Aggregation Functions: A Guide for Practitioners (Studies in by Gleb Beliakov

By Gleb Beliakov

A large advent to the subject of aggregation features is to be present in this e-book. It additionally offers a concise account of the homes and the most periods of such services. a few cutting-edge options are awarded, besides many graphical illustrations and new interpolatory aggregation services. specific cognizance is paid to identity and development of aggregation capabilities from software particular standards and empirical data.

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Extra info for Aggregation Functions: A Guide for Practitioners (Studies in Fuzziness and Soft Computing)

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Similarly, if a > e then a = f (a, . . , a, 1) ≥ f (e, . . , e, 1) = 1. 34 (Zero divisor). An element a ∈]0, 1[ is a zero divisor of an aggregation function f if for all i ∈ {1, . . , the equality f (x1 , . . , xi−1 , a, xi+1 , . . , xn ) = 0, can hold for some x > 0 with a at any position. 35. Because of monotonicity of f , if a is a zero divisor, then all values b ∈]0, a] are also zero divisors. The interpretation of zero divisors is straightforward: if one of the inputs takes the value a, or a smaller value, then the aggregated value could be zero, for some x.

Prade. A review of fuzzy set aggregation connectives. Inform. , 36:85–121, 1985. D. Dubois and H. Prade. Fundamentals of Fuzzy Sets. Kluwer, Boston, 2000. D. Dubois and H. Prade. On the use of aggregation operations in information fusion processes. Fuzzy Sets and Systems, 142:143–161, 2004. J. Fodor and M. Roubens. Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht, 1994. M. T. A. Walker. Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference. Kluwer, Dordrecht, 1995.

Then if a ≤ e, we get the contradiction a = 0, since it is a = f (a, . . , a, 0) ≤ f (e, . . , e, 0) = 0. Similarly, if a > e then a = f (a, . . , a, 1) ≥ f (e, . . , e, 1) = 1. 34 (Zero divisor). An element a ∈]0, 1[ is a zero divisor of an aggregation function f if for all i ∈ {1, . . , the equality f (x1 , . . , xi−1 , a, xi+1 , . . , xn ) = 0, can hold for some x > 0 with a at any position. 35. Because of monotonicity of f , if a is a zero divisor, then all values b ∈]0, a] are also zero divisors.

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