Pointwise Construction of Lipschitz Aggregation Operators With Specific Properties

Abstract

This paper describes an approach to pointwise construction of general aggregation operators, based on monotone Lipschitz approximation. The aggregation operators are constructed from a set of desired values at certain points, or from empirically collected data. It establishes tight upper and lower bounds on Lipschitz aggregation operators with a number of different properties, as well as the optimal aggregation operator, consistent with the given values. We consider conjunctive, disjunctive and idempotent n-ary aggregation operators; p-stable aggregation operators; various choices of the neutral element and annihilator; diagonal, opposite diagonal and marginal sections; bipolar and double aggregation operators. In all cases we provide either explicit formulas or deterministic numerical procedures to determine the bounds. The findings of this paper are useful for construction of aggregation operators with specified properties, especially using interpolation schemata.

Absorbent tuples of aggregation operators

“…In this section we construct the largest and the smallest Lipschitz aggregation operators with the desired null set, and will also identify the optimal operator. The method of construction of Lipschitz aggregation operators with the desired values at a given subset is based on the results on optimal interpolation from [4,6,5,7]. We denote by Lip(M) the set of functions with the Lipschitz constant no greater than M, and denote by Mon the set of functions monotone increasing in each argument.…”

“…Suppose that we have a set of desired values of the aggregation operator f n , D = {(z, h(z)), z ∈ , h(z) ∈ I, f n (z) = h(z)}, and the Lipschitz condition (7). The data are consistent with the Lipschitz condition and monotonicity h ∈ Lip(M) ∩ Mon.…”

Construction of Aggregation Operators With Noble Reinforcement

“…(19) where is any disjunctive aggregation operator, applied only to the components of with the indexes in .…”

Piecewise linear aggregation functions based on triangulation

“…Basically, this amounts to a parameter estimation problem. On the other hand, new construction methods are being proposed that interpolate a given data set, preferably in a simple manner [6,7]. Characteristic of such approaches is the use of splines [2,3].…”

“…doi:10.1016/j.ins.2010. 10.002 A most interesting approach is that of Beliakov et al [6]. Given a data set, they pointwisely construct an aggregation function which minimizes the maximum error w.r.t.…”