Information fusion is a broad area that studies methods to combine data or information supplied by multiple sources. Aggregation operators are some of the functions that can be used for combining data. This book is intended for those interested in methods for aggregating information and, specially, for those who need to embed such methods in applications. It constitutes an introduction to the field. The main focus is on functions that deal with numerical information although other kinds of functions (specially ones for ordinal scales) are considered as well. It is aimed at senior undergraduate and beginning graduate students of computer science, engineering, and mathematics.
This is an introductory book in the field of aggregation operators, focused on practical applications; we have tried, on the one hand, to limit the operators and results to a set of manageable size and, on the other hand, to include some descriptions and examples of such operators at work. We have also included a few computational issues. It has to be said that although for most operators no implementation details are given, their implementation is usually straightforward. Most of the operators and methods appearing in the book have been implemented by the authors (in Java). Due to our objective, results with a mainly mathematical interest are not included in the text. For example, only aggregation operators that combine a finite number of inputs have been studied in detail. Some definitions and results that can be useful for further study but are not relevant for real applications have been included in separate figures. This is the case for definitions of fuzzy integrals of continuous functions.
The book does not assume specific previous knowledge of aggregation operators, and Chapters 2 and 3 give some preliminaries to make it self-contained. Although the chapters have been written to avoid dependences as much as possible, there are some dependences between chapters. The most important relationships are enumerated here. Chapter 4 uses functional equations reviewed in Chapter 3, and Chapter 6 defines fuzzy integrals that use the fuzzy measures described in Chapter 5. Evaluation methods (Chapter 7) are based on the particular operators and the particular parameters explained in previous chapters (e.g., Shapley value for a fuzzy measure). The problem of parameter determination for a given operator (Chapter 8) naturally needs the operator under consideration (described in previous chapters). Nevertheless, to prevent the reader from going back and forth, there are minor repetitions in the text.
