

1 Fuzzy Sets and Fuzzy Logic KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur,

2 Outline traditional logic : {true,false} Crisp Logic Fuzzy Logic Fuzzy Logic Applications Conclusion

3 Crisp Logic Crisp logic is concerned with absolutes true or false, there is no inbetween. Example: Rule: If the temperature is higher than 80F, it is hot; otherwise, it is not hot. Cases: Temperature = 100F Hot Temperature = 80.1F Hot Temperature = 79.9F Not Hot Temperature = 50F Not Hot

4 Membership function A fuzzy set is a generalization of an ordinary set by by allowing a degree (or grade) of membership for each element, which varies from 0 to 1, i.e. [0, 1]. Let the people in an organization be the universe. A subset of this is a crisp set. Consider a set of young people in this organization. The youngness is not a step function from 0 to 1, for certain age, say 30.

5 A degree of youngness is associated to each element, like {Ann/0.8, Bob/0.1, Cathy/1}. Perhaps they are 28,40,23. Each element is represented as <element/degree>. The membership function of a set maps each element to its degree.

6 Membership function of crisp logic True 1 False HOT 0 80F Temperature If temperature >= 80F, it is hot (1 or true); If temperature < 80F, it is not hot (0 or false).

7 Drawbacks of crisp logic The membership function of crisp logic fails to distinguish between members of the same set.

8 Concept of Fuzzy Logic Many decision making and problem solving tasks are too complex to be defined precisely However, people succeed by using imprecise knowledge Fuzzy logic resembles human reasoning in its use of approximate information and uncertainty to generate decisions.

9 Natural Language is not crisp Consider: Joe is tall what is tall? Joe is very tall what does this differ from tall? Natural language (like most other activities in life and indeed the universe) is not easily translated into the absolute terms of 0 and 1.

10 What is Fuzzy Logic? An approach to uncertainty that combines real values [0 1] and logic operations Fuzzy logic is based on the ideas of fuzzy set theory and fuzzy set membership often found in natural (e.g., spoken) language.

11 Example: Young Example: Ann is 28, 0.8 in set Young Bob is 35, 0.1 in set Young Charlie is 23, 1.0 in set Young Unlike statistics and probabilities, the degree is not describing probabilities that the item is in the set, but instead describes to what extent the item is the set.

12 Membership function for young Membership function m(x) to be young: x M(x) 1.0 for 0<x 25 1 m x ={ } for x>25 1 x M(x) Membership function for youngness CSE Dept., MBM 0 Engg. 25 Col., 30 JNV Univ. Age->

13 Membership function of fuzzy logic DOM Fuzzy sets Degree of Membership 1 Young Middle Old Fuzzy values have associated degrees of membership in the set. Age

14 Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set

15 Crisp set vs. Fuzzy set

16 Benefits of fuzzy logic You want the value to switch gradually as Young becomes Middle and Middle becomes Old. This is the idea of fuzzy logic.

17 Fuzzy Set Operations Fuzzy union ( ): Union of two fuzzy sets is the maximum (MAX) of each element from two sets. A B={x/max(mA(x),mB(x) x U} Let a fuzzy set is A = Comfortable house for six persons. A = {(1,.2), (2,.5), (3,.8), (4, 1), (5,.7), (6,.3)} Let B = Large type of house = {(3,.2), (4,.4), (5,.6), (6,.8), (7, 1), (8, 1)} A B = {(1,.2), (2,.5), (3,.8), (4, 1), (5,.7), (6,.8), (7, 1), (8, 1)}

18 Fuzzy Set Operations.. Fuzzy intersection ( ): Intersection of two fuzzy sets is just the MIN of each element from the two sets. A B={x/min(mA(x),mB(x) x U} E.g. A = {(1,.2), (2,.5), (3,.8), (4, 1), (5,.7), (6,.3)} Let B = Large type of house B = {(3,.2), (4,.4), (5,.6), (6,.8), (7, 1), (8, 1)} A B = { (3,.2), (4,.4), (5,.6), (6,.3) }

19 Fuzzy Set Operations.. The complement of a fuzzy variable with DOM x is (1 x). Complement ( A ): The complement of a fuzzy set is composed of all elements complement. A ={x/(1 ma(x)) x U} Example. A = {(1,.2), (2,.5), (3,.8), (4, 1), (5,.7), (6,.3)} A = {(1,.8), (2,.5), (3,.2), (4, 0), (5,.3), (6,.7)}

20 Crisp Relations Ordered pairs showing connection between two sets: (a,b): a is related to b (2,3) are related with the relation < Relations are set themselves < = {(1,2), (2, 3), (2, 4),.} Relations can be expressed as matrices <

21 Fuzzy Relations Triples showing connection between two sets: (a,b,#): a is related to b with degree # Fuzzy relations are set themselves Fuzzy relations can be expressed as matrices

22 Fuzzy Relations Matrices Example: Color Ripeness relation for tomatoes R 1 (x, y) unripe semi ripe ripe green yellow Red

23 Inference rule The inference rule is rule of composition [ a Relation of composition on crisp sets: R S={(a,c) (a,b) R,(b,c) S,a A,b B,c C} Fuzzy Compositional rule: b ] [ ο x ] c d y = [ ax by ] cx dy

24 Inference rule Consider that there are two relation sets. One related the color of tomato to ripeness, and other relates ripeness to prices(hi,low). Given this find out the relation from color to price. gr ye re [ 1 u n se ri hi low.5 o ] un se ri [ ] = gr ye re hi lo [. 5 1 ]

25 Example: Fuzzy Inference Two temperature Inputs (x, y) and one output (z) Membership functions: low(t) = 1 ( t / 10 ); is o/p as function of i/p high(t) = t / 10 ; is o/p as a fn. of i/p Low High x Crisp Inputs y t Low(x) = 0.68, High(x) = 0.32, Low(y) = 0.39, High(y) = 0.61

26 Create rule base Rule 1: If x is low AND y is low Then z is high (i.e., if I/P x,y have membership of low, then the o/p z has membership of high. x and y are ANDed) Rule 2: If x is low AND y is high Then z is low Z = X Y Rule 3: If x is high AND y is low Then z is low X Z Y Rule 4: If x is high AND y is high Then z is high X Y Z

27 Inference step #1(conjunction part of composition) Rule1: low(x)=0.68, low(y)=0.39 => high(z)=min(0.68,0.39)=0.39 Rule2: low(x)=0.68, high(y)=0.61 => low(z)=min(0.68,0.61)=0.61 Rule3: high(x)=0.32, low(y)=0.39 => low(z)=min(0.32,0.39)=0.32 Rule4: high(x)=0.32, high(y)=0.61 => high(z)=min(0.32,0.61)=0.32 Conjunction is performed of x and y.

28 Inference step #2 (disjunction part of composition) Low(z) = max(rule2, rule3) = max(0.61, 0.32) = 0.61 High(z) = max(rule1, rule4) = max(0.39, 0.32) = Low High 0.61 Z low =.61 Z high = t

29 Fuzzy Resolution and other Fuzzy Resolution: L C1, L C 2 C 1 C 2 Fuzzy De Morgans: (F G) = F G and (F G) = F G

30 Other operations on fuzzy sets Concentration CON(A)={x/mA(x) 2 x U} Dilation DIL(A)={x/sqrt(mA(x) x U} Normalization NORM(A)={x/(mA(x)/Max) x U}

31 Operations on Fuzzy Crisp Input System Fuzzification Input Membership Functions Fuzzy Input Rule Evaluation Rules / Inferences Fuzzy Output Defuzzification Output Membership Functions Crisp Output

32 In what areas are fuzzy systems effective and why? Difficult cases where traditional techniques do not work Used in fuzzy control of physical or chemical characteristics such as temperature, electric current, flow of liquid/gas, motion of machines, etc. Fuzzy logic can be applied in fuzzy knowledge based systems, which uses fuzzy if then rules; fuzzy software engineering, that may incorporate fuzziness in programs and data; fuzzy databases that store and retrieve fuzzy information; fuzzy pattern recognition that deals with fuzzy visual or audio signals; applications to medicine, economics, and management problems that involve fuzzy information processing.

33 In what areas are fuzzy systems effective and why?.. Fuzzy systems are useful for approximate reasoning where mathematical model are hard to derive It allows for decision making with estimated values under incomplete information For hard systems, conventional non fuzzy systems are expensive and depend on mathematical approximation (e.g., linearization of nonlinear problems), which may lead to poor performance. Response of fuzzy systems are smoother.

34 Control System Based on rules of logic obtained from train drivers so as to model real human decisions as closely as possible Task: Controls the speed at which the train takes curves as well as the acceleration and braking systems of the train

35 Control System This system is still not perfect; humans can do better because they can make decisions based on previous experience and anticipate the effects of their decisions This led to use of fuzzy systems

36 Decision Support: Predictive Fuzzy Control Can assess the results of a decision and determine if the action should be taken Has model of the motor and break to predict the next state of speed, stopping point, and running time input variables Controller selects the best action based on the predicted states.

37 Decision Support: Predictive Fuzzy Control The results of the fuzzy logic controller for the Sendai subway (Japan) are excellent!! The train movement is smoother than most other trains Even the skilled human operators who sometimes run the train cannot beat the automated system in terms of smoothness or accuracy of stopping

38 Fuzzy Expert Systems Fuzzy expert system is a collection of membership functions and rules that are used to reason about data. Usually, the rules in a fuzzy expert system have the following form: if x is low and y is high then z is medium Classical: if x and y then z