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Unit 5 Introduction to Neuro-Fuzzy : In the field of artificial intelligence, neuro-fuzzy refers to combinations of artificial neural networks and fuzzy logic. Neuro-fuzzy hybridization results in a hybrid intelligent system that synergizes these two techniques by combining the human-like reasoning style of fuzzy systems with the learning and connectionist structure of neural networks. Neuro-fuzzy hybridization is widely termed as Fuzzy Neural Network (FNN) or Neuro-Fuzzy System (NFS) in the literature. Neuro-fuzzy system incorporates the human-like reasoning style of fuzzy systems through the use of fuzzy sets  a linguistic model consisting of a set of IF-THEN fuzzy rules. The main strength of neuro-fuzzy systems is that they are universal approximators with the ability to solicit interpretable IF-THEN rules. The strength of neuro-fuzzy systems involves two contradictory requirements in fuzzy modeling: interpretability versus accuracy. The neuro-fuzzy in fuzzy modeling research field is divided into two areas: linguistic fuzzy modeling that is focused on interpretability fuzzy modeling that is focused on accuracy Fuzzy concept A fuzzy concept is a concept of which the content, value, or boundaries of application can vary according to context or conditions, instead of being fixed once and for all. In logic, fuzzy concepts are often regarded as concepts which in their application, or formally speaking, are neither completely true or completely false, or which are partly true and partly false; 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 can be usefully described with imprecise categories (such as "high", "medium" or "low"). Application of fuzzy concepts Fuzzy concepts often play a role in the creative process of forming new concepts to understand something. In the most primitive sense, this can be observed in infants who, through practical experience, learn to identify, distinguish and generalise the correct application of a concept, and relate it to other concepts. However, fuzzy concepts may also occur in scientific, journalistic, programming and philosophical activity, when a thinker is in the process of clarifying and defining a newly emerging concept which is based on distinctions which, for one reason or another, cannot (yet) be more exactly specified or validated. Fuzzy concepts are often used to denote complex phenomena, or to describe something which is developing and changing, which might involve shedding some old meanings and acquiring new ones. •	In translation work, fuzzy concepts are analyzed for the purpose of good translation. A concept in one language may not have quite the same meaning or significance in another language, or it may not be feasible to translate it literally, or at all. Some languages have concepts which do not exist in another language, raising the problem of how one would most easily render their meaning. •	In information services fuzzy concepts are frequently encountered because a customer or client asks a question about something which could be interpreted in many different ways, or, a document is transmitted of a type or meaning which cannot be easily allocated to a known type or category, or to a known procedure. It might take considerable inquiry to "place" the information, or establish in what framework it should be understood. •	In the legal system, it is essential that rules are interpreted and applied in a standard way, so that the same cases and the same circumstances are treated equally. Otherwise one would be accused of arbitrariness, which would not serve the interests of justice. Consequently, lawmakers aim to devise definitions and categories which are sufficiently precise that they are not open to different interpretations. For this purpose, it is critically important to remove fuzziness, and differences of interpretation are typically resolved through a court ruling based on evidence. Alternatively, some other procedure is devised which permits the correct distinction to be discovered and made. •	In statistical research, it is an aim to measure the magnitudes of phenomena. For this purpose, phenomena have to be grouped and categorized so that distinct and discrete counting units can be defined. It must be possible to allocate all observations to mutually exclusive categories so that they are properly quantifiable. Survey observations do not spontaneously transform themselves into countable data; they have to be identified, categorized and classified in such a way that they are not counted twice or more. Again, for this purpose it is a requirement that the concepts used are exactly defined, and not fuzzy. There could be a margin of error, but the amount of error must be kept within tolerable limits, and preferably its magnitude should be known.

Fuzzy_set_theory extension of the classical notion of set. In classical set theory, the membership of elements in a set is assess in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the regular review of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets simplify classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.[4] In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics. A fuzzy set is a pair    where  is a set    and For each   the value     is called the grade of membership of    in          For a finite set       the fuzzy set     is often denoted by Let   Then  is called not included in the fuzzy set      if   	is called fully included if      	is called a fuzzy member if      	 The set is called the support of    and the set    is called its kernel. The function  is called the membership function of the fuzzy set membership function must be convex, normalized, at least segmentally continuous A fuzzy number is a convex, normalized fuzzy set      whose membership function is at least segmentally continuous and has the functional value    at precisely one element. object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Fuzzy set operations Standard fuzzy set operations Standard complement cA(x) = 1 − A(x) Standard intersection (A ∩ B)(x) = min [A(x), B(x)] Standard union (A ∪ B)(x) = max [A(x), B(x)] Fuzzy complements A(x) is defined as the degree to which x belongs to A. Let cA denote a fuzzy complement of A of type c. Then cA(x) is the degree to which x belongs to cA, and the degree to which x does not belong to A. (A(x) is therefore the degree to which x does not belong to cA.) Let a complement cA be defined by a function c : [0,1] → [0,1] c(A(x)) = cA(x) Axioms for fuzzy complements Axiom c1. Boundary condition c(0) = 1 and c(1) = 0 Axiom c2. Monotonicity For all a, b ∈ [0, 1], if a < b, then c(a) ≥ c(b) Axiom c3. Continuity c is continuous function. Axiom c4. Involutions c is an involution, which means that c(c(a)) = a for each a ∈ [0,1] Fuzzy intersections The intersection of two fuzzy sets A and B is specified in general by a binary operation on the unit interval, a function of the form i:[0,1]×[0,1] → [0,1]. (A ∩ B)(x) = i[A(x), B(x)] for all x. Axioms for fuzzy intersection Axiom i1. Boundary condition i(a, 1) = a Axiom i2. Monotonicity b ≤ d implies i(a, b) ≤ i(a, d) Axiom i3. Commutativity i(a, b) = i(b, a) Axiom i4. Associativity i(a, i(b, d)) = i(i(a, b), d) Axiom i5. Continuity i is a continuous function Axiom i6. Subidempotency i(a, a) ≤ a Fuzzy unions The union of two fuzzy sets A and B is specified in general by a binary operation on the unit interval function of the form u:[0,1]×[0,1] → [0,1]. (A ∪ B)(x) = u[A(x), B(x)] for all x Axioms for fuzzy union Axiom u1. Boundary condition u(a, 0) =u(0 ,a) = a Axiom u2. Monotonicity b ≤ d implies u(a, b) ≤ u(a, d) Axiom u3. Commutativity u(a, b) = u(b, a) Axiom u4. Associativity u(a, u(b, d)) = u(u(a, b), d) Axiom u5. Continuity u is a continuous function Axiom u6. Superidempotency u(a, a) ≥ a Axiom u7. Strict monotonicity a1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2) Fuzzy rule A fuzzy rule is defined as a conditional statement in the form: IF x is A THEN y is B where x and y are linguistic variables; A and B are linguistic values determined by fuzzy sets on the universe of discourse X and Y, respectively. Comparison between Boolean and fuzzy logic rules A classical IF-THEN statement uses binary logic, for instance: IF man_height is > 180cm THEN man_weight is > 50kg Comparison between computational verb and fuzzy logic rules Computational verb rules(verb rules, for short) are expressed in computational verb logic. The difference between verb and fuzzy rules is that the former using verbs other than BE in the statement while the latter using verb BE only. For example, the two fuzzy rules in above have the following corresponding computational verb counterparts:

•	IF man_height becomes tall THEN man_weight become heavy; •	•	IF man_height increase to tall THEN man_weight probably grow to heavy; •	•	IF man_height stays tall THEN man_weight could remain heavy; Fuzzy logic. Fuzzy logic is a form of many-valued logic or probabilistic logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic, true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false.[1] The reasoning in fuzzy logic is similar to human reasoning. It allows for approximate values and inferences as well as incomplete or ambiguous data (fuzzy data) as opposed to only relying on crisp data (binary yes/no choices). Fuzzy logic is able to process incomplete data and provide approximate solutions to problems other methods find difficult to solve Fuzzy logic and probabilistic logic are mathematically similar – both have truth values ranging between 0 and 1 – but conceptually distinct, due to different interpretations— Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full. The meaning of each of them can be represented by a certain fuzzy set. Then one might define the glass as being 0.7 empty and 0.3 full. Note that the concept of emptiness would be subjective and thus would depend on the observer or designer. Another designer might equally well design a set membership function where the glass would be considered full for all values down to 50 ml. Applying truth values A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. Fuzzy logic temperature In this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three "truth values"—one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as "not hot". The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold". Linguistic variables While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric linguistic variables are often used to facilitate the expression of rules and facts.[4] A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modified via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions. Example Fuzzy set theory defines fuzzy operators on fuzzy sets. The problem in applying this is that the appropriate fuzzy operator may not be known. For this reason, fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices. Rules are usually expressed in the form: IF variable IS property THEN action For example, a simple temperature regulator that uses a fan might look like this: IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain level IF temperature IS hot THEN speed up fan There is no "ELSE" – all of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different degrees. The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators. So for the fuzzy variables x and y: NOT x = (1 - truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y))