In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of L-fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (\textbf{Agop}) and the category of partially ordered groupoids with universal bounds (\textbf{Pogpu}). Moreover, the subcategories of \textbf{Agop} consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of \textbf{Pogpu} formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered.
A synthesis of recent development of regime-switching models based on aggregation operators is presented. It comprises procedures for model specification dans identification, parameter estimation and model adequacy testing. Constructions of models for real life data from hydrology and finance are presented.
The weighted average is a well-known aggregation operator that is widely applied in various mathematical models. It possesses some important properties defined for aggregation operators, like monotonicity, continuity, idempotency, etc., that play an important role in practical applications. In the paper, we reveal whether and in which way such properties can be observed also for the fuzzy weighted average operator where the weights as well as the weighted values are expressed by noninteractive fuzzy numbers. The usefulness of the obtained results is discussed and illustrated by several numerical examples.
Recently, the utilization of invariant aggregation operators, i.e., aggregation operators not depending on a given scale of measurement was found as a very current theme. One type of invariantness of aggregation operators is the homogeneity what means that an aggregation operator is invariant with respect to multiplication by a constant. We present here a complete characterization of homogeneous aggregation operators. We discuss a relationship between homogeneity, kernel property and shift-invariance of aggregation operators. Several examples are included.
An overview of multivariate modelling based on logistic and exponential smooth transition models with transition variable generated by aggregation operators and orders of auto and exogenous regression selected by information criterion separately for each regime is given. Model specification procedure is demonstrated on trivariate exchange rates time series. The application results show satisfactory improvement in fit when particular aggregation operators are used. Source code in the form of Mathematica package is provided.
We characterize some bivariate semicopulas and, among them, the semicopulas satisfying a Lipschitz condition. In particular, the characterization of harmonic semicopulas allows us to introduce a new concept of depedence between two random variables. The notion of multivariate semicopula is given and two applications in the theory of fuzzy measures and stochastic processes are given.
Generalized aggregation operators are the tool for aggregation of fuzzy sets. The apparatus was introduced by Takači in \cite{4}. T-extension is a construction method of a generalized aggregation operator and we study it in the paper. We observe the behavior of a T-extension with respect to different order relations and we investigate properties of the construction.