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11.07.2019 | Methodologies and Application Open Access

The \(\lambda \)-additive measure in a new light: the \(Q_{\nu }\) measure and its connections with belief, probability, plausibility, rough sets, multi-attribute utility functions and fuzzy operators

Soft Computing
József Dombi, Tamás Jónás
Wichtige Hinweise
Communicated by V. Loia.

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The aim of this paper is twofold. On the one hand, the \(\lambda \)-additive measure (Sugeno \(\lambda \)-measure) is revisited, and a state-of-the-art summary of its most important properties is provided. On the other hand, the so-called \(\nu \)-additive measure as an alternatively parameterized \(\lambda \)-additive measure is introduced. Here, the advantages of the \(\nu \)-additive measure are discussed, and it is demonstrated that these two measures are closely related to various areas of science. The motivation for introducing the \(\nu \)-additive measure lies in the fact that its parameter \(\nu \in (0,1)\) has an important semantic meaning as it is the fix point of the complement operation. Here, by utilizing the \(\nu \)-additive measure, some well-known results concerning the \(\lambda \)-additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the \(\nu \)-additive measure is connected with the belief-, probability- and plausibility measures. Next, it is also shown that two \(\nu \)-additive measures, with the parameters \(\nu _1\) and \(\nu _2\), are a dual pair of belief- and plausibility measures if and only if \(\nu _1+\nu _2 = 1\). Furthermore, it is demonstrated how a \(\nu \)-additive measure (or a \(\lambda \)-additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the \(\nu \)-additive measures are connected with rough sets, multi-attribute utility functions and certain operators of fuzzy logic.

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