Weak monotonicity of Lehmer and Gini means

doi:10.1016/j.fss.2015.11.006

Fuzzy Sets and Systems

Volume 299, 15 September 2016, Pages 26-40

https://doi.org/10.1016/j.fss.2015.11.006 Get rights and content

Abstract

We analyze directional monotonicity of several mixture functions in the direction $(1, 1 \dots, 1)$ , 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.

Introduction

Aggregation functions play an important role in many applications including decision making, fuzzy systems and image processing [8], [19], [27]. In particular, aggregation functions are suitable models for fuzzy connectives [16]. Averaging functions, whose prototypical examples are the arithmetic mean and the median, allow compensation between low values of some inputs and high values of the others. These functions are often used in some fuzzy logic systems [35], for example in the weighted compensative logic [17]. More sophisticated averaging functions in which the weightings of the inputs depend on their magnitude include mixture functions [21], [25], density based means [1], statistically grounded aggregation operators [23] and Bajraktarević means, which generalize quasi-arithmetic means [2], [3].

Such functions with variable weights are not necessarily monotone in all arguments, and technically do not fit the definition of the aggregation functions. Some sufficient conditions for monotonicity of mixture functions were established in [22]. A recently proposed concept of weak monotonicity [9], [13], [32], [33], [34] presumes that the value of the aggregate does not decrease when all the inputs are increased by the same value, but may actually decrease if a subset of the inputs increases. For example, when combining the consequents of the If–Then rules in fuzzy rule-based systems, we may want to discard some of the activated rules if their output is very inconsistent with the rest. 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 robust location estimators [26] and some useful classes of means, like Lehmer means, certain mixture functions, density-based means and the mode, are all weakly monotone [9], [13], [32], [33], [34]. Recently the notion of pre-aggregation functions was proposed as a generalization of aggregation functions [20], where the functions are not necessarily monotone but directionally monotone. The weakly monotone functions are hence pre-aggregation functions.

It becomes important to establish conditions under which various averaging functions are weakly monotone. In this paper, we will deal with some mixture and quasi-mixture functions, and in particular with Lehmer and Gini means. In mixture (and quasi-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. In Lehmer and Gini means the weighting function is the power function. We improve the results of [9], [33] which provide some sufficient conditions of monotonicity of Lehmer means. We also improve the results related to weak monotonicity of Gini means [5], [34].

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. Our main results are concentrated in Sections 3, 4 and 5. In Section 6, we recall duality of mixture functions. Our conclusions are presented in Section 7. The proofs of Theorem 3, Theorem 5 and Theorem 6 are presented in Appendix A.

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 $I = [a, b] \subseteq \bar{R} = [- \infty, \infty]$ and that tuples in $I^{n}$ are defined as $x = (x_{i, n} | n \in N, i \in {1, \dots, n})$ . We write $x_{i}$ as the shorthand for $x_{i, n}$ such that it is implicit that $i \in {1, \dots, n}$ . Furthermore, $I^{n}$ is ordered such that for $x, y \in I^{n}$ , $x \leq y$ implies that each component of x is no greater than the corresponding component of y. Unless otherwise stated, a

Weak monotonicity of Lehmer means

Definition 8

The mapping $L_{m} : {[0, \infty [}^{n} \to [0, \infty [$ , $m \in R$ , given by $L_{m} (x_{1}, \dots, x_{n}) = \frac{\sum_{i = 1}^{n} x_{i}^{m + 1}}{\sum_{i = 1}^{n} x_{i}^{m}}$ is called the Lehmer mean.

Note that the Lehmer mean is homogeneous (of degree one), and hence it is sufficient to establish (weak) monotonicity on some finite domain ${[0, b]}^{n}$ . The Lehmer mean is monotone for $m \in [- 1, 0]$ and hence weakly monotone in that parameter range, but it is not weakly monotone for $m \in] 0, 1 [$ , see [18], [32].

The following sufficient condition for weak monotonicity of the Lehmer mean was established in [9],

Weak monotonicity of Gini means

Previously the following condition was established in [5].

Theorem 4

The Gini mean (5) for $p, q \in R$ is weakly monotone on ${[0, \infty [}^{n}$ if either

1.
$q \leq 0$ and $p \geq q$ , or
2.
$p \leq 0$ and $p \leq q$ , or
3.
$(n - 1) {({(\frac{q}{p - 1})}^{p - 1} {(\frac{q - 1}{p})}^{q - 1})}^{\frac{1}{p - q}} \leq 1$ and $(p \geq q \geq 1 or 1 \leq p \leq q)$ .

Note that for $p = q$ the Gini mean is given by (6) which is monotone. The cases $q = 0$ or $p = 0$ result in (monotone) power means. Since $G^{p, q} = G^{q, p}$ we only consider the case $p > q$ . In the case $q \in] 0, 1 [$ the partial derivative at $x = (a, b)$ when $a \to 0^{+}$ tends to −∞, so the Gini mean is not weakly monotone for such

Weak monotonicity of mixture functions with affine weighting functions

A linear weighting function is appealing due to its simplicity, but as we have seen, $L_{1}$ is weakly monotone only for two arguments, which significantly limits its application. In this section we discuss a modification of the weighting function in $L_{1}$ which would ensure the result is weakly monotone.

Theorem 7

Let $M_{w} : {[0, 1]}^{n} \to [0, 1]$ be a mixture function defined by (4) with an affine weighting function $w (x) = x + l$ , $l \geq 0$ . Then $M_{w}$ is weakly monotone increasing for $l \geq \frac{\sqrt{2} - 1}{2} ≐ 0.207107 .$

Proof

On the basis of sufficient condition

The duals of Lehmer mean and other mixture functions

One of the most applied transformations in the context of aggregation functions is the duality transformation $φ_{d} : [0, b] \to [0, b] : φ_{d} (x) = b - x .$ By transformation of a given function F on the interval $[0, b]$ by $φ_{d}$ we get the dual function $F^{d}$ .

Definition 9

Let $F : {[0, b]}^{n} \to [0, b]$ be a function. Then the function $F^{d} : {[0, b]}^{n} \to [0, b]$ given by $F^{d} (x_{1}, \dots, x_{n}) = b - F (b - x_{1}, \dots, b - x_{n}),$ is called the dual of F.

Duality preserves monotonicity and boundary conditions of aggregation functions, idempotency, symmetry, as well as the associativity and

Conclusion

The concept of weak monotonicity of averaging functions has found a number of applications in data analysis. Aggregating data which may contain outliers is a challenging task, because the outliers need to be first identified, and then discarded so as not to corrupt the aggregate. The number of outliers is not fixed a priori, and it is the value of a particular input with respect to the rest which may identify this input as an outlier. In this context the standard monotonicity of aggregation

Acknowledgement

Jana Špirková was supported by the Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic for the Structural Funds of EU, under project ITMS: 26110230082.

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Weak monotonicity of Lehmer and Gini means

Abstract

Introduction

Section snippets

Aggregation functions

Weak monotonicity of Lehmer means

Weak monotonicity of Gini means

Weak monotonicity of mixture functions with affine weighting functions

The duals of Lehmer mean and other mixture functions

Conclusion

Acknowledgement

Inf. Sci.

Knowl.-Based Syst.

Inf. Sci.

Inf. Sci.

Eur. J. Oper. Res.

Inf. Sci.

Fuzzy Sets Syst.

Eur. J. Oper. Res.

Eur. J. Oper. Res.

Inf. Sci.

Inf. Sci.

Inf. Fusion

Sur une équation fonctionelle aux valeurs moyennes

Glasnik Mat.-Fiz. i Astronom. Drustvo Mat. Fiz. Hrvatske. Ser. II

Sur une genéralisation des moyennes quasilineaire

Publ. Inst. Math. (Belgr.)

Characterizing compactness of geometrical clusters using fuzzy measures

IEEE Trans. Fuzzy Syst.

Fuzzy measures of pixel cluster compactness

Aggregation Functions: A Guide for Practitioners