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.
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.
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.
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.
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.
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.
For this reason, one can assume that g is monotone increasing, as otherwise we can simply take \(-g\).
- 7.
Observe that the limiting cases \(\min \) and \(\max \) are not quasi-arithmetic means.
- 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.
Recall \({n \atopwithdelims ()m} = \frac{n!}{m!(n-m)!}\).
- 10.
See Definition 1.51 on p. 18.
- 11.
This is a so-called quantifier, p. 120.
- 12.
See Definition 1.43. Continuity and decomposability imply idempotency.
- 13.
See Definition 1.44.
- 14.
These measures of entropy can be obtained by relaxing the subadditivity condition which characterizes Shannon entropy [TY05].
References
J. Aczél, On mean values. Bull. Am. Math. Soc. 54, 392–400 (1948)
G. Beliakov, Shape preserving approximation using least squares splines. Approx. Theory Appl. 16, 80–98 (2000)
G. Beliakov, How to build aggregation operators from data? Int. J. Intell. Syst. 18, 903–923 (2003)
G. Beliakov, Learning weights in the generalized OWA operators. Fuzzy Optim. Decis. Making 4, 119–130 (2005)
G. Beliakov, T. Calvo, S. James, On Lipschitz properties of generated aggregation functions. Fuzzy Sets Syst. 161, 1437–1447 (2010)
G. Beliakov, J.J. Dujmovic, Extension of bivariate means to weighted means of several arguments by using binary trees, submitted (2015)
G. Beliakov, S. James, Stability of weighted penalty-based aggregation functions. Fuzzy Sets Syst. 226, 1–18 (2013)
G. Beliakov, S. James, D. Góomez, J.T. Rodríguez, J. Montero, Learning stable weights for data of varying dimension. In: Proceedings of the AGOP Conference, Katowice, Poland (2015)
G. Beliakov, R. Mesiar, L. Valaskova, Fitting generated aggregation operators to empirical data. Int. J. Uncertainty, Fuzziness Knowl. Based Syst. 12, 219–236 (2004)
G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, vol. 221, Studies in Fuzziness and Soft Computing (Springer, Berlin, 2007)
C. Bonferroni, Sulle medie multiple di potenze. Bollettino Matematica Italiana 5, 267–270 (1950)
J.M. Borwein, P.B. Borwein, PI and the AGM: A Study in Analytic Number Theory and Computational Complexity (Wiley, New York, 1987)
P.S. Bullen, Handbook of Means and Their Inequalities (Kluwer, Dordrecht, 2003)
T. Calvo, A. Kolesárová, M. Komorníková, R. Mesiar, Aggregation operators: properties, classes and construction methods. In: Aggregation Operators. New Trends and Applications, ed. by T. Calvo, G. Mayor, R. Mesiar (Physica—Verlag, Heidelberg, 2002), pp. 3–104
T. Calvo, G. Mayor, Remarks on two types aggregation functions. Tatra Mount. Math. Publ. 16, 235–254 (1999)
T. Calvo, R. Mesiar, R.R. Yager, Quantitative weights and aggregation. IEEE Trans. Fuzzy Syst. 12, 62–69 (2004)
T. Calvo et al., Generation on weighting triangles associated with aggregation functions. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 8, 417–451 (2000)
M. Cardin, M. Manzi, Supermodular and ultramodular aggregation evaluators. In: Proceedings of the 5th International Summer School of Aggregation Operators AGOP’09, July 6–11. Palma de Mallorca, Spain, 2009
A.L. Cauchy, Cours d’analyse de l’Ecole Royale Polytechnique. Analyse algébrique 1 (1821)
O. Chisini, Sul concetto di media. Periodico di Matematiche 4, 106–116 (1929)
D. Dubois, H. Prade, A review of fuzzy set aggregation connectives. Inf. Sci. 36, 85–121 (1985)
D. Dubois, H. Prade, On the use of aggregation operations in information fusion processes. Fuzzy Sets Syst. 142, 143–161 (2004)
J.J. Dujmovic, An efficient algorithm for general weighted aggregation. In: Proceedings of the AGOP Conference, Katowice, Poland (2015)
J.J. Dujmovic, Two integrals related to means. Univ. Beograd. Publ. Elektrotechn. Fak. (1973)
J.J. Dujmovic, Weighted conjunctive and disjunctive means and their application in system evaluation. Univ. Beograd. Publ. Elektrotechn. Fak. 147–158 (1974)
J.J. Dujmovic, Seven flavors of andness/orness. In: EUROFUSE Belgrade, 2005
J.J. Dujmovic, Continuous preference logic for system evaluation. IEEE Trans. Fuzzy Syst. 15, 1082–1099 (2007)
J.J. Dujmovic, G. Beliakov, Idempotent weighted aggregation based on binary aggregation trees, submitted (2015)
C. Eisenhart, The development of the concept of the best mean of a set of measurements from antiquity to the present day. In: American Statistical Association Presidential Address, http://www.york.ac.uk/depts/maths/histstat/eisenhart.pdf (1971)
J.M. Fernández Salido, S. Murakami, Extending Yager’s orness concept for the OWA aggregators to other mean operators. Fuzzy Sets Syst. 139, 515–542 (2003)
J. Fodor, J.-L. Marichal, On nonstrict means. Aequationes Mathematicae 54, 308–327 (1997)
J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support (Kluwer, Dordrecht, 1994)
C. Gini, Le Medie. Milan (Russian translation, Srednie Velichiny, Statistica, Moscow, 1970): Unione Tipografico-Editorial Torinese, 1958
D. Gómez, J. Montero, J.T. Rodríguez, K. Rojas, Stability in aggregation operators. In: Proceedings of IPMU, Catania, Italy, 2012
D. Gómez, K. Rojas, J. Montero, J.T. Rodríguez, G. Beliakov, Consistency and stability in aggregation operators: an application to missing data problems. Int. J. Comput. Intell. Syst. 7, 595–604 (2014)
A. Kishor, A.K. Singh, N.R. Pal, Orness measure of OWA operators: a new approach. IEEE Trans. Fuzzy Syst. 22, 1039–1045 (2014)
M. Komorníková, Aggregation operators and additive generators. Int. J. Uncertainty Fuzziness Knowl Based Syst. 9, 205–215 (2001)
X. Liu, An orness measure for quasi-arithmetic means. IEEE Trans. Fuzzy Syst. 14, 837–848 (2006)
R.A. Marques Pereira, R.A. Ribeiro, Aggregation with generalized mixture operators using weighting functions. Fuzzy Sets Syst. 137, 43–58 (2003)
J. Matkowski, A mean-value theorem and its applications. J. Math. Anal. Appl. 373, 227–234 (2011)
G. Mayor, T. Calvo, Extended aggregation functions. In: IFSA’97. vol. 1, Prague, 1997, pp. 281–285
J. Merikoski, Extending means of two variables to several variables. J. Inequal. Pure Appl. Math. 5, Article 65 (2004)
R. Mesiar, J. Spirková, Weighted means and weighting functions. Kybernetik 42, 151–160 (2006)
R. Mesiar, J. Spirková, L. Vavríková, Weighted aggregation operators based on minimization. Inf. Sci. 178, 1133–1140 (2008)
S. Mustonen, Logarithmic Mean for Several Arguments, http://www.survo.fi/papers/logmean.pdf (2002)
E. Neuman, The weighted logarithmic mean. J. Math. Anal. Appl. 188, 885–900 (1994)
A.O. Pittenger, The logarithmic mean in n variables. Am. Math. Monthly 92, 99–104 (1985)
A. Pradera, E. Trillas, T. Calvo, A general class of triangular norm-based aggregation operators: quasi-linear T-S operators. Int. J. Approx. Reasoning 30, 57–72 (2002)
J.T. Rickard, J. Aisbett, New classes of threschold aggregation funcitons based upon the Tsallis q-exponential with applications to perceptual computing. IEEE Trans. Fuzzy Syst. 22, 672–684 (2014)
K. Rojas, D. Gómez, J. Montero, J.T. Rodríguez, Strictly stable families of aggregation operators. Fuzzy Sets Syst. 228, 44–63 (2013)
E. Rubin, Quantitative commentary on thucydides. Am. Stat. 25, 52–54 (1971)
V. Torra, Learning weights for the quasi-weighted means. IEEE Trans. Fuzzy Systems 10, 653–666 (2002)
L. Troiano, R.R. Yager, A meaure of dispersion for OWA operators. In: Proceedings of the 11th IFSA World Congress, ed. by Y. Liu, G. Chen, M. Ying (Tsinghua University Press, Springer, Beijing China, 2005), pp. 82–87
R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Syst. Man Cybern. 18, 183–190 (1988)
R.R. Yager, Connectives and quantifiers in fuzzy sets. Fuzzy Sets Syst. 40, 39–76 (1991)
R.R. Yager, Measures of entropy and fuzziness related to aggregation operators. Inf. Sci. 82, 147–166 (1995)
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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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