Fuzzy Systems (1/2) Francesco Masulli

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1 (1/2) Francesco Masulli DIBRIS - University of Genova, ITALY & S.H.R.O. - Sbarro Institute for Cancer Research and Molecular Medicine Temple University, Philadelphia, PA, USA francesco.masulli@unige.it MLCI 2018 Francesco Masulli (1/2)

2 Outline 1 2 Francesco Masulli (1/2)

3 Fuzzy Sets Linguistic variable Francesco Masulli (1/2)

4 Fuzzy Sets Linguistic variable Definition (Linguistic variable) A linguistic variable is characterized by a quintuple (x, U, T (x), G, M) in which x is the name of variable; U is the universe of discourse; T (x) is the term set of x, that is, the set of names of linguistic values of x with each value being a fuzzy set defined on U; G is a syntactic rule for generating the names of values of x; M is a semantic rule for associating with each value its meaning M(x), i.e., defines the fuzzy set in U. Francesco Masulli (1/2)

5 Fuzzy Sets Fuzzy Hedges G (syntactic rule) and M(semantic rule) can be tables or algorithms. Definition (Linguistic Modifier (or Fuzzy Hedge) ) Mathematical operation modifing the meaning of a term Examples: concentration: µ con(a) (u) = (µ A (u)) 2 dilatation: µ dil(a) (u) = (µ A (u)).5 contrast intensification: { 2(µA (u)) µ int(a) (u) = 2 if (µ A (u)) [0, 5] 1 2(1 µ A (u)) otherwise. Francesco Masulli (1/2)

6 Fuzzy Sets Fuzzy Hedges Definition (Some Fuzzy Hedges: ) very A = con(a) more or less A = dil(a) plus A = A 1.25 etc. Francesco Masulli (1/2)

7 Structured Linguistic Variable Let us reconsider the linguistic variable "Age." The term set shall be assumed to be T (Age) = {old, very old, very very old,...} The term set can now be generated recursively by using the following syntactic rule G (algorithm): T i+1 = {old}u{very T i } that is, etc. T 0 = T 1 = {old} T 2 = {old, very old} T 3 = {old, very old, very very old}, Francesco Masulli (1/2)

8 Structured Linguistic Variable For the semantic rule M, we only need to know the meaning of "old" and the meaning of the modifier "very" in order to determine the meaning of an arbitrary term of the term set. If one defines "very" as the concentration very(.) = con(.), then the terms of the term set of the structured linguistic variable "Age" can be determined, given that the membership function of the term "old" µ old (.) is known. Francesco Masulli (1/2)

9 Fuzzy implication FUZZY LOGIC IS A PRECISE WAY OF REASONING USING IMPRECISE INFORMATION (Zadeh) A fuzzy if-then rule (fuzzy rule, fuzzy implication, or fuzzy conditional statement ) assumes the form if x is A then y is B (or A B) where A and B are linguistic values defined by fuzzy sets on universes of discourse X and Y, respectively. Often "x is A" is called the antecedent or premise, while "y is B" is called the consequence or conclusion. Francesco Masulli (1/2)

10 Fuzzy implication The linguistic knowledge on a problem is given in form of fuzzy rules If blood pressure is above the target and decreasing slowly then reduce drug infusion. If pressure is high then volume is small. If the road is slippery then driving is dangerous. If a tomato is red then it is ripe. If the speed is high then apply the brake a little. Francesco Masulli (1/2)

11 Fuzzy implication A fuzzy implication A B describes a relation between two variables x and y; this suggests that a fuzzy if-then rule be defined as a binary fuzzy relation R on the product space XxY Francesco Masulli (1/2)

12 Fuzzy implication A binary fuzzy relation R is an extension of the classical Cartesian product, where each element (x, y) XxY is associated with a membership grade denoted by µ R (x, y). Alternatively, a binary fuzzy relation R can be viewed as a fuzzy set with universe X x Y, and this fuzzy set is characterized by a two-dimensional membership function µ R (x, y). Francesco Masulli (1/2)

13 Fuzzy implication If we interpret a fuzzy implication F = A B as "A coupled with B" then µ A B (x, y) = µ A (x) µ B (y), where is a t-norm ("fuzzy and"). Francesco Masulli (1/2)

14 Fuzzy Reasoning (Approximate Reasoning) Fuzzy reasoning (also known as approximate reasoning) is an inference procedure used to derive conclusions from a set of fuzzy if-then rules and one or more conditions. Francesco Masulli (1/2)

15 Compositional rule of inference Francesco Masulli (1/2)

16 Compositional rule of inference Francesco Masulli (1/2)

17 Compositional rule of inference To find the resulting fuzzy set B: 1 we construct a cylindrical extension c(a) with base A (that is, we expand the domain of A from X to X x Y to get c(a)). µ c(a) (x, y) = µ A (x) 2 The intersection of c(a) and F forms the analog of the region of intersection µ c(a) F(x,y) = min[µ c(a) (x, y), µ F (x, y)] = min[µ (A) (x), µ F (x, y)] 3 By projecting c(a) F onto the y-axis, we infer y as a fuzzy set B on the y-axis µ B (y) = max{min[µ (A) (x), µ F (x, y)]} = max{µ (A) (x) µ F (x, y)} x x sup-star composition: B = A F (compositional rule of inference) Francesco Masulli (1/2)

18 Compositional rule of inference (sup-star composition) To find the resulting fuzzy set B apply the compositional rule of inference B = A F (sup-star composition) B = A F = max {µ (A)(x) µ F (x, y)} x Two possibile implementations of the sup-star composition: max-min composition: min for fuzzy AND and max for fuzzy OR µ B (y) = max min[µ (A) (x), µ F (x, y)] x max-product composition: product for fuzzy AND and max for fuzzy OR µ B (y) = max [µ (A)(x) µ F (x, y)] x Francesco Masulli (1/2)

19 Fuzzy reasoning Using the compositional rule of inference, we can formalize an inference procedure, called fuzzy reasoning, upon a set (bank) of fuzzy if-then rules. The basic rule of inference in traditional two-valued logic is modus ponens, according to which we can infer the truth of a proposition B from the truth of A and the implication A B Francesco Masulli (1/2)

20 Modus ponens (MP) If A is identified with "the tomato is red" and B with "if the tomato is red then it is ripe", then if it is true that "the tomato is red", it is also true that "the tomato is ripe". Francesco Masulli (1/2)

21 Generalized Modus Ponens (GMP) Generalized Modus Ponens (GMP) or fuzzy reasoning or approximate reasoning where A is close to A and B is close to B If we have the same implication rule "if the tomato is red then it is ripe" and we know that "the tomato is more or less red", then we may infer that "the tomato is more or less ripe" NOTE: Generalized Modus Ponens has Modus Ponens as a special case. Francesco Masulli (1/2)

22 Fuzzy Reasoning Based on Max-Min Composition Let A, A, and B be fuzzy sets of X, X, and Y, respectively. Assume that the fuzzy implication A B is expressed as a fuzzy relation R on X x Y. Then the fuzzy set B induced by "x is A " and the fuzzy rule "if x is A then y is B" is defined by or, equivalentely, µ B (y) = max min[µ (A x )(x), µ R (x, y)] B = A R = A (A B) Francesco Masulli (1/2)

23 Fuzzy Reasoning Based on Max-Min Composition Choices: max-min composition AND fuzzy (T-norm) min OR fuzzy (S-norm) + max µ B (y) = max x min[µ A (x), µ R (x, y)] = x [µ A (x) µ R (x, y)] Francesco Masulli (1/2)

24 Fuzzy Reasoning Based on Max-Min Composition Choice: R = A B = A B (fuzzy conjunction) µ B (y) = x [µ A (x) µ R (x, y)] = x [µ A (x) (µ A (x) µ B (y))] = w µ B (y) where w = x [µ A (x) µ A (x)] (constant) Francesco Masulli (1/2)

25 Fuzzy Reasoning Based on Max-Min Composition A B rule A fact B conclusion Note: if A = A w = 1 B = B i.e. GMP MP Francesco Masulli (1/2)

26 Multiple Antecedents We interpret the rule as R = A B C (conjunction) µ R (x, y, z) = µ (A B) C (x, y, z) = µ A (x) µ B (y) µ C (z) (conjunction) then C = (A B ) (A B C) Francesco Masulli (1/2)

27 Multiple Antecedents µ C (z) = x,y[µ A (x) µ B (y)] [µ A (x) µ B (y) µ C (z)] = = { x,y[µ A (x) µ B (y) µ A (x) µ B (y)} µ C (z)] where = w 1 w 2 µ C (z) w 1 = x[µ A (x) µ A (x)] (constant) w 2 = y[µ B (x) µ B (x)] (constant) Francesco Masulli (1/2)

28 Multiple Rules with Multiple Antecedents We interpret R 1 = A 1 B 1 C 1 and R 2 = A 2 B 2 C 2 (conjunction) then C = (A B ) (R 1 R 2 ) = [(A B ) R 1 ] [(A B ) R 2 ] = C 1 C 2 Francesco Masulli (1/2)

29 Multiple Rules with Multiple Antecedents Using min and max operators as fuzzy AND and OR Francesco Masulli (1/2)

30 Fuzzy inference systems Pure fuzzy inference systems fuzzy expert systems (FuzzyCLIPS ) Francesco Masulli (1/2)

31 Fuzzy inference systems Fuzzy inference systems with fuzzifier and de-fuzzifier Francesco Masulli (1/2)

32 Fuzzy inference systems Fuzzy inference systems with fuzzifier and defuzzifier Many different names: fuzzy inference system fuzzy model fuzzy-rule-based system fuzzy expert system fuzzy associative memory fuzzy logic controller fuzzy system Applications: automatic control, data classification, decision analysis, expert systems, computer vision, bioinformatics Francesco Masulli (1/2)

33 Fuzzy inference systems Fuzzifier singleton non-singleton (fuzzy number) Francesco Masulli (1/2)

34 Defuzzifiers The outputs (conclusions) of a fuzzy reasoning are fuzzy sets µ j B (y) (for rule Rj ) We are interested to crisp values of y. Types of Defuzzifiers: Maximum Defuzzifier (MAX) Mean of Maxima Defuzzifier (MOM) Center of Area Defuzzifier (COA) Height Defuzzifier (HD) Francesco Masulli (1/2)

35 Defuzzifiers Types of Defuzzifiers: MAX: where MOM: Mean of Maxima y = arg sup(µ B (y)) y V µ B (y) = µ B 1(y) µ B N (y) Francesco Masulli (1/2)

36 Defuzzifiers COA (Center of Areas) or Centroid, Center of Gravity, Barycenter. if Y continous S y c = y µ B (y)dy S µ B (y)dy where S Y support of b, if Y continous if Y discrete I i=1 y c = y i µ B (y i ) I i=1 µ B (y i) (heavy to compute) Francesco Masulli (1/2)

37 Defuzzifiers Height y h = J j=1 ȳ j µ B j (ȳ j ) j j=1 µ B j (ȳ j ) ȳ j is the MAX of the output fuzzy set B j of rule R j ȳ j = arg max µ B j (y) y and can be computed before the inference. Note that with Height defuzzifier the shape of the output membership is not relevant: we use ȳ j of rule R j only. Francesco Masulli (1/2)

38 Fuzzy inference systems Mamdani fuzzy inference system (max-min composition) Using min and max operators as fuzzy AND and OR; singleton fuzzifier, COA defuzzifier Francesco Masulli (1/2)

39 Fuzzy inference systems Mamdani fuzzy inference system (max-prod composition) prod and max operators as fuzzy AND and OR; singleton fuzzifier, COA defuzzifier fuzzy AND: product compositon: max-prod C = (A B ) (A B C) µ C (z) = x,yµ A B (x, y) µ A B C(x, y, z) = x,y[µ A (x) µ B (y)] [µ A (x) µ B (y) µ C (z)] = { x,y[µ A (x) µ B (y) µ A (x) µ B (y)} µ C (z)] where = w 1 w 2 µ C (z) w 1 = x[µ A (x) µ A (x)] w 2 = y[µ B (x) µ B (x)] (constants) Francesco Masulli (1/2)

40 Fuzzy inference systems Mamdani fuzzy inference system (max-prod composition) Using prod and max operators as fuzzy AND and OR; singleton fuzzifier, COA defuzzifier Francesco Masulli (1/2)

41 Fuzzy inference systems Tagagi-Sugeno fuzzy inference system R k : IF (x 1 is F k 1 and x n is F k n ) THEN y k = f k (x 1 x n ) y(x) = ω k y k ω k with ω k = n i=1 µ F k i (x i ) Francesco Masulli (1/2)

42 Fuzzy inference systems Tagagi-Sugeno fuzzy inference system (sup-star composition) Francesco Masulli (1/2)

43 Fuzzy inference systems Tagagi-Sugeno fuzzy inference system The premise of a rule IF (x 1 is F k 1 and x n is F k n ) perfoms a fuzzy segmentation of the rule input space (universe of the discourse) Francesco Masulli (1/2)

44 Fuzzy inference systems Tsukamoto fuzzy inference system (sup-star composition) Francesco Masulli (1/2)

45 Fuzzy inference systems Simplified rule (Jang) IF x is A and y is B THEN Z is b where b is a crisp value / a singleton fuzzy set Functional identity: Mamdami: b singleton Sugeno: fist order polynomial Tsukamoto: constant belief networks mixture of experts Francesco Masulli (1/2)

46 Fuzzy inference systems Fuzzy rules - Air conditioning motor speed controller bank of rules should cover the universe of discourse Francesco Masulli (1/2)

47 Fuzzy inference systems Fuzzy Rules Input space (universe of the discourse) partitioning induced by the bank of fuzzy rules: (a) grid partitioning; (b) tree partitioning; (c) scatter partitioning bank of rules should cover the universe of discourse Francesco Masulli (1/2)

48 Fuzzy inference systems Rules bank of rules should cover the universe of discourse rules are given by an expert of the application in linguistic way rule can be learned from data after learning rules can be extracted from the fuzzy system Francesco Masulli (1/2)

49 Fuzzy inference systems Possibilities [Mendel, 1995] at least = different FLS s!!: We must decide on type of fuzzification (singleton or nonsingleton) functional forms for membership functions (triangular, trapezoidal, Gaussian, piecewise linear) parameters of membership functions (fixed ahead of time, tuned during a training procedure), composition (max-min, max-product) inference (minimum, product) defuzzifier (centroid, height, modified height). Francesco Masulli (1/2)

50 Fuzzy inference systems How to select "our fuzzy systewm? Criteria: computational cost empirical fit assiomatic force simplicity in modifification/adaptaption capability of learning from data universal function approximation property... Francesco Masulli (1/2)