On Aggregation in a Qualitative Setting Lluís Godo1, Vicenç Torra2 1 Institut d'Investigació en Intel.ligència Artificial (IIIA), CSIC
Campus UAB s/n, 08193 Bellaterra (Spain). e-mail:
[email protected] 2 Departament Enginyeria Informàtica (ETSE), Universitat Rovira i Virgili,
Carretera de Salou s/n, 43006 Tarragona (Spain). e-mail:
[email protected]
Abstract In many applications values to be aggregated are qualitative. In this case, if one wants to compute an average value in a pure qualitative setting, one is basically restricted to weighted versions of max - min combinations. In this paper we introduduce a qualitative counterpart of the weighted mean operator without having to necessarily use some kind of numerical interpretation of the values.
Keywords: aggregation operators, qualitative aggregation, weighted mean
1. Introduction. Aggregation operators have been traditionally defined in the numerical setting (notorius exceptions are [3, 6]). Most notorious numerical aggregation operations are those based on the weighted mean (WM) [1] and those based on the ordered weighted average (OWA) [9, 10]. However, in many applications, values to be aggregated are qualitative, or ordinal, rather than quantitative. In such a cases, the usual technique is to map the qualitative values and weights into a numerical scale, and then perform the aggregation of those numerical values, and optionally get back to a value in an ordinal scale, if needed. If one wants to strictly remain in a pure qualitative setting, where the ordering of the values and weights is what only matters, then one is basically led to weighted versions of max or min combinations (Sugeno integrals), provided that the domain of values U and the domain of weights W are commensurate scales. For instance, if we take U = W, the weighted-max and weighted-min of a set of values {x1, ..., xn } according to a set of weights {w1, ..., wn } are respectively defined as x+ = max(min(x1, w1), ..., min(xn , wn ))) x- = min(max(x1, n(w1)), ..., max(xn , n(wn ))) where n is the order reversing involution in W and the normalization condition max(w1, ..., wn ) = 1W is assumed to hold. Then, it can be shown that the inequalities min(x1, ..., xn ) < x- < x+ < max(x1, ..., xn ) always hold. The above qualitative aggregations are usual in possibility and fuzzy sets theory. The expressions x+ and x- respectively correspond to the possibility and the necessity
measures of a fuzzy event A over a (finite) domain D = {d1, ..., dn }, with membership function µA : D ∅ [0, 1], given some possibility distribution π: D ∅ [0, 1]. Namely Posπ(A) and Necπ(A) are nothing but the weighted-max and weighted-min of the membership values xi = µA (di) according to the weights wi = π(di), and the involution n(wi) = 1 – wi. But even in a qualitative setting, very often a notion of average is also needed. In this paper we explore the possibility of defining a qualitative counterpart of the weighted mean operator without having to necessarily use some kind of numerical interpretation of the values.
2. From a quantitative to a qualitative setting In more details, our problem we want to deal with is the following: to average values from an ordinal and finite scale (U,
