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:

1
Tall
0.62
0
1.8
Height (m)
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
~
A = {((x,u), 1) | x  X, u  Jx, Jx  [0,1]}




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)
Interval Type-2 Fuzzy Sets

Returning to Tall
~
Tall

1
Upper MF Tall
Type -1 MF
= FOU
0
Height (m)
Lower MF Tall
Interval Type-2 Fuzzy Sets

Fuzzification:
~

1
Tall
0.78
~ (1.8) = [0.42,0.78]
Tall
0.42
0
1.8
Height (m)
Interval Type-2 Fuzzy Sets

Defuzzification – two stages:



Type-reduction
Interval centroid
Type-reduction (centroid):
GC =  1Jx …  1Jx
1
N
1
Ni=1 xii
/
N
i=1
i
= [Cl, Cr]
(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:
~

1
Tall
0
Cl
Cr
Height (m)
Centroid
Interval Type-2 Fuzzy Sets
Cl
Cl switch point
X
Interval Type-2 Fuzzy Sets
Centroid
Cr
Cr switch point
X
Interval Type-2 Fuzzy Sets





These properties are exploited by KarnikMendel algorithm
Converges in at most N steps
3-4 steps typical
Widely used
Hardware implementation
Interval Type-2 Fuzzy Systems
Rules
Output processing
Defuzzifier
Crisp
inputs
Fuzzifier
Type-reducer
Inference
Type-2 Interval FIS
Crisp
outputs
Typereduced
outputs
(interval)
Interval Type-2 Fuzzy Systems
Mamdani or TSK systems
We’ll only look at Mamdani
Example rule base:



1.
2.
~
~
~
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[G~ 1(zi)  RA1, G~ 1(zi)  RA1]
~’
Rule 2: G 2 = i..n[G~2(zi)  RA2, G~ 2(zi)  RA1]
~’
Interval Type-2 Fuzzy Systems

Consequent combination:
Gc = i..n [G~1’ (gi) V G~2’ (gi) , G~1’ (gi) V G~2’ (gi) ]
~
Where V is a t-conorm, generally max
Interval Type-2 Fuzzy Systems
A~

1

0
1
0
~
C

B~
1

0
1
0
~
D
1

0
1
0
~
G
1
~
G
2
Interval Type-2 Fuzzy Systems
A~

1

0
~
C
1
0

B~
1
1

0
1
~
D

1
0
x
0
0
y
~
G
1
~
G
2
Interval Type-2 Fuzzy Systems
~
~
(min)
~
A
B
G




1

1
1
0
~
C
1
0

0
1
1
~
D

1
0
x
0
0
y
~
G
2
Interval Type-2 Fuzzy Systems
~
~
(min)
~
A
B
G




1

1
1
0
~
C
1
0

0
1
1
~
D

1
0
x
0
0
y
~
G
2
Interval Type-2 Fuzzy Systems
~
~
(min)
~
A
B
G




1

1
1
0
~
C
1
0

0
1
1
~
D

1
0
x
0
0

1
y
~
G
2
~
GC
max
0
Cl Cr
Interval Type-2 Fuzzy Systems
~
~
(prod)
~
A
B
G




1

1
1
0
~
C
1
0

0
1
1
~
D

~
G
2
1
0
x
0
0
~
GC

1
y
max
0
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
~
A = {((x,u), A~ (x,u)) | x  X, u  Jx, Jx  [0,1]}




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]
Generalised Type-2 Fuzzy Sets


Representation theorem (Mendel and John)
Represent generalised type-2 fuzzy sets and
operations as collection of embedded fuzzy
sets
~
A=
n
 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
(x)
(x,u) 1
1
~
A
X
Generalised Type-2 Fuzzy Sets

Fuzzification
(x)
(x,u) 1
1
~
A
x
X
Generalised Type-2 Fuzzy Sets

Fuzzification
(x)
(x,u) 1
1
~
A
x
X
Generalised Type-2 Fuzzy Sets

Fuzzification
(x)
(x,u) 1
1
~
A
x
(x,u)
1
~A(x)
(x)
X
1
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

(x,u)
1
Join and meet under min:
f
(x,u)
1
g
meet
join
(x)
1
(x)
1
Generalised Type-2 Fuzzy Sets

(x,u)
1
Join and meet under prod:
f
(x,u)
1
g
meet
join
(x)
1
(x)
1
Generalised Type-2 Fuzzy Sets


More efficient join and meet operations:
Apex points 1 and 2
(x,u)
1
f
1
g
2
1
(x)
Generalised Type-2 Fuzzy Sets

More efficient join and meet operations:
f g (u) =
f g (u) =
{
{
f(u) Λ g(u),
u<1
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(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))
~
zZ
~ (z)
G
Generalised Type-2 Fuzzy Sets


Combination of Consequents:
Join all consequent sets at every point in the
in the consequent domain:
G =  G~1(z) G~2(z)
~
zZ
…
G~n(z)
Generalised Type-2 Fuzzy Sets


Type-reduction (centroid)
Gives a type-1 fuzzy set:
GC =  1Jz …  1Jz
1

N
N
i=1
G(zii)
~
/
N
i=1
zii
Ni=1 i
(Karnik and Mendel)
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
(z,u) 1
1
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1

1
(z,u) 1
Z
Z
Generalised Type-2 Fuzzy Sets

Type-reduction
(z)
1
CZ

1
(z,u) 1
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:



Signal Processing:



Robot navigation (Hagras, Coupland, Castillo)
Plant (Castillo, Chaoui, Hsiao)
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/