1. Linguistic Summarization Using IF-THEN Rules Authors: Dongrui Wu, Jerry M. Mendel and Jhiin Joo

2. Introduction Type I & Type II Fuzzy Systems Dataset Description Linguistic Descriptions Implementation

3. Type-1 Fuzzy Sets Crisp sets, where x  A or x  A Membership is a continuous grade  [0,1] Membership a value 1.77 0 1  Height (m) Degree of “Tall-ness” 0.6

4. Interval Type-2 Fuzzy Sets Interval type-2 fuzzy sets - interval membership grades X is primary domain J x is the secondary domain All secondary grades (  A (x,u)) equal 1  A (x) is the secondary membership function at x (vertical slice representation) A = {((x,u), 1) | x  X, u  J x, J x  [0,1]} ~ ~

5. Interval Type-2 Fuzzy Sets Tall 0 1  Height (m) ~ Upper Membership Function Lower MF  Tall Type -1 MF = FOU (explained in next slide) Membership no longer crisp

6. ~ Interval Type-2 Fuzzy Sets Fuzzification: 1.8 0.42 Tall 0 1  Height (m) ~ 0.78  Tall (1.8) = [0.42,0.78]

7. Interval Type-2 Fuzzy Sets FOU Vertical slice of a Type 2 membership function – Indicating 3D structure of Type 2 Mendel Jerry M. and. Bob John Robert I, “Type-2 Fuzzy Sets Made Simple.” IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002.

8. Haberman’s survival Data set - UIUC From a study conducted between 1958 and 1970 at the University of Chicago's Billings Hospital on the survival of patients who had undergone surgery for breast cancer. Attribute Information: 1. Age of patient at time of operation (numerical) 2. Patient's year of operation (year - 1900, numerical) 3. Number of positive axillary nodes detected (numerical) 4. Survival status (class attribute) -- 1 = the patient survived 5 years or longer -- 2 = the patient died within 5 year

9. Linguistic Summarizations: IF - THEN Type 1: IF AGE is 35 AND YEAR is 1962, THEN SURVIVAL is YES Type 2: IF AGE is around 35 AND YEAR is around 1962, THEN SURVIVAL is YES

10. Some parameters T – Degree of Truth; an assessment of Validity – T increases as more data satisfying antecedent also satisfy consequent

11. Some parameters C – Degree of Sufficient Coverage – Determines if sufficient data satisfies a rule (trigger) – C=f(rc) U – Degree of Usefulness – Indicates how useful a rule is – A rule is useful iff it has high degree of truth: most of the data satisfy the rule’s antecedents as well as its consequent It has sufficient coverage: enough data are described by it. – U=min(T,C) It depends on the parameters described earlier

12. Some parameters O – Degree of Outlier – Indicates if a rule describes the outliers instead of most of the data – If T=0, O=0 since no data is described by the rule – Described by the complement of T & C since they both depend on the data (not outlier)

13. Some parameters S - Degree of Simplicity Determined by the length of the summary L = number of antecedents Simplest rule: S=1 (one antecedent and one consequent)

14. MAMC Rules Multi Antecedent Multi consequent

15. Implementation Each case represented as a piecewise linear curve Blue – strength of supporting rule Red- cases violating given rule Black- Irrelevant Figure shows if C is used for ranking, T may/may not be high

16. Implementation Figure shows if U is used for ranking, high U indicates high T & C : useful rule

17. Conclusions An important method of ranking rules using the parameters: – Degree of Truth – Degree of Sufficient Coverage – Degree of Usefulness – Degree of Outlier – Degree of Simplicity