Fuzzy approach and uncertainty processing

Field characteristic

Copulas and Risk Analysis

We have studied limit properties of transformations of copulas. Limit copulas are (if they exist) closely related to maximum attractors copulas and thus their application in risk analysis (e.g., in finance, hydrology, etc.) is expected. Observe that copulas are Schur concave functions, what is not the case of triangular norms. Thus our interest was turned also to characterization of triangular norms which are Schur concave. Also some other relaxed properties of triangular norms, especially of their sections, have been studied. Application of attained results is expected in several optimization tasks dealing with triangular norms. Another group of our results concerns the algebraic structure of triangular norms. Especially, we have introduced and studied the Archimedean components of triangular norms, construction of triangular norms with prescribed Archimedean components, etc. An important role of triangular sub-norms in this setting was stressed. Comparison meaningful functions are an indispensable tool in measurement evaluation and subsequent decision making. Till now, only continuous comparison meaningful functions have been characterized. We have succeeded to characterize all comparison meaningful functions. In this characterization, a key role is played by the lattice polynomials. Among other results, a mention of the building preference proposal structures utilizing the tools of fuzzy set and fuzzy logic theories was obtained.

Vagueness and Uncertainty

Parallel run with the continuation of the investigation of fuzzy coalitional games, continuing from the previous periods, a new topic was open. The formal presentation of geographical data is realized via specific objects similar to $n$-dimensional vectors, $n>2$, where the first two components are real numbers (geographical coordinates) and the remaining components reflect various phenomena connected with the coordinated location. They are mathematically represented by real or integer numbers, other discrete sets, or also by logical variables. Each of the above components can be vague, what means that it can be processed by means of fuzzy set theoretical tools. The vagueness is usually reduced in the terrestrial geography, but it becomes significant in the geographical representation of remote or on the spot realized sensing of other planets or moons. The research of this topic is in its opening period. It can use the previous results derived for vector fuzzy quantities, multi-criterion fuzzy ordering, and fuzzy logical operations. The aim is to derive synthetical methods for parallel processing of all relevant components respecting their vagueness and uncertainty.

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Selected publications

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