Lower integrals and upper integrals with respect to nonadditive set functions

Abstract

Nonadditive set functions have been used as a proper mathematical model for describing the contribution rates, as well as their interaction, from predictive attributes toward some target. In information fusion and data mining, a mathematical tool is needed to aggregate contributions from each predictive attribute to the target. Essentially, it is a mapping from a higher dimensional space (sample space) to a lower dimensional space (decision space). The mapping, of course, depends on the nonadditive set function. Such a tool is generally called an integral. Due to the nonadditivity of the set function, the integral is a nonlinear functional defined on the set of the so-called nonnegative measurable functions, where each function is a point in the sample space. The upper integral, the lower integral, and the Choquet integral are some special types of nonlinear integrals. When the set of attributes is finite, all nonlinear integrals are generally called indeterminate integrals. In this case, the upper integral and the lower integral are two extreme specified indeterminate integrals. A way for calculating the newly defined integrals is shown in this paper when the universal set (the set of attributes) is finite. Recalling the concept of indeterminate integral, the value of any specified indeterminate integral of a given function can be expressed as a convex combination of the values of its upper integral and lower integral. In such a point of view, the inverse problem of the indeterminate integral with respect to a given nonadditive set function is rather easy to be solved.