Type-2 Fuzzy Sets and Systems

1. Type-2 Fuzzy Sets and Systems

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

3. History

4. 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.

5. 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

6. 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.

7. Collect Data froma Group of Subjects • “ On a scale of 0–10 locate the end points of an interval for some eye contact”

8. Collect Data froma Group of Subjects “ On a scale of 0–10 locate the end points of an interval for some eye contact”

9. 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

10. Fill-er-in and Some New Terms • UMF: Upper membership function (MF) • LMF: Lower MF • Shaded region: Footprint of uncertainty (FOU)

11. Weighting the FOU • Non-uniform secondary MF: General T2 FS • Uniform secondary MF: Interval T2 FS

12. More Terms

13. 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 Quite tall e.g. Tallness e.g. Joe Bloggs

14. We are only interested In type 2 for now 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

15. 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

16. 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. μA(x,u) ~ The amplitude of the ‘sticks’ is called a secondary grade

17. 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)

18. 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.

19. Type-2 Fuzzy Sets - Notation x,u intersection somewhere in the FOU For all u contained in our primary memberships /domain

20. Type-2 Fuzzy Sets- Notation

21. Important Representationsof an IT2 FS: 1 • Vertical Slice Representation—Very useful and widely used for computation

22. Important Representationsof an IT2 FS: 2 • Wavy Slice Representation—Very useful and widely used for theoretical developments

23. Example

24. 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

25. 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

26. 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

27. Set-Theoretic Operations Centroid of type-2 fuzzy sets

28. 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

29. 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

30. Other FOUs

31. 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.

32. Interval valued type-2 fuzzy sets i.e. = 1

33. Interval valued type-2 fuzzy sets A lot of researchers use IVFS as a way of resolving computational expense issue. IVFS in 2-dimensions

34. Type-2 person FS

35. Type-2 person FS

36. Type-2 fuzzy system - overview So this is extra Compared to type-1

37. Fuzzification Fuzzifying in type-1 (fairly easy) Fuzzifying in type-2 (not so easy)

38. 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

39. Interpretation for an IT2 FLS • A T2 FLS is a collection of T1 FLSs

40. IT2 FLS Inference for One Rule

41. IT2 FLS Inference to Output for Two Fired Rules

42. Output Processing • Defuzzification is trivial once type-reduction has been performed