This book offers an easy-to-use and practice-oriented reference guide to mathematical averages. It presents different ways of aggregating input values given on a numerical scale, and of choosing and/or constructing aggregating functions for specific applications. Building on a previous monograph by Beliakov et al. published by Springer in 2007, it outlines new aggregation methods developed in the interim, with a special focus on the topic of averaging aggregation functions. It examines recent advances in the field, such as aggregation on lattices, penalty-based aggregation and weakly monotone averaging, and extends many of the already existing methods, such as: ordered weighted averaging (OWA), fuzzy integrals and mixture functions. A substantial mathematical background is not called for, as all the relevant mathematical notions are explained here and reported on together with a wealth of graphical illustrations of distinct families of aggregation functions. The authors mainly focus on practical applications and give central importance to the conciseness of exposition, as well as the relevance and applicability of the reported methods, offering a valuable resource for computer scientists, IT specialists, mathematicians, system architects, knowledge engineers and programmers, as well as for anyone facing the issue of how to combine various inputs into a single output value.

Presents the state-of-art in aggregation methods, with a special focus on averaging functions Discusses generalizations as well as prototypical applications Includes all the relevant mathematical concepts and formal definitions Includes supplementary material: sn.pub/extras

Autorentext

Gleb Beliakov is a Professor in Mathematics at Deakin University, Australia. He completed his PhD in 1992 in Moscow and since then worked at various universities, for the last 25 years at Deakin University. His research is focused on computational mathematics, numerical optimisation and aggregation functions. He co-authored three monographs and numerous research papers in this area. Simon James is an Associate Professor in Mathematics at Deakin University. He completed his Ph.D. there on the topic of aggregation functions in 2010 under the supervision of Gleb Beliakov and held an academic position there since 2011. His main research areas of aggregation and capacities (or fuzzy measures) are most commonly applied in computational intelligence and machine learning as prediction and analysis tools. Jian-Zhang Wu was a Research Fellow at Deakin University. He received his PhD in Management Science and Engineering in 2011 from Beijing Institute of Technology and was previously a professor at Ningbo University. He has also worked as a chief data scientist in the finance industry. His research focuses on machine learning, decision-making, and explainable AI, especially using Choquet capacities and fuzzy integrals. He has led several national and provincial research projects in China.

Inhalt
Notations and Abbreviations.- Classical Averaging Functions.- Ordered Weighted Averaging.- Fuzzy Integrals.- Penalty Based Averages.- More Types of Averaging and Construction Methods.- Non-monotone Averages.- Averages on Lattices.