Weakly Monotone Averaging Functions

Abstract

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 non-monotonic 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 non-monotonic means are actually weakly monotonic averaging functions. Specifically we provide sufficient conditions for weak monotonicity of the Lehmer mean and generalised 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 (monotone) aggregation functions and non-monotone averaging functions can be related within the same framework, allowing us to bridge the gap between these previously disparate areas of research.

On the weak monotonicity of Gini means and other mixture functions

“…The authors of [32,33] proved some useful properties of weakly monotone averaging aggregations. We present some of the results here that we rely on later.…”

“…In [32] the authors introduced weakly monotone averaging functions. They were motivated by the following observations: first, that many averages that appear in the literature are in fact not monotone -the mode, Gini means, Bajraktarevic means [2,3,11] and mixture operators [19,24] are particularly well known cases; and, second, that some applications in fact require non-monotone averaging for good performance.…”

“…In this work we extend the results of [10,32] to encompass more general classes of means, in particular Gini means, as well as other mixture functions. Analysis of monotonicity of mixture and quasi-mixture functions has been performed recently in [19,24].…”

“…Lehmer means, certain mixture functions, density-based means and the mode, are all weakly monotone [10,14,32,34]. The relation to the form of penalty functions for shift-invariance and hence weak monotonicity was established.…”

“…In [10,14,32,34] a weaker condition than monotonicity was proposed for averaging functions, in which the value of the aggregate does not decrease when all the inputs are increased by the same value, but may decrease if a subset of the inputs increases. Two principal justifications for this weaker monotonicity were given: (a) many important, existing averages are not actually monotone, but rather are weakly monotone (for example the mode and robust estimators of location); and (b) weak monotonicity is very useful when calculating representative values of clusters of data in the presence of outliers.…”