Elsevier

Information Sciences

Volume 300, 10 April 2015, Pages 70-84
Information Sciences

On the weak monotonicity of Gini means and other mixture functions

https://doi.org/10.1016/j.ins.2014.12.030Get rights and content

Abstract

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.

Introduction

Growing interest in the field of aggregation functions [9], [16], [29] has led to the development of several new mathematical techniques, which have subsequently fostered the creation and analysis of new families of aggregation functions. One such technique is based on minimizing penalties that describe disagreement between the inputs [13], [21], such that the output of an aggregation procedure is a value that is representative of the inputs in terms of the smallest penalty. It is well known that the arithmetic mean and the median are values that minimize the sum of squared and absolute deviations from the inputs respectively. Many other aggregation functions can be represented in terms of minima of particular penalty functions and all averaging functions can be represented as penalty based functions [13].

The class of penalty based functions is extensive and includes idempotent functions that are not monotone; for example the mode. 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. Indeed, cluster structure may change when only some inputs are increased (or decreased), but it does not change when all inputs are changed by the same value. It was shown that location estimators [27] and some useful classes of means, like 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 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]. Mixture functions are a subclass of Bajraktarevic means, which generalize quasi-arithmetic means [2], [3]. In mixture functions the inputs are averaged as in a weighted mean, but the weights depend on the inputs. Weights can thus be chosen so as to alternatively emphasize or de-emphasize the small or large inputs. When applied to averaging functions measuring the distance between the inputs, such means allow for de-emphasizing contributions from outliers. An equivalent class of functions was also presented in [23] under the name of statistically grounded aggregation operators. It was shown in [20] that mixture operators can be represented as penalty based functions.

However, as mixture functions are generally not monotone, the conditions on the weighting functions which guarantee monotonicity of the aggregation are important. Several sufficient, but not necessary conditions for monotonicity were established in [19], [24]. These conditions are rather restrictive; for example, the weighting function must be monotone. As weak monotonicity has been proposed as a relaxation on the definition of averaging aggregation functions, it makes sense to study sufficient conditions that guarantee the weak monotonicity of mixture functions. A better understanding of this class of functions will support their broader application and this is precisely the goal of this paper. Among other contributions we will identify sufficient conditions for the weak monotonicity of Gini means.

The remainder of this article is structured as follows. In Section 2 we provide the necessary mathematical foundations that underpin aggregation functions and means, which we rely on in subsequent sections. The recent contribution of weak monotonicity and known properties are also presented. In Section 3 we provide new contributions to the understanding of weakly monotone averaging aggregation functions by providing further properties. Within Section 4 we examine several non-monotone mixture functions and prove conditions under which they are weakly monotone. In Section 5 we present sufficient conditions for the weak monotonicity of Gini means. Our conclusions are presented in Section 6.

Section snippets

Aggregation functions

In this article we make use of the following notations and assumptions. Without loss of generality we assume that the domain of interest is any closed, non-empty interval I=[a,b]R¯=[-,] and that tuples in In are defined as x=(xi,n|nN,i{1,,n}). We write xi as the shorthand for xi,n such that it is implicit that i{1,,n}. Furthermore, In is ordered such that for x,yIn,xy implies that each component of x is no greater than the corresponding component of y. Unless otherwise stated, a

Preservation of weak monotonicity under ϕ-transforms

We extend the work in [32], [34] by determining whether or not a nonlinear ϕ-transform of a weakly monotone averaging function preserves weak monotonicity. We know that standard monotonicity is preserved under the ϕ-transform when ϕ is a monotone bijection, and in [32] it was shown that linear transformations preserve weak monotonicity. However this is no longer the result for weak monotonicity in the general case, although of course for specific weakly monotone functions this property will be

Examples of weakly monotone mixture functions

Let us consider mixture functions given by Eq. (3) and three cases for ensuring they are monotone increasing functions. In each case consider the weights w:[0,1][0,[. Then sufficient conditions for monotonicity are:

  • 1.

    ww for any increasing, piecewise differentiable weighting function [19];

  • 2.

    w(x)w(x)(1-x) for all x[0,1]: [20], [21]; or, if we fix the dimension n of the domain,

  • 3.

    w2(x)(n-1)w(1)+w(x)w(x)(1-x),x[0,1],n>1: [20].

Analogous results have been obtained for decreasing weighting

Weak monotonicity of Gini means

The Gini means are an interesting class of the means of Bajraktarevic Mwg where the weighting functions w and the scaling function g are power functions. In this section we extend the weak monotonicity results obtained in [32] for Lehmer means, which states that:

Theorem 3

The Lehmer mean of n arguments,Lq(x)=i=1nxiq+1i=1nxiq,qR(0,1),is weakly monotone on [0,)n if q0 or if q1 and n1+q+1q-1q-1.

Theorem 4

The Gini mean Gp,q(x1,,xn)=i=1nxipi=1nxiq1p-q for p,qR is weakly monotone on[0,)n if either a) q0 and

Conclusion

In this article we have extended the understanding of averaging aggregation by examining further properties of weakly monotone averaging functions.

Two issues prompted the study of weakly monotone averages. First, that there exist several important classes of averages that fall outside the current definition of aggregation functions (which requires monotonicity in all arguments), and second, that many important applications where the main concern is the effect of noise and outliers on the

Acknowledgement

T. Calvo wishes to acknowledge support from the project TIN2013-42795-P of the National Science Foundation of Spain.

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