An Introduction to Type-2 Fuzzy Sets and Systems Dr Simon Coupland [email protected] Centre for Computational Intelligence De Montfort University Leicester

An Introduction toType-2 Fuzzy Sets

and Systems

Dr Simon Coupland

[email protected] for Computational Intelligence

De Montfort UniversityLeicester

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:

1.8

0.62

Tall

0

1

Height (m)


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?


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]}~

~


Returning to TallTall

0

1

Height (m)

~ Upper MF Tall

Lower MF Tall

Type -1 MF= FOU

~


Fuzzification:

1.8

0.42

Tall

0

1

Height (m)

~

0.78Tall (1.8) = [0.42,0.78]


Defuzzification – two stages: Type-reduction Interval centroid

Type-reduction (centroid):

GC = 1Jx1… 1JxN

1 = [Cl, Cr]/

i=1 xii

i=1 i

N

N

(Karnik and Mendel)


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}


Defuzzification:Tall

0

1

Height (m)

~

Cl Cr


XCl switch point

Cen

troi

d

Cl


XCr switch point

Cen

troi

d

Cr


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

Fuzzifier

Defuzzifier

Rules

Inference

Type-reducer

Crispinputs

Crispoutputs

Type-reducedoutputs

(interval)

Output processing

Type-2 Interval FIS


Mamdani or TSK systems We’ll only look at Mamdani Example rule base:

1. If x is A and y is B then z is G1

2. If x is C and y is D then z is G2

~

~

~

~

~

~


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

~

~

~

~

~

~

~

~


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]

~ ~

~ ~

~

~


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 SystemsA

0

1

0

1

0

1

0

1

0

1

0

1

C

B

D

G1

G2~

~

~

~

~

~


x

0

1

0

1

0

1

0

1

0

1

0

1

y

A

C

B

D

G1

G2~

~

~

~

~

~


x

0

1

0

1

0

1

0

1

0

1

0

1(min)

y

A

C

B

D

G1

G2~

~

~

~

~

~


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~

~

~

~


x

0

1

0

1

0

1

0

1

0

1

0

1

0

1

(min)

y

max

Cl Cr

G1

G2~

~

GC

~


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

~


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]}~

~

~

~


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)


Fuzzification

X(x,u)

(x)

1

1

~A


Fuzzification

X(x,u)

(x)

1

1 x

~A


Fuzzification

X(x,u)

(x)

1

1 x

~A


Fuzzification

X(x,u)

(x)

1

1 x (x,u)

(x)1

1

~A ~A(x)


Antecedent ‘and’ – the meet Two SMF’s: f = i / vi and g = j / wj

The meet:

f g = i j / vi wj

(Zadeh)


Antecedent ‘or’ – the join Two SMF’s: f = i / vi and g = j / wj

The join:

f g = i j / vi V wj

(Zadeh)


Join and meet under min:(x,u)

(x)1

1(x,u)

(x)1

1joinmeetf g


Join and meet under prod:(x,u)

(x)1

1(x,u)

(x)1

1joinmeetf g

Generalised Type-2 Fuzzy Sets More efficient join and meet operations: Apex points 1 and 2

(x,u)

(x)1

1f g

1 2

Generalised Type-2 Fuzzy Sets More efficient join and meet operations:

f g (u) = f(u) Λ g(2), 1u<2

f(u) Λ g(u), u2

{f g (u) = f(1) Λ g(u), 1u<2

(f(u) V g(u)) Λ (f(1) Λ g(2)), u2

f(u) Λ g(u), u<1{ (f(u) V g(u)) Λ (f(1) Λ g(2)), u<1

(Karnik and Mendel), (Coupland and John)


Implication: Meet every point in consequent with

antecedent value:

A(x) B(y) G = (A(x) B(y)) G(z)zZ

~~~ ~ ~ ~


Combination of Consequents: Join all consequent sets at every point in the

in the consequent domain:

G = G1(z) G2

(z) … Gn(z)

zZ

~~ ~ ~


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)


Type-reduction

Z(z,u)

(z)

1

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1


Type-reduction

Z(z,u)

(z)

1

1

Z

1

CZ

CZ

Show again


Type-reduction – number embedded sets:


Computational complexity is a huge problem Inference complexity relates to join and meet Type-reduction is not a sensible approach


Geometric approach (Coupland and John): Model membership functions as geometric

objects Operations become geometric Run geometric model


Let the generalised type-2 fuzzy set A consist of n triangles:


The centroid of A is the weighted average of the area and centroid of each triangle:


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





Criticisms: No ‘measure of uncertainty’ Problems with rotational symmetry

On the plus side: Computes in a reasonable time Interesting potential implementations


Summary: Rich model – membership grades are fuzzy

numbers High computational complexity

Inference problems solved Type-reduction partly solved (geometric approach)


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 DirectionsMendel, J.M.

http://www.cse.dmu.ac.uk/~simonc/eldertech/ http://www.cci.dmu.ac.uk/

Documents

An Introduction to Type-2 Fuzzy Sets and Systems Dr Simon Coupland [email protected] Centre for Computational Intelligence De Montfort University Leicester