An Introduction to Type-2 Fuzzy Sets and Systems Dr Simon Coupland simonc@dmu.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 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 Applications

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 Yuan)

Why do we need type-2 fuzzy sets? Type-1 fuzzy sets do not model uncertainty: Tall  1 0.62 Height (m) 1.8

Why do we need type-2 fuzzy sets? So, a person x, who’s 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 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 Fuzzy Sets Interval type-2 fuzzy sets - interval membership grades X 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 to Tall ~ Upper MF Tall Tall  1 Type -1 MF = FOU Height (m) Lower MF Tall

Interval Type-2 Fuzzy Sets Fuzzification: ~ Tall  1 0.78 Tall (1.8) = [0.42,0.78] ~ 0.42 Height (m) 1.8

Interval Type-2 Fuzzy Sets Defuzzification – two stages: Type-reduction Interval centroid Type-reduction (centroid): GC =  1Jx1…  1JxN 1 = [Cl, Cr] N / i=1 xii N i=1 i (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 Type-2 Fuzzy Sets Defuzzification: ~ Tall  1 Height (m) Cl Cr

Interval Type-2 Fuzzy Sets Centroid Cl X Cl switch point

Interval Type-2 Fuzzy Sets Cr Centroid X Cr switch point

Interval Type-2 Fuzzy Sets These properties are exploited by Karnik-Mendel algorithm Converges in at most N steps 3-4 steps typical Widely used Hardware implementation

Interval Type-2 Fuzzy Systems Output processing Rules Crisp outputs Defuzzifier Crisp inputs Fuzzifier Type- reduced outputs (interval) Type-reducer Inference Type-2 Interval FIS

Interval Type-2 Fuzzy Systems Mamdani or TSK systems We’ll only look at Mamdani 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 calculation: Rule 1: G’1 = i..n[G1(zi)  RA1, G1(zi)  RA1] Rule 2: G’2 = i..n[G2(zi)  RA2, G2(zi)  RA1] ~ ~ ~ ~ ~ ~

Interval Type-2 Fuzzy Systems Consequent combination: Gc = i..n [G1’ (gi) V G2’ (gi) , G1’ (gi) V G2’ (gi) ] Where V is a t-conorm, generally max ~ ~ ~ ~ ~

Interval Type-2 Fuzzy Systems ~ ~ ~ A B G1    1 1 1 ~ ~ ~  C  D  G2 1 1 1

Interval Type-2 Fuzzy Systems ~ ~ ~ A B G1    1 1 1 ~ ~ ~  C  G2 D  1 1 1 y x

Interval Type-2 Fuzzy Systems  ~ ~ (min) ~ A B G1    1 1 1 ~ ~ ~  C   G2 D 1 1 1 y x

Interval Type-2 Fuzzy Systems  ~ ~ (min) ~ A B G1    1 1 1 ~ ~ ~  C   G2 D 1 1 1 y x

Interval Type-2 Fuzzy Systems  ~ ~ (min) ~ A B G1    1 1 1 ~ ~ ~  C  D  G2 1 1 1 ~  GC y 1 x max Cl Cr

Interval Type-2 Fuzzy Systems  ~ ~ (prod) ~ A B G1    1 1 1 ~ ~ ~  C  D  G2 1 1 1 ~  GC y 1 1 x max Cl Cr

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

Generalised Type-2 Fuzzy Sets Generalised type-2 fuzzy sets – type-1 fuzzy numbers for membership grades X 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]} ~ ~ ~

Generalised Type-2 Fuzzy Sets Representation theorem (Mendel and John) Represent generalised type-2 fuzzy sets and operations as collection of embedded fuzzy sets n ~ A =  Ae ~ j j = 1 ~ Ae = {(x, (u, A(x,u)) | x  X, u  Jx, Jx  [0,1]} ~ Only used for theoretical working (to date)

Generalised Type-2 Fuzzy Sets Fuzzification ~ A (x) 1 1 (x,u) X

Generalised Type-2 Fuzzy Sets Fuzzification ~ A (x) 1 x 1 (x,u) X

Generalised Type-2 Fuzzy Sets Fuzzification ~ A (x) 1 x 1 (x,u) X

Generalised Type-2 Fuzzy Sets Fuzzification ~ A(x) A ~ (x) 1 x (x,u) 1 (x) 1 1 (x,u) X

Generalised Type-2 Fuzzy Sets Antecedent ‘and’ – the meet Two SMF’s: f =  i / vi and g =  j / wj The meet: f g =   i j / vi  wj (Zadeh)

Generalised Type-2 Fuzzy Sets Antecedent ‘or’ – the join Two SMF’s: 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,u) f g meet join 1 1 (x) (x) 1 1

Generalised Type-2 Fuzzy Sets Join and meet under prod: (x,u) (x,u) f g meet join 1 1 (x) (x) 1 1

Generalised Type-2 Fuzzy Sets More efficient join and meet operations: Apex points 1 and 2 (x,u) f g 1 1 2 1 (x)

Generalised Type-2 Fuzzy Sets More efficient join and meet operations: { f(u) Λ g(u), u<1 f g (u) = f(1) Λ g(u), 1u<2 (f(u) V g(u)) Λ (f(1) Λ g(2)), u2 { (f(u) V g(u)) Λ (f(1) Λ g(2)), u<1 f g (u) = f(u) Λ g(2), 1u<2 f(u) Λ g(u), u2 (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) ~ ~ ~ ~ ~ zZ

Generalised Type-2 Fuzzy Sets Combination of Consequents: Join all consequent sets at every point in the in the consequent domain: ~ G =  G1(z) G2(z) … Gn(z) ~ ~ ~ zZ

Generalised Type-2 Fuzzy Sets Type-reduction (centroid) Gives a type-1 fuzzy set: GC =  1Jz1…  1JzN i=1 G(zii) N N / i=1 zii ~ N i=1 i (Karnik and Mendel)

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1 1 (z,u) Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) 1  1 1 (z,u) Z Z

Generalised Type-2 Fuzzy Sets Type-reduction (z) CZ 1  1 1 (z,u) Z Z 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

Generalised 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 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 Type-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 grades 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, 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 Type-1 fuzzy sets can’t 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/