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Symmetric Fuzzy Measures: OWA

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Discrete Fuzzy Measures

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 382))

Abstract

The focus of this chapter is on the ordered weighted averaging (OWA) functions, which are Choquet integrals with respect to symmetric fuzzy measures. The formal definitions and the main properties of OWA are presented, with some extensions discussed in detail, in particular the weighted OWA functions. Various methods of fitting OWA functions to empirical data are also presented. This chapter ends with a discussion of median functions and order statistics as special cases of OWA functions.

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Notes

  1. 1.

    I.e., Q is a monotone increasing function \([0,1]\rightarrow [0,1]\), \(Q(0)=0,Q(1)=1\) whose value Q(t) represents the degree to which t satisfies the fuzzy concept represented by the quantifier.

  2. 2.

    This is an informal definition. The proper definition involves the concepts of distributions and measures, see, e.g., [Rud91].

  3. 3.

    Note that in OWA, weighted median and ordinal OWA, \(x_{(k)}\) denotes the kth largest element of \(\mathbf {x}\).

  4. 4.

    Note that in this case the weights need not sum to 1.

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  1. Deakin University, Burwood, VIC, Australia

    Gleb Beliakov & Simon James

  2. Ningbo University, Ningbo, Zhejiang, China

    Jian-Zhang Wu

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  1. Gleb Beliakov
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Beliakov, G., James, S., Wu, JZ. (2020). Symmetric Fuzzy Measures: OWA. In: Discrete Fuzzy Measures. Studies in Fuzziness and Soft Computing, vol 382. Springer, Cham. https://doi.org/10.1007/978-3-030-15305-2_6

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