Erich Peter Klement

Johannes Kepler University of Linz | JKU · Institute of Knowledge-based Mathematical Systems

A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels. The proofs are based on embedding theorems as well as on probability techniques in Banach space.

The Choquet and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. However, their applicability is restricted because of the special operations used in the construction of these integrals. Therefore, we provide a concept of integrals generalizing both the Choquet an...

We approach the problem whether left-continuous triangular norm-based valuations (called T-measures or T-probability measures) defined on triangular normbased tribes of the unit cube can be disintegrated by Markov kernels. We prove that each T-measure based on a “fundamental” triangular norm (these triangular norms T, together with their correspond...

Ordinal sums of semigroups in the sense of [3] leading to triangular norms [29] are studied, generalizing the ordinal sum of triangular norms [30]. The summands of these general ordinal sums are fully characterized, and the ordinal irreducibility of triangular norms is investigated, inducing a natural partition of the class of all triangular norms.

The Archimedean components of triangular norms (which turn the closed unit interval into anabelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimed...

Comprehensive families of copulas including the three basic copulas (at least as limit cases) are useful tools to model countermonotonicity, independence, and comonotonicity of pairs of random variables on the same probability space. In this contribution, we study how the transition from a (basic) copula to a copula modeling a different dependence...

The Seventeenth International Conference on Fuzzy Set Theory and Applications Liptovský Ján, Slovak Republic January 28 - February 2, 2024

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method...

Copulas are functions that link an n-dimensional distribution function with its one-dimensional margins. In this contribution we show how n-variate copulas with given values at two arbitrary points can be constructed. Thereby, we also answer a so far open question whether lower and upper bounds for n-variate copulas with given value at a single arb...

Following a historical overview of the development of ordinal sums, a presentation of the most relevant results for ordinal sums of triangular norms and copulas is given (including gluing of copulas, orthogonal grid constructions and patchwork operators). The ordinal sums of copulas considered here are constructed not only by means of the comonoton...

A prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb gener...

Bivariate polynomial copulas of degree 5 (containing the family of Eyraud-Farlie-Gumbel-Morgenstern copulas) are in a one-to-one correspondence to certain real parameter triplets (a, b, c), i.e., to some set of polynomials in two variables of degree 1: p(x, y) = ax + by + c. The set of the parameters yielding a copula is characterized and visualize...

Starting with [Goguen, J.A. 1967. Journal of Mathematical Analysis and Applications], several generalizations of the original definition of a fuzzy set have been proposed. In one popular case, one considers as truth values the points in the lower left triangle of the unit square, where their first coordinate is interpreted as “degree of membership”...

Several extensions of the family of (bivariate) Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM copulas) are considered. Some of them are well-known from the literature, others have recently been suggested (copulas based on quadratic constructions, based on some forms of convexity, and polynomial copulas). For each of these extensions we analyze whi...

We start with some binary (“outer”) copula, apply it to an arbitrary binary (“inner”) copula and its dual (the latter being transformed by some real function) and ask under which conditions the result is again a binary copula. Sufficient convexity conditions for the transformation function and for the “outer” copula (ultramodularity and Schur conca...

Picture fuzzy sets, recently introduced by B. C. Cuong and V. Kreinovich, are a special case of L-fuzzy sets. We discuss the set of truth values for these fuzzy sets as well as aggregation functions for these truth values, paying special attention to t-norms and t-conorms. The important role of representable t-norms and t-conorms is emphasized.

We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each res...

In this paper we study the smallest and the greatest M-Lipschitz continuous n-ary aggregation functions with a given diagonal section. We show that several properties that were studied for the smallest and the greatest 1-Lipschitz continuous binary aggregation functions with a given diagonal section extend naturally to higher dimensions while consi...

We discuss several types of ordinal sums for conjunctive operations for an infinite set of truth values (modeled by the real unit interval). In some cases, they can be seen as both a construction method and a representation (for example, when considering copulas), this is no more true for the product-based ordinal sums when considering quasi-copula...

Based on a general construction method by means of bivariate ultramodular copulas we construct, for particular settings, special bivariate conic, extreme value, and Archimax copulas. We also show that the sets of copulas obtained in this way are dense in the sets of all conic, extreme value, and Archimax copulas, respectively.

We study a generalized Frank functional equation in the broader framework of associative aggregation functions and show that, up to the two projections, we obtain exactly the same set of solutions as in the original paper by M. J. Frank (Aequationes Mathematicae 19, 1979).

We discuss and stress the role of ultramodularity and Schur concavity in special types of constructions of copulas. After recalling some known ultamodularity-based results, we focus on the so-called D-product of a copula and its dual. We show that for each copula D which is ultramodular and Schur concave on the left upper triangle of the unit squar...

Six different functions measuring the defect of a quasi-copula, i. e., how far away it is from a copula, are discussed. This is done by means of extremal non-positive volumes of specific rectangles (in a way that a zero defect characterizes copulas). Based on these defect functions, six transformations of quasi-copulas are investigated which give r...

Generalizing a first approach by L. A. ZADEH (J. Math. Anal. Appl. 23, 1968), expected values of fuzzy events are studied which are (up to standard boundary conditions) only required to be monotone. They can be seen as an extension of capacities, i.e., monotone set functions satisfying standard boundary conditions. Some of these expected values can...

This paper presents a novel approach for detecting cracked or broken reciprocating compressor valves under varying load conditions. The main idea is that the time frequency representation of vibration measurement data will show typical patterns depending on the fault state. The problem is to detect these patterns reliably. For the detection task, w...

An overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set (called here monotone measures). Our survey includes not only the Choquet integral (1954) [10], the Shilkret integral (1971) [66] and the Sugeno integral (1974) [71] and some of their properties, but also some more general a...

In the last two decades, multi-objective evolutionary algorithms (MOEAs) have become ever more used in scientific and industrial decision support and decision making contexts the require an a posteriori articulation of preference. The present work is focused on a comparative analysis of the performance of two master–slave parallelization (MSP) meth...

We present a novel approach to detecting leaking reciprocating compressor valves based on the idea that a leaking valve affects the shape of the pressure-volume diagram (pV diagram). This effect can be observed when the valves are closed. To avoid disturbances due to the load control, we concentrate on the expansion phase, and linearize it using th...

The theory of classical measures and integral reflects the genuine property of several quantities in standard physics and/or geometry, namely the σ-additivity. Though monotone measure not assuming σ-additivity appeared naturally in models extending the classical ones (for example, inner and outer measures, where the related integral was considered...

This paper examines robustness issues of fault detection methods for reciprocating compressor valves. The authors have previously proposed two independent fault detection approaches for reciprocating compressor valves. One method is based on vibration analysis of accelerometer data, the other one on analyzing pV diagrams. Based on real world data,...

A hierarchical family of integrals based on a fixed copula is introduced and discussed. The extremal members of this family correspond to the inner and outer extension of integrals of basic functions, the copula under consideration being the corresponding multiplication. The limits of the members of the family are just copula-based universal integr...

We describe a hybrid and adaptive coevolutionary optimization method that can efficiently solve a wide range of multi-objective optimization problems (MOOPs) as it successfully combines positive traits from three main classes of multi-objective evolutionary algorithms (MOEAs): classical approaches that use Pareto-based selection for survival criter...

Ultramodular binary copulas are discussed, i.e., copulas of a random vector whose components are mutually stochastically decreasing with respect to each other. The additive generators of Archimedean ultramodular binary copulas are fully characterized. Finally, a new construction method for binary copulas based on n-ary ultramodular aggregation func...

This paper presents a novel data-driven approach for detecting broken reciprocating compressor valves that is based on the idea that a broken valve will affect the shape of the pressurevolume (pV) diagram. This effect can be observed when the valves are closed. To avoid disturbances due to the load control we concentrate on the expansion phase line...

Performance optimization of electrical drives implies a lot of degrees of freedom in the variation of design parameters, which in turn makes the process overly complex and sometimes impossible to handle for classical analytic optimization approaches. This, and the fact that multiple non-independent design parameter have to be optimized synchronousl...

The task of designing electrical drives is a multi-objective optimization problem (MOOP) that remains very slow even when using state-of-the-art approaches like particle swarm optimization and evolutionary algorithms because the fitness function used to assess the quality of a proposed design is based on time-intensive finite element (FE) simulatio...

This paper presents a novel data-driven approach to detecting broken reciprocating compressor valves that is based on the idea that a broken valve will affect the shape of the pressure-volume (pV) diagram. This effect can be observed when the valves are closed. To avoid disturbances due to the load control, we concentrate on the expansion phase, li...

A hierarchical family of copula-based integrals is introduced and discussed. When considering the product copula, a family of decomposition integrals independently introduced by Even and Lehrer, and by Mesiar and Stupňanová, is recovered. Boundary members are distinguished universal integrals introduced by Klement at al. in 2010.

This paper is focused on a comparative analysis of the performance of two master-slave parallelization methods, the basic generational scheme and the steady-state asynchronous scheme. Both can be used to improve the convergence speed of multi-objective evolutionary algorithms (MOEAs) that rely on time-intensive fitness evaluation functions. The imp...

A common criticism to simple majority voting rule is the slight support that such rule demands to declare an alternative as a winner. Among the distinct majority rules used for diminishing this handicap, we focus on majorities based on difference in ...

While machine learning is most often concerned with learning from humans, the fact that human behavior systematically differs for (groups of) people with different gender, age, education or cultural background is widely ignored. Obviously, such differences are reflected in the training humans provide to machine learning algorithms that in turn affe...

We propose a 2-population cooperative coevolutionary optimization method that can efficiently solve multi-objective optimization problems as it successfully combines positive traits from classic multi-objective evolutionary algorithms and from newer optimization approaches that explore the concept of differential evolution. A key part of the algori...

An axiomatic approach to universal integrals based on level dependent capacities is presented. Based on a given semicopula, two corresponding extremal universal integrals based on level dependent capacities are given. If the underlying semicopula is even a copula, another class of universal integrals based on level dependent capacities is studied,...

Several discrete universal integrals on finite universes are discussed from an axiomatic point of view. We start from the first attempt due to B. Riemann and cover also most recent approaches based on level dependent capacities. Our survey includes, among others, the Choquet and the Sugeno integral and general copula-based integrals.

This paper presents a novel data-driven approach for detecting cracks in reciprocating compressor valves by analyzing vibration data. The main idea is that the timefrequency representation will show typical patterns, depending on the fault state and other variables. The problem of detecting these patterns reliably is solved by taking a detour via t...

This paper presents a visual approach for detecting cracked or broken reciprocating compressor valves. The main idea is that the time frequency representation of vibration measurement data shows typical patterns depending on the fault state. As the patterns differ from those induced by changing load levels, the method is independent of the load lev...

Fuzzy integrals can be seen as expressions for the expectation of fuzzy events. Based on the general concept of universal integrals, we present universal fuzzy integrals on abstract spaces. In particular, we discuss their discrete versions linked to finite universes. An axiomatic approach to discrete universal fuzzy integrals is also given.

A copula-based method to integrate a real vector-valued function, obtaining a single real number, is discussed. Special attention is paid to the case when the underlying universe is finite. The integral considered here is shown to be an extension of [0,1]-valued copula-based universal integrals.

We study the cross-migrativity of triangular norms. The classes of continuous triangular norms, which are cross-migrative with respect to some strict or nilpotent triangular norm, respectively, are completely characterized, as well as those which are cross-migrative with respect to the greatest and smallest triangular norms, respectively. As a by-p...

Mechanical material testing combined with optical coherence tomography (OCT) allows for the first time the immediate detection of inner structural changes along with a qualitative observation of the local strain distribution in sur- face near bulk regions of semitransparent and translucent specimens. In addition to a 3D full field strain analysis (...

Ultramodular aggregation functions are investigated and discussed, including a study of structural properties and the proposal of some construction methods.

Among manufacturing companies there is a widespread consensus that women are better suited to perform visual quality inspection, having higher endurance and making decisions with better reproducibility. Up to now gender-differences in visual inspection decision making have not been thoroughly investigated. We propose a machine learning approach to...

AbstractAntipollution legislation in automotive internal combustion engines requires active control and prediction of pollutant formation and emissions. Predictive emission models are of great use in the system calibration phase, and also can be integrated for the engine control and on-board diagnosis tasks. In this paper, fuzzy modelling of the NO...

Ultramodular aggregation functions are investigated and discussed, including a study of structural properties and the proposal of some construction methods.

An OWA-operator (ordered weighted averaging aggregation operator) can be seen as a discrete Choquet integral with respect to a symmetric monotone measure. Based on this representation and using universal integrals, several modifications of OWA-operators are introduced and discussed.

Among manufacturing companies there is a wide-spread consensus that women are better suited to perform visual quality inspection, having higher endurance and making decisions with better reproducibility. Up to now gender-differences in visual inspection decision making have not been thoroughly investigated. We propose a machine learning approach to...

The Choquet and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. However, their applicability is somehow restricted because of the special operations used in the construction of these integrals. This survey presents the main ideas and results concerning the constr...

We introduce a set of axioms for measures of non-exchangeability for bivariate vectors of continuous and identically distributed random variables and give some examples together with possible applications in statistical models based on the copula function. KeywordsCopula-Dependence properties-Exchangeability

Following the ideas of stronger forms of monotonicity for unary real functions and for capacities, k-monotone and strongly k-monotone aggregation functions are introduced and discussed. In the special case k = 2 also some applications are given.

New emission abatement technologies for the internal combustion engine, like selective catalyst systems or diesel particulate filters, need of accurate, predictive emission models. These models are not only used in the system calibration phase, but can be integrated for the engine control and on-board diagnosis tasks. In this paper, we are investig...

Bioanalytical chip-based assays have been enormously improved in sensitivity in the recent years; detection of trace amounts of substances down to the level of individual fluorescent molecules has become state-of-the-art technology. The impact of such detection methods, however, has yet not fully been exploited, mainly due to a lack of appropriate...

The stability of discrete universal integrals based on copulas is discussed and examined, both with respect to the norms L1 (Lipschitz stability) and L∞ (Chebyshev stability). Each of these integrals is shown to be 1-Lipschitz. Exactly the discrete universal integrals based on a copula which is stochastically increasing in its first coordinate turn...

This paper surveys the present state of knowledge on propositional fuzzy logics extending SBL with an additional involutive negation. The involutive negation is added as a new propositional connective in order to improve the expressive power of the standard mathematical fuzzy logics based on continuous triangular norms.

Uninorms as binary operations on the unit interval have been widely applied in the fuzzy set theory. This paper presents some properties of uninorm-like operations for which the underlying operations are given by ordinal sums. If the underlying operations ...

Following the ideas of stronger forms of monotonicity for unary real functions and for capacities, k-monotone and strongly k-monotone aggregation functions are introduced and discussed. In the special case k = 2 also some applications are given.

Special attention is paid to bivariate ultramodular aggregation functions, where modular functions and copulas play a substantial role, but also Archimedean ultramodular copulas are characterized and a new method for constructing bivariate copulas based on ultramodular aggregation functions is proposed and exemplified.

For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures, including as special cases Choquet and Sugeno integrals. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the great...

Three different types of universal integral based on level dependent capacities are introduced and discussed. Two ex- tremal types are based on Caratheodory's idea of inner and outer measures, while the third type is introduced for copula-based univer- sal integrals only. Keywords— Choquet integral, Copula, Fuzzy measure, Level de- pendent capaciti...

We determine two constructions that, starting with two bivariate copulas, give rise to new bivariate and trivariate copulas, respectively. These constructions are used to determine pointwise upper and lower bounds for the class of all trivariate copulas with given bivariate marginals.

This chapter describes some highlights of successful research focusing on knowledge-based and data-driven models for industrial and decision processes. This research has been carried out during the last ten years in a close cooperation of two research institutions in Hagenberg: - the Fuzzy Logic Laboratorium Linz-Hagenberg (FLLL), a part of the Dep...