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 is it 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 from a Group of Subjects “ On a scale of 0–10 locate the end points of an interval for some eye contact”

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

14. 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

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) 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) ~

17. 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) μ A (x,u) ~

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 3 rd 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 Representations of an IT2 FS: 1 Vertical Slice Representation—Very useful and widely used for computation

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

26. 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

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 IVFS in 2-dimensions A lot of researchers use IVFS as a way of resolving computational expense issue.

34. 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