Linguistic Summarization Using IF-THEN Rules

Authors: Dongrui Wu, Jerry M. Mendel and Jhiin Joo

Introduction

• Type I & Type II Fuzzy Systems• Dataset Description• Linguistic Descriptions• Implementation

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

Interval Type-2 Fuzzy Sets

• Interval type-2 fuzzy sets - interval membership grades

• X is primary domain• Jx 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 Jx, Jx [0,1]}~

~

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

~

Interval Type-2 Fuzzy Sets

• Fuzzification:

1.8

0.42

Tall

0

1

Height (m)

~

0.78Tall (1.8) = [0.42,0.78]

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.

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

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

Some parameters

• T – Degree of Truth; an assessment of Validity

– T increases as more data satisfying antecedent also satisfy consequent

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

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)

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)

MAMC Rules

• Multi Antecedent Multi consequent

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

Implementation

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

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