Three types of monotonicity of averaging functions

Beliakov, Gleb; Calvo, Tomasa; Wilkin, Tim

doi:10.1016/j.knosys.2014.08.028

Cited by 32 publications

(18 citation statements)

References 27 publications

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“…Note that directional monotonicity in this context was also discussed in [12], monotonicity of mixture functions was studied in [30] and cone monotonicity was investigated in [6], where the name weak monotonicity was justified.…”

Section: Weak Monotonicitymentioning

confidence: 99%

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On the weak monotonicity of Gini means and other mixture functions

Beliakov

Calvo

Wilkin

2015

Information Sciences

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Section: Weak Monotonicitymentioning

confidence: 99%

“…Unlike the conditions from [20,21], it appears there is no simple way to characterize weighting functions w which satisfy (6). We shall now look at some special but prototypical cases of the weighting functions for mixture operators, see [14,34].…”

Section: Examples Of Weakly Monotone Mixture Functionsmentioning

confidence: 99%

On the weak monotonicity of Gini means and other mixture functions

Beliakov

Calvo

Wilkin

2015

Information Sciences

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“…14 and monotonicity with respect to cones of directions was further investigated in Ref. 15. We stress herein the special role of weak monotonicity within the class of directionally monotone functions.…”

Section: Introductionmentioning

confidence: 99%

Weakly Monotonic Averaging Functions

Wilkin

Beliakov

2014

Int. J. Intell. Syst.

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119

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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.

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“…(2) The idea of directional monotonicity (Bustince et al 2015), which generalizes weak monotonicity considering increasingness or decreasingness along a ray which can be defined by any vector. When a function is directionally increasing with respect to a set of vectors that form a cone, it is called cone-monotone (Beliakov, Calvo and Wilkin 2014). This ideas have led to the notion of preaggregation functions, which is a function fulfilling the same boundary conditions as an aggregation function but which is just directionally increasing along some ray.…”

Section: Introductionmentioning

confidence: 99%

Curve-based monotonicity: a generalization of directional monotonicity

Hierro

Sesma‐Sara

Špirková

et al. 2019

International Journal of General Systems

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In this work we propose a generalization of the notion of directional monotonicity. Instead of considering increasingness or decreasingness along rays, we allow more general paths defined by curves in the n-dimensional space. These considerations lead us to the notion of α-monotonicity, where α is the corresponding curve. We study several theoretical properties of α-monotonicity and relate it to other notions of monotonicity, such as weak monotonicity and directional monotonicity.

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Three types of monotonicity of averaging functions

Cited by 32 publications

References 27 publications

On the weak monotonicity of Gini means and other mixture functions

On the weak monotonicity of Gini means and other mixture functions

Weakly Monotonic Averaging Functions

Curve-based monotonicity: a generalization of directional monotonicity

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