Directional and ordered directional monotonicity of mixture functions

© Springer International Publishing AG 2018. In this contribution, we discuss the concepts of so-called fusion functions, pre-aggregation functions and their directional and ordered directional monotonicity in the context of mixture functions. Mixture functions represent a special class of weighted averaging functions whose weights are determined by continuous weighting functions which depend on the input values. They need not be monotone, in general. If they are monotone increasing, they also belong to the important class of aggregation functions. If the are directionally monotone, they belong to the class of pre-aggregation functions. Currently there is increased interest in studying generalized forms of monotonicity such as weak, directional or ordered directional monotonicity due to their possible application in fields such classification or image processing. This paper discusses properties of selected mixture functions with special emphasis on their directional and ordered directional monotonicity. The concept of directional and ordered directional monotonicity of mixture functions is investigated with respect to linear and quadratic weighting functions.