By Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo
This ebook bargains an easy-to-use and practice-oriented reference consultant to mathematical averages. It provides other ways of aggregating enter values given on a numerical scale, and of selecting and/or developing aggregating capabilities for particular functions. construction on a prior monograph through Beliakov et al. released through Springer in 2007, it outlines new aggregation tools constructed in the intervening time, with a different concentrate on the subject of averaging aggregation services. It examines fresh advances within the box, similar to aggregation on lattices, penalty-based aggregation and weakly monotone averaging, and extends the various already latest tools, equivalent to: ordered weighted averaging (OWA), fuzzy integrals and combination capabilities. a considerable mathematical heritage isn't really referred to as for, as the entire suitable mathematical notions are defined the following and stated on including a wealth of graphical illustrations of detailed households of aggregation capabilities. The authors normally concentrate on sensible purposes and provides vital significance to the conciseness of exposition, in addition to the relevance and applicability of the mentioned tools, delivering a necessary source for machine scientists, IT experts, mathematicians, process architects, wisdom engineers and programmers, in addition to for a person dealing with the difficulty of ways to mix a number of inputs right into a unmarried output value.
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Extra info for A Practical Guide to Averaging Functions
This is because every conjunctive function coincides with min(x) for those x that have at least one zero component, and min is not strictly monotone (similarly, disjunctive aggregation functions coincide with max(x) for those x that have at least one component xi = 1). However, strict monotonicity on the semi-open set ]a, b]n (respectively [a, b[n ) is often considered for conjunctive (disjunctive) aggregation functions. Of course, there are plenty of strictly increasing averaging aggregation functions, such as arithmetic means.
For example, by presenting a user of a computer system with some information and recording their actions or decisions. In the most typical case, the data comes in pairs (x, y), where x ∈ [0, 1]n is the input vector and y ∈ [0, 1] is the desired output. There are several pairs, which will be denoted by a subscript k: (xk , yk ), k = 1, . . , K . , the interval [y k , y k ]). In fitting an aggregation function to the data, we will distinguish interpolation and approximation problems. In the case of interpolation, our aim is to fit the specified output values exactly.
6 How to Choose an Aggregation Function 33 Third, the data could be collected in an experiment, by asking a group of lay people or experts about their input and output values, but without associating these values with some aggregation rule. For example, an interesting experiment reported in [Zim96, ZZ80] consisted in asking a group of people about the membership values they would use for different objects in the fuzzy sets “metallic”, “container”, and then in the combined set “metallic container”.
A Practical Guide to Averaging Functions by Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo