1、An Introduction to Type-2 Fuzzy Sets and Systems,Dr Simon Couplandsimoncdmu.ac.uk Centre for Computational Intelligence De Montfort University Leicester United Kingdom www.cci.dmu.ac.uk,Contents,My background Motivation Interval Type-2 Fuzzy Sets and Systems Generalised Type-2 Fuzzy Sets and Systems
2、 An Example Application Mobile Robotics,My Background,Research Fellow from the UK Here on a collaborative grant with Prof. Keller Worked in type-2 fuzzy logic for 5 years Awarded PhD “Geometric Type-2 Fuzzy Systems” in 2006 Working on: Computational problems of generalised type-2 fuzzy logic Applica
3、tions,My Background,Created and maintain type2fuzzylogic.org Information, experts, publications (450), news and events 600 members 70 countries,Type-2 Publications,Type-1 Fuzzy Sets,Extend crisp sets, where x A or x A Membership is a continuous grade 0,1 Describe vagueness not uncertainty (Klir and
4、Yuan),Why do we need type-2 fuzzy sets?,Type-1 fuzzy sets do not model uncertainty:,1.8,0.62,Tall,0,1,Height (m),Why do we need type-2 fuzzy sets?,So, a person x, whos height is 1.8 metres is Tall to degree 0.62 (Tall(1.8) = 0.62) Improvement on Tall or not Tall Vagueness, but no uncertainty How do
5、we model uncertainty?,Why do we need type-2 fuzzy sets?,We need, x is Tall to degree about 0.62 But how to model about 0.62? Two schools of thought: Interval type-2 fuzzy sets about 0.62 is a crisp interval Generalised type-2 fuzzy sets about 0.62 is a fuzzy set Run blurring example,Interval Type-2
6、Fuzzy Sets,Interval type-2 fuzzy sets - interval membership gradesX is primary domain Jx is the secondary domain All secondary grades (A(x,u) equal 1 Fully characterised by upper and lower membership functions (Mendel and John),A = (x,u), 1) | x X, u Jx, Jx 0,1,Interval Type-2 Fuzzy Sets,Returning t
7、o Tall,Tall,0,1,Height (m),Upper MF Tall,Lower MF Tall,Type -1 MF,= FOU,Interval Type-2 Fuzzy Sets,Fuzzification:,1.8,0.42,Tall,0,1,Height (m),0.78,Tall (1.8) = 0.42,0.78,Interval Type-2 Fuzzy Sets,Defuzzification two stages: Type-reduction Interval centroid Type-reduction (centroid):GC = 1Jx1 1JxN
8、1 = Cl, Cr,/,i=1 xii,i=1 i,N,N,(Karnik and Mendel),Interval Type-2 Fuzzy Sets,Only need to identify two embedded fuzzy sets Only Jx1 and JxN will belong to those sets Identify two switch points on X Switch point against X is a convex function Mendel and Liu showed switch point = C where l,r,Interval
9、 Type-2 Fuzzy Sets,Defuzzification:,Tall,0,1,Height (m),Cl,Cr,Interval Type-2 Fuzzy Sets,X,Cl switch point,Centroid,Cl,Interval Type-2 Fuzzy Sets,X,Cr switch point,Centroid,Cr,Interval Type-2 Fuzzy Sets,These properties are exploited by Karnik-Mendel algorithm Converges in at most N steps 3-4 steps
10、typical Widely used Hardware implementation,Interval Type-2 Fuzzy Systems,Fuzzifier,Defuzzifier,Rules,Inference,Type-reducer,Crisp inputs,Crisp outputs,Type- reduced outputs (interval),Output processing,Type-2 Interval FIS,Interval Type-2 Fuzzy Systems,Mamdani or TSK systems Well only look at Mamdan
11、i Example rule base:If x is A and y is B then z is G1 If x is C and y is D then z is G2,Interval Type-2 Fuzzy Systems,Antecedent calculation:Rule 1: RA1 = A(x) B(y), A(x) B(y)Rule 2: RA2 = C(x) D(y), C(x) D(y)where is a t-norm, generally min or prod,Interval Type-2 Fuzzy Systems,Consequent calculati
12、on:Rule 1: G1 = inG1(zi) RA1, G1(zi) RA1Rule 2: G2 = inG2(zi) RA2, G2(zi) RA1,Interval Type-2 Fuzzy Systems,Consequent combination:Gc = in G1 (gi) V G2 (gi) , G1 (gi) V G2 (gi) Where V is a t-conorm, generally max,Interval Type-2 Fuzzy Systems,A,0,1,0,1,0,1,0,1,0,1,0,1,C,B,D,G1,G2,Interval Type-2 Fu
13、zzy Systems,x,0,1,0,1,0,1,0,1,0,1,0,1,y,A,C,B,D,G1,G2,Interval Type-2 Fuzzy Systems,x,0,1,0,1,0,1,0,1,0,1,0,1,(min),y,A,C,B,D,G1,G2,Interval Type-2 Fuzzy Systems,x,0,1,0,1,0,1,0,1,0,1,0,1,(min),y,A,C,B,D,G1,G2,A,C,B,D,Interval Type-2 Fuzzy Systems,x,0,1,0,1,0,1,0,1,0,1,0,1,0,1,(min),y,max,Cl,Cr,G1,G
14、2,GC,Interval Type-2 Fuzzy Systems,x,0,1,0,1,0,1,0,1,0,1,0,1,0,1,(prod),y,0,1,max,Cl,Cr,B,D,A,C,G1,G2,GC,Interval Type-2 Fuzzy Systems,Summary: Membership grades are crisp intervals Two parallel type-1 systems (up to defuzzification) Defuzzification in two stages: Type-reduction (KM) Defuzzification
15、,Generalised Type-2 Fuzzy Sets,Generalised type-2 fuzzy sets type-1 fuzzy numbers for membership gradesX is primary domain Jx is the secondary domain A(x) is the secondary membership function at x (vertical slice representation) All secondary grades (A(x,u) 0,1,A = (x,u), A(x,u) | x X, u Jx, Jx 0,1,
16、Generalised Type-2 Fuzzy Sets,Representation theorem (Mendel and John) Represent generalised type-2 fuzzy sets and operations as collection of embedded fuzzy sets,Ae = (x, (u, A(x,u) | x X, u Jx, Jx 0,1,A = Ae,j,j = 1,n,Only used for theoretical working (to date),Generalised Type-2 Fuzzy Sets,Fuzzif
17、ication,X,(x,u),(x),1,1,Generalised Type-2 Fuzzy Sets,Fuzzification,X,(x,u),(x),1,1,x,Generalised Type-2 Fuzzy Sets,Fuzzification,X,(x,u),(x),1,1,x,Generalised Type-2 Fuzzy Sets,Fuzzification,X,(x,u),(x),1,1,x,(x,u),(x),1,1,A,A(x),Generalised Type-2 Fuzzy Sets,Antecedent and the meet Two SMFs: f = i
18、 / vi and g = j / wj The meet:,f g = i j / vi wj,(Zadeh),Generalised Type-2 Fuzzy Sets,Antecedent or the join Two SMFs: f = i / vi and g = j / wj The join:,f g = i j / vi V wj,(Zadeh),Generalised Type-2 Fuzzy Sets,Join and meet under min:,(x,u),(x),1,1,(x,u),(x),1,1,join,meet,f,g,Generalised Type-2
19、Fuzzy Sets,Join and meet under prod:,(x,u),(x),1,1,(x,u),(x),1,1,join,meet,f,g,Generalised Type-2 Fuzzy Sets,More efficient join and meet operations: Apex points 1 and 2,(x,u),(x),1,1,f,g,1,2,Generalised Type-2 Fuzzy Sets,More efficient join and meet operations:,f g (u) = f(u) g(2), 1u2,f(u) g(u), u
20、2,f g (u) = f(1) g(u), 1u2,(f(u) V g(u) (f(1) g(2), u2,f(u) g(u), u1,(f(u) V g(u) (f(1) g(2), u1,(Karnik and Mendel), (Coupland and John),Generalised Type-2 Fuzzy Sets,Implication: Meet every point in consequent with antecedent value:,A(x) B(y) G = (A(x) B(y) G(z),zZ,Generalised Type-2 Fuzzy Sets,Co
21、mbination of Consequents: Join all consequent sets at every point in the in the consequent domain:,G = G1(z) G2(z) Gn(z),zZ,Generalised Type-2 Fuzzy Sets,Type-reduction (centroid) Gives a type-1 fuzzy set:GC = 1Jz1 1JzN i=1 G(zii),/,i=1 zii,i=1 i,N,N,N,(Karnik and Mendel),Generalised Type-2 Fuzzy Se
22、ts,Type-reduction,Z,(z,u),(z),1,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Genera
23、lised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z
24、,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fu
25、zzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,
26、Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-r
27、eduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,Generalised
28、 Type-2 Fuzzy Sets,Type-reduction,Z,(z,u),(z),1,1,Z,1,CZ,CZ,Show again,Generalised Type-2 Fuzzy Sets,Type-reduction number embedded sets:,Generalised Type-2 Fuzzy Sets,Computational complexity is a huge problem Inference complexity relates to join and meet Type-reduction is not a sensible approach,G
29、eneralised Type-2 Fuzzy Sets,Geometric approach (Coupland and John): Model membership functions as geometric objects Operations become geometric Run geometric model,Generalised Type-2 Fuzzy Sets,Let the generalised type-2 fuzzy set A consist of n triangles:,Generalised Type-2 Fuzzy Sets,The centroid
30、 of A is the weighted average of the area and centroid of each triangle:,Generalised Type-2 Fuzzy Sets,The centroid of a triangle is the mean of the x component of the three vertices The area of a triangle is half the cross product of any two edge vectors,Generalised Type-2 Fuzzy Sets,Generalised Ty
31、pe-2 Fuzzy Sets,Generalised Type-2 Fuzzy Sets,Generalised Type-2 Fuzzy Sets,Criticisms: No measure of uncertainty Problems with rotational symmetry On the plus side: Computes in a reasonable time Interesting potential implementations,Generalised Type-2 Fuzzy Sets,Summary: Rich model membership grade
32、s are fuzzy numbers High computational complexity Inference problems solved Type-reduction partly solved (geometric approach),Generalised Type-2 Fuzzy Sets,Applications: Control: Robot navigation (Hagras, Coupland, Castillo) Plant (Castillo, Chaoui, Hsiao) Signal Processing: Classification (Mendel,
33、John, Liang) Prediction (Rhee, Mendez, Castillo) Perceptual reasoning: Perceptual computing (Mendel) Modelling perceptions (John),Generalised Type-2 Fuzzy Sets,Example Application: Robot control and navigation:,Generalised Type-2 Fuzzy Sets,Example Application: Robot control and navigation:,Summary,
34、Type-1 fuzzy sets cant model uncertainty Interval type-2 fuzzy sets crisp interval Generalised type-2 fuzzy sets fuzzy set Interval systems fast, simple computation Generalised high computational complexity Outperformed type-1 growing applications,Further Reading,http:/www.type2fuzzylogic.org/ Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions Mendel, J.M. http:/www.cse.dmu.ac.uk/simonc/eldertech/ http:/www.cci.dmu.ac.uk/,
