Fuzzy Systems. Fuzzy Systems in Knowledge Engineering. Chapter 4. Christian Jacob. 4. Fuzzy Systems. Fuzzy Systems in Knowledge Engineering

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1 Chapter 4 Fuzzy Systems Knowledge Engeerg Fuzzy Systems Christian Jacob jacob@cpsc.ucalgary.ca Department of Computer Science University of Calgary [Kasabov, 1996] Fuzzy Systems Knowledge Engeerg [Kasabov, 1996] What Does Fuzzy Logic Mean? Fuzzy Logic: Just Human... Fuzzy logic was troduced by Lotfi Zadeh (UC Berkeley) Fuzzy logic attempts to formalize approximate knowledge and approximate reasong. Fuzzy logic is based on fuzzy set theory, an extension of classical set theory. Fuzzy logic did not attract any attention until the 1980s (fuzzy controller applications) Humans primarily use fuzzy terms: large, small, fast, slow, warm, cold, We say: If the weather is nice and I have a little time, I will probably go for a hike the Rockies. We don t say: If the temperature is above 24 degrees and the cloud cover is less than 10% and I have 3 hours time, I will go for a hike with a probability of 0.47.

2 Fuzzy Logic: Lotfi Zadeh: Make use of the leeway of fuzzess. Fuzzess as a prciple of economics: Precision is expensive. Only apply as much precision to a problem as necessary. Example (1): Backg to a parkg space How long would it take if we had to park the car with a precision of ±2 mm? Example (2): Temperature control How much effort would be volved controllg the temperature of the water flowg to your bathtub by ±1 C? Basics of Fuzzy Sets Basics of Fuzzy Sets (2) Example: the set of young people Fuzzy set theory offers a variable notion of membership: A person of age 21 could still belong to the set of young people, but only to a degree of less than one, maybe 0.9. We can also defe a characteristic function for this set: Now the set young contas people with ages between 20 and 30 with a learly decreasg degree of membership. Fuzzy Membership Function Lguistic Variables Coverg the doma of a variable with several fuzzy sets, together with a correspondg semantics, defes a lguistic variable. Lguistic variable age

3 Lguistic Variables (2) Fuzzy Granules Usg fuzzy sets allows us to corporate the fact that no sharp boundaries between these groups exist. The correspondg fuzzy sets overlap certa areas, formg non-crisp or fuzzy boundaries. This way of defg fuzzy sets over the doma of a variable is referred to as granulation, contrast to the division to crisp sets (quantization). Granulation results a groupg of objects to imprecise clusters of fuzzy granules. The objects formg a granule are drawn together by similarity. This can be seen as a form of fuzzy data compression. Often granulation is obtaed manually through expert terviews. Fdg Fuzzy Granules Shapes for Membership Functions If expert knowledge on a doma is not available, an automatic granulation approach can be used. Triangle: [a,b,c] Trapezoid: [a,b,c,d] Standard granulation usg an odd number of membership functions: NL: negative large, NM: negative medium, NS: negative small,z: zero,... Gaussian: [a,θ] Sgleton: [a,m] Parameters of Fuzzy Membership Fcts. Membership Functions: Support s A := { x : µ A (x) > 0 } Support: s A := { x : µ A (x) > 0 } The area where the membership function is greater than zero. Core: c A := { x : µ A (x) = 1 } The area for which elements have maximum degree of membership to the fuzzy set A. α-cut: A α := { x : µ A (x) = α } The cut through the membership function of A at height α. Height: h A := max x { µ A (x) } The maximum value of the membership function of A. Triangle: [a,b,c] Gaussian: [a,θ] Trapezoid: [a,b,c,d] Sgleton: [a,m]

4 Membership Functions: Core c A := { x : µ A (x) = 1 } Membership Functions: α-cut A α := { x : µ A (x) = α } α α Triangle: [a,b,c] Trapezoid: [a,b,c,d] Triangle: [a,b,c] Trapezoid: [a,b,c,d] m=1 α α Sgleton: [a,m] Sgleton: [a,m] Gaussian: [a,θ] Gaussian: [a,θ] Membership Functions: Height h A := max x { µ A (x) } Height = 1 Height = 1 Triangle: [a,b,c] Trapezoid: [a,b,c,d] Height = 1 Height = m Sgleton: [a,m] Gaussian: [a,θ] Fuzzy Numbers Fuzzy Numbers as Fuzzy Sets Real-world measurements are always imprecise nature. Usually such measurements are modeled through a crisp number x, denotg the most typical value, together with an terval describg the amount of imprecision. In a lguistic sense, this could be described as ab x. Usg fuzzy sets we can corporate this formation directly. Fuzzy numbers are a special type of fuzzy sets, restrictg the possible types of membership functions: µ A must be normalized (i.e., the core is non-empty, c A ). µ A must be sgular, i.e., there is precisely one pot which lies side the core, modelg the typical value (= modal value) of the fuzzy number. µ A must be monotonically creasg left of the core and monotonically decreasg on the right. This makes sure that there is only one peak, and therefore only one typical value exists.

5 Fuzzy Numbers: Example Addg Numbers Typically triangular membership functions are chosen for fuzzy numbers. Let us first consider the classical crisp version of addition. Ab 2 Ab Ab Operations on Fuzzy Numbers Fuzzy Addition on Crisp Numbers Usg the so-called extension prciple, we extend classical operators (addition, multiplication) to their fuzzy counterparts, such that we can also handle termediate degrees of membership. For an arbitrary bary operator : We check whether the fuzzy addition is consistent with normal addition on crisp numbers. For a value x a degree of membership is derived which is the maximum of m{µ A (y), µ B (z)} over all possible pairs of y,z for which y z = x holds. Fuzzy Addition Fuzzy Addition

6 Fuzzy Addition and Multiplication Fuzzy Number Operations: How to... For practical purposes we can calculate the result of applyg a monotonical operation on fuzzy numbers as follows: Subdivide the membership functions µ A (x) and µ B (x) to monotonically creasg and decreasg parts. Then perform the operation jotly on the creasg (decreasg) parts of numbers A and B. Plateaus can be dealt with a sgle computation step. Fuzzy Number Operations: How to (2) Let A and B be fuzzy numbers and a strongly monotonical operation. Let [a 1, a 2 ] and [b 1, b 2 ] be the tervals which µ A (x) and µ B (x) are monotonically creasg (decreasg). Now if there exist subtervals [α 1, α 2 ] [a 1, a 2 ] and [β 1, β 2 ] [b 1, b 2 ] such that µ A + B (t) = 1.0 t =... µ A (x A ) = µ B (x B ) = λ x A [α 1, α 2 ], x B [β 1, β 2 ], then: µ A B (t) = λ t [α 1 β 1, α 2 β 2 ], µ A + B (t) = 1.0 t = = 110 µ A + B (t) = 0.4 t [...,...]

7 µ A + B (t) = 0.4 t [20+64, 30+64] µ A + B (t) = 0.4 t [20+64, 30+64] = [84, 94] µ A + B (t) = 0.4 t =... µ A + B (t) = 0.4 t = = 122 µ A + B (t) = 0.4 t = = 122 Support: s A + B = [...,...]

8 Support: s A + B = [10+60, 50+80] = [70, 130] and the rest by lear terpolation: Done! Fuzzy Addition and Multiplication Applyg a Function to a Fuzzy Number µ looses its triangular shape! Examples from the Mathematica Fuzzy Logic Package

9 Intersection, Union, and Complement Fuzzy Union and Intersection Let A and B be fuzzy sets. Intersection (conjunction): Union (disjunction): Complement: Fuzzy Union and Intersection (2) Fuzzy Set Laws (M-Max Operators) Commutativity: A B = B A A B = B A Associativity: (A B) C = A (B C) (A B) C = A (B C) Distributivity: A (B C) = (A B) (A C) A (B C) = (A B) (A C) Fuzzy Set Laws (M-Max Operators) (2) Union and Intersection: Alternatives Adjunctivity: A (B A) = A A (B A) = A De Morgan Laws: (A B) = A B (A B) = A B However, the laws of complementarity do not hold: A A A A 1

10 Fuzzy As the complementarity laws do not hold fuzzy logic, we can not simply use laws from classical logic to derive other operators, such as implication. Consequently, implication has to be defed rather than derived: (motivated by A B = A (A B) ) Generalized Modus Ponens Generalized Modus Ponens In classical logic, conclusions can be drawn from known facts and implications based on these facts. In fuzzy logic, this process of ference can be extended also to partial true facts: x IS A IF x IS A THEN y IS B y IS B In classical logic, conclusions can be drawn from known facts and implications based on these facts. In fuzzy logic, this process of ference can be extended also to partial true facts: x IS A µ A (x) IF x IS A THEN y IS B A B y IS B µ B (y) Generalized Modus Ponens (2) The terpretation of GMP depends on the defition of implication. One possible GMP terpretation: Jot Constrat: Usg m-max norms, the implication can be seen as formg a constrat on a jot variable (x, y) A B. Cartesian product: B can then be obtaed through B = A (A B), or

11 . Max-M IF THEN coolg valve = almost open. IF THEN coolg Max-M Max-M 1. Input of crisp value IF THEN coolg 1. Input of crisp value 2. Fuzzification IF THEN coolg Max-M Max-M 1. Input of crisp value 2. Fuzzification 3. IF THEN coolg 1. Input of crisp value 2. Fuzzification IF THEN coolg Output set Fuzzy OR

12 Max-M Max-M 1. Input of crisp value 2. Fuzzification IF THEN coolg Output set Fuzzy OR 1. Input of crisp value 2. Fuzzification Output set 5. Defuzzification IF THEN coolg Center of gravity Another Example Another Example (2) IF age IS young AND car power IS high THEN risk IS high. IF age IS middle aged AND car power IS medium THEN risk IS medium. Extractg Fuzzy Models: Graphs representation of functions, contours, and relations:

13 Global Granulation of Input Space Extractg Fixed-Grid Fuzzy Rules Buildg Adaptive Grid- Based Fuzzy Rules References Berthold, M. and Hand, D. J. (1999). Intelligent Data Analysis. Berl, Sprger. Kasabov, N. (1996). Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engeerg. Cambridge, MA, MIT Press. Bothe, H. (1995). Fuzzy Logic Eführung Theorie und Anwendungen. Berl, Sprger.