Introduction to Fuzzy Set Theory Weldon A. Lodwick

1. Introduction to Fuzzy Set Theory Weldon A. Lodwick OBJECTIVES 1. To introduce fuzzy sets and how they are used 2. To define some types of uncertainty and study what methods are used to with each of the types. 3. To define fuzzy numbers, fuzzy logic and how they are used 4. To study methods of how fuzzy sets can be constructed 5. To see how fuzzy set theory is used and applied in cluster analysis I. INTRODUCTION: Math Clinic Fall 2003

2. OUTLINE I. INTRODUCTION – Lecture 1 A. Why fuzzy sets 1. Data/complexity reduction 2. Control and fuzzy logic 3. Pattern recognition and cluster analysis 4. Decision making B. Types of uncertainty 1. Deterministic, interval, probability 2. Fuzzy set theory, possibility theory C. Examples – Tejo river, landcover/use, surfaces I. INTRODUCTION: Math Clinic Fall 2003

3. II. BASICS – Lecture 2 A. Definitions 1. Sets – classical sets, fuzzy sets, rough sets, fuzzy interval sets, type-2 fuzzy sets 2. Fuzzy numbers B. Operations on fuzzy sets 1. Union 2. Intersection 3. Complement I. INTRODUCTION: Math Clinic Fall 2003

4. BASICS (continued) C. Operations on fuzzy numbers 1. Arithmetic 2. Relations, equations 3. Fuzzy functions and the extension principle I. INTRODUCTION: Math Clinic Fall 2003

5. III. FUZZY LOGIC – Lecture 3 A. Introduction B. Fuzzy propositions C. Fuzzy hedges D. Composition, calculating outputs E. Defuzzification/action IV. FUZZY SET METHODS Cluster analysis – Lecture 4 I. INTRODUCTION: Math Clinic Fall 2003

6. I. INTRODUCTION – Lecture 1 • Fuzzy sets are sets that have gradations of belongingEXAMPLES: GreenBIG Near • Classical sets, either an element belongs or it does not EXAMPLES: Set of integers – a real number is an integer or not You are either in an airplane or not Your bank account is x dollars and y cents I. INTRODUCTION: Math Clinic Fall 2003

7. A. Why fuzzy sets? - Modeling with uncertainty requires more than probability theory - There are problems where boundaries are gradual EXAMPLES: What is the boundary of the USA? Is the boundary a mathematical curve? What is the area of USA? Is the area a real number? Where does a tumor begin in the transition? What is the habitat of rabbits in 20km radius from here? What is the depth of the ocean 30 km east of Myrtle Beach? 1. Data reduction – driving a car, computing with language 2. Control and fuzzy logic a. Appliances, automatic gear shifting in a car b. Subway system in Sendai, Japan (control outperformed humans in giving smoother rides) I. INTRODUCTION: Math Clinic Fall 2003

8. Temperature control in NASA space shuttles IF x AND y THEN z is A IF x IS Y THEN z is A … etc. If the temperature is hot and increasing very fast then air conditioner fan is set to very fast and air conditioner temperature is coldest. There are four types of propositions we will study later.3. Pattern recognition, cluster analysis - A bank that issues credit cards wants to discover whether or not it is stolen or being illegally used prior to a customer reporting it missing - Given a cat-scan determine the organs and their position; given a satellite imagine, classify the land/cover use - Given a mobile telephone, send the signal to/from a particular receiver to/from the telephone I. INTRODUCTION: Math Clinic Fall 2003

9. 4. Decision making - Locate mobile telephone receptors/transmitters to optimally cover a given area - Locate recycling bins to optimally cover UCD - Position a satellite to cover the most number of mobile phone users - Deliver sufficient radiation to a tumor to kill the cancerous cells while at that same time sparing healthy cells (maximize dosage up to a limit at the tumor while minimizing dosage at all other cells) - Design a product in the following way: I want the product to be very light, very strong, last a very long time and the cost of production is the cheapest. I. INTRODUCTION: Math Clinic Fall 2003

10. Introduction B. Types of Uncertainty 1. Deterministic – the difference between a known real number value and its approximation is a real number (a single number). Here one has error. For example, if we know the answer x must be the square root of 2 and we have an approximation y, then the error is x-y (or if you wish, y-x). 2. Interval – uncertainty is an interval. For example, measuring pi using Archimedes’ approach. 3. Probabilistic – uncertainty is a probability distribution function 4. Fuzzy – uncertainty is a fuzzy membership function 5. Possibilistic - uncertainty is a possibility distribution function, generated by nested sets (monotone) I. INTRODUCTION: Math Clinic Fall 2003

11. Types of sets (figure from Klir&Yuan) I. INTRODUCTION: Math Clinic Fall 2003

12. Introduction (figure from Klir&Yuan) I. INTRODUCTION: Math Clinic Fall 2003

13. Error, uncertainty - information/data is often imprecise, incoherent, incomplete DEFINITION: The error is the difference between the exact value (a real number) and a value at hand (an approximation). As such, when one talks about error, one presupposes that there exists a “true” (real number) value. The precision is the maximum number of digits that are used to measure an approximation. It is the property of the instrument that is being used to measure or calculate the (exact) value. When a subset is being used to measure/calculate, it corresponds to subset that can no longer be subdivided. It depends on the granularity of the input/output pairs (object/value pairs) or the resolution being used. EXAMPLE – satellite imagery at 1meter by 1 meter I. INTRODUCTION: Math Clinic Fall 2003

14. DEFINITION:Accuracy is the number of correct digits in an approximation. For example, a gps reading is (x,y) +/- … DEFINITION: Item of information – is a quadruple (attribute, object, value, confidence) (definition is from Dubois&Prade, Possibility Theory) Attribute: a function that attaches value to the object; for example: area, position, color; it’s the recipe that tells us how to obtain an output (value) from an input (object) Object: the entity (domain or input); for example, Sicily for area or my shirt for color or room 4.2 for temperature. Value: the assignment or output of the attribute; for example 211,417.6 sq. km. for Sicily or green for shirt Confidence: reliability of the information I. INTRODUCTION: Math Clinic Fall 2003

15. AMBIGUITY: a one to many relationship; for example, she is tall, he is handsome. There are a variety of alternatives 1. Non-specificity: Suppose one has a heart blockage and is prescribed a treatment. In this case “treatment” is a non-specificity in that it can be an angioplasty, medication, surgery (to name three alternatives) 2. Dissonance/contradiction: One physician says to operate and another says go to Myrtle Beach. I. INTRODUCTION: Math Clinic Fall 2003

16. VAGUENESS – lack of sharp distinction or boundaries, our ability to discriminate between different states of an event, undecidability (is a glass half full/empty) SET THEORY PROBABILITY POSSIBILITY THEORY FUZZY SET DEMPSTER/SHAFER THEORY THEORY I. INTRODUCTION: Math Clinic Fall 2003

17. EXAMPLES Cidalia Fonte will go over in more detail the ideas introduced here at a later time. Example 1. Tejo River - The problem The dimension of water bodies, and consequently their position, is subject to variation over time, especially in regions which are frequently flooded or subject to tidal variations, creating considerable uncertainty in positioning these geographical entities. River Tejo is an example, since frequent floods occur in several places along its bed. The region near the village of Constância, where rivers Tejo and Zezere meet, was the chosen for this example. A fuzzy geographical entity corresponding to rivers Tejo and Zezere is considered a fuzzy set. To generate this fuzzy entity, the membership function has to be constructed. This was done using a Digital Elevation Model of the region, created from the contours of the 1:25 000 map of the Army Geographical Institute of Portugal and information regarding the daily means of the river water level registered in the hydrometric station of Almourol, located in the vicinity, from 1982 to 1990. The variation of the water level during these year are on the next slide: I. INTRODUCTION: Math Clinic Fall 2003

18. Example 1 (figures from Cidalia Fonte & Lodwick) The membership function of points to the fuzzy set is given by: I. INTRODUCTION: Math Clinic Fall 2003

19. Example (figure from Cidalia Fonte & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003

20. Example 2 – Landcover/use (figures from Cidalia Fonte & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003

21. Example 2 – Landcover/use continued I. INTRODUCTION: Math Clinic Fall 2003

22. GIS - Display I. INTRODUCTION: Math Clinic Fall 2003

23. Example 3 – Surface modeling 3. Surface models - The problem: Given a set of reading of the bottom of the ocean whose values are uncertain, generate a surface that explicitly incorporates this uncertainty mathematically and visually - The approach: Consistent fuzzy surfaces - Here with just introduce the associated ideas I. INTRODUCTION: Math Clinic Fall 2003

24. Imprecision in Points: Fuzzy Points (figures from Jorge dos Santos) 2D 3D I. INTRODUCTION: Math Clinic Fall 2003

25. Transformation of real-valued functions to fuzzy functions Instead of a real-valued function or let’s now consider a fuzzy function or where every element x or (x,y) is associated with a fuzzy number . Statement of the Interpolation Problem Knowing the values { } of a fuzzy function over a finite set of points {xi} or {(xi,yi)}, interpolate over the domain in question to obtain a (nested) set of surfaces that represent the uncertainty in the data. . I. INTRODUCTION: Math Clinic Fall 2003

26. Computing surfaces Given a data set of fuzzy numbers: I. INTRODUCTION: Math Clinic Fall 2003

27. Computing surfaces – Example I. INTRODUCTION: Math Clinic Fall 2003

28. Consistent Fuzzy Surfaces (curves) • The surfaces (curves) are defined enforcing the following properties: • The surfaces are defined analytically via the fuzzy functions; that is, model directly the uncertainty using fuzzy functions or • All fuzzy surfaces maintain the characteristics of the generating method. That is, if splines are being used then all generated fuzzy surfaces have the continuity and smoothness conditions associated with the splines being used. I. INTRODUCTION: Math Clinic Fall 2003

29. Fuzzy Interpolating Polynomial - (figure from Jorge dos Santos & Lodwick) Utilizing alpha-levels to obtain fuzzy polynomials, we have: I. INTRODUCTION: Math Clinic Fall 2003

30. 60 50 40 30 20 10 0 -10 -50 0 50 100 150 200 2-D Example (from Jorge dos Santos & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003

31. 20 15 10 5 10 15 20 25 30 50 0 -50 -100 0 20 40 60 80 100 120 140 160 180 200 60 40 20 0 -20 0 20 40 60 80 100 120 140 160 180 200 Fuzzy Curves (figures from Jorge dos Santos & Lodwick) P. Lagrange Spline linear I. INTRODUCTION: Math Clinic Fall 2003

32. 50 40 30 20 10 0 -10 0 20 40 60 80 100 120 140 160 180 200 50 40 30 20 10 0 -10 0 20 40 60 80 100 120 140 160 180 200 Fuzzy Curves (figures from Jorge dos Santos & Lodwick) Cubic Spline Consistent Cubic Spline I. INTRODUCTION: Math Clinic Fall 2003

33. 33 32 31 30 29 28 27 155 160 165 170 Details of the Consistent Fuzzy Cubic Spline (figures from Jorge dos Santos & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003

34. 200 180 160 140 120 100 80 60 40 20 0 -50 0 50 100 150 200 250 3-D Example (from Jorge dos Santos & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003

35. Another Representation/View of the Fuzzy Points(figure from Jorge dos Santos & Lodwick) 35 30 25 20 15 200 10 150 5 -50 100 0 50 50 100 150 0 200 I. INTRODUCTION: Math Clinic Fall 2003

36. Fuzzy Surface via Triangulation (figure from Jorge dos Santos & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003

37. Fuzzy Surfaces via Linear Splines (figure from Jorge dos Santos & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003

38. Fuzzy Surfaces via Cubic Splines (figure from Jorge dos Santos & Lodwick) I. INTRODUCTION: Math Clinic Fall 2003