Fuzzy Rules and Fuzzy Reasoning (chapter 3)

Transcription

1 9/4/00 Fuzz ules and Fuzz easoning (chapter ) Kai Goebel, ill Cheetham GE Corporate esearch & Development goebel@cs.rpi.edu cheetham@cs.rpi.edu (adapted from slides b. Jang) Fuzz easoning: The ig Picture Useful for: utomatic Control Epert Sstems Pattern ecognition Time Series Prediction Data Classification There are different schemes to accomplish these goals depending on our understanding and boundar conditions of the world. Etending: crisp domains to fuzz domains: Etension Principle n-ar fuzz relations: Fuzz elations fuzz domains to fuzz domains: Fuzz Inference (fuzz rules, compositional rules of inference) Outline Etension principle Fuzz relations Fuzz IF-THEN rules Compositional rule of inference Fuzz reasoning Etension Principle Etends crisp domains of mathematical epressions to fuzz domains is a fuzz set on : = ( ) / + ( ) / + + ( ) / n n The image of under f is a fuzz set :, i.e., =f() = ( ) / + ( ) / + + ( ) / n n where i = f(i), i = to n. 4 If f is a man-to-one mapping, then ( ) = ma ( ) = f ( ) Etension Principle: Eample =0./-+0.4/-+0.8/0+0.9/+0./ Etension Principle, continuous vars. ( ) = bell( ;. 5,, 0. 5) and ( ) if 0 f = if < 0 f()= - =0./+0.4/-+0.8/-+0.9/-+0./ =0.8/-+(0.4v0.9)/-+(0.v0.)/ =0.8/-+0.9/-+0./ 5 6 Page

2 9/4/00 Fuzz elations Ma-Min Composition 7 fuzz relation is a D MF: = {((, ), (, )) (, ) } Eamples: depends on ( and are events) If is large, then is small ( is an observed reading and is a corresponding action) is much greater than ( and are numbers) (, ) = + + if 0 if < If ={,4,5} and ={,4,5,6,7} Epress fuzz relation as a relation matri = The ma-min composition of two fuzz relations (defined on and ) and (defined on and ) is (, z ) = [ (, ) (, z ) Note: calculation ver similar to matri multiplication Properties: ssociativit: ( S T ) = ( S ) T Distributivit over union: ( S T ) = ( S ) ( T ) Week distributivit over intersection: ( S T ) ( S ) ( T ) Monotonicit: S T ( S ) ( T ) Ma-min Composition: Eample Ma-min Composition: Eample = is relevant to = is relevant to z where = ={ = } ={a b} Calculate: is relevant to a = ={ } ={a b} = (, ) =, m a , 0. 0., , a ( a) = ( ) 9 0 Ma-Star Composition Ma-product composition: (, ) [ (, ) (, ) z = z In general, we have ma-* composition: (, z ) = [ (, ) * where * is a T-norm operator. (, z ) Even more generall, we have (S-norm)-(T-norm) compositions Linguistic Variables (knowledge rep.) Precision vs. significance numerical variable can take on numerical (ge = 65) or linguistic values (ge is old) linguistic value is characterized b the variable name (age) a fuzz set. ll linguistic values form a term set: T(age) = {oung, not oung, ver oung,... middle aged, not middle aged,... old, not old, ver old, more or less old,... not ver oung and not ver old,...} The sntactic rule refers to how the linguistic values are generated. The semantic rule defines the membership value of each linguistic variable. Page

3 9/4/00 Linguistic Values (Terms) Operations on Linguistic Values Concentration: Dilation: Contrast Intensification: CON = DIL = 0.5,. INT = 0 05 ( ), Fuzz IF-THEN ules General format: IF is THEN is if <antecedent> then <consequent> Eamples: IF pressure is high THEN volume is small. IF the road is slipper THEN driving is dangerous. IF a tomato is red THEN it is ripe. IF the speed is high THEN appl the brake a little. Fuzz IF-THEN ules Two was to interpret IF is THEN is : coupled with entails 5 6 Fuzz If-Then ules Two was to interpret If is then is : coupled with : ( and ) = = = ( ) ~ ( ) (, ) entails : (not or ) - Material implication Fuzz IF-THEN ules Fuzz implication function: (, ) = f (, ) = f ( a, b) coupled with - Propositional calculus ( ) 7 - Etended propositional calculus ( ) - Generalization of modus ponens ~, = su p c * c a n d 0 c { } Note: these all reduce to not or in two-valued logic (, ) = min(, ) 8 (, ) = (, ) = 0 ( + ) ( ) if =, = if = 0 otherwise Page

4 9/4/00 Fuzz IF-THEN ules coupled with = bell ; 4,, 0 = bell ; 4,, 0 Fuzz IF-THEN ules entails (not an ehaustive list) min algebraic product bounded product drastic product ( ) ( ( ) (, ) = ma min,, ) (, ) = min, + (, ) = min, ( (, ) = ma, 9 0 Fuzz IF-THEN ules Compositional ule of Inference entails = bell ; 4,, 0 = bell ; 4,, 0 Same idea as ma-min composition Derivation of = b from = a and = f(): b b adeh s arithmetic rule adeh s ma-min rule oolean Fuzz Implication Gougen s Fuzz Impl. = f() = f() a a and b: points = f() : a curve a a and b: intervals = f() : an interval-valued function Compositional ule of Inference, contd. ecall: Clindrical Etension To find the resulting interval =b (which corresponds to =a) construct a clindrical etension of a find intersection with curve project intersection to -ais ase set Clindrical Et. of c (, ) = 4 Page 4 4

5 9/4/00 ecall: D MF Projection Compositional ule of Inference Two-dimensional MF Projection onto Projection onto a is a fuzz set and = f() is a fuzz relation: 5 (, ) = = ma (, ) ma (, ) 6 Putting it together: Comp. rule of Inf. Clindrical etension with base c (, ) = Intersection of c() with F (, ) = min (, ), c F c F (, ) = min, (, ) [ [ F Projection onto -ais = ma min [, F (, ) = [ F (, ) Crisp easoning Modus Ponens: Fact: is ule: IF is THEN is Conclusion: is 7 epresentation: = F Note relation to etension principle 8 Fuzz easoning Single rule with single antecedent Fact: is ule: if is then is Conclusion: is (Generalized Modus Ponens) Graphic epresentation: F - Single ule, Single ntecedent Graphical epresentation: - find degree of match w between () and () intuitivel: degree of belief for antecedent which gets propagated; result should be not greater than w w 9 0 is is Page 5 5

6 9/4/00 Fuzz easoning: Single ntecedent Fuzz easoning,, and be fuzz sets of,, and, respectivel. ssumption: the fuzz implication -> is epressed as a fuzz relation on The fuzz set induced b fact: is and premise: IF is then is = ma min [, (, ) = (, ) or: [ = = ( ) Single rule with multiple antecedents Facts: is and is ule: if is and is then z is C Conclusion: z is C [ [ [ C z, { [ } [ ( z) = z, C C = C = [ ( C) [ ( C) { } C = z w w F - Single ule, Multiple ntecedents Fuzz easoning Graphical epresentation - w denotes degree of compatibilit between and and w between and - w w is degree of fulfilment of the rule T-norm C v w w w Multiple rules with multiple antecedent Fact: is and is ule : if is and is then z is C ule : if is and is then z is C Conclusion: z is C Graphic epresentation: (net slide) C is is z is C 4 Fuzz easoning Graphics representation: C w C w last slide T-norm C is is z is C 5 6 Page 6 6