G. Beliakov – A Practical Guide to Averaging Functions (2016)

Автор: G. Beliakov
Название книги: A Practical Guide to Averaging Functions
Формат: PDF
Жанр: Приборостроение
Страницы: 365
Качество: Изначально компьютерное, E-book

Averaging is ubiquitous in many sciences, engineering, and everyday practice. The notions of the arithmetic, geometric, and harmonic means developed by the ancient Greeks are in widespread use today. When thinking of an average, most people would use arithmetic mean, “the average”, or perhaps its weighted version in order to associate the inputs with the degrees of importance. While this is certainly the simplest and most intuitive averaging function, its use is often not warranted. For example, when averaging the interest rates, it is the geometric and not the arithmetic mean which is the right method. On the other hand, the arithmetic mean can also be biased for a few extreme inputs, and hence can convey false meaning. This is the reason why real estate markets report the median and not the average prices (which could be biased by one or a few outliers), and why judges’ marks in some Olympic sports are trimmed of the smallest and the largest values.
The world of averages (also called means) is very rich in both mathematical and practical senses. There exist averages that allow one to incorporate not only the weights of importance but also various interactions among the inputs; averages that are robust to a few, or even many outlying values; averages that model various types of majority, necessary, desirable, and sufficient inputs; averages that model a representative (in various senses) input, and so on.
The theory of aggregation functions, which includes most averages, became an established area of research in the last 30 years. Theoretical advances are complemented by numerous applications in decision sciences, artificial intelligence, fuzzy systems, and image processing. Several monographs and edited volumes dedicated to this topic provide a comprehensive analysis of both theory and applications, and regular conferences and special sessions on aggregation provide a forum for presenting the latest achievements.
However, it has been 7 years since the publication of the most recent monograph in the field of aggregation, and we think it is time to provide an update on the most recent developments in this area. Our specific focus is on averaging functions. These functions, whose prototypical example is the arithmetic mean, are most often used in decision sciences as they provide compensatory properties: low values of some inputs are compensated by high values of the others, and the output is always bounded by the smallest and the largest input. The result of the averaging is a value representative of the inputs.
The target audience of this book is computer scientists, system architects, knowledge engineers, and programmers, as well as decision scientists and mathematicians, who face a problem of combining various inputs into a single output. Our intent is to provide these people with an easy-to-use guide about possible ways of averaging input values given on a numerical scale, and ways of choosing/ constructing aggregation functions for their specific applications. All relevant mathematical notions are explained in the book (in the introduction or as footnotes).
Review of Aggregation Functions
Classical Averaging Functions
Ordered Weighted Averaging
Fuzzy Integrals
Penalty Based Averages
More Types of Averaging and Construction Methods
Non-monotone Averages
Averages on Lattices