Human Focused Summarizing Statistics Using OWA Operators

Ronald R. Yager (Iona College, USA)
DOI: 10.4018/978-1-60566-858-1.ch009

Abstract

The ordered weighted averaging (OWA) operator is introduced and the author discusses how it can provide a basis for generating summarizing statistics over large data sets. The author further notes how different forms of OWA operators, and hence different summarizing statistics, can be induced using weight-generating functions. The author shows how these weight-generating functions can provide a vehicle with which a data analyst can express desired summarizing statistics. Modern data analysis requires the use of more human focused summarizing statistics then those classically used. The author’s goal here is to develop to ideas to enable a human focused approach to summarizing statistics. Using these ideas we can envision a computer aided construction of the weight generating functions based upon a combination of graphical and linguistic specifications provided by a data analyst describing his desired summarization.
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Owa Operators

The Ordered Weighted Averaging (OWA) operator of dimension n is a mapping F:Rn → R such that F(a1, ..., an) = where bj is the jthlargest of the ai. The wj are weights such that wj ∈ [0, 1] and . An alternative representation of the OWA operator can be had by letting dj be the jthsmallest of the arguments, dj = bn+1-j. Using this we get F(a1, ..., an) = = .Letting vj = wn+1-j we can express F(a1, ..., an) = vj dj. Here vj is the weight associated with jth smallest of the arguments. We shall find it intuitively more satisfying to use this representation of the OWA operator. Collectively we can represent the vj by an n-dimension vector V called the weighting vector. In this vector the weights associated with the smaller arguments are at the top. A further notational convenience can be had if we let D be the n–dimensional vector whose components are the dj, we call D the ordered argument vector. Using this we get F(a1, ..., an) = VTD.

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