Construction of aggregation operators for automated decision making via optimal interpolation and global optimization

This paper examines methods of point wise construction of aggregation operators via optimal interpolation. It is shown that several types of application-specific requirements lead to interpolatory type constraints on the aggregation function. These constraints are translated into global optimization problems, which are the focus of this paper. We present several methods of reduction of the number of variables, and formulate suitable numerical algorithms based on Lipschitz optimization.

Absorbent tuples of aggregation operators

We generalize the notion of an absorbent element of aggregation operators. Our construction involves tuples of values that decide the result of aggregation. Absorbent tuples are useful to model situations in which certain decision makers may decide the outcome irrespective of the opinion of the others. We examine the most important classes of aggregation operators in respect to their absorbent tuples, and also construct new aggregation operators with predefined sets of absorbent tuples.

Construction of aggregation operators with noble reinforcement

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements.

Fitting ST-OWA operators to empirical data

The OWA operators gained interest among researchers as they provide a continuum of aggregation operators able to cover the whole range of compensation between the minimum and the maximum. In some circumstances, it is useful to consider a wider range of values, extending below the minimum and over the maximum. ST-OWA are able to surpass the boundaries of variation of ordinary compensatory operators. Their application requires identification of the weighting vector, the t-norm, and the t-conorm. This task can be accomplished by considering both the desired analytical properties and empirical data.

Constructions of aggregation operators that preserve ordering of the data

We address the issue of identifying various classes of aggregation operators from empirical data, which also preserves the ordering of the outputs. It is argued that the ordering of the outputs is more important than the numerical values, however the usual data fitting methods are only concerned with fitting the values. We will formulate preservation of the ordering problem as a standard mathematical programming problem, solved by standard numerical methods.

Guest editorial: foreword to the special issue on aggregation operators

The fourteen papers in this special section are devoted to aggregation operators with respect to knowledge based systems.

Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.
With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.

Pointwise construction of Lippschitz aggregation operators with specific properties

This paper describes an approach to pointwise construction of general aggregation operators, based on monotone Lipschitz approximation. The aggregation operators are constructed from a set of desired values at certain points, or from empirically collected data. It establishes tight upper and lower bounds on Lipschitz aggregation operators with a number of different properties, as well as the optimal aggregation operator, consistent with the given values. We consider conjunctive, disjunctive and idempotent n-ary aggregation operators; p-stable aggregation operators; various choices of the neutral element and annihilator; diagonal, opposite diagonal and marginal sections; bipolar and double aggregation operators. In all cases we provide either explicit formulas or deterministic numerical procedures to determine the bounds. The findings of this paper are useful for construction of aggregation operators with specified properties, especially using interpolation schemata.

Interpolatory type construction of general aggregation operators

In this chapter we examine a number of methods to construct aggregation operators of interpolatory type for specific applications. The construction is based on the desired values of the aggregation operator at certain prototypical points, and on other desired properties, such as, conjuctive, disjunctive or averaging behaviour, symmetry and marginals.

Identification of weights in aggregation operators

This chapter provides a review of various techniques for identification of weights in generalized mean and ordered weighted averaging aggregation operators, as well as identification of fuzzy measures in Choquet integral based operators. Our main focus is on using empirical data to compute the weights. We present a number of practical algorithms to identify the best aggregation operator that fits the data.

OWA operators in linear regression and detection of outliers

We consider the use of Ordered Weighted Averaging (OWA) in linear regression. Our goal is to replace the traditional least squares, least absolute deviation, and maximum likelihood criteria with an OWA function of the residuals. We obtain several high breakdown robust regression methods as special cases (least median, least trimmed squares, trimmed likelihood methods). We also present new formulations of regression problem. OWA-based regression is particularly useful in the presence of outliers.

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.

Generalized Bonferroni mean operators in multi-criteria aggregation

In this paper we provide a systematic investigation of a family of composed aggregation functions which generalize the Bonferroni mean. Such extensions of the Bonferroni mean are capable of modeling the concepts of hard and soft partial conjunction and disjunction as well as that of k-tolerance and k-intolerance. There are several interesting special cases with quite an intuitive interpretation for application.

Image reduction with local reduction operators

In this work we propose an image reduction algorithm based on weak local reduction operators. We use several averaging functions to build these operators and we analyze their properties. We present experimental results where we apply the algorithm and weak local reduction operators in procedures of reduction, and later, reconstruction of images. We analyze these results over natural images and noisy images.