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Classical Averaging Functions

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A Practical Guide to Averaging Functions

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 329))

Abstract

This chapter presents the classical means, starting with the weighted arithmetic and power means, and then continuing to the quasi-arithmetic means. The topics of generating functions, comparability and weights selection are covered. Several interesting classes of non-quasi-arithmetic means are presented, including Gini, Bonferroni, logarithmic and Bajraktarevic means. Methods of extension of symmetric bivariate means to the multivariate case are also discussed.

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Notes

  1. 1.

    In some languages there are no distinct terms that refer separately to the average and the mean, e.g. moyenne (Fr), medie (It.), promedio (or media) (Sp.), sredniaya (Ru.), Gemiddelde (Dut.), Gjennomsnitt (No.), whereas in others there are, e.g. Durchschnitt or Mittelwerte (Ger.). For etymology of the words mean and average see [Eis71]. We will use both terms synonymously.

  2. 2.

    A function g is convex if and only if \(g(\alpha t_1+(1-\alpha )t_2) \le \alpha g(t_1)+(1-\alpha ) g(t_2)\) for all \(t_1,t_2 \in Dom(g)\) and \(\alpha \in [0,1]\).

  3. 3.

    It is easy to check that

    $$ \int _{[0,1]^n} M(\mathbf {x}) d \mathbf {x} = \frac{1}{n}\left( \int _{0}^1 x_1dx_1+\cdots +\int _{0}^1 x_ndx_n\right) =\frac{n}{n}\int _{0}^1 tdt=\frac{1}{2}. $$

    Substituting the above value in (2.1) we obtain \(orness(M)=\frac{1}{2}\). Following, for a weighted arithmetic mean we also obtain

    $$ \int _{[0,1]^n} M_{\mathbf {w}}(\mathbf {x}) d \mathbf {x} = w_1\int _{0}^1 x_1dx_1+\cdots +w_n\int _{0}^1 x_ndx_n=\sum _{i=1}^nw_i\int _{0}^1 tdt=\frac{1}{2}. $$

    .

  4. 4.

    We shall use square brackets in the notation \(M_{[r]}\) for power means to distinguish them from quasi-arithmetic means \(M_g\) (see Sect. 2.3), where parameter g denotes a generating function rather than a real number. The same applies to the weighted power means.

  5. 5.

    The limiting cases \(\min \) (\(r=-\infty \)) and \(\max \) (\(r=\infty \)) which have neutral elements \(e=1\) and \(e=0\) respectively, are not themselves power means.

  6. 6.

    For this reason, one can assume that g is monotone increasing, as otherwise we can simply take \(-g\).

  7. 7.

    Observe that the limiting cases \(\min \) and \(\max \) are not quasi-arithmetic means.

  8. 8.

    This concept of supermodularity of a function on \({\mathbb I}^n\) is different from supermodularity and submodularity of fuzzy measures in Definition 4.14.

  9. 9.

    Recall \({n \atopwithdelims ()m} = \frac{n!}{m!(n-m)!}\).

  10. 10.

    See Definition 1.51 on p. 18.

  11. 11.

    This is a so-called quantifier, p. 120.

  12. 12.

    See Definition 1.43. Continuity and decomposability imply idempotency.

  13. 13.

    See Definition 1.44.

  14. 14.

    These measures of entropy can be obtained by relaxing the subadditivity condition which characterizes Shannon entropy [TY05].

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Authors and Affiliations

  1. School of Information Technology, Deakin University, Burwood, VIC, Australia

    Gleb Beliakov

  2. Departamento de Automática y Computación, Universidad Pública de Navarra, Pamplona, Spain

    Humberto Bustince Sola

  3. Departamento de Ciencias de la Computación, Universidad de Alcalá, Alcalá de Henares, Madrid, Spain

    Tomasa Calvo Sánchez

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Correspondence to Gleb Beliakov .

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Beliakov, G., Bustince Sola, H., Calvo Sánchez, T. (2016). Classical Averaging Functions. In: A Practical Guide to Averaging Functions. Studies in Fuzziness and Soft Computing, vol 329. Springer, Cham. https://doi.org/10.1007/978-3-319-24753-3_2

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