ORDERED WEIGHTED AVERAGING OPERATORS

In this work, we propose the use of topic classifiers as part of a filtering language. The user is able to formulate logical rules combining the available topics, e.g., (Topic1 AND Topic2) OR Topic3, in order to retrieve or filter related documents in an incoming document stream. Under this assumption, we are interested in finding the operators that provide the best filtering performance. We can either aggregate the final decisions or the estimated probabilities of the classifiers. In the first (Boolean) case, the final decision is influenced by the selection of the decision thresholds, and it provides no ordering. The second (fuzzy) case provides ordering and a means to optimize the selection of the operators.

Ordered weighted averaging operators (OWA) (Yager, 1994) is a family of mean like operators that can adjust the degree of "AND-ing" and "OR-ing" in an aggregation. OWA have been used in many applications, including machine leaning (see Yager, 1997; Cho, 1995).

More formally , an OWA operator of dimension n is a mapping f : R ⁿ ’ R that has an associated vector w = [w ₁ w ₂ w _n ], such that (1) w _i ˆˆ [0,1] and , be the membership values to be aggregated, then with b _i the i ^th largest of the ± _i . Therefore, the weight w _i is not associated with a value ± _i but with the i ^th ordered position, imposing nonlinearity in the aggregation.

The classical Min, Max, and Average aggregations are special cases of OWA operators:

1. F*( ± ₁ , , ± _n ) = Max _i ( ± _i ), with associated vector W ^* = [1 0 0]

2. F _* ( ± ₁ , , ± _n ) = Min _i ( ± _i ), with associated vector W _* = [0 0 1]

3. , with associated vector

By appropriate choice of the weighting vector, we can move continuously from AND (Min) to OR (Max) type aggregation. A special family of OWA operators, called the S-OWA-OR (OR-like) and S-OWA-AND (AND-like) aggregations, are:

where ± _i are the numbers in the unit interval to be aggregated. As we can see, for b ˆˆ [0,1], the S-OWA-OR operator is between the mean and the maximum of numbers ± _i , while for ± ˆˆ [0,1], the S-OWA-AND operator is between the minimum and the mean of the numbers ± _i .