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.

Aggregation Functions: A Guide for Practitioners

Developers of computer codes, analysts who use the codes, and decision makers who rely on the results of the analyses face a critical question: How should confidence in modeling and simulation be critically assessed? Verification and validation (V&V) of computational simulations are the primary methods for building and quantifying this confidence. Briefly, verification is the assessment of the accuracy of the solution to a computational model. Validation is the assessment of the accuracy of a computational simulation by comparison with experimental data. In verification, the relationship of the simulation to the real world is not an issue. In validation, the relationship between computation and the real world, i.e., experimental data, is the issue.

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Verification, Validation, and Predictive Capability in Computational Engineering and Physics

This report summarizes a variety of the most useful and commonly applied methods for obtaining Dempster-Shafer structures, and their mathematical kin probability boxes, from empirical information or theoretical knowledge. The report includes a review of the aggregation methods for handling agreement and conflict when multiple such objects are obtained from different sources.

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Constructing Probability Boxes and Dempster-Shafer Structures

http://www.isahp.org/2007Proceedings/Papers/Master%20Sessions/Fuzzifying%20Numerical%20Judgements%20in%20AHP.pdf

On the invalidity of fuzzifying numerical judgments in the Analytic Hierarchy Process

Several simple test problems are used to explore the following approaches to the representation of the uncertainty in model predictions that derives from uncertainty in model inputs: probability theory, evidence theory, possibility theory, and interval analysis. Each of the test problems has rather diffuse characterizations of the uncertainty in model inputs obtained from one or more equally credible sources. These given uncertainty characterizations are translated into the mathematical structure associated with each of the indicated approaches to the representation of uncertainty and then propagated through the model with Monte Carlo techniques to obtain the corresponding representation of the uncertainty in one or more model predictions. The different approaches to the representation of uncertainty can lead to very different appearing representations of the uncertainty in model predictions even though the starting information is exactly the same for each approach. To avoid misunderstandings and, potentially, bad decisions, these representations must be interpreted in the context of the theory/procedure from which they derive. Published by Elsevier Ltd.

An exploration of alternative approaches to the representation of uncertainty in model predictions.

A considerable amount of research has been done on the notions of pseudo complement, intersection and union of fuzzy sets [1], [4], [11]. Most of this work consists of generalizations or alternatives of the basic concepts introduced by L. A. Zadeh in his famous paper [13]: generalization of the unit interval to arbitrary complete and completely distributive lattices or to Boolean algebras [2]; alternatives to union and intersection using the concept of t-norms [3], [10]; alternative complements in the unit interval in [4], [11], a.s.o. Although it is usually accepted that I := [0,1] is a natural generalization of {0,1} and that a --> 1 - a is a natural generalization of the Boolean complement on {0,1}, we do however not find canonical and mathematical justification for this fact, which nevertheless lies at the heart of the definition of fuzzy sets. It is the purpose of this note to present a canonical way of obtaining I and L. A. Zadeh's pseudocomplement.
Moreover, if we consistently use this canonical machine we shall see that also other set-concepts can, and maybe should have been extended to fuzzy sets. Further we also give a possible generalization involving a choice of arbitrary t-norms.

Robert Lowen

Papers

On the fundamentals of fuzzy sets.