On the individual ergodic theorem in -posets of fuzzy sets
Calculus for observables in a space of functions from an abstract set to the unit interval is developed and then the individual ergodic theorem is proved.
On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application
In this paper, we use a new method to obtain the necessary and sufficient condition guaranteeing the validity of the Minkowski-Hölder type inequality for the generalized upper Sugeno integral in the case of functions belonging to a wider class than the comonotone functions. As a by-product, we show that the Minkowski type inequality for seminormed fuzzy integral presented by Daraby and Ghadimi [11] is not true. Next, we study the Minkowski-Hölder inequality for the lower Sugeno integral and the...
-symmetric bi-capacities
Bi-capacities have been recently introduced as a natural generalization of capacities (or fuzzy measures) when the underlying scale is bipolar. They allow to build more flexible models in decision making, although their complexity is of order , instead of for fuzzy measures. In order to reduce the complexity, the paper proposes the notion of -symmetric bi- capacities, in the same spirit as for -symmetric fuzzy measures. The main idea is to partition the set of criteria (or states of nature,...
Possibilistic alternatives of elementary notions and relations of the theory of belief functions
The elementary notions and relations of the so called Dempster–Shafer theory, introducing belief functions as the basic numerical characteristic of uncertainty, are modified to the case when probabilistic measures and basic probability assignments are substituted by possibilistic measures and basic possibilistic assignments. It is shown that there exists a high degree of formal similarity between the probabilistic and the possibilistic approaches including the role of the possibilistic Dempster...
Probabilistic propositional calculus with doubled nonstandard semantics
The classical propositional language is evaluated in such a way that truthvalues are subsets of the set of all positive integers. Such an evaluation is projected in two different ways into the unit interval of real numbers so that two real-valued evaluations are obtained. The set of tautologies is proved to be identical, in all the three cases, with the set of classical propositional tautologies, but the induced evaluations meet some natural properties of probability measures with respect to nonstandard...
Program for generating fuzzy logical operations and its use in mathematical proofs
Fuzzy logic is one of the tools for management of uncertainty; it works with more than two values, usually with a continuous scale, the real interval . Implementation restrictions in applications force us to use in fact a finite scale (finite chain) of truth degrees. In this paper, we study logical operations on finite chains, in particular conjunctions. We describe a computer program generating all finitely-valued fuzzy conjunctions (-norms). It allows also to select these -norms according to...
Pseudo-Delta Functions and Sequences in the Optimization Theory
Radon-Nikodym derivatives and conditioning in fuzzy measure theory.
In the last twenty years many papers have appeared dealing with fuzzy theory. In particular, fuzzy integration theory had its origin in the well-known Thesis of Sugeno [7]. More recently, some authors faced this topic by means of some binary operations (see for instance [3], [8] and references): a fuzzy measure must be additive with respect to one of them, an the integral is to define in a way, which is very similar to the construction of the Lebesgue integral. On the contrary, we are interested...
Remark on an integral of M. Matloka
-measures, -measures and distinguished classes of fuzzy measures
-measures are special fuzzy measures decomposable with respect to some fixed t-conorm . We investigate the relationship of -measures with some distinguished properties of fuzzy measures, such as subadditivity, submodularity, belief, etc. We show, for example, that each -measure is a plausibility measure, and that each -measure is submodular whenever is 1-Lipschitz.
Selection and correction of weighted rules based on Łukasiewicz's fuzzy logic with evaluated syntax
The core of the expert knowledge is typically represented by a set of rules (implications) assigned with weights specifying their (un)certainties. In the paper, a method for hierarchical selection and correction of expert's weighted rules is described particularly in the case when Łukasiewicz's fuzzy logic with evaluated syntax for dealing with weights is used.
Software cost estimation with fuzzy inputs: Fuzzy modelling and aggregation of cost drivers
Parametric software cost estimation models are well-known and widely used estimation tools, and several fuzzy extensions have been proposed to introduce a explicit handling of imprecision and uncertainty as part of them. Nonetheless, such extensions do not consider two basic facts that affect the inputs of software cost parametric models: cost drivers are often expressed through vague linguistic categories, and in many cases cost drivers are better expressed in terms of aggregations of second-level...
Some remarks about fuzzy integration.
Symmetry vs. asymmetry.
The cancellation law for pseudo-convolution
Cancellation law for pseudo-convolutions based on triangular norms is discussed. In more details, the cases of extremal t-norms and , of continuous Archimedean t-norms, and of general continuous t-norms are investigated. Several examples are included.
The Choquet integral as Lebesgue integral and related inequalities
The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions...
The entropy on -quantum spaces
The Jordan decomposition of the null-additive signed fuzzy measures.
The Lebesgue decomposition theorem for generalized measures.
The stability of parameter estimation of fuzzy variables
Recently, the parameter estimations for normal fuzzy variables in the Nahmias’ sense was studied by Cai [4]. These estimates were also studied for general -related, but not necessarily normal fuzzy variables by Hong [10] In this paper, we report on some properties of estimators that would appear to be desirable, including unbiasedness. We also consider asymptotic or “large-sample” properties of a particular type of estimator.