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Weakly monotonic averaging functions

journal contribution
posted on 2015-02-01, 00:00 authored by Tim WilkinTim Wilkin, Gleb BeliakovGleb Beliakov
Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. It is also a limiting property that results in many important nonmonotonic averaging functions being excluded from the theoretical framework. This work proposes a definition for weakly monotonic averaging functions, studies some properties of this class of functions, and proves that several families of important nonmonotonic means are actually weakly monotonic averaging functions. Specifically, we provide sufficient conditions for weak monotonicity of the Lehmer mean and generalized mixture operators. We establish weak monotonicity of several robust estimators of location and conditions for weak monotonicity of a large class of penalty-based aggregation functions. These results permit a proof of the weak monotonicity of the class of spatial-tonal filters that include important members such as the bilateral filter and anisotropic diffusion. Our concept of weak monotonicity provides a sound theoretical and practical basis by which (monotonic) aggregation functions and nonmonotonic averaging functions can be related within the same framework, allowing us to bridge the gap between these previously disparate areas of research.

History

Journal

International Journal of Intelligent Systems

Volume

30

Pagination

144-169

Location

NJ, United States

ISSN

0884-8173

eISSN

1098-111X

Language

eng

Publication classification

C Journal article, C1 Refereed article in a scholarly journal

Copyright notice

2014, John Wiley & Sons

Issue

2

Publisher

John Wiley & Sons