CHAPTER-1INTRODUCTION1.1 MEASURE:In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size (by Ref.1 [1]). In this sense, a measure is a generalization of the concept of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space (according to Ref.1 [1,3]).Measure theory was developed in successive stage during the late 19th and early 20th centuries by mile Borel, Henri Lebesgue, Johann Radon and Maurice Frchet, among others. (by Ref.1 [2,5,6]). The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorovs axiomatisation of probability theory and in ergodic theory.Ref.1[4]. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral (see Ref.1 [1]). Probability theory considers measures that assign to the whole set of size 1, and considers measurable subsets to be events whose probability is given by the measure (by Ref.1 [3]). Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system (Ref.1 [4]). Technically, a measure is a function that assigns a non-negative real number or to certain subsets of a set X(Ref.1 [4]). It must assign 0 to the empty set and be countably additive: the measure of a large subset that can be decomposed into a countable number of smaller disjoint subset, is the sum of the measures of the smaller subsets by(Ref.1 [8]). In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure(Ref.1 [2]). This problem was resolved by defining measure only on a sub-collection of all subsets; the so called measurable subsets, which are required to form a algebra (Ref.1[5]), this means that countable unions, countable intersections and complements of measurable subsets are measurable. 1.1.1Definition:Let X be a set and a algebra over X. A function from to the extended real number line i.e. : is called a measure if it satisfies the following properties: (according to Ref.1 [9])(a) Non-negativity: for all (b) Null empty set:(c) Countable additivity (or - additivity):For all countable collections of pairwise disjoint sets in :

One may require that at least one set E has finite measure. Then the null set automatically has measure zero because of countable additivity, because

is finite if and only if the empty set has measure zero. If only the second and third conditions of the definition of measure above are met, and takes on at most one of the values , then is called a signed measure by(Ref.1 [1,5]).1.1.2PropertiesSeveral further properties can be derived from the definition of a countably additive measure by(Ref.1 [6, 8, 9]).(a) Monotonicity: A measure is monotonic: If and are measurable sets with then, . (1.3)(b) Measures of infinite unions of measurable sets:A measure is countably subadditive: If E1, E2, E3, . is a countable sequence of sets in , not necessarily disjoint, then

A measure is continuous from below: If E1, E2, E3, . are measurable sets and En is a subset of En+1 for all n, then the union of the sets En is measurable, and

(c) Measures of infinite intersections of measurable sets:A measure is continuous from above: If E1, E2, E3, . are measurable sets and En+1 is a subset of En for all n, then the intersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then

This property is false without the assumption that at least one of the En has finite measure. For instance, for each , let , which all have infinite Lebesgue measure, but the intersection is empty.1.1.3.Some measures are defined here by (Ref.1 [2, 5, 8, 9])ISigma-finite measure: A measure space (X, , ) is called finite if (X) is a finite real number. It is called - finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has finite measure if it is a countable union of sets with finite measure.For example, the real numbers with the standard Lebesgue measure are finite but not finite. Consider the closed intervals [k, k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of real numbers i.e. number of points in the set. This measure space is not finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such set to cover the entire real line. The finite measure spaces have some very convenient properties; finiteness can be compared in this respect to the Lindelf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have uncountable measure.IIComplete measure: A measurable set X is called a null set if (X). A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete measure if every negligible set is measurable.A measure can be extended to a complete one by considering the algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set.III - Additive measure: A measure on is additive if for any and any family the following conditions hold;

Note that the second condition is equivalent to the statement that the ideal of null sets is - complete. Some other important measures are listed here by (Ref.1[1, 3, 8])IVLebesgue Measure: The Lebesgue measure on R is a complete translation-invariant measure on a -algebra containing the intervals in R such that ; and every other measure with these properties extends Lebesgue measure.The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets. The Haar measure for a locally compact topological group is also a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties. Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.Other measures used in various theories include: Borel measure, Jordan measure, Ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure and so on (by Ref.1 [6,9]).1.1.4. Generalizations: For certain purposes, it is useful to have a measure whose values are not restricted to the non-negative real or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively (by Ref.1 [4]). A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures, which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures(Ref.1 [2]).Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Finite additive measures are connected with notions such as Banach limits, the dual of and the Stone-Cech compactification (by Ref.1 [8]). All these are linked in one way or another to the axiom of choice.

1.2.FUZZY:Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadehs seminal work Fuzzy Sets (by Ref.2[1]). A fuzzy subset A of a set X is a function , where L is the interval [0,1]. This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for . More generally, one can use a complete lattice L in a definition of a fuzzy subset A (Ref.2 [2]).The evolution of the fuzzification of mathematical concepts can be broken down into three stages: (by Ref.2 [10])(a)straightforward fuzzification during the sixties and seventies,(b)the explosion of the possible choices in the generalization process during the eighties,(c)the standardization, axiomatization and L-fuzzification in the nineties.Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let A and B be two fuzzy subsets of X. Intersection and union are defined as follows:

Instead of min and max one can use t-norm and t-conorm, respectively, (by Ref.2 [11]) for example, min (a,b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on min and max operations because in this case more properties of traditional mathematics can be extended to the fuzzy case.A very important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on X. The closure property for fuzzy subsets A of X is that for all Let (G, *) be a group and A a fuzzy subset of G. Then A is a fuzzy subgroup of G if for all x, y in G, (by Ref.2 [11]).A fuzzy is a concept of which the meaningful content, value, or boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all (Ref.2[3]). This generally means the concept is vague, lacking a fixed, precise meaning, without however being meaningless altogether (Ref.2[4]). It has a meaning, or multiple meaning. But these can become clearer only through further elaboration and specification, including a closer definition of the context in which they are used. Fuzzy concepts lack clarity and are difficult to test or operationalize (Ref.2[5]).In logic, fuzzy concepts are often regarded as concepts which in their application, or formally speaking, are neither completely true nor completely false, or which are partly true or partly false; they are ideas which require further elaboration, specification or qualification to understand their applicability (Ref.2 [7]) (the conditions under which they truly make sense).In mathematics and statistics, a fuzzy variable (such as the temperature, hot or cold) is a value which could lie in a probable range defined by quantitative limits or parameters, and which can be usefully described with imprecise categories (such as high, medium of low).In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called degree-theoretic semantics by logicians and philosophers (Ref.2 [6]), but more usual term is fuzzy logic or many-valued logic. The basic idea is, that a real number is assigned to each statement written in a language, within a range from 0 to 1, where 1 means that the statement is completely true, and 0 means that the statement is completely false, while values less than 1 but greater than 0 represent that the statements are partly true, to a given, quantifiable extent. This makes it possible to analyze a distribution of statement for their truth-content, identify data patterns, make inferences and predictions, and model how processes operate.Fuzzy reasoning (i.e. reasoning with graded concepts) has many practical uses (by Ref.2 [12]). It is nowadays widely used in the programming of vehicle and transport electronics, household appliances, video games, language filters, robotics, and various kinds of electronic equipment used for pattern recognition, surveying and monitoring(such as radars). Fuzzy reasoning is also used in artificial intelligence and virtual intelligence research.(Ref.2 [13]) Fuzzy risk scores are used by project managers and portfolio managers to express risk assessments. Ref.2 [14].A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are very often represented by real-number values ranging in the closed interval between 0 and 1(by Ref.2 [9] on page 4-5). Thus, a fuzzy set representing our concept of sunny might assign a degree of membership of 1 to a cloud cover of 0%, 0.8 to a cloud cover of 20%, 0.4 to a cloud cover of 30%, and 0 to a cloud cover of 75%. These grades signify the degree to which each percentage of cloud cover approximate our subjective concept of sunny, and the set itself models the semantic flexibility inherent in such a common linguistic term. Because full membership and full non-membership in the fuzzy set can still be indicated by the values of 1 and 0, respectively, we can consider the concept of a crisp set to be a restricted case of the more general concept of a fuzzy set for which only these two grades of membership are allowed( Ref.2 [9]). 1.3.UNCERTAINTYFuzzy concepts can generate uncertainty because they are imprecise (especially if they refer to a process in motion or a process of transformation where something is in the process of turning into something else). In that case, they do not provide a clear orientation for action or decision-making, reducing fuzziness, perhaps by applying fuzzy logic, would generate more certainty (Ref.2 [7]). A fuzzy concept may indeed provide more security, because it provides a meaning for something when an exact concept is unavailable which is better than not being able to denote it at all (Ref.2 [8]).It is generally agreed that an important point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by (Ref.2 [1]), even though some ideas presented in the paper were envisioned some few years earlier by (Ref.2 [15]). In this paper, Zadeh introduced a theory whose objects fuzzy sets are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of a degree (Ref.2 [9])The significance of (Ref.2 [1]) was that it challenged not only probability theory as the sole agent for uncertainty, but the very foundations upon which probability theory is based: Aristotelian two-valued logic. When A is a fuzzy set and x is a relevant object, the proposition x is a member of A is not necessarily either true or false, as required by two-valued logic, but it may be true only to some degree, the degree to which x is actually a member of A. it is most common, but not required, to express degrees of membership in fuzzy sets as well as degree of truth of the associated propositions by numbers in the closed unit interval [0,1]. The extreme values in this interval, 0 and 1, then represent, respectively, the total denial and affirmation of the membership in a given fuzzy set as well as the falsity and truth of the associated proposition.For instance, suppose we are trying to diagnose an ill patient. In simplified terms, we may be trying to determine whether this patient belongs to the set of people with, say, pneumonia, bronchitis, emphysema, or a common cold. A physical examination may provide us with helpful yet inconclusive evidence (by Ref.2 [9] on page 179). For example, we might assign a high values, say 0.75, to our best guess, bronchitis, and a lower value to the other possibilities, such as 0.45 for the set consisting of pneumonia and emphysema and 0 for a common cold. These values reflect the degrees to which the patients symptoms provide evidence for the individual diseases or set of diseases; their collection constitutes a fuzzy measure representing the uncertainty associated with several well-defined alternatives. It is important to realize that this type of uncertainty, which results from information deficiency, is fundamentally different from fuzziness, which results from the lack of sharp boundaries.Uncertainty-based information was first conceived in terms of classical set theory and in terms of probability theory. The term information theory has almost invariably been used to a theory based upon the well known measure of probabilistic uncertainty established by (Ref.2 [16]). Research on a broader conception of uncertainty-based information, liberated from the confines of classical set theory and probability theory, began in the early eighties. The name generalized information theory was coined for a theory based upon this broader conception.The ultimate goal of generalized information theory is to capture properties of uncertainty-based information formalized within any feasible mathematical framework. Although this goal has not been fully achieved as yet, substantial progress has been made in this direction. In addition to classical set theory and probability theory, uncertainty-based information is now well understood in fuzzy set theory, possibility theory and evidence theory.Three types of uncertainty are now recognized in the five theories, in which measurement of uncertainty is currently well established. These three uncertainty types are: nonspecificity (or imprecision), which is connected with sizes (cardinalities) of relevant sets of alternatives; fuzziness (or vagueness), which results from imprecise boundaries of fuzzy sets; and strife (or discord), which expresses conflicts among the various sets of alternatives.1.3.1Nonspecificity (or Imprecision):Measurement of uncertainty and associated information was first conceived in terms of classical set theory. It was shown by (Ref.2 [17]) that using a function from the class of functions (1.7)where denotes the cardinality of a finite nonempty set A, and b, c are positive constants , is the only sensible way to measure the amount of uncertainty associated with a finite set of possible alternatives. Each choice of values of the constants b and c determines the unit in which uncertainty is measured. When b=2 and c=1, which is most common choice, uncertainty is measured in bits, and obtain (1.8)One bit of uncertainty is equivalent to the total uncertainty regarding the truth or falsity of one proposition. The set function U defined by equation (1.8) be called a Hartley function. Its uniqueness as a measure of uncertainty associated with sets of alternatives can also be proven axiomatically.A natural generalization of the Hartley function from classical set theory to fuzzy set theory was proposed in the early eighties under the name U- uncertainty. For any nonempty fuzzy set A defined on a finite universal set X, the generalized Hartley function (Ref.2[17]) has the form (1.9)

where denotes the cardinality of the - cut of A and h(A) is the height of A. Observe that U(A), which measures nonspecificity of A, is a weighted average of values of the Hartley function for all distinct - cuts of the normalized counterpart of A, defined by A(x)/h(A) for all xX. Each weight is a difference between the values of of a given -cut and the immediately preceding -cut. Fuzzy sets are equal when normalized have the same nonspecificity measured by function U (Ref.2 [17]).

1.3.2.Fuzziness (or Vagueness):The second type of uncertainty that involves fuzzy sets (but not crisp sets) is fuzziness (or vagueness). In general, a measure of fuzziness is a function where P(X) denotes the set of all fuzzy subsets of X (fuzzy power set). For each fuzzy set A, this function assign a nonnegative real number f(A) that express the degree to which the boundary of A is not sharp.(Ref.2 [9]) One way is to measure fuzziness of any set A by a metric distance between its membership grade function and the membership grade function (or characteristic function) of the nearest crisp set.(Ref.2 [1]) Even when committing to this conception of measuring fuzziness, the measurement is not unique. To make it unique, we have to choose a suitable distance function.Another way of measuring fuzziness, which seems more practical as well as more general, is to view the fuzziness of a set in terms of the lack of distinction between the set and it complement. Indeed, it is precisely the lack of distinction between the sets and their complements that distinguishes fuzzy sets from crisp sets.(Ref.2 [1]) The less a set differs from its complement, the fuzzier it is. Let us restrict our discussion to this view of fuzziness, which is currently predominant in the literature.1.3.3.Strife (or discord):This type of uncertainty which is connected with conflicts among evidential claims has been far more controversial. Although it is generally agreed that a measure of this type of uncertainty must be a generalization of the well-established Shannon entropy (Ref.2[17]) from probability theory, which all collapse to the Shannon entropy within the domain of probability theory, is the right generalization. As argued in the companion paper (Ref.2 [18]), either of these measures is deficient in some trails. To overcome these deficiencies, another measure was prepared in the companion paper(Ref.2 [18]), which is called a measure of discord. This measure is expressed by a function D and defined by the formula,

The rationale for choosing this function is explained as follows. The term,

In equation (1.10) expresses the sum of conflicts of individual evidential claim m(B) for all with respect to the evidential claim m(A) focusing on a particular set A; each individual conflict is properly scaled by the degree to which the subsethood relation is violated. The function , which is employed in equation (1.10), is monotonic increasing with Con (A) and, consequently, it represents the same quantity as Con(A), but on the logarithmic scale. The use of the logarithmic scale is motivated in the same way as in the case of the Shannon entropy (Ref.2 [17]). Function D is clearly a measure of the average conflict among evidential claims within a given body of evidence. Consider, as an example, incomplete information regarding the age of a person X. Assume that the information is expressed by two evidential claims pertaining to the age of X: X is between 15 and 17 years old with degree m(A), where A = {15, 17}, and X is a teenager with degree m(B), where B = {13, 19}. Clearly, the weaker second claim does not conflict with the stronger first claim. Assume that , in this case, the situation is inverted: the claim focusing on B is not implied by the claim focusing on A and, consequently, m(B) dose conflict with m(A) to a degree proportional to number of elements in A that are not covered by B. This conflict is not captured by function Con since in this case.It follows from these observations that the total conflict of evidential claims within a body of evidence {F, m} with respect to a particular claim m(A) should be expressed by function

rather than function Con given by equation (1.11). Replacing Con (A) with CON (A), we obtain a new function, which is better justified as a measure of conflict in evidence theory than function D. This new function, which is called strife and denoted by S, is defined by the form,

In ordered possibility distributions , the form of function S (strife) in possibility theory is defined by:

Where U(r) is the measure of possibilistic nonspecificity (U-uncertainty). The maximum value of possibilistic strife, given by equation (1.14), depends on n in exactly the same way as the maximum value of possibilistic discord: it increases with n and converges to a constant, estimated as 0.892 as (Ref.2 [19]). However the possibility distributions for which the maximum of possibilistic strife are obtained (one for each value of n) are different from those for possibilistic discord. This property and the intuitive justification of function S make this function better candidate for the entropy-like measure in Dempster-shafer theory (DST) than any of the previously considered functions.1.3.4.Applications of Measures of Uncertainty:Uncertainty measures becomes well justified, they can be used in many other contexts as for managing uncertainty and the associated information. For example, they can be used for extrapolating evidence, assessing the strength of relationship between given groups of variables, assessing the influence of given input variables on given output variables, measuring the lost of information when a system is simplified, and the like. In many problem situations, the relevant measures of uncertainty are applicable only in their conditional or relative terms. The use of relevant uncertainty measures is as broad as the use of any relevant measuring instrument.(Ref.2[24]) Three basic principles of uncertainty were developed to guide the use of uncertainty measures in different situations. These principles are: a principle of minimum uncertainty, a principle of maximum uncertainty, and a principle of uncertainty invariance.The principle of minimum uncertainty is an arbitration principle. It is used for narrowing down solutions in various systems problems that involve uncertainty. It guides the selection of meaningful alternatives from possible solutions of problems in which some of the initial information is inevitably lost but in different solutions it is lost in varying degrees. The principle states that we should accept only those solutions with the least loss of information i.e. whose uncertainty is minimal. Application of the principle of minimum uncertainty is the area of conflict-resolution problems. For some development of this principle see(Ref.2 [23]). For example when we integrate several overlapping models into one larger model, the models may be locally inconsistent. It is reasonable to require that each of the models be appropriately adjusted in such a way that the overall model becomes consistent. It is obvious that some information contained in the given models is inevitably lost by these adjustments. This is not desirable. Hence, we should minimize this lost of information. That is, we should accept only those adjustments for which the total increase of uncertainty is minimal.The principle of maximum uncertainty is essential for any problem that involves applicative reasoning. Applicative reasoning is indispensable to science in a variety of ways. For example, when we want to estimate microstates from the knowledge of relevant microstates and partial information regarding the microstates, we must resort to applicative reasoning. Applicative reasoning is also common and important in our daily life where the principle of maximum uncertainty is not always adhered to. Its violation leads almost invariable to conflicts in human communication, as well expressed by (Ref.2 [26]) ..whenever you find yourself getting angry about a difference in opinion, be on your guard; you will probably find, on examination, that your belief is getting beyond what the evidence warrants. This is reasoning in which conclusions are not entailed in the given premises. The principle may be expressed by the following requirement: in any applicative inference, use all information available, but make sure that no additional information is unwittingly added. The principle of maximum uncertainty is applicable in situations in which we need to go beyond conclusions entailed by verified premises. (Ref.2 [24]) The principle states that any conclusion we make should maximize the relevant uncertainty within constraints given by the verified premises. In other words, the principle guides us to utilize all the available information but at the same time fully recognize our ignorance. This principle is useful, for example, when we need to reconstruct an overall system from the knowledge of some subsystems. The principle of maximum uncertainty is well developed and broadly utilized within classical information theory,(Ref.2 [25]) where it is called the principle of maximum entropy.(Ref.2 [9]) The last principle, the principle of uncertainty invariance, is of relatively recent origin (Ref.2 [21]). Its purpose is to guide meaningful transformations between various theories of uncertainty. The principle postulates that the amount of uncertainty should be preserved in each transformation of uncertainty from one mathematical framework to another. The principle was first studied in the context of probability-possibility transformations (Ref.2 [20]). Unfortunately, at the time, no well justified measure of uncertainty was available. Due to the unique connection between uncertainty and information, the principle of uncertainty invariance can also be conceived as a principle of information invariance or information preservation.(Ref.2 [22]).1.4.FUZZY MEASURE: In mathematics,fuzzy measure considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership functionand the classical probability measures. In the fuzzy measure theory, the conditions are precise, but the information about an element alone is insufficient to determine which special classes of measure should be used (Ref.3[1]). The central concept of fuzzy measure theory is the fuzzy measure which was introduced by Choquet in 1953(by Ref.3[1]) and independently defined by Sugeno in 1974 (by Ref.3[2]) in the context of fuzzy integrals.Consider, however, the jury members for a criminal trial who are uncertain about the guilt or innocence of the defendant. The uncertainty in this situation seems to be of a different type; the set of people who are guilty of the crime and the set of innocent people are assumed to have very distinct boundaries. The concern, therefore, is not with the degree to which the defendant is guilty, but with the degree to which the evidence proves his membership in either the crisp set of guilty people or the crisp set of innocent people. We assume that perfect evidence would point to full membership in one and only one of these sets. However, our evidence is rarely, if ever, perfect, and some uncertainty usually prevails. In order to represent this type of uncertainty, we could assign a value to each possible crisp set to which the element in question might belong. This value would indicate the degree of evidence or certainty of the elements membership in the set. Such a representation of uncertainty is known as a fuzzy measure.In classical, two-valued logic, we would have to distinguish cold from not cold by fixing a strict changeover point. We might decide that anything below 8 degrees Celsius is cold, and anything else is not cold. This can be rather arbitrary. Fuzzy logic lets us avoid having to choose an arbitrary changeover point essentially by allowing a whole spectrum of degrees of coldness. A set of temperature like hot or cold is represented by a function. Given a temperature, the function will return a number representing the degree of membership of that set. This number is called a fuzzy measures.For example:Some fuzzy measure for the set cold:Temperature in 0CFuzzy Measure

-2731 Cold

-401 Cold

00.9 Not quite cold

50.7On the cold side

100.3A bit cold

150.1Barely cold

1000Not cold

10000Not cold

The basis of the idea of coldness may be how people use the word cold perhaps. 30% of people think that 100C is cold (function value 0.3) and 90% think that 00C is cold (function value 0.9). Also it may depend on the context. In terms of the weather, cold means one thing. In terms of the temperature of the coolant in a nuclear reactor cold may means something else, so we would need to have a cold function appropriate to our context.Definition:Given a universal set X and a non empty family of subsets of X, a fuzzy measure on X, is a function that satisfies the following requirements (by Ref.2[9]):[1]Boundary requirements: [2]Monotonicity:If .[3]Continuity from below: For any increasing sequence

[4]Continuity from above: For any decreasing sequence

The boundary requirements [1] state that the element in question definitely does not belong to the empty set and definitely does belong to the universal set. The empty set does not contain any element hence it cannot contain the element of our interest, either; the universal set contains all elements under consideration in each particular context; therefore it must contain our element as well.Requirement [2] states that the evidence of the membership of an element in a set must be at least as great as the evidence that the element belongs to any subset of that set. Indeed with some degree of certainty that the element belongs to a set, then our degree of certainty that is belongs to a larger set containing the former set can be greater or equal, but it cannot be smaller. Requirements [3] and [4] are clearly applicable only to an infinite universal set. They can therefore be disregarded when the universal set is finite. Fuzzy measures are usually defined on families that satisfy appropriate properties (rings, semirings, - algebras, etc.). In some cases, consists of the full power set P(X) (Ref.2[9]).Three additional remarks regarding this definition are needed. First, functions that satisfy [1], [2], and either [3] or [4] are equally important in fuzzy measure theory as functions that satisfy all four requirements. These functions are called semicontinuous fuzzy measure; they are either continuous from below or continuous from above. Second, it is sometimes needed to generalize fuzzy measures by extending the range of function from [0, 1] to the set of nonnegative real numbers and by excluding the second boundary requirement, . These generalizations are not desirable for our purpose. Third, fuzzy measures are generalizations of probability measures or generalizations of classical measures. The generalization is obtained by replacing the additivity requirement with the weaker requirements of monotonicity and continuity or, at least, semicontinuity (Ref.3[5]). 1.4.1.Properties of fuzzy measures: For any , a fuzzy measure is1.Additive:if ;2.super additive:if;3.sub additive:if;4.super modular:if5.sub modular:if ;6.Symmetric:7.Boolean:Let A and B are any two sets then . The monotonicity of fuzzy measures that every fuzzy measures g satisfies the inequality for any three sets ,(1.15)Similarly for any two sets, the monotonicity of fuzzy measures implies that every fuzzy measure g satisfies the inequality for any three sets ,(1.16)Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integralor Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral (Ref.3[4]). In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging(OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral (Ref.3[2]).Since fuzzy measures are defined on the power set(or, more formally, on the - algebraassociated withX), even in discrete cases the number of variables can be quite high. For this reason, in the context of multi-criteria decision analysisand other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure isadditive, it will hold that

and the values of the fuzzy measure can be evaluated from the values onX. Similarly, asymmetricfuzzy measure is defined uniquely by values. Two important fuzzy measures that can be used are the Sugeno--fuzzy measure and -additive measures, introduced by Sugenoand Grabisch respectively (Ref.3[3]&[10]). Fuzzy measure theory is of interest of its three special branches: probability theory, evidence theory and possibility theory. Although our principle interest is in possibility theory and its comparison with probability theory, evidence theory will allow us to examine and compare the two theories from a broader perspective.1.4.2.Probability Theory:Probability represents a unique encoding of incomplete information. The essential task of probability theory is to provide methods for translating incomplete information into this code. The code is unique because it provides a method, which satisfies the following set of properties for any system.(Ref.3[12])1.If a problem can be solved in more than one way, all ways must lead to the same answer.2.The question posed and the way the answer is found must be totally transparent. There must be no laps of faith required to understand how the answer followed from the given informations.3.The methods of solution must not be ad-hoc. They must be general and admit to being used for any problem, not just a limited class of problems. Moreover the applications must be totally honest. For example, it is not proper for someone to present incomplete information, develop an answer and then prove that the answer is incorrect because in the light of additional information a different answer is obtained.4.The process should not introduce information that is not present in the original statement of the problem.Basic Terminologies Used in Probability Theory:(I)The Axioms of Probability: The language systems, contains a set of axioms that are used to constrain the probabilities assigned to events (Ref.3[12]). Four axioms of probability are as follows:1.All values of probabilities are between zero and one i.e. 2.Probabilities of an event that are necessarily true have a value of one, and those that are necessarily false have a value of zero i.e. P(True) = 1 and P(False) = 0.3.The probability of a disjunction is given by:4.A probability measure, Pro is required to satisfy the equation .This requirement is usually referred to as the additivity axiom of probability measures.[Ref.3(16)].An important result of these axioms is calculating the negation of a probability of an event. i.e..(II)The Joint Probability Distribution: The joint probability distribution is a function that specifies a probability of certain state of the domain given the states of each variable in the domain. Suppose that our domain consists of three random variables or atomic events [Ref.3(12)].Then an example of the joint probability distribution of this domain where would be same value depending upon the values of different probabilities of. The joint probability distribution will be one of the many distributions that can be calculated using a probabilistic reasoning system.[Ref.2(20)].(III)Conditional Probability and Bayes Rule:Suppose a rational agent begins to perceive data from its world, it stores this data as evidence. This evidence is used to calculate a posterior or conditional probability which will be more accurate than the probability of an atomic event without this evidence, known as a prior or unconditional probability.[Ref.3(12)].Example:The reliability of a particular skin test for tuberculosis (TB) is as follows:If the subject has TB then the sensitivity of the test is 0.98.If the subject does not have TB then the specificity of the test is 0.99.From a large population, in which 2 in every 10,000 people have TB, a person is selected at random and given the test, which comes back positive. What is the probability that the person actually has TB?Lets define event A as the person has TB and event B as the person tests positive for TB. It is clear that the prior probability.The conditional probability,the probability that the person will test positive for TB given that the person has TB. This was given as 0.98. The other value we need , the probability that the person will test positive for TB given that the person does not have TB. Since a person who does not have TB will test negative 99% (given) of the time, he/she will test positive 1% of the time and therefore . By Bayes Rule as:

We might find this hard to believe, that fewer than 2% of people who test positive for TB using this test actually have the disease. Ever though the sensitivity and specificity of this test are both high, the extremely low incidence of TB in the population has a tremendous effect on the tests positive predictive value, the population of people who test positive that actually have the disease. To see this, we might try answering the same question assuming that the incidence of TB in the population is 2 in 100 instead of 2 in 10,000.(IV)Conditional Independence:When the result of one atomic event, A, does not affect the result of another atomic event, B, those two atomic events are known to be independent of each other and this helps to resolve the uncertainty[Ref.3(13)]. This relationship has the following mathematical property: . Another mathematical implication is that: Independence can be extended to explain irrelevant data in conditional relationship.Disadvantages with Probabilistic Method:Probabilities must be assigned even if no information is available and assigns an equal amount of probability to all such items. Probabilities require the consideration of all available evidence, not only from the rules currently under consideration [Ref.2(20)]. Probabilistic methods always require prior probabilities which are very hard to found out apriority [Ref.3(16)]. Probability may be inappropriate where as the future is not always similar to the past. In probabilistic method independence of evidences assumption often not valid and complex statements with conditional dependencies cannot be decomposed into independent parts. In this method relationship between hypothesis and evidence is reduced to a number [Ref.3(13)]. Probability theory is an ideal tool for formalizing uncertainty in situations where class frequencies are known or where evidence is based on outcomes of a sufficiently long series of independent random experiments [Ref.3(12)].1.4.3.Evidence Theory:Dempster-Shafer theory (DST) is a mathematical theory of evidence. The seminal work on the subject is done by Shafer [Ref.3(8)], which is an expansion of previous work done by Dempster [Ref.3(6)]. In a finite discrete space, DST can be interpreted as a generalization of probability theory where probabilities are assigned to sets as opposed to mutually exclusive singletons. In traditional probability theory, evidence is associated with only one possible event. In DST, evidence can be associated with multiple possible events, i.e. sets of events. DST allows the direct representation of uncertainty. There are three important functions in DST by[Ref.3(6) &(8)]: (1) The basic probability assignment function (bpa or m), (2) The Belief function (Bel) and (3) The Plausibility function (Pl). The theory of evidence is based on two dual non-additive measures: (2) and (3).(1) Basic Probability Assignment:Basic probability assignment does not refer to probability in the classical sense. The basic probability assignment, represented by m, defines a mapping of the power set to the interval between 0 and 1, s.t. satisfying the following properties:[Ref.2(20)](a) , and

The value of p(A) pertains only to the set A and makes no additional claim about any subset of A. Any further evidence on the subset of A would be represented by another basic probability assignment. The summation of the basic probability assignment of all the subsets of the power set is 1. As such, the basic probability assignment cannot be equated with a classical probability in general [Ref.3(13)].(2) Belief Measure: Given a measurable space, a belief measure is a function satisfying the following properties:(a) ,(b) , and

Due to the inequality (1.18), belief measures are called superadditive. When X is infinite, function Bel is also required to be continuous from above. For each is defined as the degree of belief, which is based on available evidence [Ref.3(6)], that a given element of X belongs to the set Y. The inequality (1.18) implies the monotonicity requirement [2] of fuzzy measure. Let Applying now to (1.18), we get Since , we have

Let in (1.18) for n=2. Then we have,

(1.19)Inequality (1.19) is called the fundamental property of belief measures.(3) Plausibility Measure: Given a measurable space, a plausibility measure is a function satisfying the following properties:(a) ,(b) , and

Due to the inequality (1.19), plausibility measures are called subadditive. When X is infinite, function Pl is also required to be continuous from below [Ref.3(8)]. Let in (1.20) for n=2. Then we have,

(1.21)According to inequality (1.19) and (1.21) we say that each belief measure, Bel, is a plausibility measure, Pl, i.e. the relation between belief measure and plausibility measure is defined by the equation[Ref.3(7)](1.22)(1.23)1.4.4.Possibility Theory:Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Possibility theory is an uncertainty theory devoted to the handling of incomplete information. It is comparable to probability theory because it is based on set-functions. Possibility theory has enabled a typology of fuzzy rules to be laid bare, distinguishing rules whose purpose is to propagate uncertainty through reasoning steps, from rules whose main purpose is similarity-based interpolation [Ref.3(18)]. The name Theory of Possibility was coined by Zadeh [Ref.3(19)], who was inspired by a paper by Gaines and Kohout [Ref.3(20)]. In Zadeh's view, possibility distributions were meant to provide a graded semantics to natural language statements. Possibility theory was introduced to allow a reasoning to be carried out on imprecise or vague knowledge, making it possible to deal with uncertainties on this knowledge. Possibility is normally associated with some fuzziness, either in the background knowledge on which possibility is based, or in the set for which possibility is asserted [Ref.3(16)].Let S be a set of states of affairs (or descriptions thereof), or states for short. A possibility distribution is a mapping from S to a totally ordered scale L, with top 1 and bottom 0, such as the unit interval. The function represents the state of knowledge of an agent (about the actual state of affairs) distinguishing what is plausible from what is less plausible, what is the normal course of things from what is not, what is surprising from what is expected (Ref.3[15]). It represents a flexible restriction on what is the actual state with the following conventions (similar to probability, but opposite of Shackle's potential surprise scale):(1) (s) = 0 means that state s is rejected as impossible;(2) (s) = 1 means that state s is totally possible (= plausible).For example, imprecise information such as Xs height is above 170cm implies that any height h above 170 is possible any height equal to or below 170 is impossible for him. This can be represented by a possibility measure defined on the height domain whose value is 0 if and 1 if (0 = impossible and 1 = possible).when the predicate is vague like in X is tall, the possibility can be accommodate degrees, the largest the degree, the largest the possibility. For consonant body of evidence, the belief measure becomes necessity measure and plausibility measure becomes possibility measure (Ref.3[19]).Hence Becomes And Becomes And also Some other important measures on fuzzy sets and relations are defined here.1.4.2.Additive Measure:Let X, be a measurable space. A function is an - additive measure when the following properties are satisfied: 1. 2. If n = 1, 2, ... is a set of disjoint subsets of then

The second property is called -additivity, and the additive property of a measurable space requires the -additivity in a finite set of subsets . A well-known example of -additive is the probabilistic space (X, , p) where the probability p is an additive measure such that

for all subsets . Other known examples of -additive measure are the Lebesgue measures defined that are an important base of the XX century mathematics (Ref.3[9]). The Lebesgue measures generalise the concept of length of a segment, and verify that if

Other measures given by Lebesgue are the exterior Lebesgue measures and interior Lebesgue measures. A set A is Lebesgue measurable when both interior and exterior Lebesgue measures are the same (Ref.3[11]). Some examples of Lebesgue measurable sets are the compact sets, the empty set and the real numbers set R.1.4.3. - additive fuzzy measure:A discrete fuzzy measure on a setXis called -additive if its Mbius representation verifies, whenever for any , and there exists a subset F with elements such that.The -additive fuzzy measure limits the interaction between the subsets to size. This drastically reduces the number of variables needed to define the fuzzy measure, and as can be anything from 1 to, it allows for a compromise between modelling ability and simplicity (Ref.3[3]).1.4.4.Normal Measure:Let X, be a measurable space. A measure is a normal measure if there exists a minimal set and a maximal set in such that:1. 2. For example, the measures of probability on a space X, are normal measures with and . The Lebesgue measures are not necessarily normal[Ref.3(9)]. 1.4.5.Sugeno Fuzzy Measure:Let be an -algebra on a universe X. A Sugeno fuzzy measure is verifying by (Ref.3[14]):1. 2.If .3.If and then

Property 3 is called Sugenos convergence. The Sugeno measures are monotone but its main characteristic is that additivity is not needed. Several measures on finite algebras, as probability, credibility measures and plausibility measures are Sugeno measures. The possibility measure on possibility distributions introduced by Zadeh (Ref.3[7]) gives Sugeno measures. 1.4.6.Fuzzy Sugeno -Measure:Sugeno (Ref.3[2]) introduces the concept of fuzzy -measure as a normal measure that is -additive. So the fuzzy -measures are fuzzy measures. Let and let X, be a measurable space. A function is a fuzzy -measure if for all disjoint subsets A, B in , For example, if then the fuzzy -measure is an additive measure.1.5.NULL ADDITIVE FUZZY MEASURE:The concept of fuzzy measure does not require additivity , but it requires monotonicity related to the inclusion of sets. Additivity can be very effective and convenient in some applications, but can also be somewhat inadequate in many reasoning environments of the real world as in approximate reasoning, fuzzy logic, artificial intelligence, game theory, decision making, psychology, economy, data mining, etc., that require the definition of non-additive measures and large amount of open problems. For example, the efficiency of a set of workers is being measured; the efficiency of the some people doing teamwork is not the addition of the efficiency of each individual working on their own. The theory of non-additive set functions had influences on many parts of mathematics and different areas of science and technology. Recently, many authors have investigated different type of non-additive set functions, as sub-measures, k-triangular set functions, - decomposable measures, null-additive set functions and others. The range of the null-additive set functions, introduced by (Ref.4[6]). Suzuki introduced and investigated atoms of fuzzy measures (Ref.4[4]) and Pap introduced and discussed atoms of null-additive set functions (Ref.4[3]).Definition:Let X be a set and a algebra over X. A set function m, is called null-additive by (Ref.4[1]), if: It is obvious that for null-additive set function m we have whenever there exists such that Examples:1. - decomposable measure with respect to a t-conorm (Ref.4[7]), such that and whenever and is always null-additive.2.A generalization of -decomposable measure from example 1 is the -decomposable measure . Namely, let [a, b] be a closed real interval, where . The operation (pseudo-addition) is a function which is commutative, non-decreasing, associative and has a zero element, denoted by 0 which is either a or b (Ref.4[2]). A set function is a -decomposable measure if there hold and whenever and . Then m is null-additive.3.Let whenever . Then m is null-additive.4.Let and define m in the following way: Then m is not null-additive.1.5.1.Saks decomposition:Let m be a null-additive set function. A set with is called an atom provided (Ref.4[2]) that if then1.,or2.Remarks: 1.For null-additive set functions the condition 2 implies , what is analogous to the property of the usual -finite measures (for this case the opposite implication is also true).2.For -finite measure m, each of its atom has always finite measure.1.5.2.Darboux property:A set function m is called autocontinuous from above (below) if for every and every there exists such that(resp. whenever holds (Ref.4[5])Remark:If m is a fuzzy measure, then we can omit in the preceding definition the supposition (resp. ).The set function m is a null-additive non fuzzy measure which is autocontinuous from above and from below. The autocontinuity from above (below) in the sense of Wang (Ref.4[5]), if whenever , is called W-autocontinuity. If m is a finite fuzzy measure, then m is autocontinuous from above (below) iff m is W-autocontinuous from above (below). If m is a finite fuzzy measure, then m is autocontinuous from below iff it is autocontinuous from below (Ref.4[5]).Each set function which is autocontinuous from above is null-additive but there exists null-additive fuzzy measure which is not autocontinuous from above (Example 1). If the set X is countable, then null-additivity of a finite fuzzy measure is equivalent to the autocontinuity (Ref.4[6]).26