ARTIFICIAL INTELLIGENCE. Uncertainty: fuzzy systems

Transcription

1 INFOB2KI Utrecht University The Netherlands ARTIFICIAL INTELLIGENCE Uncertainty: fuzzy systems Lecturer: Silja Renooij These slides are part of the INFOB2KI Course Notes available from

2 Outline Boolean logic and crisp sets Fuzzy logic & sets Inferences with fuzzy rules

3 Reasoning models Classical models (logic, exact) IfPressure = 10 atm then Volume = 2.5 cc Imprecise models IfPressure 5 atm then Volume 6 cc Probabilistic models IfPressure 5 atm then P(Volume = 6 cc) = 0.9 Vague models IfPressure = HIGH then Volume = LOW

4 Boolean logic & Crisp sets I Boolean logic uses sharp distinctions. It forces us to draw lines between members of a class and non members. tall men For instance, we may say, Tom is tall because his height is 181 cm. If we drew a line at 180 cm, we would find that David, who is 179 cm, is small.

5 Boolean Logic & Crisp sets II Consider a universe of discourse (interest) X and its elements x. In classical set theory, a crisp subset A of X is defined by the characteristic function f A (x) of A: f A (x) : X {0, 1}, where f A ( x) 1, if 0, if x x A A

6 The trouble with crisp sets Sorites Paradox: the paradox of the heap Consider a heap of sand from which grains are individually removed. One might construct the following argument: 1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2) Repeated application of Premise 2 eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand at some point it must stop being a heap!

7 Fuzzy reasoning Experts rely on common sense when they solve problems. How can we represent expert knowledge that uses vague and ambiguous terms in a computer? Fuzzy logic is based on the idea that all things admit of degrees. Temperature, height, speed, distance, beauty all come on a sliding scale. The motor is running really hot. Tom is a very tall guy.

8 Fuzzy, or multi valued logic: A bit of History First introduced in the 1930s by Jan Lukasiewicz (Polish philosopher) work led to an inexact reasoning technique often called possibility theory. extended into a formal system of mathematical logic by Lofti Zadeh in 1965 new logic for representing and manipulating fuzzy terms was called fuzzy logic. Unlike other logical systems, it deals with imprecise or uncertain knowledge.

9 Fuzzy logic and Fuzzy sets Fuzzy logic is multi valued, unlike Boolean logic; it deals with degrees of membership and degrees of truth. uses the continuum between 0 (completely false) and 1 (completely true), accepting that things can be partly true and partly false at the same time. tall men, to a certain degree

10 Fuzzy sets Consider a universe of discourse (interest) X and its elements x. In fuzzy set theory, a fuzzy subset A of X is defined by the membership function µ A (x) of A: µ A (x) : X [0,1], where µ A (x) = 1 if x is totally in A; µ A (x) = 0 if x is not in A; 0 < µ A (x) < 1 if x is partly in A.

11 Fuzzy set notations Consider a (discrete) universe of discourse (interest) X and its elements x. Different notations are used to describe fuzzy (sub)set A of X : A = {(x 1, 0.4), (x 2, 0.3), (x 3, 1), (x 4, 0.6)} i.e. A = {(x, µ A (x)) x in X} Or, using Zadeh s notation: A = 0.4/x /x 2 + 1/x /x 4 i.e. A = x µ A (x)/x (not fractions!)

12 Fuzzy Sets vs Crisp sets Name Chris Frank John Tom David Mike Bob Steven Bill Peter Height, cm Degree of Membership Crisp Fuzzy

13 Fuzzy sets vs Crisp sets Degree of Membership Degree of Membership 1.0 Crisp Sets Short Average Short Tall Tall Men Fuzzy Sets Height, cm Short Average Tall Tall

14 Fuzzy Set Representation Typical functions that can be used to represent a fuzzy set are sigmoid, Gaussian and Pi. However, these functions increase the time of computation. in practice, most applications use linear fit functions, or just a number of ordered pairs mapping x to µ (discrete) X Fuzzy Subset A (x) 1 Crisp Subset A 0 Fuzziness Fuzziness x

15 Fuzzy sets and probability Take care! Fuzzy membership values may look like probabilities, but have a totally different meaning!

16 Operators Complement /negation Intersection / AND Union / OR containment / implication Equality Cardinality Empty set

17 Boolean Logic/Crisp Sets: ops ( x A) Not A A B A x A x B Complement Containment x A x B A B A B x A x B Intersection Union

18 Fuzzy ops: complement/negation Boolean/Crisp: Who does not belong to the set? Fuzzy: How much do elements not belong to the set? Standard negation: Complement A of fuzzy set A can be found as follows: A (x) = 1 A (x) Example: consider set X and fuzzy subset A X = {a, b, c, d, e} A = 1/a + 0.3/b + 0.2/c + 0.8/d + 0/e A = 0/a + 0.7/b + 0.8/c + 0.2/d + 1/e

19 Negation: alternatives Minimum negation A (x) = 1 if A (x)=0 = 0 otherwise Maximum negation A (x) = 0 if A (x)=1 = 1 otherwise

20 Fuzzy ops: Intersection/AND Boolean/Crisp: Which element belongs to both sets? Fuzzy: How much of the element is in both sets? Standard intersection: (minimum) Intersection of two fuzzy subsets A and B of X is the lower membership in both sets of each element: A B (x) = min [ A (x), B (x)] = A (x) B (x) Example: consider set X and two fuzzy subsets A,B X = {a, b, c, d, e} A = 1/a + 0.3/b + 0.2/c + 0.8/d + 0/e B = 0.6/a + 0.9/b + 0.1/c + 0.3/d + 0.2/e A B = 0.6/a + 0.3/b + 0.1/c + 0.3/d + 0/e

21 Intersection: alternatives Product A B (x) = A (x) B (x) Lukasiewicz A B (x) = max[ A (x)+ B (x) 1, 0 ] Drastic A B (x) = min[ A (x), B (x) ], if A (x)=1 or B (x)=1 = 0, otherwise

22 Fuzzy operations: Union/OR Boolean/Crisp: Which element belongs to either set? Fuzzy: How much of the element is in either set? Standard union: (maximum) reverse of intersection union of two fuzzy subsets A and B of X is the largest membership value of each element in either set: A B (x) = max [ A (x), B (x)] = A (x) B (x) Example: consider set X and two fuzzy subsets A,B X = {a, b, c, d, e} A = 1/a + 0.3/b + 0.2/c + 0.8/d + 0/e B = 0.6/a + 0.9/b + 0.1/c + 0.3/d + 0.2/e A B =1/a + 0.9/b + 0.2/c + 0.8/d + 0.2/e

23 Probabilistic sum A B (x) = A (x) + B (x) A (x) B (x) Union: alternatives Lukasiewicz A B (x) = min[ A (x)+ B (x), 1 ] Drastic A B (x) = max[ A (x), B (x) ], if A (x)=0 or B (x)=0 = 1, otherwise

24 Fuzzy ops: containment(inclusion)/implication Boolean/Crisp: Which sets belong to which other sets? Fuzzy: Which sets belong to other sets? Inclusion: each element can belong less to the subset than to the larger set. Fuzzy set A X is included in (is a subset of) another fuzzy set, B X: A (x) B (x), x X Example: Consider X = {1, 2, 3} and fuzzy subsets A,B A = 0.3/ / /3 B = 0.5/ / /3 then A is a subset of B, or A B

25 Fuzzy Equality Fuzzy set A is considered equal to a fuzzy set B, IF AND ONLY IF (iff): A (x)= B (x), x X Example: A = 0.3/ / /3 B = 0.3/ / /3 therefore A = B

26 Fuzzy cardinality Crisp Sets: How many elements belong to the set? Fuzzy Sets: What is the total membership value of the set? Cardinality (aka sigma count): card A = A (x 1 ) + A (x 2 ) + A (x n ) = Σ A (x i ) for i=1..n Example: Consider X = {1, 2, 3} and fuzzy subsets A and B A = 0.3/ / /3 B = 0.5/ / /3 card A = 1.8 card B = 2.05

27 Empty Fuzzy Set A fuzzy set A is empty IF AND ONLY IF: A (x) = 0, x X Example: Consider X = {1, 2, 3} and fuzzy set A A = 0/1 + 0/2 + 0/3 then A is empty

28 Fuzzy to Crisp operations cuts (alpha cuts) Support core

29 Fuzzy to crisp: alpha cut An cut or level set of a fuzzy set A X is a crisp set A X, such that: A ={x X A (x) } Example: Consider X = {1, 2, 3} and set A A = 0.3/ / /3 then A 0.5 = {2, 3}, A 0.1 = {1, 2, 3}, A 1 = {3}

30 Fuzzy to Crisp: Core and Support Consider a fuzzy subset A of X the support of A is the crisp subset of X consisting of all elements with positive membership grade: supp(a) = {x X A (x) 0} the core of A is the crisp subset of X consisting of all elements with membership grade 1: core(a) = {x X A (x)= 1} Example: Consider two fuzzy subsets of the set X = {a, b, c, d, e} A = 1/a + 0.3/b + 0.2/c + 0.8/d + 0/e and B = 0.6/a + 0.9/b + 0.1/c + 0.3/d + 0.2/e Support: supp(a) = {a, b, c, d} and supp(b) = {a, b, c, d, e} Core: core(a) = {a} and core(b) = {}

31 Fuzzy Inference

32 Knowledge base: fuzzy rules A fuzzy rule can be defined as a conditional statement in the form: IF x is A THEN y is B where x and y are linguistic variables (e.g. length, weight); A and B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively (e.g. short, heavy)

33 Classical vs Fuzzy Rules A classical IF THEN rule uses binary logic Rule: 1 Rule: 2 IF speed > 100 IF speed < 40 THEN stopping_distance = 300 THEN stopping_distance = 40 variable speed: any numerical value between 0 and 220 km/h variable stopping_distance: numbers 40,, 300 Stopping distance rules in a fuzzy form: Rule: 1 Rule: 2 IF speed = fast IF speed = slow THEN stopping_distance = long THEN stopping_distance = short Linguistic variable speed: range (universe of discourse) still between 0 and 220 km/h, but includes fuzzy sets, such as slow, medium and fast. variable stopping_distance: between 0 and 300m and may include such fuzzy sets as short, medium and long.

34 Firing Fuzzy Rules: example I Fuzzy rules fire partially and relate fuzzy sets to each other Fuzzy sets can provide the basis for a weight estimation model which captures the relationship between a man s height and his weight: IF height is tall THEN weight is heavy Degree of Membership Tall men Degree of Membership 1.0 Heavy men Height, cm Weight, kg

35 Firing Fuzzy Rules: example II The value of the output, i.e. the truth membership grade of the rule consequent can be estimated: directly from a corresponding truth membership grade in the antecedent. This form of fuzzy inference uses a method called monotonic selection. Degree of Membership Tall men Degree of Membership Heavy men Height, cm Weight, kg

36 Boiling fuzzy eggs

37 Mamdani Fuzzy Inference The Mamdani style fuzzy inference process is performed in four steps: 1. Fuzzification of the input variables 2. Rule evaluation (inference) 3. Aggregation of the rule outputs (composition) 4. Defuzzification.

38 Mamdani Fuzzy Inference We examine a simple two input one output problem that includes 3 rules, 3 variables and 8 fuzzy sets. Rule: 1 Rule: 1 IF x is A3 IF project_funding is adequate OR y is B1 OR project_staffing is small THEN z is C1 THEN risk is low Rule: 2 Rule: 2 IF x is A2 IF project_funding is marginal AND y is B2 AND project_staffing is large THEN z is C2 THEN risk is normal Rule: 3 Rule: 3 IF x is A1 IF project_funding is inadequate THEN z is C3 THEN risk is high

39 Step 1: Fuzzification Take the crisp inputs, and determine the degree to which these inputs belong to each of the appropriate fuzzy sets. Example: x1 = 35% and y1 = 60% (project funding and project staffing) Crisp Input x1 Crisp Input y A1 A2 A B1 B x1 X 0 y1 Y (x = A1) = 0.5 (y = B1) = 0.1 (x = A2) = 0.2 (y = B2) = 0.7 (x=a3) = 0.0

40 Step 2: Rule Evaluation Take the fuzzified inputs and apply them to the antecedents of the fuzzy rules. For fuzzy rules with multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation. This number is then applied to the consequent membership function, using clipping or scaling

41 Rule Evaluation: clipping Clipping (related to alpha cut): cut the consequent membership function at the level of the antecedent truth the most common method Less complex and faster mathematics; generates an aggregated output surface that is easier to defuzzify. Degree of Membership 1.0 C Since the top of the membership function is sliced, the clipped fuzzy set loses some information. Z

42 Rule Evaluation: scaling Scaling better approach for preserving the original shape of the fuzzy set. Adjust original membership function of the rule consequent by multiplying all its membership degrees by the truth value of the rule antecedent. Degree of Membership 1.0 C Z

43 Step 2: Rule Evaluation - example 1 1 A3 B OR 0.1 (max) 0 x1 X 0 y1 Y 0 Rule 1: 1 A2 0 x1 X Rule 2: IF x is A3 (0.0) OR y is B1 (0.1) THEN z is C1 (0.1) B2 0 y1 Y AND (min) IF x is A2 (0.2) AND y is B2 (0.7) THEN z is C2 (0.2) C1 C1 C2 C2 C3 Z C3 Z 1 A C1 C2 C3 0 x1 X 0 Rule 3: IF x is A1 (0.5) THEN z is C3 (0.5) Z

44 Step 3: Aggregating rule outputs Aggregation is the process of unification of the outputs of all rules. We take the membership functions of all rule consequents previously clipped or scaled and combine them into a single fuzzy set. the output is one fuzzy set for each output variable C1 z is C1 (0.1) Z C2 z is C2 (0.2) Z C3 z is C3 (0.5) Z Z

45 Step 4: Defuzzification Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number. Several defuzzification methods; most popular is the centroid technique: find the point where a vertical line would slice the aggregate set into two equal masses (mathematically: centre of gravity (COG)) COG b a b a A A x x x dx dx (approximate with summations) ( x ) a A b 210 X

46 Step 4: Defuzzification (example) Degree of Membership Z COG ( ) 0.1 ( ) 0.2 ( )

47 Conclusions Fuzzy sets can be used to model vague terms Important when crisp distinctions would lead to arbitrary cut offs Inference with fuzzy logic possible, but not as clean as with first order logic

48 Applications Automatically parking cars Video cameras (noise reduction, steady image) Image processing Classifying situations in games Household appliances (see Fuzzy in Appliances: Japanese subway (Sendai Subway 1000 series)