Choquet Capacities and Fuzzy Integrals

Choquet capacities, which provide the weighting mechanism for the Choquet and other fuzzy integrals, model synergistic and antagonistic interactions between variables by assigning value to all subsets rather than individual inputs.

While the flexibility of capacities (also referred to as fuzzy measures and cooperative games) comes at the expense of an exponentially increasing number of parameters, the ability to explain their behavior using various value and interaction indices makes them appealing for applications requiring transparency and interpretability. As well as a number of useful indices that in some way capture the extent to which positive and negative interactions occur, significant progress has been made in addressing the scalability issues that arise in applications. This book provides a detailed overview of the background concepts relating to capacities and their role in fuzzy integration and aggregation, then presents specialised chapters on most recent results in learning, random sampling and optimization that involve Choquet capacities.

Topics and features:

- Fundamentals of Choquet capacities (fuzzy measures) and their use in modeling importance and interaction between variables

- Definitions, properties and mappings between alternative representations that allow capacities and fuzzy integrals to be interpreted and applied in different settings

- Various simplification assumptions, from k-additive, p-symmetric and l-measures to more recent concepts such as k-interactive and hierarchical frameworks

- Capacity learning formulations that allow the diverse types to be elicited from datasets or according to user-specified requirements

- Recent findings related to random sampling and optimisation with Choquet integral objectives

This book includes illustrative examples and guidance for implementation, including an appendix detailing functions found in the pyfmtools software library. It aims to be useful for practitioners and researchers in decision and data-driven fields, or those who wish to apply these emerging tools to new problems.

The authors are all affiliated with the School of Information Technology at Deakin University, Australia. Gleb Beliakov is a professor, Simon James is an Associate Professor, and Jian-Zhang Wu is a Research Fellow.

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.