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1997 | Buch

Kolmogorov’s Theorem and Its Impact on Soft Computing Kapitel lesen Erstes Kapitel lesen

The Ordered Weighted Averaging Operators

Theory and Applications

herausgegeben von: Ronald R. Yager, Janusz Kacprzyk

Verlag: Springer US

Enthalten in: Professional Book Archive

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Über dieses Buch

Aggregation plays a central role in many of the technological tasks we are faced with. The importance of this process will become even greater as we move more and more toward becoming an information-cent.ered society, us is happening with the rapid growth of the Internet and the World Wirle Weh. Here we shall be faced with many issues related to the fusion of information. One very pressing issue here is the development of mechanisms to help search for information, a problem that clearly has a strong aggregation-related component. More generally, in order to model the sophisticated ways in which human beings process information, as well as going beyond the human capa­ bilities, we need provide a basket of aggregation tools. The centrality of aggregation in human thought can be be very clearly seen by looking at neural networks, a technology motivated by modeling the human brain. One can see that the basic operations involved in these networks are learning and aggregation. The Ordered Weighted Averaging (OWA) operators provide a parameter­ ized family of aggregation operators which include many of the well-known operators such as the maximum, minimum and the simple average.

Inhaltsverzeichnis

Frontmatter

Basic Issues in Aggregation

Frontmatter
Kolmogorov’s Theorem and Its Impact on Soft Computing
Abstract
In this chapter, we describe various applications of the Kolmogorov’s theorem on representing continuous functions of several variables (as superpositions of functions of one and two variables) to soft computing. Kolmogorov’s theorem leads to a theoretical justification, as well as to design methodologies, for neural networks. In the design of intelligent systems, Kolmogorov’s theorem is used to show that general logical operators can be expressed in terms of basic fuzzy logic operations.
In the area of reliable computing (i.e., computing that takes into consideration the accuracy of the input data), an extended version of Kolmogorov’s theorem justifies the need to use operations with three or more operands in soft computing. Such operations have already been actively used in soft computing; the simplest (and, so far, most used) of such operations are ordered weighted averaging (OWA) operators proposed by R. R. Yager.
Hung T. Nguyen, Vladik Kreinovich
Possibility and Necessity in Weighted Aggregation
Abstract
Yager [14] discussed the issue of weighted min and max aggregations and provided for a formalization of the process of importance weighted transformation. Generalizing Yager’s principle we suggest the use of fuzzy implication operators for importance weighted transformation.
Christer Carlsson, Robert Fullér, Szvetlana Fullér
OWA operators and an extension of the contrast model
Abstract
A framework was proposed (Bouchon-Meunier et al. 1996) to classify and to generate measures of comparison. It is based on concepts analogous to those developed by Tversky for his contrast model (Tversky 1977) (Suppes et al. 1990). This framework distinguishes, in the set of measures of comparison, measures of similitude and measures of dissimilarity. A measure of similitude can, in particular, depend on its use and thus on required properties.
Bernadette Bouchon-Meunier, Maria Rifqi
Equivalence of Changes in Proportions at Crossroads of Mathematical Theories
Abstract
After restating certain ‘reasonable’ requirements for changes in proportions to be equivalent, we determine all relations which satisfy them (there are uncountably many). The proof makes use of the theories of webs, of iteration groups, of concave functions and of functional equations.
János Aczél, Günter Rote, Jens Schwaiger

Fundamental Aspects of OWA Operators

Frontmatter
On the Inclusion of Importances in OWA Aggregations
Abstract
In this work we concentrate on the issue of the inclusion of importances in the OWA aggregation process. We first look at the ways in which importances have been included in some notable examples of OWA operators. We see that in these cases the importance is used to transform the scores into some effective score. Using this knowledge we build, using a fuzzy systems model, a general transform operator for the inclusion of importances. A second approach for the inclusion of importances is also discussed this approach requires the existence of a linguistic quantifier to generate the OWA weights modified by the importances.
Ronald R. Yager
On the Linguistic OWA Operator and Extensions
Abstract
A summary on the linguistic OWA operators existing in the literature is presented. To deal with linguistic information with equal importance the LOWA and ordinal OWA operators are studied. To deal with weighted linguistic information two extensions of the LOWA operator are analyzed, the LWA operator when the weights have linguistic nature and the L-WOWA operator if they have numerical nature.
F. Herrera, E. Herrera-Viedma
Alternative Representations of OWA Operators
Abstract
It is known that OWA operators appear to be particular cases of Choquet integral with respect to a suitable fuzzy measure. Recently, it has been shown that fuzzy measures can be expressed in three different, completely equivalent forms (called representations), among which the so-called interaction representation. It has been shown that only the interaction representation makes sense for a decision maker in multicriteria or multiattribute or multiperson problems, with close links to the Shapley value in cooperative game theory. In this paper we will give the three representations of an OWA operator, with emphasis on the interaction representation, in order to give new insights into its meaning in aggregation. We will also give best linear and 2nd order approximations of OWA operators.
Michel Grabisch

Mathematical Issues and OWA Operators

Frontmatter
Useful Tools for Aggregation Procedures: Some Consequences and Applications of Strassen’s Measurable Hahn-Banach-Theorem
Abstract
We state a celebrated theorem due to Strassen (1965) and derive from it Meyer’s (1966) characterization of dilation kernels. It is shown that the latter result and thus Strassen’s theorem provides a useful tool for deriving characterizations of certain orderings. As an example we prove a famous result due to Hardy/Littlewood/Pólya 1934, 1952. Finally we state some applications in the field of OWA-operators.
Heinz J. Skala
OWA Specificity
Abstract
A comprehensive model for evaluating specificity of fuzzy sets is presented. It is designed in terms of possibility values, independent of the domain of discourse. For a discrete distribution π = (p1 ≥ p2 ≥ …) two measures are defined exponential logarithmic I(π) =∑(pi - Pi+1)log i Measure Sp(π) is derived from a few intuitively plausible properties of specificity; measure I(π) is dual to nonspecificity in Dempster-Shafer theory.
The resulting model has a natural OWA structure, which follows necessarily from the basic assumptions. This leads to an inverse problem, one of developing, within the general OWA framework, the features successfully employed in specificity and uncertainty models. We suggest some directions in the concluding section.
Arthur Ramer
Ordered Continuous Means and Information
Abstract
Possibilistic measures of information have been characterized as a form of OWA operators on the possibility values. This correspondence can be carried on to the continuous domains and distributions.
Continuous information measures have been previously discussed only once in the open literature [12], while continuous OWA’s were only hinted at in conference papers [17, 18]. This chapter presents a method of defining such functions on a continuous universe of discourse — a domain which is a measurable space of measure 1. The method is based on the concept of rearangement [5] of a function, used in lieu of sorting for the discrete possibility values.
For a continuous distribution, represented by a measurable function f(x) on the domain of discourse X, first a decreasing rearrangement—f(x) on [0, μ(X)]—is constructed. Then, depending on μ(X), one of two definitions is appropriate
For technical reasons the quantification of uncertainty must be in the form of information distance [6, 13] measuring the departure from the most ‘uninformed’ distribution (constant possibility 1). The final form of the information content for possibility distribution f, defined on domain X, μ(X) = 1, is given by the continuous OWA operato
Relationship with the discrete OWA’s, and especially the discrete uncertainty measures, is discussed and various limit properties and approximations are established. Lastly, an investigation of continuous OWA as an integral transform is indicated.
Arthur Ramer

OWA Operators in Decision Analysis

Frontmatter
OWA Operators in Decision Making with Uncertainty and Nonnumeric Payoffs
Abstract
We consider the problem of decision making in environments in which there exists some uncertainty about the state of nature. A general approach to the representation of uncertainty using the Dempster-Shafer belief structure is presented. A comprehensive methodology for evaluating the worth of each of the alternatives using the OWA operators to model the decision makers attitude is described. We then consider the situation in which the payoffs are nonnumeric values. Here we consider only the existence of a linear ordering on the allowable values. It is pointed out that in these nonnumeric environments a need arises for an operation to replace the weighted average. We show that in the case of only a linear ordering on the payoffs we can use a weighted median operation to replace the weighted average.
Ronald R. Yager, Maria Teresa Lamata
On the Role of Immediate Probability in Various Decision Making Models
Abstract
Recently, new approaches have been presented which contribute to decision analysis modeling. These methods provide a means of handling various ways of representing events, decision makers’ attitudes and payoffs. We will discuss how the new approaches may be used with each other as well as in conjunction with traditional methods. The role of immediate probability in various decision making models will be highlighted.
Kurt J. Engemann, Ronald R. Yager
Risk Management Using Fuzzy Logic and Genetic Algorithms
Abstract
Risk management is a complex and subjective task. It involves identifying risk factors, setting risk thresholds, and determining appropriate actions to reduce risk [Bernstein, 1995]. Fuzzy logic techniques are especially well suited to subjective problems of this type. Traditionally, fuzzy techniques have relied on user supplied data to define fuzzy functional parameters. However, in real life people are notoriously inaccurate and unreliable in reporting their preferences, especially as the complexity and uncertainty of the problem increases. In this paper, we describe a methodology which uses a genetic algorithm to automate and validate derivation of fuzzy functional parameters.
Teresa C. Rubinson, Georgette Geotsi
OWA Operators for doctoral student selection problem
Abstract
Yager [7] introduced a new aggregation technique based on the ordered weighted averaging (OWA) operators. In this article we illustrate the applicability of OWA operators to a doctoral student selection problem at the Graduate School of Turku Centre for Computer Science.
Christer Carlsson, Robert Fullér, Szvetlana Fullér

OWA Operators in Multicriteria and Multiperson Decision Making

Frontmatter
Beyond Min Aggregation in Multicriteria Decision: (Ordered) Weighted Min, Discri-Min, Leximin
Abstract
Conjunctive aggregation based on min operation provides too crude a ranking of the possible alternatives in multiple criteria aggregation, since decisions are only compared on the basis of the worst-rated criteria, and also since the levels of importance of the different criteria or constraints are not taken into account. Various types of weighted min operations are distinguished. Two refinements of the minbased ordering (and of the Pareto ordering which corresponds to a fuzzy set inclusion) are presented and relations between them are laid bare. These refinements aim to increase the discriminating power of the min-based aggregation, yet keeping its noncompensatory nature. A relationship between the leximin ordering and ordered weighted averages (OWA) is also discussed. Lastly, ordered weighted min operations are introduced and are shown to be of interest when only most of the criteria have to be taken into account in the evaluation.
Didier Dubois, Hélène Fargier, Henri Prade
OWA Operators in Group Decision Making and Consensus Reaching Under Fuzzy Preferences and Fuzzy Majority
Abstract
We discuss the use of Yager’s (1988) ordered weighted averaging (OWA) operators with importance qualification (Yager, 1993, 1996) for dealing with a fuzzy majority, meant as a linguistic quantifier (most, almost all, … ), in group decision making and consensus formation under fuzzy preferences. We show how new solution concepts in group decision making, and new “soft” degrees of consensus can be defined.
Janusz Kacprzyk, Mario Fedrizzi, Hannu Nurmi
Applications of the Linguistic OWA Operators in Group Decision Making
Abstract
Assuming a group decision making problem where the experts express their opinions by means of linguistic preference relations, the application of the Linguistic OWA operator guided by fuzzy majority is analyzed. Two different perspectives of the use of the LOWA operator are presented: (i) in the selection process, to aggregate individual linguistic preference relations in a collective one and to calculate different linguistic choice degrees of the alternatives, and (ii) in the consensus reaching process to obtain the linguistic consensus measures. In all cases, the concept of fuzzy majority is represented by means of a fuzzy linguistic quantifier used to obtain the weights that the LOWA operator needs in its aggregation way.
F. Herrera, E. Herrera-Viedma, J. L. Verdegay
Aggregation Rules in Committee Procedures
Abstract
Very often, decision procedures in a committee compensate potential manipulations by taking into account the ordered profile of qualifications. It is therefore rejected the standard assumption of an underlying associative binary connective allowing the evaluation of arbitrary finite sequences of items by means of a one-by-one sequential process. In this paper we develop a mathematical approach for non-associative connectives allowing a sequential definition by means of binary fuzzy connectives. It will be then stressed that a connective rule should be understood as a consistent sequence of binary connective operators. Committees should previously decide about which connective rule they will be condidering, not just about a single operator.
Javier Montero, Vincenzo Cutello

OWA Operators in Querying and Information Retrieval

Frontmatter
Quantified Statements and Some Interpretations for the OWA Operator
Abstract
The interpretation of fuzzy quantified statements of the type “Q X are A” (where Q is a fuzzy quantifier and A is a fuzzy predicate) thanks to the OWA operator is the main topic of this paper. A meaning in terms of α-cuts of fuzzy sets is proposed and the relationships between this approach and fuzzy integrals on the one hand and Dempster-Shafer theory on the other hand, is investigated. The use of fuzzy quantified statements for database querying purposes is also illustrated.
Patrick Bosc, Ludovic Liétard
Using OWA Operator in Flexible Query Processing
Abstract
The use of OWA operators has been widely used in database contexts to process flexible queries where linguistic quantifiers are involved. This paper is devoted to comparing these operators’ use with other ways of addressing the same problems such as the Zadeh’s approach. Furthermore a new proposal is presented in order to avoid some inconveniences which the former approaches have in a wider class of problems. Finally, the paper analyzes the different types of queries with linguistic quantifiers which may appear in connection with relational databases and proposes the most appropiate way to solving them,in each case.
Maria-Amparo Vila, Juan-Carlos Cubero, Juan-Miguel Medina, Olga Pons
Application of OWA Operators to Soften Information Retrieval Systems
Abstract
In this contribution an overview of the use of OWA operators to model a soft retrieval activity in Information Retrieval Systems (IRSs) is presented. In particular some extensions of the Boolean document representation and of the Boolean query language are introduced, and a possible evolution of these approaches is outlined.
Gloria Bordogna, Gabriella Pasi
Implementation of OWA Operators in Fuzzy Querying for Microsoft Access
Abstract
We present an implementation of the OWA operators in FQUERY for Access (Kacprzyk and Zadrozny, 1994a, b; 1995a, b; 1997), an add-on for Microsoft Access supporting fuzzy queries of the type ”find all records such that the values of, e.g. most of the important fields are as specified (possibly fuzzily”. The OWA operators are used as general aggregation operators for fuzzy linguistic quantifier quided aggregation of atomic conditions in the query.
Janusz Kacprzyk, Slawomir Zadrożny

OWA Operators in Learning and Classification

Frontmatter
OWA — Based Computing: Learning Algorithms
Abstract
The paradigm of knowledge-based neurocomputing imposes an imperative requirement on the functional elements used in such computational architectures. What has been lacking in standard neurocomputing is an ability of the networks exploited therein to encapsulate all pieces of domain knowledge that are usually available in advance. Any successful symbiosis calls for the satisfaction of several fundamental functional postulates [2]:
  • emerging topologies should easily encapsulate any prior and sometimes qualitative or imprecise domain knowledge
  • an interpretation of the emerging network needs to be straightforward.
Witold Pedrycz
OWA Operators in Machine Learning from Imperfect Examples
Abstract
We show how Yager’s (1988) ordered weighted averaging (OWA) operators can be employed in (inductive) learning from examples which are assumed to be imperfect in the sense of errors, misclassifications, classifications to a degree, etc. We formulate the problem as to find a concept decription covering, say, almost all of the positive examples and almost none of the negative examples. Thus, by neglecting some examples, those errors are somehow “masked”.
Janusz Kacprzyk
An Application of OWA Operators to the Aggregation of Multiple Classification Decisions
Abstract
The paper considers a classification scheme made up by pooling together multiple classifiers and aggregating their decisions. The individual decisions are treated as degrees of membership assigned by the classifier to the object to be classified. We are interested in how the OWA operators compare to simple voting, linear and logarithmic techniques. In general, all the aggregation schemes appear to be of the same quality, superior to the single classifiers. It was found that OWA operators tend to generalize better than their competitors when the individual classifiers are overtrained. The idea is illustrated on a real and on an artificial data set.
Ludmila I. Kuncheva
Backmatter
Metadaten
Titel
The Ordered Weighted Averaging Operators
herausgegeben von
Ronald R. Yager
Janusz Kacprzyk
Copyright-Jahr
1997
Verlag
Springer US
Electronic ISBN
978-1-4615-6123-1
Print ISBN
978-1-4613-7806-8
DOI
https://doi.org/10.1007/978-1-4615-6123-1