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 is it 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 from a Group of Subjects

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

some eye contact”

Collect Data from a 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. Tallness Quite tall e.g. Joe Bloggs

Type-n Fuzzy Sets

In type 2 for now • 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

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)

μ

~

A (x,u)

j x is the set of possible u values, i.e.

j 3 = [0.6, 0.8] J x 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

μ

~

A (x,u)

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

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 3 rd dimension on top of it.

Type-2 Fuzzy Sets - Notation

x,u

intersection somewhere in the FOU For all u contained in our primary memberships /domain

Type-2 Fuzzy Sets - Notation

Important Representations of an IT2 FS: 1

•

Vertical Slice Representation

—Very useful and widely used for computation

Important Representations of 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 N is discretisation of x M is discretisation of u i.e. the union of all the embedded sets Indicates how to calculate the number of embedded sets

example on previous slide

Important Representations of 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 expensive (but..) 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

 ~

A

(

x

,

u

) i.e.

 ~

A

(

x

,

u

) = 1

Interval valued type-2 fuzzy sets

A lot of researchers use IVFS as a way of resolving computational expense issue.

IVFS in 2-dimensions

Type-2 person FS

Type-2 person FS

Type-2 fuzzy system overview

So this is extra Compared 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