Marek Gagolewski

Nearest piecewise linear approximation of fuzzy numbers

Abstract The problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers is discussed. By using 1-knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study. They might be also perceived as a generalization of the trapezoidal approximations. Moreover, these approximations possess some desirable properties. Apart from theoretical considerations approximation algorithms that can be applied in practice are also given.

Monotone measures and universal integrals in a uniform framework for the scientific impact assessment problem

The Choquet, Sugeno, and Shilkret integrals with respect to monotone measures, as well as their generalization - the universal integral, stand for a useful tool in decision support systems. In this paper we propose a general construction method for aggregation operators that may be used in assessing output of scientists. We show that the most often currently used indices of bibliometric impact, like Hirschs h, Woegingers w, Egghes g, Kosmulskis MAXPROD, and similar constructions, may be obtained by means of our framework. Moreover, the model easily leads to some new, very interesting functions.

Bibliometric impact assessment with R and the CITAN package

In this paper CITAN, the CITation ANalysis package for R statistical computing environment, is introduced. The main aim of the software is to support bibliometricians with a tool for preprocessing and cleaning bibliographic data retrieved from SciVerse Scopus and for calculating the most popular indices of scientific impact.

Aggregating different paper quality measures with a generalized h-index

The process of assessing individual authors should rely upon a proper aggregation of reliable and valid papers’ quality metrics. Citations are merely one possible way to measure appreciation of publications. In this study we propose some new, SJR- and SNIP-based indicators, which not only take into account the broadly conceived popularity of a paper (manifested by the number of citations), but also other factors like its potential, or the quality of papers that cite a given publication. We explore the relation and correlation between different metrics and study how they affect the values of a real-valued generalized h-index calculated for 11 prominent scientometricians. We note that the h-index is a very unstable impact function, highly sensitive for applying input elements’ scaling. Our analysis is not only of theoretical significance: data scaling is often performed to normalize citations across disciplines. Uncontrolled application of this operation may lead to unfair and biased (toward some groups) decisions. This puts the validity of authors assessment and ranking using the h-index into question. Obviously, a good impact function to be used in practice should not be as much sensitive to changing input data as the analyzed one.

Spread measures and their relation to aggregation functions

The theory of aggregation most often deals with measures of central tendency. However, sometimes a very different kind of a numeric vector’s synthesis into a single number is required. In this paper we introduce a class of mathematical functions which aim to measure spread or scatter of one-dimensional quantitative data. The proposed definition serves as a common, abstract framework for measures of absolute spread known from statistics, exploratory data analysis and data mining, e.g. the sample variance, standard deviation, range, interquartile range (IQR), median absolute deviation (MAD), etc. Additionally, we develop new measures of experts’ opinions diversity or consensus in group decision making problems. We investigate some properties of spread measures, show how are they related to aggregation functions, and indicate their new potentially fruitful application areas.

Scientific impact assessment cannot be fair

In this paper we deal with the problem of aggregating numeric sequences of arbitrary length that represent e.g. citation records of scientists. Impact functions are the aggregation operators that express as a single number not only the quality of individual publications, but also their authors productivity.

On the relationship between symmetric maxitive, minitive, and modular aggregation operators

In this paper the relationship between symmetric minitive, maxitive, and modular aggregation operators is considered. It is shown that the intersection between any two of the three discussed classes is the same. Moreover, the intersection is explicitly characterized. It turns out that the intersection contains families of aggregation operators such as OWMax, OWMin, and many generalizations of the widely-known Hirschs h-index, often applied in scientific quality control.

Axiomatic characterizations of (quasi-) L-statistics and S-statistics and the Producer Assessment Problem

Two classes of aggregation functions: L-statistics and S-statistics and their generalizations called quasi-L-statistics and quasi-S-statistics are considered. Some interesting characterizations of these families of operators are given. The aforementioned functions are useful for various applications. In particular, they are very helpful for modeling the socalled Producer Assessment Problem.

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.

OM3: Ordered maxitive, minitive, and modular aggregation operators – Axiomatic and probabilistic properties in an arity-monotonic setting

The recently-introduced OM3 aggregation operators fulfill three appealing properties: they are simultaneously minitive, maxitive, and modular. Among the instances of OM3 operators we find e.g. OWMax and OWMin operators, the famous Hirsch h-index and all its natural generalizations. In this paper the basic axiomatic and probabilistic properties of extended, i.e. in an arity-dependent setting, OM3 aggregation operators are studied. We illustrate the di culties one is inevitably faced with when trying to combine the quality and quantity of numeric items into a single number. The discussion on such aggregation methods is particularly important in the information resources producers assessment problem, which aims to reduce the negative e ects of information overload. It turns out that the Hirsch-like indices of impact do not fulfill a set of very important properties, which puts the sensibility of their practical usage into question. Moreover, thanks to the probabilistic analysis of the operators in an i.i.d. model, we may better understand the relationship between the aggregated items’ quality and their producers’ productivity.