Search

When an application needs a fusion mechanism, the developer has to solve an essential problem: the construction of the appropriate model. This corresponds to (i) the selection of an aggregation operator and (ii) the determination of its parameters. This process should take into account several factors. Some of them are highlighted here. [ABSTRACT FROM AUTHOR]

This chapter reviews some of the existing tools for evaluating aggregation methods and their parameters. We focus on some indices for fuzzy measures (Shapley and Banzhaf), an interaction index, and the degree of disjunction. Other methods exist. The influence function and other tools such as grosserror sensitivity and local-shift sensitivity developed in robust statistics (see Section 2.2.6) are of interest here. The tools permit us to have some knowledge on how a particular estimator might behave when embedded in a real system. In particular, we have seen that the influence function of the arithmetic mean is unbounded while that of the median is bounded. [ABSTRACT FROM AUTHOR]

In this chapter we review some aggregation operators for numerical information. While in Chapter 4 description was centered on functional equations, and operators were introduced as a natural consequence of some basic properties (unanimity, positive homogeneity, and so on), here, operators are introduced for greater modeling capabilities and generality. This progression into general aggregation operators leads to a review of operators that are particular cases of Choquet and Sugeno integrals. On the one hand, the Choquet integral generalizes not only arithmetic mean and weighted mean (the most widely used and well-known aggregation operators), but also OWA operators. On the other hand, the Sugeno integral generalizes weighted minimum, weighted maximum, and median operators. In the rest of this chapter we will use Choquet integral family to refer to aggregation operators that are generalized by the Choquet integral. In the same way, the Sugeno integral family will refer to aggregation operators that the Sugeno integral generalizes. [ABSTRACT FROM AUTHOR]

Most aggregation operators use some kind of parameterization to express additional information about the objects that take part in the aggregation process. Applying the jargon of artificial intelligence, we can say that the parameters are used to represent the background knowledge. For example, it is well known that in the case of the weighted mean, the weights — i.e., the weighting vector — play this role. In an application, we can use them to express the reliability of the information sources (sensors, experts, and so on). For example, when fusing data from sensors, we can express wich sensor is more likely to give data of better quality and which is more likely to give erroneous data. In a similar way, other aggregation functions use other parameterizations. [ABSTRACT FROM AUTHOR]

In this chapter we study some aggregation operators for numerical information. The description is focused on results based on functional equations. Therefore, not only are the operators given, but also, at least for some of them, their characterization. We refer to these results as syntheses of judgements. Although the term could be used for any aggregation operator, we restrict its use to the case of characterizations using functional equations. [ABSTRACT FROM AUTHOR]

Functional equations are equations where the unknowns are functions. A well-known example of functional equation is the following Cauchy equation: (3.1)$$ \phi (x + y) = \phi (x) + \phi (y). $$ A function φ is a solution of this equation if, for any two values x and y, the application of φ to x + y equals the addition of the application of φ to x and to y. Therefore, the equation establishes conditions that functions φ have to satisfy. Typical solutions of this Cauchy equation are the functions φ(x) = αx for an arbitrary value for α. [ABSTRACT FROM AUTHOR]

In this chapter we will review some of the concepts that are needed later in the book. In particular, we focus on measurement theory and some basic elements of probability theory and fuzzy sets theory. [ABSTRACT FROM AUTHOR]

Information fusion techniques, in general, and aggregation operators (or aggregation functions), in particular, are extensively used in several fields of human knowledge. They are used to produce the most comprehensive and specific datum about an entity from data supplied by several information sources (or the same source at different periods of time). They are used in systems to reduce some type of noise, increase accuracy, summarize information, extract information, make decisions, and so on. To illustrate this, we consider below some examples in different fields. Some of the typical applications are also included. [ABSTRACT FROM AUTHOR]