Fitting Generated Aggregation Operators to Empirical Data

Abstract

This paper treats the problem of fitting general aggregation operators with unfixed number of arguments to empirical data. We discuss methods applicable to associative operators (t-norms, t-conorms, uninorms and nullnorms), means and Choquet integral based operators with respect to a universal fuzzy measure. Special attention is paid to k-order additive symmetric fuzzy measures.

Learning Weights in the Generalized OWA Operators

“…In the same fashion as in (Grabisch et al (1995), Beliakov et al (2004a, b)), we can add further conditions on the fuzzy measure, such as k-additivity (Grabisch (1997)), sub-or super-additivity, substitutivity of certain variables, and so on, which all translate into linear restrictions on the values of v. Furthermore, following (Beliakov et al (2004b)) we can study symmetric k-additive generated fuzzy measures (whose coefficients are defined by a generating function similar to BUM), after linearizing the data.…”

“…. ;-variate OWA operators, as the vector of weights of any dimension can be generated from f. This is explored in (Beliakov et al (2004b)), where such families (called generalized aggregation operators (Calvo et al (2002)) are identified.…”

“…In the context of learning aggregation operators from data, identification of Choquet aggregation operator is equivalent to identification of the fuzzy measure vðTÞ, described by 2 n coefficients. This problem was addressed in (Grabisch et al (1995), Sicilia et al (2003), Beliakov et al (2004b)). …”

“…We provide two formulations of the problem of learning, one which is reduced to linear programming, and the second one which is reduced to a quadratic programming problem. We use similar techniques as those of (Beliakov (2002(Beliakov ( , 2003b, Beliakov et al (2004b)), designed for ordinary OWA operators and Choquet integrals.…”

Construction of Aggregation Operators With Noble Reinforcement

“…Such tuples (and their subsets) correspond to some easily understood prototypical situations, which characterize the desired aggregation operator. Interpolatory techniques are very flexible and are used in various contexts, for instance, fitting parameters of a t-conorm [5], fitting weights of ordered weighed averaging (OWA) operators [8], [9], fitting coefficients of a fuzzy measure [10]- [12], fitting generator functions [13], [14], and fitting general aggregation operators with desired properties to data [15].…”