Three types of monotonicity of averaging functions

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions, which are one of the basic tools in knowledge-based systems. The functions known as means (or averages) are idempotent and typically are monotone, however there are many important classes of means that are non-monotone. Weak monotonicity was recently proposed as a relaxation of the monotonicity condition for averaging functions. In this paper we discuss the concepts of directional and cone monotonicity, and monotonicity with respect to majority of inputs and coalitions of inputs. We establish the relations between various kinds of monotonicity, and illustrate it on various examples. We also provide a construction method for cone monotone functions.

Weakly monotonic averaging functions

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.

On the weak monotonicity of Gini means and other mixture functions

Weak monotonicity was recently proposed as a relaxation of the monotonicity condition for averaging aggregation, and weakly monotone functions were shown to have desirable properties when averaging data corrupted with outliers or noise. We extended the study of weakly monotone averages by analyzing their ϕ-transforms, and we established weak monotonicity of several classes of averaging functions, in particular Gini means and mixture operators. Mixture operators with Gaussian weighting functions were shown to be weakly monotone for a broad range of their parameters. This study assists in identifying averaging functions suitable for data analysis and image processing tasks in the presence of outliers.

Weak monotonicity of Lehmer and Gini means

We analyze directional monotonicity of several mixture functions in the direction (1,1...,1), called weak monotonicity. Our particular focus is on power weighting functions and the special cases of Lehmer and Gini means. We establish limits on the number of arguments of these means for which they are weakly monotone. These bounds significantly improve the earlier results and hence increase the range of applicability of Gini and Lehmer means. We also discuss the case of affine weighting functions and find the smallest constant which ensures weak monotonicity of such mixture functions.

Characteristics evaluation of a μ-Synthesis H∞ controller for a grid-connected three-phase photovoltaic system

This paper presents a μ-Synthesis H∞ Controller for regulating the switching signal of the inverter connected with a three-phase photovoltaic (PV) system. To facilitate the control design, the system is represented in terms of state space realization with uncertainties. The control design involves selecting proper weighting functions and performing synthesis. The controller order is reduced by Henkel-norm method. Simulations are carried out to evaluate the characteristics of the controller under parametric uncertainties. It is found out that the proposed controller is inherently stable, possesses significantly small tracking error, and preserves nominal performance, robust stability and robust performance for the grid-connected three-phase PV system.

Characteristics evaluation of an LMI-synthesis H∞ controller for a superconducting magnetic energy systems applied in power systems

© 2015 IEEE.This paper presents an H« controller synthesised based on linear matrix inequalities (LMI) for a current source converter based superconducting magnetic energy systems (SMESs) connected to a node of power systems where the regulation of grid current has considered as a control objective. To facilitate the control design, the system is represented in terms of state space realization with uncertainties. The control design involves selecting proper weighting functions and performing LMI-synthesis. The controller order is reduced by Henkel-norm method. Simulations are carried out to evaluate the characteristics of the controller under parametric uncertainties. It is found out that the proposed controller is inherently stable, possesses significantly small tracking error, and preserves robust performance for the SMES.

Fitting aggregation functions to data: part I-linearization and regularization

The use of supervised learning techniques for fitting weights and/or generator functions of weighted quasi-arithmetic means – a special class of idempotent and nondecreasing aggregation functions – to empirical data has already been considered in a number of papers. Nevertheless, there are still some important issues that have not been discussed in the literature yet. In the first part of this two-part contribution we deal with the concept of regularization, a quite standard technique from machine learning applied so as to increase the fit quality on test and validation data samples. Due to the constraints on the weighting vector, it turns out that quite different methods can be used in the current framework, as compared to regression models. Moreover, it is worth noting that so far fitting weighted quasi-arithmetic means to empirical data has only been performed approximately, via the so-called linearization technique. In this paper we consider exact solutions to such special optimization tasks and indicate cases where linearization leads to much worse solutions.

Molecular basis of copper transport: cellular and physiological functions of Menkes and Wilson disease proteins (ATP7A and ATPB)

Eukaryotic cells prevent copper-induced, free radical damage to cell components by employing copper-binding proteins and transporters that minimize the likelihood of free copper ions existing in the cell. In the cell, copper is actively transported from the cytoplasm during the biosynthesis of secreted coppercontaining proteins and, as a protective measure, when there is an excess of copper. In humans, this is accomplished by two related copper-transporting ATPases (ATP7A and ATP7B), which are the affected genes in two distinct human genetic disorders of copper transport, Menkes disease (copper deficiency) and Wilson disease (copper toxicosis). The study of these ATPases has revealed their molecular mechanisms of copper transport and their roles in physiological copper homeostasis. Both ATP7A and ATP7B are expressed in specific brain regions and neurological abnormalities are important clinical features in Menkes and Wilson disease.

Comparison of Two Methods for Caluculating the Partition Functions of Various Spatial Statistical Models

Likelihood computation in spatial statistics requires accurate and efficient calculation of the normalizing constant (i.e. partition function) of the Gibbs distribution of the model. Two available methods to calculate the normalizing constant by Markov chain Monte Carlo methods are compared by simulation experiments for an Ising model, a Gaussian Markov field model and a pairwise interaction point field model.

Interpolation of Lipschitz functions

This paper describes a new computational approach to multivariate scattered data interpolation. It is assumed that the data is generated by a Lipschitz continuous function f. The proposed approach uses the central interpolation scheme, which produces an optimal interpolant in the worst case scenario. It provides best uniform error bounds on f, and thus translates into reliable learning of f. This paper develops a computationally efficient algorithm for evaluating the interpolant in the multivariate case. We compare the proposed method with the radial basis functions and natural neighbor interpolation, provide the details of the algorithm and illustrate it on numerical experiments. The efficiency of this method surpasses alternative interpolation methods for scattered data.

A comparison of search functions and other navigational tools provided on government websites

This paper reports the outcomes of a study that evaluated the effectiveness of search functions compared to other navigational tools available on government websites. The study used an observation exercise triangulated with a post observation interview. Results suggest that while there wasn't any significant difference in effectiveness between search functions and other navigational tools, the skill with which the search function is implemented and participants' familiarity with the website, are fundamental determinants of users' opinions. Implications of the findings for research and practice are discussed.

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the computer enumeration of bent functions by Meng et al.