Book•
Aggregation Functions: A Guide for Practitioners
01 Jan 2007-
TL;DR: A broad introduction into the topic of aggregation functions, and provides a concise account of the properties and the main classes of such functions, including classical means, medians, ordered weighted averaging functions, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums.
Abstract: Aggregation of information is of primary importance in the construction of knowledge based systems in various domains, ranging from medicine, economics, and engineering to decision-making processes, artificial intelligence, robotics, and machine learning. This book gives a broad introduction into the topic of aggregation functions, and provides a concise account of the properties and the main classes of such functions, including classical means, medians, ordered weighted averaging functions, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It also presents some state-of-the-art techniques, many graphical illustrations and new interpolatory aggregation functions. A particular attention is paid to identification and construction of aggregation functions from application specific requirements and empirical data. This book provides scientists, IT specialists and system architects with a self-contained easy-to-use guide, as well as examples of computer code and a software package. It will facilitate construction of decision support, expert, recommender, control and many other intelligent systems.
Citations
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TL;DR: It is noted that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade, and introduces a general class of sets called q-rung orthopair fuzzy sets in which the sum of the ${\rm{q}}$th power of the support against is bonded by one.
Abstract: We note that orthopair fuzzy subsets are such that that their membership grades are pairs of values, from the unit interval, one indicating the degree of support for membership in the fuzzy set and the other support against membership. We discuss two examples, Atanassov's classic intuitionistic sets and a second kind of intuitionistic set called Pythagorean. We note that for classic intuitionistic sets the sum of the support for and against is bounded by one, while for the second kind, Pythagorean, the sum of the squares of the support for and against is bounded by one. Here we introduce a general class of these sets called q-rung orthopair fuzzy sets in which the sum of the ${\rm{q}}$ th power of the support for and the ${\rm{q}}$ th power of the support against is bonded by one. We note that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade. We investigate various set operations as well as aggregation operations involving these types of sets.
1,056 citations
Cites background from "Aggregation Functions: A Guide for ..."
...We recall that Sugμ(F) ∈ [0, 1] and that Sug(F) is an aggregation operator [25], [34]....
[...]
..., an) = n j=1 b wj j where each wj ∈ [0, 1] and Σjwj = 1 [25], [28]....
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...shown in [24] and [25], many common operators are aggregation functions....
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...We now define the dual of an aggregation operator [25]....
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Book•
22 Nov 2010TL;DR: A comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations that are described by partial differential and integral equations and the simulations that result from their numerical solution.
Abstract: Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.
966 citations
TL;DR: It is shown that Pythagorean membership grades are a subclass of complex numbers called Π‐i numbers, and the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction is looked at.
Abstract: We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π-i numbers. We investigate operations that are closed under Π-i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π-i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.
953 citations
05 Nov 2008
TL;DR: There is given a short overview of the monograph rdquoAgg functionsrdquo (M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap).
Abstract: There is given a short overview of the monograph rdquoAggregation Functionsrdquo (M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap), Cambridge University Press (in press).
468 citations
Cites background from "Aggregation Functions: A Guide for ..."
...…as characterizations of various families of aggregation functions), and many connections have been done with either related fields or former works (such as triangular norms in probabilistic metric spaces, theory of means and averages, etc.), see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]....
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01 Apr 2011
TL;DR: An intuitionistic fuzzy Bonferroni mean (IFBM) is developed and applied to multicriteria decision making and its variety of special cases are discussed.
Abstract: The Bonferroni mean (BM) was originally introduced by Bonferroni and then more recently generalized by Yager. The desirable characteristic of the BM is its capability to capture the interrelationship between input arguments. Nevertheless, it seems that the existing literature only considers the BM for aggregating crisp numbers instead of any other types of arguments. In this paper, we investigate the BM under intuitionistic fuzzy environments. We develop an intuitionistic fuzzy BM (IFBM) and discuss its variety of special cases. Then, we apply the weighted IFBM to multicriteria decision making. Some numerical examples are given to illustrate our results.
406 citations
References
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01 Jan 1985
TL;DR: A mathematical tool to build a fuzzy model of a system where fuzzy implications and reasoning are used is presented and two applications of the method to industrial processes are discussed: a water cleaning process and a converter in a steel-making process.
Abstract: A mathematical tool to build a fuzzy model of a system where fuzzy implications and reasoning are used is presented. The premise of an implication is the description of fuzzy subspace of inputs and its consequence is a linear input-output relation. The method of identification of a system using its input-output data is then shown. Two applications of the method to industrial processes are also discussed: a water cleaning process and a converter in a steel-making process.
18,803 citations
"Aggregation Functions: A Guide for ..." refers methods in this paper
...formulate the problem first using crisp sets, and then fuzzify them using TSK (Takagi-Sugeno-Kang) methodology [231]....
[...]
...Later we will fuzzify this interval by using TSK methodology [231], but at the moment we concentrate on crisp intervals....
[...]
Book•
01 Jan 1976
TL;DR: This book develops an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called "monotone of order of infinity." and Dempster's rule for combining such set functions.
Abstract: Both in science and in practical affairs we reason by combining facts only inconclusively supported by evidence. Building on an abstract understanding of this process of combination, this book constructs a new theory of epistemic probability. The theory draws on the work of A. P. Dempster but diverges from Depster's viewpoint by identifying his "lower probabilities" as epistemic probabilities and taking his rule for combining "upper and lower probabilities" as fundamental. The book opens with a critique of the well-known Bayesian theory of epistemic probability. It then proceeds to develop an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called "monotone of order of infinity." and Dempster's rule for combining such set functions. This rule, together with the idea of "weights of evidence," leads to both an extensive new theory and a better understanding of the Bayesian theory. The book concludes with a brief treatment of statistical inference and a discussion of the limitations of epistemic probability. Appendices contain mathematical proofs, which are relatively elementary and seldom depend on mathematics more advanced that the binomial theorem.
14,565 citations
"Aggregation Functions: A Guide for ..." refers background in this paper
...Belief and plausibility measures constitute the basis of Dempster and Shafer Evidence Theory [222]....
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TL;DR: Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
13,376 citations
"Aggregation Functions: A Guide for ..." refers background in this paper
...Such aggregation functions are useful when aggregating membership values of interval-valued and intuitionistic fuzzy sets [8]....
[...]
TL;DR: Chapter 11 includes more case studies in other areas, ranging from manufacturing to marketing research, and a detailed comparison with other diagnostic tools, such as logistic regression and tree-based methods.
Abstract: Chapter 11 includes more case studies in other areas, ranging from manufacturing to marketing research. Chapter 12 concludes the book with some commentary about the scienti c contributions of MTS. The Taguchi method for design of experiment has generated considerable controversy in the statistical community over the past few decades. The MTS/MTGS method seems to lead another source of discussions on the methodology it advocates (Montgomery 2003). As pointed out by Woodall et al. (2003), the MTS/MTGS methods are considered ad hoc in the sense that they have not been developed using any underlying statistical theory. Because the “normal” and “abnormal” groups form the basis of the theory, some sampling restrictions are fundamental to the applications. First, it is essential that the “normal” sample be uniform, unbiased, and/or complete so that a reliable measurement scale is obtained. Second, the selection of “abnormal” samples is crucial to the success of dimensionality reduction when OAs are used. For example, if each abnormal item is really unique in the medical example, then it is unclear how the statistical distance MD can be guaranteed to give a consistent diagnosis measure of severity on a continuous scale when the larger-the-better type S/N ratio is used. Multivariate diagnosis is not new to Technometrics readers and is now becoming increasingly more popular in statistical analysis and data mining for knowledge discovery. As a promising alternative that assumes no underlying data model, The Mahalanobis–Taguchi Strategy does not provide suf cient evidence of gains achieved by using the proposed method over existing tools. Readers may be very interested in a detailed comparison with other diagnostic tools, such as logistic regression and tree-based methods. Overall, although the idea of MTS/MTGS is intriguing, this book would be more valuable had it been written in a rigorous fashion as a technical reference. There is some lack of precision even in several mathematical notations. Perhaps a follow-up with additional theoretical justi cation and careful case studies would answer some of the lingering questions.
11,507 citations
Additional excerpts
...These methods are described elsewhere [4, 122, 225]....
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Book•
01 Jan 1983
TL;DR: The Calculus of Variations as discussed by the authors is a generalization of the calculus of variations, which is used in many aspects of analysis, such as generalized gradient descent and optimal control.
Abstract: 1. Introduction and Preview 2. Generalized Gradients 3. Differential Inclusions 4. The Calculus of Variations 5. Optimal Control 6. Mathematical Programming 7. Topics in Analysis.
9,498 citations
"Aggregation Functions: A Guide for ..." refers background or methods in this paper
...Generalizations of the notion of gradient (like Clarke’s subdifferential, or quasi-differential [60, 70]) are applied....
[...]
...For non-differentiable functions the generalizations of the gradient (subgradient, quasi-gradient [60, 70]) are often used....
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