On the definition of penalty functions in data aggregation
Introduction
Aggregation functions [1], [2] are mainly used for obtaining a single output value from several input values. This procedure is indispensable in many applications, such as fuzzy rule based systems and classification systems [3], [4], [5], pattern recognition, image processing [6] or decision making [7], [8]. Examples of aggregation functions are t-norms and t-conorms, uninorms, overlap and grouping functions, weighted quasi-arithmetic means, ordered weighted averaging (OWA) functions, Choquet and Sugeno integrals (see, e.g., [3], [4], [9], [10], [11]).
In particular, averaging aggregation functions [1], [12] provide output values that are bounded by the minimum and maximum of the inputs, representing a consensus value of the inputs. They include a large class of functions (e.g., quasi-arithmetic means, medians, OWA functions and fuzzy integrals), which are often used in preference aggregation, aggregation of expert opinions, judgments in sports competitions. See [12] and the references therein.
Penalty functions have been defined as a measure of deviation from such a consensus value, or a penalty for not having a consensus, and have been studied by several authors, e.g., [8], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Examples of functions that minimize some penalties, called penalty-based functions, are the weighted arithmetic and geometric means, the median and the mode.
After the early works by Yager [13], Yager and Rybalov [14] and Calvo et al. [15], Calvo and Beliakov [11] provided a general definition of penalty function (and the corresponding penalty-based function) in 2010 and tried to show that every averaging function can be represented as a penalty-based function, studying a large class of such averaging aggregation functions. This definition did not encompass all the required features to agree with the intuition carried by the concept of penalty, and for this reason, other different definitions of penalty functions appeared in the literature (see, e.g., [8], [16], [17], [18]).
However, we have found several problems in those different definitions, as well as in their related results, which prevent their use in real world applications and also in further theoretical research. Then, the objectives of this paper are:
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To shortly describe the evolution of the idea of using penalty functions in aggregation/fusion processes, by presenting the different definitions of penalty functions found in the literature, pointing out several problems we have encountered;
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To analyze and discuss such problems, proposing a new standard definition of penalty functions that overcomes all of them, that is, encompassing the most important compatible properties of the existing definitions, but suppressing the controversial ones;
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To present some important results related to averaging aggregation functions, as the characterization of penalty-based averaging aggregation functions;
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To present some examples, as the continuous penalty functions based on spread measures [21], which formalize the measures of absolute spread known from statistics, exploratory data analysis and data mining (e.g., the sample variance, the standard deviation and the range), and are able to measure the deviation that exists among the different inputs (and so, they are conceptually related to penalty functions);
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To sketch the definition of quasi-penalty functions, in order to deal with non-monotonic (or weakly/directionally monotonic) averaging functions, (e.g., the mode, which is weakly monotone, robust estimators of location and the least median of squares estimators) [12], [18], [22].
The paper is organized as follows. Section 2 presents some preliminary concepts, including new results that are necessary for the development of the paper. Section 3 recovers the evolution of the definition of penalty functions since it was initially proposed, analyzing and discussing the problems we have encountered. Our proposal for a new standard definition of penalty functions, and the related results, are introduced in Section 4, which also discusses the use of spread measures to obtain continuous penalty functions and mentions the role of quasi-penalty functions. Finally, Section 5 is the Conclusion.
Section snippets
Preliminaries
For the sake of completeness, we recall here some mathematical notions and results that will be useful for our subsequent developments. We also introduce some results that are necessary for the development of the paper, and fix some notations.
We denote by a closed subinterval of the extended real line, i.e., . We start by recalling the notion of convex and quasi-convex function.
Definition 2.1 A function is convex if for every and for every the inequality
Different definitions of penalty functions
Yager [13] in 1993 has attempted to present the initial ideas related to the use of a penalty function to help aggregation processes. In that paper, Yager introduced the notion of a penalty cost that may be attributed to some input datum , whenever one disregards it by concluding as a result of the aggregation process some value y that conflicts with .
Later, in 1997, Yager and Rybalov [14] have considered the idea of using the minimization of a penalty function as a method for obtaining a
A proposal for the definition of penalty functions and related results
Taking into account all the problems in the definitions of penalty function existing in the literature, as we discussed in the previous section, in the following we propose a new definition that encompasses all conditions for it to satisfy the desired properties of the notion of penalty, allowing for its use in applications.
Denote by any closed real interval.
Definition 4.1 For any closed interval , the function is a penalty function if and only if there exists such that: , for
Conclusion
In this paper, we recovered the evolution of the definition of penalty functions since it was initially proposed, analyzing different definitions as well their related results, and discussing their properties and problems. We then provided a new definition of penalty function, which requires the conditions that are compatible with what one means intuitively by penalty function, but suppressing the theoretical gaps we have found in the previous definitions, so allowing its application in
Acknowledgements
This work was partially supported by the Spanish Ministry of Science and Technology under the project TIN2013-40765-P, by the Brazilian funding agency CNPq under Processes 481283/2013-7, 306970/2013-9, 232827/2014-1 and 307681/2012-2, and by APVV-14-0013.
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