OWA operators in regression problems

We consider an application of fuzzy logic connectives to statistical regression. We replace the standard least squares, least absolute deviation, and maximum likelihood criteria with an ordered weighted averaging (OWA) function of the residuals. Depending on the choice of the weights, we obtain the standard regression problems, high-breakdown robust methods (least median, least trimmed squares, and trimmed likelihood methods), as well as new formulations. We present various approaches to numerical solution of such regression problems. OWA-based regression is particularly useful in the presence of outliers, and we illustrate the performance of the new methods on several instances of linear regression problems with multiple outliers.

Use of aggregation functions in decision making

A key component of many decision making processes is the aggregation step, whereby a set of numbers is summarised with a single representative value. This research showed that aggregation functions can provide a mathematical formalism to deal with issues like vagueness and uncertainty, which arise naturally in various decision contexts.

Non-monotone averaging aggregation

We advance the theory of aggregation operators and introduce non-monotone aggregation methods based on minimization of a penalty for inputs disagreements. The application in mind is processing data sets which may contain noisy values. Our aim is to filter out noise while at the same time preserve signs of unusual values. We review various methods of robust estimators of location, and then introduce a new estimator based on penalty minimisation.

Aggregation for Atanassov's intuitionistic and interval valued fuzzy sets: the median operator

Atanassov's intuitionistic fuzzy sets (AIFS) and interval valued fuzzy sets (IVFS) are two generalizations of a fuzzy set, which are equivalent mathematically although different semantically. We analyze the median aggregation operator for AIFS and IVFS. Different mathematical theories have lead to different definitions of the median operator. We look at the median from various perspectives: as an instance of the intuitionistic ordered weighted averaging operator, as a Fermat point in a plane, as a minimizer of input disagreement, and as an operation on distributive lattices. We underline several connections between these approaches and summarize essential properties of the median in different representations.

Fuzzy logic for decision support in chronic care

Computerized clinical guidelines can provide significant benefits in terms of health outcomes and costs, however, their effective computer implementation presents significant problems. Vagueness and ambiguity inherent in natural language (textual) clinical guidelines makes them problematic for formulating automated alerts or advice. Fuzzy logic allows us to formalize the treatment of vagueness in a decision support architecture. In care plan on-line (CPOL), an intranet-based chronic disease care planning system for general practitioners (GPs) in use in South Australia, we formally treat fuzziness in interpretation of quantitative data, formulation of recommendations and unequal importance of clinical indicators. We use expert judgment on cases, as well as direct estimates by experts, to optimize aggregation operators and treat heterogeneous combinations of conjunction and disjunction that are present in the natural language decision rules formulated by specialist teams.

Construction of k-Lipschitz triangular norms and conorms from empirical data

This paper examines the practical construction of k-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based on k-convex additive generators and translate k-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guarantees k-Lipschitz property of the resulting triangular norms and conorms.

On migrative means and copulas

In this short work we extend the results of J.Fodor and I.J. Rudas [6] characterizing migrative triangular norms, to quasi-arithmetic means. We use idempotisation construction to obtain quasi-arithmetic means migrative with respect to fixed parameter a. We also obtain the necessary and sufficient condition for a migrative triangular norm to be a copula.

On generalized bonferroni means

In this paper we investigate modeling capabilities of Bonferroni means and their generalizations. We will show that weighted Bonferroni means can model the concepts of hard and soft partial conjunction, and lead to several interesting special cases, with quite an intuitive interpretation.

Using choquet integrals for evaluating citation indices in journal ranking

The use of citation-based indices to evaluate the quality of journals is becoming increasingly widespread. Recently, ISI Web of Knowledge has begun to include three new indices in their journal statistics, including a 5-year impact factor. Here, we continue our earlier research which modeled the behavior of journal assessors based on some of these indices and the Choquet integral. We interpret the obtained fuzzy measures of many new datasets toward understanding the importance of these newly published indices and how indicative they may be of a journal's quality. The problem is one of ordinal classification, and the values of the best-fitting fuzzy measures can be obtained using the FMtools software package.

Image reduction using means on discrete product lattices

We investigate the problem of averaging values on lattices, and in particular on discrete product lattices. This problem arises in image processing when several color values given in RGB, HSL, or another coding scheme, need to be combined. We show how the arithmetic mean and the median can be constructed by minimizing appropriate penalties, and we discuss which of them coincide with the Cartesian product of the standard mean and median. We apply these functions in image processing. We present three algorithms for color image reduction based on minimizing penalty functions on discrete product lattices.

One image reduction algorithm for RGB color images

We investigate the problem of combining or aggregating several color values given in coding scheme RGB. For this reason, we study the problem of averaging values on lattices, and in particular on discrete product lattices. We study the arithemtic mean and the median on product lattices. We apply these aggregation functions in image reduction and we present a new algorithm based on the minimization of penalty functions on discrete product lattices.

On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts

Extensions of aggregation functions to Atanassov orthopairs (often referred to as intuitionistic fuzzy sets or AIFS) usually involve replacing the standard arithmetic operations with those defined for the membership and non-membership orthopairs. One problem with such constructions is that the usual choice of operations has led to formulas which do not generalize the aggregation of ordinary fuzzy sets (where the membership and non-membership values add to 1). Previous extensions of the weighted arithmetic mean and ordered weighted averaging operator also have the absorbent element 〈1,0〉, which becomes particularly problematic in the case of the Bonferroni mean, whose generalizations are useful for modeling mandatory requirements. As well as considering the consistency and interpretability of the operations used for their construction, we hold that it is also important for aggregation functions over higher order fuzzy sets to exhibit analogous behavior to their standard definitions. After highlighting the main drawbacks of existing Bonferroni means defined for Atanassov orthopairs and interval data, we present two alternative methods for extending the generalized Bonferroni mean. Both lead to functions with properties more consistent with the original Bonferroni mean, and which coincide in the case of ordinary fuzzy values.

On some properties of weighted averaging with variable weights

Density-based means have been recently proposed as a method for dealing with outliers in the stream processing of data. Derived from a weighted arithmetic mean with variable weights that depend on the location of all data samples, these functions are not monotonic and hence cannot be classified as aggregation functions. In this article we establish the weak monotonicity of this class of averaging functions and use this to establish robust generalisations of these means. Specifically, we find that as proposed, the density based means are only robust to isolated outliers. However, by using penalty based formalisms of averaging functions and applying more sophisticated and robust density estimators, we are able to define a broader family of density based means that are more effective at filtering both isolated and clustered outliers. © 2014 Elsevier Inc. All rights reserved.