Type-2 Fuzzy Sets and Systems

Outline

• Introduction• Type-2 fuzzy sets.• Interval type-2 fuzzy sets• Type-2 fuzzy systems.

History

What is a T2 FS and How isit Different From a T1 FS?

• T1 FS: crisp grades of membership

• T2 FS: fuzzy grades of membership, a fuzzy-fuzzy set.

Type-2 fuzzy sets

• Blur the boundaries of a T1 FS

• Possibility assigned – could be non-uniform

• Clean things up• Choose uniform possibilities

– interval type-2 FS

Where Does a T2 FS Come From?

• Consider a FS as a model for a word• Words mean different things to different

people.• So, we need a FS model that can capture

the uncertainties of a word.• A T2 FS can do this.• Let’s see how.

Collect Data froma Group of Subjects

• “ On a scale of 0–10 locate the end points of an interval for some eye contact”

Collect Data froma Group of Subjects

“ On a scale of 0–10 locate the end points of an interval for some eye contact”

Create a Multitude of T1 FSs• Choose the shape of the MF, as we do for T1

FSs, e.g. symmetric triangles• Create lots of such triangles that let us cover the

two intervals of uncertainty

Fill-er-in and Some New Terms

• UMF: Upper membership function (MF)• LMF: Lower MF• Shaded region: Footprint of uncertainty (FOU)

Weighting the FOU

• Non-uniform secondary MF: General T2 FS• Uniform secondary MF: Interval T2 FS

More Terms

Type 2 fuzzy sets• Imagine blurring a type 1 membership function.• There is no longer a single value for the

membership function for any x value, there are a few

e.g. TallnessQuite tall

e.g. Joe Bloggs

Type-n Fuzzy Sets

• A fuzzy set is of type n, n = 2, 3, . . . if its membership function ranges over fuzzy sets of type n-1. The membership function of a fuzzy set of type-1 ranges over the interval [0,1].

Zadeh, L.A., The Concept of a Linguistic Variable and its Application to Approximate Reasoning - I, Information Sciences, 8,199–249, 1975

We are only interested In type 2 for now

Type 2 fuzzy sets• These values need not all be the same• We can therefore assign an amplitude distribution to all of the

points• Doing this creates a 3-D membership function i.e. a type 2

membership function • This characterises a fuzzy set

Example of a type 2 membership function. The shaded area is called the ‘Footprint of

Uncertainty’ (FOU)jx is the set of possible u values, i.e.

j3 = [0.6, 0.8]

Jx is called the primary membership of x and is the domain of the secondary membership function.

The amplitude of the ‘sticks’ is called a secondary grade

μA(x,u)~

Referring to the diagram, the secondary membership function at x = 1 is a /0 + b /0.2 + c /0.4.

Its primary membership values at x = 1 are u = 0, 0.2, 0.4, and their associated secondary grades are a, b and c respectively.

(Mendel, 2001, p85)

μA(x,u)~

FOU continued

• The FOU is the union of all primary memberships

• It is the region bounded by all of the ‘j’ values i.e. the red shaded region on the earlier slide.

• FOU is useful because:– Focuses our attention on uncertainties (blurriness!)– Allows us to depict a type 2 fuzzy set graphically in

2 dimensions instead of 3.– The shaded FOUs imply the 3rd dimension on top

of it.

Type-2 Fuzzy Sets - Notation

x,u intersection somewhere in the FOU

For all u contained inour primary memberships

/domain

Type-2 Fuzzy Sets - Notation

Important Representationsof an IT2 FS: 1

• Vertical Slice Representation—Very useful and widely used for computation

Important Representationsof an IT2 FS: 2

• Wavy Slice Representation—Very useful and widely used for theoretical developments

Example

Calculating the number of embedded sets

In the above example there would be:

5 x 5 x 2 x 5 x 5 = 1250

So there are 1250 embedded sets

More formally:Fundamental Decomposition Theorem

i.e. the union of all the embedded sets

Indicates how tocalculate the numberof embedded sets

example on previous slide

N is discretisation of x

M is discretisation of u

Important Representationsof an IT2 FS

• Wavy Slice: Also known as “Mendel- John Representation Theorem (RT)”– Importance: All operations involving IT2 FSs

can be obtained using T1 FS mathematics• Interpretation of the two

representations: Both are covering theorems, i.e., they cover the FOU

Set-Theoretic Operations

Centroid of type-2 fuzzy sets

Comparison with type-1• Type-1 fuzzy sets are two dimensional• Type-2 fuzzy sets are three dimensional

• Type one membership grades are in [0,1]• We can have linguistic grades with type-2

• Type-1 fuzzy systems are computationally cheap• Type-2 fuzzy systems are computationally

expensive (but..)Various approaches being taken

To solve/get round this

Interval T2 FSs• Rest of tutorial focuses exclusively on IT2 FSs

– Computations using general T2 FSs are very costly– Many computations using IT2 FSs involve only

interval arithmetic– All details of how to use IT2 FSs in a fuzzy logic

system have been worked out– Software available– Lots of applications have already occurred

Other FOUs

Interval Valued type-2 fuzzy sets

When the amplitudes of of the secondary membership function all equal 1, we have an interval valued fuzzy set.

Interval valued type-2 fuzzy sets

),(~ uxA

i.e. = 1),(~ uxA

Interval valued type-2 fuzzy sets

IVFS in 2-dimensions

A lot of researchersuse IVFS as a way

of resolvingcomputationalexpense issue.

Type-2 person FS

Type-2 person FS

Type-2 fuzzy system - overview

So this is extraCompared to type-1

Fuzzification

Fuzzifying in type-1 (fairly easy)

Fuzzifying in type-2(not so easy)

Interval Type-2 FLS

• Rules don’t change, only the antecedent and consequent FS models change

• Novel Output Processing: Going from a T2 fuzzy output set to a crisp output—type-reduction + defuzzification

Interpretation for an IT2 FLS

• A T2 FLS is a collection of T1 FLSs

IT2 FLS Inference for One Rule

IT2 FLS Inference to Output for Two Fired Rules

Output Processing

• Defuzzification is trivial once type-reduction has been performed