Generalized forms of monotonicity in the data aggregation framework

Zusammenfassung

The process of aggregation addresses the problem of combining a collection of numerical values into a single representative number and the functions that perform this operation are called aggregation functions. Aggregation functions are required to satisfy two boundary conditions and to be monotone with respect to all their arguments. One of the trends in the research area of aggregation functions is the relaxation of the condition of monotonicity. In that attempt, various relaxed forms of monotonicity have been introduced. This is the case of weak, directional and cone monotonicity. However, all these relaxed forms of monotonicity are based on the idea of increasing, or decreasing, along a fixed ray defined by a real vector. There is no notion of monotonicity allowing the direction of increasingness to depend on the specific values to aggregate, nor there exists any other notion that considers increasingness along more general paths, such as curves. Additionally, another trend in the theory of aggregation is the extension to handle more general scales than real numbers and there is no relaxation of monotonicity available in that general context. In this dissertation, we propose a collection of new relaxed forms of monotonicity for which the directions of monotonicity may vary from one point of the domain to another. Specifically, we introduce the concepts of ordered directional monotonicity, strengthened ordered directional monotonicity and pointwise directional monotonicity. Based on the concept of ordered directionally monotone functions, we propose an edge detection algorithm that justifies the applicability of these concepts. Furthermore, we generalize the concept of directional monotonicity so that, instead of directions in Rn, more general paths are considered: we define curve-based monotonicity. Finally, combining both trends in the theory of aggregation, we generalize the concept of directional monotonicity to functions that are defined on more general scales than real numbers.