Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /

Conjuction and Disjunction in Fuzzy Logic

Tato práce se věnuje triangulárním normám a jejich zobecněním - uninormám, zejména jejich transformacím pomocí jednotkové funkce, a také modelování fuzzy logické konjunkce a disjunkce na základě empirických dat. Byly stanoveny podmínky, při nichž transformací určitých uninorem vznikne opět uninorma. Taktéž byly určeny podmínky invariantnosti transformace pro třídu určitých uninorem. Transformace uninorem rozšiřuje třídu možných spojek a jejich studium nám ulehčilo hledání podmínek při modelování. Byl také nalezen a analyzován algoritmus pro modelování fuzzy logických spojek. Na základě programu implementujícího tento algoritmus byl vyhodnocen experiment, který měl za cíl modelovat fuzzy konjunkci užívanou lidmi.This thesis deals with triangular norms and their generalizations - uninorms, in particular with their transformation using a unit function and also with modeling of fuzzy conjunction and disjunction based on empirical data. The conditions under which transformations of certain uninorms give again uninorms are established. Conditions of invariance transformations for certain class of uninorms are found. Transformations of uninorms extend class of connectives. The study of these transformations faciliates the search of modeling conditions. An algorithm for modeling of fuzzy connectors is also proposed and analyzed. An experiment based on implementation of the algorithm is evaluated. This experiment aims at modeling of fuzzy conjunction for human use.