Magda Komorníková

Aggregation operators: properties, classes and construction methods

Aggregation (fusion) of several input values into a single output value is an indispensable tool not only of mathematics or physics, but of majority of engineering, economical, social and other sciences. The problems of aggregation are very broad and heterogeneous, in general. Therefore we restrict ourselves in this contribution to the specific topic of the aggregation of finite number of real inputs only. Closely related topics of aggregating infinitely many real inputs [23,109,64,52,43,42,44,99], of aggregating inputs from some ordinal scales [41,50], of aggregating complex inputs (such as probability distributions [107,114], fuzzy sets [143]), etc., are treated, among others, in the quoted papers, and we will not deal with them. In this spirit, if the number of input values is fixed, say n, an aggregation operator is a real function of n variables. This is still a too general topic. Therefore we restrict our considerations regarding inputs as well as outputs to some fixed interval (scale) I = [a, b] ⊑ [-∞, ∞]. It is a matter of rescaling to fix I = [0,1].

Triangular norm-based iterative compensatory operators

Abstract Aggregation operators based on a fixed t-norm and on suitable transformations of processed data are introduced. A special attention is paid to the class of iterative compensatory operators containing also the classical arithmetic, geometric and harmonic means. Several properties of introduced operators are studied, e.g., the symmetry, the associativity, the idempotency, the anihilation, etc. Several examples are given, including continuous idempotent bisymmetric iterative compensatory operators with zero anihilator.

Aggregation functions on bounded partially ordered sets and their classification

Several classes of aggregation functions on bounded partially ordered sets are introduced and discussed. Considering their conjunctive and/or disjunctive attitude, some classification approaches are proposed, including several types of averaging aggregation functions. Aggregation functions on special cases of lattices and chains are discussed, too.

A Review of Aggregation Functions

Several local and global properties of (extended) aggregation functions are discussed and their relationships are examined. Some special classes of averaging, conjunctive and disjunctive aggregation functions are reviewed. A special attention is paid to the weighted aggregation functions, including some construction methods

Triangular Norm-Based Aggregation of Evidence Under Fuzziness

Several types of aggregation operators on fuzzy sets are discussed, all of them based on triangular norms and triangular conorms. The main stress is given to the operators allowing to compensate a low input value by another high input value, i. e., to the compensatory operators. We discuss only the pointwise aggregation and the crisp output can be obtained afterwards by any defuzzification procedure. The properties of exponential compensatory operators, convex-linear compensatory operators, convex non-linear compensatory operators, uni-norms and generated aggregative operators are investigated.

Aggregation Functions on Bounded Posets

A general concept of aggregation on bounded posets is introduced and discussed. Up to general results, several particular results due to E.E. Kerre and his research group are recalled in the light of our approach. A special attention is paid to triangular norms on bounded posets.

Perturbation of bivariate copulas

New types of constructions of bivariate copulas are introduced, discussed and exemplified. Based on a given copula C, we look for its perturbation into another copula C H , possibly close to C. A special stress is put on the perturbation of the basic copulas M, W, ?. Our results generalize several methods known from the literature, such as the Farlie-Gumbel-Morgenstern copula family, for example. An illustrative example when fitting copulas to real data is also added.

On Some Construction Methods for Bivariate Copulas

We propose a rather general construction method for bivariate copulas, generalizing some construction methods known from the literature. In some special cases, the constraints ensuring the output of the proposed method to be a copula are given. Our approach opens several new problems in copula theory.

Refining Aggregation Operator-Based Orderings in Multifactorial Evaluation—Part I: Continuous Scales

Aggregation operators are often needed when building preference relations in multicriteria decision making problems. Most existing approaches have limitations due to incomparability between decisions or ties due to the use of some aggregation operations that produce a ranking. The natural way of overcoming the lack of discrimination power is to refine the obtained ranking. We bring an overview of methods that enable aggregation-based rankings to be refined, generalizing concepts like discrimin (max), leximin (max), and Lorentz orderings that refine such aggregation operations like the minimum (the maximum) and the sum.

Analysis of Nitrate Concentrations Using Nonlinear Time Series Models

Analysis of Nitrate Concentrations Using Nonlinear Time Series Models This study examines two long-term time series of nitrate-nitrogen concentrations from the River Ouse and Stour situated in the Eastern England. The time series of monthly averages were decomposed into trend, seasonal and cyclical components and residuals to create a simple additive model. Residuals were then modelled by linear time series models represented by models of the ARMA (autoregressive moving average) class and nonlinear time series models with multiple regimes represented by SETAR (self-exciting threshold autoregressive) and MSW (Markov switching) models. The analysis showed that, based on the minimal value of residual sum of squares (RSS) of one-step ahead forecast in both datasets, SETAR and MSW models described the time series better than models ARMA. However, the relative improvement of SETAR models against ARMA models was low ranging between 1% and 4% with the exception of the three-regime model for the River Stour where the improvement was 48.9%. In comparison, the relative improvement of MSW models was between 44.6% and 52.5 for two-regime and from 60.4% to 75% for three-regime models. However, the visual assessment of models plotted against original datasets showed that despite a high value of RSS, some ARMA models could describe the analyzed time series better than AR (autoregressive), MA (moving average) and SETAR models with lower values of RSS. In both datasets MSW models provided a very good visual fit describing most of the extreme values. The results of this work could be used as a base for construction of other time series models used to describe or predict nitrate-nitrogen concentrations. Analýza Koncentrácií Dusičnanov Pomocou Nelineárnych Modelov Časových Radov. Štúdia sa zaoberá analýzou dlhých časových radov koncentrácií dusičnanového dusíka v rieke Ouse a Stour vo Východnom Anglicku. Časové rady priemerných mesačných koncentrácií dusičnanov boli rozložené na trendovú, sezónnu a cyklickú zložku a reziduá pripočítané k sebe a tvoriace jednoduchý aditívny model. Reziduá boli ďalej modelované zložitejŠími lineárnymi modelmi reprezentovanými modelmi triedy ARMA a nelineárnymi viacrežimovými modelmi SETAR a MSW. Výsledky analýzy ukázali, že na základe minimálnej hodnoty sumy Štvorcov reziduí (SSR) jednokrokovej predpovede, v oboch prípadoch SETAR aj MSW modely opísali časové rady lepŠie ako modely triedy ARMA. Vo väčŠine prípadov relatívne zlepŠenie modelov SETAR oproti jednoduchým AR(1) modelom bolo malé v rozmedzí od 1 do 4 % s výnimkou trojrežimového modelu pre rieku Stour, kde to bolo až 48,9 %. Naopak, relatívne zlepŠenie modelov MSW oproti AR(1) modelom bolo v rozmedzí 44,6 až 52,5 % pre dvojrežimové a 60,4 až 75 % pre trojrežimové modely. Vizuálne posúdenie jednotlivých modelov vŠak ukázalo, že napriek vysokým hodnotám SSR, niektoré ARMA modely dokázali lepŠie opísať dané časové rady ako modely AR, MA a SETAR s nižŠími hodnotami SSR. V oboch prípadoch MSW modely dokázali dostatočne dobre opísať aj extrémne hodnoty oboch časových radov. Výsledky práce môžu byť použité pri tvorbe iných opisných alebo predpovedných modelov koncentrácie dusičnanového dusíka vo vodách.