Vector valued information measures and integration with respect to fuzzy vector capacities

Abstract

Integration with respect to vector-valued fuzzy measures is used to define and study information measuring tools. Motivated by some current developments in Information Science, we apply the integration of scalar functions with respect to vector-valued fuzzy measures, also called vector capacities. Bartle–Dunford–Schwartz integration (for the additive case) and Choquet type integration (for the non-additive case) are considered, showing that these formalisms can be used to define and develop vector-valued impact measures. Examples related to existing bibliometric tools as well as to new measuring indices are given.

Introduction

Classical Lebesgue integration of scalar functions has provided some fundamental tools in several areas of the Information Science, including the definition of indices for measuring some aspects of information items. For instance, a great part of mathematical definitions of impact indices for scientific journals can be modeled by means of integrals. Some current research has also pointed out that a natural vector-valued integration of scalar functions with respect to vector measures—the so called Bartle–Dunford–Schwartz integration—may be used to generalize some scalar theoretical settings (scalar-valued impact measures) to vector-valued settings (multi-valued impact measures). Vector-valued integration theory has appeared in the context of pure mathematics and until now this theory has found a lot of applications in Mathematical Analysis and Operator Theory. However, it can be used as an adequate framework for the analysis of problems in other scientific disciplines, since it provides a natural way of representing multi-valued mean properties of scalar functions by the simple rule of “putting each value in a different direction of the space”—that is, vector-valued integration—[4], [12].

In this paper we are concerned with suitable applications of the non-additive extensions of this integration theory to some open investigations in Information Science. An exhaustive study of the spaces of integrable functions that are integrable with respect to a vector-valued Choquet type integration theory has recently been published in [7]. The present paper can be considered a continuation of this line of research. Our aim now—after establishing a general framework of theoretical results, is to give examples and concrete definitions of new indices with detailed explanations of several models for information measures.

There are two facts that must be taken into account in relation with the definition and application of new impact parameters for measuring the tandem quantity/quality of scientific publications. On the one hand, and in the context of the new non-standard measures of information that are called altmetrics [24], there is increasing interest in the design of multi-valued indices. Indeed, decision making on research assessment based on several indices is an outstanding challenge in Information Science (see [1], [10], [12], [32] and references therein). The scientific community agrees on the fact that several (scalar) indices must be jointly used for research evaluation: the suitable mathematical setting for representing this idea is to consider vector-valued impact indices.

On the other hand, impact measures that are not defined by additive functions appear in almost all aspects of measurement of research activity. A relevant example is the h-index which measures a rate among the number of publications with a certain number of citations, and is not given by the usual integral. Integration with respect to (scalar) fuzzy measures is being used nowadays as a standard measure in information science (see for instance [3], [13], [17], [29]). As we said, we will analyze a vector-valued version of these integrals in order to enrich our knowledge about the design of new information measures.

Several matters appear immediately. A non-linear integral of scalar functions with respect to non-finitely additive set functions is needed. In the scalar case, the Choquet integral provides such a tool, although much of the examples recently introduced are not Choquet integrals [13], [17], [29]. However, it seems to be the first natural generalization, and so vector-valued Choquet integrals and related spaces of integrable functions for vector-valued capacities has recently been studied from the formal point of view by several authors [16], [30], [31], including some of the authors of the present paper [7]. Applications of this theory to other fields are also being currently developed (see for example [5]). Thus, we are interested in analyzing vector-valued information measures—impact indices—using this theory, since this is the better-known integration with respect to non-additive measures in the scalar case. This may show the way to a general analysis of non additive vector-valued information measures. In a sense, the present paper must be understood as a continuation of the research of one of the authors of the present paper, who presented a complete theory of the spaces of Choquet integrable functions with respect to a fuzzy capacity in the article [7]. The main theoretical results and the framework of the present work can be found in this paper [7]. However, it must be said that in [7] the problem of integration with respect to a vector-valued capacity is studied in its full generality, and so we have changed notation and consider more restricted types of integrals in the present paper for the aim of simplicity. Roughly speaking, in the integration model for impact measures in Information Science, each integrable function provides such an index. This relation will become clear in the present article.

Let us present now a picture of the “state of the art” regarding the subjects involved in this article. Integration with respect to general set functions has become a very active current research topic, mainly due to its potential applications—not only in Information Science. Concepts such as fuzzy (scalar) measures, pseudo-additive measures, null-additive set functions and non-monotonic measures cover different aspects of this nonlinear theory. The interested reader can find information about these integration theories in a lot of classical and current sources; see for example [6], [8], [19], [20], [22], [23], [28] and references therein. Several authors have also recently paid attention to the second aspect that we want to point out in the paper—the vector-valued generalizations; we may mention here the papers by Kawabe (see for example [14], [15], [16] and references therein), as well as by some other authors [30], [31]). For this vector measure case, the relation between integrable functions and weakly integrable functions with the Bocher, Dunford and Pettis integrability of the corresponding distribution functions have recently been studied in the interesting paper [11] by Fernández, Mayoral and Naranjo. Though we will assume some strong requirements on the vector-valued capacity, countable additivity is not one of them. Of course, some assumptions must be made on the vector-valued fuzzy measure for assuring a reasonable behavior of the integrals (see [14], [16] and references therein). Essentially, integration with respect to a Riesz space valued monotone capacity is defined and analyzed in these papers and the ones to which they refer. Our aim is to use a Lebesgue type integral being free of the order structure of the Banach space in which the capacity takes its values. In [26], such a kind of Bartle–Dunford–Schwartz integral for vector measures on quasi-Banach spaces is considered, but in this case the order properties of quasi-Banach lattices are also strongly used. In the case of the present paper, we do not take into account any lattice order in the Banach space where the capacity takes its values for the construction of our Choquet–Lebesgue type integral. Of course, Banach-lattice-valued capacities will be considered in examples and applications.