Type of Uncertainty 1.vagueness: associated with the difficulty of making sharp or precise distinctions in the world.  the concept of fuzzy set provides a basic mathematical framework for dealing with vagueness. 2.ambiguity: associated with one-to-many relations.  the concept of fuzzy measure provides a general framework for dealing ambiguity.

3.types of ambiguity: a.nonspecificity in evidence: connected with the size of the subsets that are designated by a fuzzy measure as prospective locations of the element in questions. The larger the subsets, the less specific the characterization. b.dissonance in evidence: exhibited by disjoint subsets of X that are designated by the given fuzzy measure as prospective locations of the element of concern. c.confusion in evidence: associated with the number of subsets of X that are designated by a fuzzy measure as prospective locations of the element under consideration and that do not overlap or overlap only partially.

Imprecision 的不同面目 1.Measurement imprecision/inaccuracy 2.Intrinsic imprecision: associated with a description of the properties of a phenomenon and not with our measurement of the properties using some external device. 3.Sometimes both are conflict.

Ambiguity 1.Semantic ambiguity: “Do I turn left here?”  Ans. Right! “The food is hot.” Not fuzzy! It deals with the interpretation of disjoint and discrete concepts. Fuzzy! Attach an universal discourse and define the membership of each element to one of possible interpretation.

Ambiguity (cont.) 2. Visual (perceptual) ambiguity: Möbius strip; GUI: the use of color, icons, dialogues can introduce ambiguous visual stimuli. 3. Structural ambiguity: existing in the relationships between components of a system can reflect a high-level of ambiguity.  multiple inheritance in an OO systems

Undecidability 1.The ability to discriminate between different states of an event. 2.Generally stems from the properties of the model itself, not from any lack of knowledge on our part. 3.Nekker cube 4.Kaniza Square 5.Half-full v.s. Half-empty glass

Measure of Fuzziness

INTRODUCTION

Basic characteristics of fuzzy sets 1.Vertical dimension: height, normalization (maximum form: at least one element with 1.0 membership and one element with membership 0). 2.Horizontal dimension: universal discourse, support sets, alpha cuts. 3.Representation schemes: membership function, ordered set of pairs, polynomial- like (integral-like).

BASIC CONCEPTS

Types of Membership Functions 1.Linear type: linear/step functions. Approximating an unknown or poorly understood concept that is not a fuzzy number. (often expressed as shouldered sets) 2.Triangular type: often used to model process control systems.

Types of Membership Functions (cont.) when you decompose a variable into fuzzy sets, the amount of overlap must vary between 10% to 50%. In modeling dynamic systems, it can approximate their behaviors to nearly any degree of precision.

Types of Membership Functions (cont.) 3. Trapezoidal Type:

Types of Membership Functions (cont.) 4. Sigmoid/Logistic Type: * modeling population dynamics where the sampling of individual values approximates a continuous random variable (e.g., 硬碟 MTBF,…) * frequency (proportional) representation: usually, most, always.

Types of Membership Functions (cont.) 1.Fuzzy numbers and around representation: around, close to, few, some. 2.PI, Beta, Gaussian fuzzy sets (bell-shaped): the slope and width of the bell curves indicate the degree of compactness associated with the fuzzy number. 3.PI curves: is not asymptotic. Zero point is at a discrete and specified point.

Types of Membership Functions (cont.) 4. Beta curves: more tightly compacted than PI. 5. Similar to beta curves, but the slope goes to zero very quickly with a very short tail. 6. Irregularly shaped and arbitrary fuzzy sets: (domain value, membership) pairs  linear interpolation.

FUZZY LOGIC

7. Linguistic hedges: linguistic terms that can be used to modify fuzzy sets, such as very, extremely, quite, etc. 8. In general, fuzzy quantifiers are represented in fuzzy logic by fuzzy numbers.

Hedges 1.Modify the shape of a fuzzy set’s surface causing a change in the membership function.  become a new fuzzy set. 2.Represented by a linguistic-like construct. (intensification: very, extremely. dilation: somewhat, rather, quite) 3.Play the same role in fuzzy production rules that adjectives and adverbs play in English sentences. 4.Without mathematical theory. 5.Multiple hedges can be applied to a single fuzzy region.

Approximation hedges 1.Not only broaden or restrict existing bell- shaped fuzzy regions, but convert scalar values into bell-shaped fuzzy regions. 2.Basic approximation hedges: about, around, near, roughly; all produce the same PI curve. 3.Decide the spread of PI curve: essential issue. a. implicitly determined from context, b. explicitly domain statement.

Approximation Hedges (cont.) 4. About: creates a space that is proportional to the height and width of the generated fuzzy space. (slightly broadens the fuzzy region) 5. In the vicinity: constructs a very wide fuzzy region  much broader than the region created by “about.”  If several closely allied sets share the same underlying domain, this approximation may lead to an unforeseen degree of ambiguity. 6. Close to (neighboring): narrowly approximated regions.

Restricting a fuzzy region 1.Modify the shape of directional or bell-shaped fuzzy regions. 2.Below: can only be applied to a fuzzy membership function that increases as the domain moves from left to right. 3.Above: can only be applied to a fuzzy membership function that declines as the domain moves from left to right. 4.Can also be applied to unhedged scalars  produces a classical step function space.

Restricting a fuzzy region (cont.) 5. To prevent this rather crisp representation of what should intuitively be a fuzzy space, the scalar should be approximated as the restriction hedge is applied; such as above around 40. below close to our direct-costs.

Intensifying and diluting fuzzy regions 1.Intensification: very, extremely. 2.Dilution: somewhat, quite, rather, sort of. 3.Very:  domain independent: rescales the fuzzy set surface but maintains the normalization space for the set.  domain dependent:

Intensifying and diluting fuzzy regions 4. Concentrator hedge in Zadeh: 5. Somewhat (rather, quite): dilute the force of the membership function in a particular fuzzy region. a. effect: to increase the candidate space within the fuzzy region with a characteristic function relationship of

Intensifying and diluting fuzzy regions b. domain independent: c. context dependent: k provides a context sensitive adjusting control on the strength of the dilator.

Intensifying and diluting fuzzy regions d. generalization of dilator hedge e. power function should not exceed 0.7 to avoid too close to base set to effectively be applied in the model. f. very and somewhat are the only naturally idempotent members of the hedge set, and are the only commutative hedges.  somewhat very tall = very somewhat tall

Contrast Intensification and Diffusion 1.Change the nature of fuzzy regions by either making the region less fuzzy (intensification) or more fuzzy (diffusion). 2.The idea behind contrast hedging is related to the concept of fuzzy entropy and intrinsic ambiguity. 3.At the extremes of the fuzzy set, membership is less ambiguous  the fuzziness is minimized. 4.The values that are centered around the midpoint in the fuzzy truth function, are in an area of maximum fuzziness.

Contrast Intensification and Diffusion 5. Positively hedge (absolutely, definitely): a. changes the fuzzy surface by raising all the truth function values above 0.5 and decreasing all the truth function values below 0.5. b. reduce the overall fuzziness of the region.

Contrast Intensification and Diffusion 6. Generally hedge: diffusion modifier a. changes the fuzzy surface by reducing all the truth function values above 0.5 and increasing all the truth function values below 0.5. b. increase the overall fuzziness of the region. c. allow us to make decisions based on whether or not a truth value is broadly within the fuzzy set or definitely within the fuzzy set.

OPERATIONS ON FUZZY SETS *Possibility Theory *Fuzzy Complement 1.axiomatic skeleton [Axiom c1]: (boundary conditions) c(0)=1 and c(1)=0, that is, c behaves as the ordinary complement for crisp sets. [Axiom c2]: For all a,b [0,1], if a < b, then c(a) ≧ c(b), that is, c is monotonic nonincreasing. E.g.,threshold-type complements

5.Fuzzy set operations of union, intersection, and continuous complement that satisfy the law of excluded middle and the law of contradiction are not idempotent or distributive. 6.Given two of the three operation u, i, c, it is sometimes desirable to determine the third operation in such a way that Demorgan’s laws are satisfied.

TABLE 2.2 SOME CLASSES OF FUZZY SET UNIONS AND INTERSECTIONS ReferenceFuzzy UnionsFuzzy Intersections Range of Parameter Schweizer & Sklar [1961] Hamacher [1978] Frank [1979] Yager [1980] Dubois & Prade [1980] Dombi [1982]

FUZZY RELATIONS

Fuzzy Propositions 1. A proposition whose truth value is a matter of degree. (in the range of [0,1]) 2. Unconditional and unqualified propositions: p: ν is F (temperature is high) T(p) = F(v) (= r F (v)) 3. p: ν(i) is F (ν: I  V)

Unconditional and Qualified Propositions 1. p: ν is F is S (truth-qualified) 2. p: Pro{ν is F}is P (probability-qualified) 3. Both S (fuzzy truth qualifier) and P (fuzzy probability qualifier) are presented by fuzzy sets on [0,1]. 4. T(p) = S(F(v)) 5. ν is G

Unconditional and Qualified Propositions (cont.) Unqualified propositions are special truth-qualified propositions, in which the truth qualifier S is assumed to be true. (S(F(v)) =F(v)) Pro{ν is F} = T(p) = P( ) p: Pro{temperature t (at given place and time) is around 75 o F} is likely.

Conditional and Unqualified Propositions p: If χ is A, then y is B is R R(x,y) = δ[A(x), B(y)] (fuzzy implication)

Conditional and Unqualified Propositions (cont.) p: If χ is A, then y is B is S p: Pro{χ is A| y is B} is P