What is all the Fuzz about?

Transcription

1 What is all the Fuzz about? Fuzzy Systems CPSC 433 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary

2 Fuzzy Systems in Knowledge Engineering

3 Fuzzy Systems 1. Motivation 2. Fuzzy Sets 3. Fuzzy Numbers 4. Fuzzy Sets and Fuzzy Rules 5. Extracting Fuzzy Models from Data 6. Examples of Fuzzy Systems

4 What does Fuzzy Logic mean? Fuzzy logic was introduced by Lotfi Zadeh (UC Berkeley) in Fuzzy logic is based on fuzzy set theory, an extension of classical set theory. Fuzzy logic attempts to formalize approximate knowledge and reasoning. Fuzzy logic did not attract any attention until the 1980s (fuzzy controller applications).

5 Humans primarily use fuzzy terms: large, sma!, fast, slow, warm, cold,... We say: If the weather is nice and I have a little time, I will probably go for a hike along the Bow. We don t say: Fuzzy is Just Human 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.

6 Fuzzy is Economical Zadeh: Make use of the leeway of fuzziness. Fuzziness as a principle of economics: Precision is expensive. Only apply as much precision to a problem as necessary. Example: Backing into a parking space - How long would it take if we had to park a car with a precision of ±2 mm?

7 Fuzzy Systems 1. Motivation 2. Fuzzy Sets 3. Fuzzy Numbers 4. Fuzzy Sets and Fuzzy Rules 5. Extracting Fuzzy Models from Data 6. Examples of Fuzzy Systems

8 Basics of Fuzzy Sets Example: the set of young people young = {x P age(x) 20} We can define a characteristic function for this set: { 1 : age(x) 20 µ young (x) = 0 : 20 < age(x)

9 Basics of Fuzzy Sets Fuzzy set theory offers a variable notion of membership: A person of age 25 could still belong to the set of young people, but only to a degree of less than one, say : age(x) 20 µ young (x) = 1 age(x) : 20 < age(x) 30 0 : 30 < age(x) Now the set of young contains people with ages between 20 and 30, with a linearly decreasing degree of membership.

10 Fuzzy Membership Function µ young (x) = 1 : age(x) 20 1 age(x) : 20 < age(x) 30 0 : 30 < age(x)

11 Shapes for Membership Fcts. Triangle: [a,b,c] Trapezoid: [a,b,c,d] Singleton: [a,m] Gaussian: [a,θ]

12 Parameters of FMFs Support: s A := { x : μ A (x) > 0 } The area where the membership function is positive. Core: c A := { x : μ A (x) = 1 } The area for which elements have a maximum degree of membership to the fuzzy set A.

13 Parameters of FMFs α-cut: α A := { x : μ A (x) = α } The cut through the membership function of A at height a. Height: h A := max x { μ A (x) } The maximum value of the membership function of A.

14 Support: s A := { x : μ A (x) > 0 } Triangle: [a,b,c] Trapezoid: [a,b,c,d] Singleton: [a,m] Gaussian: [a,θ]

15 Core: c A := { x : μ A (x) = 1 } Triangle: [a,b,c] Trapezoid: [a,b,c,d] m=1 Singleton: [a,m] Gaussian: [a,θ]

16 α-cut: α A := { x : μ A (x) = α } α α Triangle: [a,b,c] Trapezoid: [a,b,c,d] α α Singleton: [a,m] Gaussian: [a,θ]

17 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 Singleton: [a,m] Gaussian: [a,θ]

18 Linguistic Variables The covering of a variable domain with several fuzzy sets, together with a corresponding semantics, defines a linguistic variable. Example: linguistic variable ag"

19 Granulation Using fuzzy sets, we can incorporate the fact that no sharp boundaries between groups, such as young, middle-aged, and old, exist. The corresponding membership functions overlap in certain areas, forming non-crisp (fuzzy) boundaries. This compositional way of defining fuzzy sets over a domain of a variable is called granulation.

20 Fuzzy Granules Granulation results in a grouping of objects into imprecise clusters of fuzzy granules. The objects forming a granule are drawn together by similarity. This can be seen as a form of fuzzy data compression.

21 Finding Fuzzy Granules If expert knowledge on a domain is not available, an automatic granulation is used. Standard granulation using an odd number of membership functions: NL: negative large, NM: negative medium, NS: negative small, Z: zero,...

22 Fuzzy Systems 1. Motivation 2. Fuzzy Sets 3. Fuzzy Numbers 4. Fuzzy Sets and Fuzzy Rules 5. Extracting Fuzzy Models from Data 6. Examples of Fuzzy Systems

23 Fuzzy vs. Crisp Numbers Real-world measurements are always imprecise. Usually, such imprecise measurements are modeled through a crisp number x, denoting the most typical value, together with an interval, describing the amount of imprecision. In a linguistic sense: about x

24 Fuzzy Numbers as Fuzzy Sets Fuzzy numbers are a special type of fuzzy sets with specific membership functions: μ A must be normalized (c A ). μ A must be singular. There is precisely one point which lies inside the core, modeling the typical value (= modal value) of the fuzzy number. μ A must be monotonically increasing left of the core and monotonically decreasing on the right (only one peak!).

25 Fuzzy Number Example Typically, triangular membership functions are chosen for fuzzy numbers. About 2 About About 0

26 Fuzzy Number Operations For practical purposes, we can calculate the result of applying a monotonical operation on fuzzy numbers as follows: Subdivide μ A (x) and μ B (x) into monotonically increasing and decreasing parts. Perform the operation jointly on the increasing (decreasing) parts of numbers A and B. Plateaus can be dealt with in a single computation step.

27 Fuzzy Number Operations Let A and B be fuzzy numbers, and a strongly monotonical operation. Let [a 1, a 2 ] and [b 1, b 2 ] be the intervals in which μ A (x) and μ B (x) are monotonically increasing (decreasing). If there exist subintervals [α 1, α 2 ] [a 1, a 2 ] and [β 1, β 2 ] [b 1, b 2 ], such that - x A [ 1, 2 ], x B [ 1, 2 ]: A (x A ) = B (x B ) = then - t [ 1 1, 2 2 ]: A B (t) =

28 Fuzzy Addition µ A + B (t) = 1.0 t =...

29 Fuzzy Addition µ A + B (t) = 1.0 t = = 110

30 Fuzzy Addition µ A + B (t) = 0.4 t [...,...]

31 Fuzzy Addition µ A + B (t) = 0.4 t [20+64, 30+64]

32 Fuzzy Addition µ A + B (t) = 0.4 t [20+64, 30+64] = [84, 94]

33 Fuzzy Addition µ A + B (t) = 0.4 t =...

34 Fuzzy Addition µ A + B (t) = 0.4 t = = 122

35 Fuzzy Addition µ A + B (t) = 0.4 t = = 122

36 Fuzzy Addition Support: s A + B = [...,...]

37 Fuzzy Addition Support: s A + B = [10+60, 50+80] = [70, 130]

38 Fuzzy Addition and the rest by linear interpolation: Done!

39 Monotonic Function µ f (A) (y) = max{µ A (x) x : f (x) = y}

40 Fuzzy Systems 1. Motivation 2. Fuzzy Sets 3. Fuzzy Numbers 4. Fuzzy Sets and Fuzzy Rules 5. Extracting Fuzzy Models from Data 6. Examples of Fuzzy Systems

41 Operations on Fuzzy Sets Let A and B be fuzzy sets. Intersection (conjunction): μ A B (x) = min { A (x), B (x)} Union (disjunction): μ A B (x) = max { A (x), B (x)} Complement: μ A (x) = 1 - A (x)

42 Fuzzy Union & Intersection μ A B (x) = max { A (x), B (x)} μ A B (x) = min { A (x), B (x)}

43 Fuzzy Union & Intersection

44 Union & Intersection: Alternatives µ A B (x) = min{µ A (x),µ B (x)} µ A B (x) = max{µ A (x),µ B (x)} µ A B (x) = µ A (x) µ B (x) µ A B (x) = min{µ A (x) + µ B (x),1}

45 Fuzzy Rules IF temperature = low THEN cooling valve = half open. IF temperature = medium THEN cooling valve = almost open.

46 Max-Min Inference Max-Min Inference temperature = low valve = half open Approximate Reasoning IF temperature = low THEN cooling valve = half open temperature = medium valve = almost open IF temperature = medium THEN cooling valve = almost open in out

47 Max-Min Inference Max-Min Inference temperature = low valve = half open Approximate Reasoning IF temperature = low THEN cooling valve = half open temperature = medium valve = almost open 1. Input of crisp value IF temperature = medium THEN cooling valve = almost open in out

48 Max-Min Inference Max-Min Inference temperature = low valve = half open Approximate Reasoning IF temperature = low THEN cooling valve = half open temperature = medium valve = almost open 1. Input of crisp value 2. Fuzzification IF temperature = medium THEN cooling valve = almost open in out

49 Max-Min Inference Max-Min Inference temperature = low valve = half open Approximate Reasoning IF temperature = low THEN cooling valve = half open temperature = medium valve = almost open 1. Input of crisp value 2. Fuzzification 3. Inference IF temperature = medium THEN cooling valve = almost open in out

50 Max-Min Inference Max-Min Inference temperature = low valve = half open Approximate Reasoning IF temperature = low THEN cooling valve = half open temperature = medium valve = almost open 1. Input of crisp value 2. Fuzzification IF temperature = medium THEN cooling valve = almost open 3. Inference 4. Output set Fuzzy OR in out

51 Max-Min Inference Max-Min Inference temperature = low valve = half open Approximate Reasoning IF temperature = low THEN cooling valve = half open temperature = medium valve = almost open 1. Input of crisp value 2. Fuzzification IF temperature = medium THEN cooling valve = almost open 3. Inference 4. Output set Fuzzy OR in out

52 Max-Min Inference Max-Min Inference temperature = low valve = half open Approximate Reasoning IF temperature = low THEN cooling valve = half open temperature = medium valve = almost open 1. Input of crisp value 2. Fuzzification 3. Inference 4. Output set 5. Defuzzification IF temperature = medium THEN cooling valve = almost open Center of gravity in out

53 Fuzzy Systems 1. Motivation 2. Fuzzy Sets 3. Fuzzy Numbers 4. Fuzzy Sets and Fuzzy Rules 5. Extracting Fuzzy Models from Data 6. Examples of Fuzzy Systems

54 Extracting Fuzzy Models: Graphs Approximate representation of functions, contours, and relations

55 Global Granulation of Input Space

56 Fixed-Grid Rules Extraction

57 Adaptive-Grid Rules Extraction

58 Fuzzy Systems 1. Motivation 2. Fuzzy Sets 3. Fuzzy Numbers 4. Fuzzy Sets and Fuzzy Rules 5. Extracting Fuzzy Models from Data 6. Examples of Fuzzy Systems

59 References Berthold, M., and Hand, D. J. (2003). Inte!igent Data Analysis. Berlin, Springer. Bothe, H. (1995). Fuzzy Logic Einführung in die Theori" und Anwendungen. Berlin, Springer. Kasabov, N. (1996). Foundations of Neural Networks, Fuzzy Systems, amd Knowledge Engineering. Cambridge, MA, MIT Press.