Fuzzy numbers and fuzzification of the Choquet integral

To deal with linguistic attributes in databases, the integrand of the Choquet integral should be allowed to assume fuzzy numbers. This paper provides a detailed discussion on one fuzzification of Choquet integral, in which the integrand as well as the integration result is fuzzy numbers, based on the extension principle. It is a generalized Choquet integral for fuzzy-valued integrand, interval-valued integrand, as well as the crisp-valued integrand. Two numerical methods are provided to calculate the membership function of the integration value of this generalized Choquet integral with respect to the fuzzy measure and the signed fuzzy measure, respectively. The presented generalized Choquet integral with respect to signed fuzzy measure can act as an aggregation tool which is especially useful in many information fusing and data mining problems (such as regression and classification) where not only crisp data but also heterogeneous fuzzy data are involved.