By Gleb Beliakov
Aggregation of knowledge is of fundamental value within the building of data dependent structures in quite a few domain names, starting from medication, economics, and engineering to decision-making methods, man made intelligence, robotics, and computing device studying. This ebook supplies a wide advent into the subject of aggregation services, and offers a concise account of the homes and the most periods of such features, together with classical capability, medians, ordered weighted averaging services, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It additionally provides a few cutting-edge innovations, many graphical illustrations and new interpolatory aggregation services. a specific recognition is paid to id and development of aggregation features from software particular specifications and empirical information. This ebook presents scientists, IT experts and approach architects with a self-contained easy-to-use advisor, in addition to examples of computing device code and a software program package deal. it's going to facilitate development of selection aid, professional, recommender, regulate and lots of different clever systems.
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Extra info for Aggregation Functions: A Guide for Practitioners
6 How to choose an aggregation function There are inﬁnitely many aggregation functions. They are grouped in various families, such as means, triangular norms and conorms, Choquet and Sugeno integrals, and many others. The question is how to choose the most suitable aggregation function for a speciﬁc application. Is one aggregation function 32 1 Introduction enough, or should diﬀerent aggregation functions be used in diﬀerent parts of the application? There are two components to the answer. First of all, the selected aggregation function must be consistent with the semantics of the aggregation procedure.
16 (Symmetry). , f (x1 , x2 , . . , xn ) = f (xP (1) , xP (2) , . . , xP (n) ), for every x and every permutation P = (P (1), P (2), . . , P (n)) of (1, 2 . . , n). The semantical interpretation of symmetry is anonymity, or equality. 8 On the other hand, in shareholders meetings the votes are not symmetric as they depend on the number of shares each voter has. 17. 15 are symmetric aggregation functions. A weighted arithmetic mean with non-equal weights w1 , w2 , . . wn , that are non-negative and add to one is not symmetric, n f (x) = wi xi = w1 x1 + w2 x2 + .
XP (n) . 8 It is frequently interpreted as anonymity criterion: anonymous ballot papers can be counted in any order. 19. In fuzzy sets literature, the notation x() = (x(1) , . . , x(n) ) is often used to denote both x and x , depending on the context. 20. We can express the symmetry property by an equivalent statement that for every input vector x f (x) = f (x ) ( or f (x) = f (x )), rather than f (x) = f (xP ) for every permutation. This gives us a shortcut for calculating the value of a symmetric aggregation function for a given x by using sort() operation (see Fig.