By Janusz T. Starczewski
This booklet generalizes fuzzy good judgment platforms for various different types of uncertainty, together with - semantic ambiguity because of constrained conception or lack of understanding approximately certain club services - loss of attributes or granularity bobbing up from discretization of genuine information - vague description of club features - vagueness perceived as fuzzification of conditional attributes. as a result, the club uncertainty will be modeled by way of combining equipment of traditional and type-2 fuzzy common sense, tough set idea and chance idea. specifically, this ebook presents a few formulae for imposing the operation prolonged on fuzzy-valued fuzzy units and provides a few uncomplicated constructions of generalized doubtful fuzzy good judgment platforms, in addition to introduces a number of of tips on how to generate fuzzy club uncertainty. it truly is fascinating as a reference publication for under-graduates in larger schooling, grasp and physician graduates within the classes of desktop technological know-how, computational intelligence, or fuzzy keep an eye on and category, and is mainly devoted to researchers and practitioners in undefined.
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Additional info for Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty
The extension of the Lukasiewicz tnorm partially leads to unexpected results. 2. Consider two fuzzy truth intervals with triangular membership F L mF +ΔF R −u and g (v) = , functions deﬁned as, f (u) = min u−mΔFF+Δ ΔF R L min v−mG +ΔGL mG +ΔGR −v , ΔGL ΔGR . 36) w (μ) = max (0, − (ΔF R +ΔGR ) μ + mF + mG + ΔF R +ΔGR − 1) . 37) Consequently, the extended Lukasiewicz t-norm based on the minimum is characterized by the two following cases presented in Fig. 5. 5 Fig. 38) otherwise μT˜L min (F,G) (w) = 1 w=0 mF +mG −1+ΔF R +ΔGR −w ΔF R +ΔGR elsewhere.
Rα (A) (x) ≥ β ⇐⇒ ∀y : (x, y) ≥ Rα and μA (y) ≥ β. Note that the original deﬁnition of a fuzzy-rough set is valid for single fuzzy relation R. Extending this, many fuzzy equivalence relations Ri forms a fuzzy partition Φ on X, which is put up by fuzzy sets Fi . , inf max μFi (x) > 0 . 62) Therefore, a fuzzy-rough set may also be interpreted as a pair Φ (A) , Φ (A) of the lower and upper approximations of A by Φ which may be derived in the spirit of Nakamura’s deﬁnition. Now, fuzzy partition sets Fi can be decomposed into α-cuts allowing the fuzzy-rough set to be an α-composition of rough-fuzzy sets.
Semantic ambiguity may also apply to gradual properties. This type of uncertainty can be handled by type-2 fuzzy sets, and consequently, their use should follow from limited perception or lack of knowledge about the exact membership function. Fuzzy set of this type are characterized by ill-known memberships in the truth interval. Therefore the main source for type-2 fuzzy sets is an incomplete knowledge of the membership description. The particular kind of these type-2 fuzzy sets — an interval-valued fuzzy set is characterized by memberships of ill-known boundaries on the truth scale.
Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty by Janusz T. Starczewski