Bernard De Baets

Fuzzy rule-based models for decision support in ecosystem management

To facilitate decision support in the ecosystem management, ecological expertise and site-specific data need to be integrated. Fuzzy logic can deal with highly variable, linguistic, vague and uncertain data or knowledge and, therefore, has the ability to allow for a logical, reliable and transparent information stream from data collection down to data usage in decision-making. Several environmental applications already implicate the use of fuzzy logic. Most of these applications have been set up by trial and error and are mainly limited to the domain of environmental assessment. In this article, applications of fuzzy logic for decision support in ecosystem management are reviewed and assessed, with an emphasis on rule-based models. In particular, the identification, optimisation, validation, the interpretability and uncertainty aspects of fuzzy rule-based models for decision support in ecosystem management are discussed.

The functional equations of Frank and Alsina for uninorms and nullnorms

The aim of this work is to study the functional equations of Frank and Alsina for two classes of commutative, associative and increasing binary operators. The first one is the class of uninorms introduced by Yager and Rybalov. The second one is the class of nullnorms arising from our study of the Frank equation for uninorms. Both classes contain t-norms and t-conorms as special cases. Moreover, the structure of the other uninorms and nullnorms is closely related to t-norms and t-conorms. These observations are the motivation for studying some generalizations of the Frank and Alsina equations. However, it is shown that all considerations lead back to the already known t-norm and t-conorm solutions. Important consequences in fuzzy preference modelling are pointed out.

Triangular norms on product lattices

Abstract In this paper, triangular norms (t-norms) are studied in the general setting of bounded partially ordered sets, with emphasis on finite chains, product lattices and the real unit square. The sets of idempotent elements, zero divisors and nilpotent elements associated to a t-norm are introduced and related to each other. The Archimedean property of t-norms is discussed, in particular its relationship to the diagonal inequality. The main subject of the paper is the direct product of t-norms on product posets. It is shown that the direct product of t-norms without zero divisors is again a t-norm without zero divisors. A weaker version of the Archimedean property is presented and it is shown that the direct product of such pseudo-Archimedean t-norms is again pseudo-Archimedean. A generalization of the cancellation law is presented, in the same spirit as the definition of the set of zero divisors. It is shown that the direct product of cancellative t-norms is again cancellative. Direct products of t-norms on a product lattice are characterized as t-norms with partial mappings that show some particular morphism behaviour. Finally, it is shown that in the case of the real unit square, transformations by means of an automorphism preserve the direct product structure.

A sub‐pixel mapping algorithm based on sub‐pixel/pixel spatial attraction models

Soft classification techniques avoid the loss of information characteristic to hard classification techniques when handling mixed pixels. Sub‐pixel mapping is a method incorporating benefits of both hard and soft classification techniques. In this paper an algorithm is developed based on sub‐pixel/pixel attractions. The design of the algorithm is accomplished using artificial imagery but testing is done on artificial as well as real synthetic imagery. The algorithm is evaluated both visually and quantitatively using established classification accuracy indices. The resulting images show increased accuracy when compared to hardened soft classifications.

Fast and accurate center of gravity defuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions

In this article three methods are presented to perform the center of gravity (COG) defuzzification method in the context of linguistic fuzzy models with t-norm-based inference: one well-known method, the discretisation method, and two new methods, the slope-based method and the modified transformation function method. The methods are worked out for trapezoidal membership functions forming a fuzzy partition in the sense of Ruspini. Experimental results show that the newly introduced methods exhibit excellent accuracy at an extremely low computational cost compared to the widely applied discretisation method.

A Historical Account of Types of Fuzzy Sets and Their Relationships

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used.

Bacterial species identification from MALDI-TOF mass spectra through data analysis and machine learning

At present, there is much variability between MALDI-TOF MS methodology for the characterization of bacteria through differences in e.g., sample preparation methods, matrix solutions, organic solvents, acquisition methods and data analysis methods. After evaluation of the existing methods, a standard protocol was developed to generate MALDI-TOF mass spectra obtained from a collection of reference strains belonging to the genera Leuconostoc, Fructobacillus and Lactococcus. Bacterial cells were harvested after 24h of growth at 28°C on the media MRS or TSA. Mass spectra were generated, using the CHCA matrix combined with a 50:48:2 acetonitrile:water:trifluoroacetic acid matrix solution, and analyzed by the cell smear method and the cell extract method. After a data preprocessing step, the resulting high quality data set was used for PCA, distance calculation and multi-dimensional scaling. Using these analyses, species-specific information in the MALDI-TOF mass spectra could be demonstrated. As a next step, the spectra, as well as the binary character set derived from these spectra, were successfully used for species identification within the genera Leuconostoc, Fructobacillus, and Lactococcus. Using MALDI-TOF MS identification libraries for Leuconostoc and Fructobacillus strains, 84% of the MALDI-TOF mass spectra were correctly identified at the species level. Similarly, the same analysis strategy within the genus Lactococcus resulted in 94% correct identifications, taking species and subspecies levels into consideration. Finally, two machine learning techniques were evaluated as alternative species identification tools. The two techniques, support vector machines and random forests, resulted in accuracies between 94% and 98% for the identification of Leuconostoc and Fructobacillus species, respectively.

Characterizable fuzzy preference structures

In this paper, we study the existence, construction and reconstruction of fuzzy preferencestructures. Starting from the definition of a classical preference structure, we propose anatural definition of a fuzzy preference structure, merely requiring the fuzzification of theset operations involved. Upon evaluating the existence of these structures, we discover thatthe idea of fuzzy preferences is best captured when fuzzy preference structures are definedusing a L Ú ukasiewicz triplet. We then proceed to investigate the role of the completenesscondition in these structures. This rather extensive investigation leads to the proposal of astrongest completeness condition, and results in the definition of a one-parameter class offuzzy preference structures. Invoking earlier results by Fodor and Roubens, the constructionof these structures from a reflexive binary fuzzy relation is then easily obtained. Thereconstruction of such a structure from its fuzzy large preference relation inevitable toobtain a full characterization of these structures in analogy to the classical case is morecumbersome. The main result of this paper is the discovery of a non-trivial characterizingcondition that enables us to fully characterize the members of a two-parameter class offuzzy preference structures in terms of their fuzzy large preference relation. As a remarkableside-result, we discover three limit classes of characterizable fuzzy preference structures,traces of which are found throughout the preference modelling literature.

THE FUNDAMENTALS OF FUZZY MATHEMATICAL MORPHOLOGY PART 1: BASIC CONCEPTS

Fuzzy mathematical morphology provides an alternative extension of the binary morphological operations to gray-scale images based on the theory of fuzzy sets. This paper introduces the basic concepts of fuzzy mathematical morphology, starting from the original definitions of the morphological operations by Serra. More specifically, the fuzzy dilation, erosion, closing and opening operations are introduced. Their basic properties such as monotonicity and interaction with union and intersection are discussed in detail. Some important relationships between the fuzzy erosion and fuzzy dilation are established.

Fuzzy preference structures without incomparability

Abstract In this paper, we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability. This study is carried out for the fuzzy preference structures introduced recently by De Baets, Van de Walle and Kerre. A set of remarkable theorems gives detailed insight in the relationships between the sup- T transitivity of the fuzzy preference and indifference relations and the sup- T transitivity of the fuzzy large preference relation. Several paths of thought, involving t-norms with or without zero-divisors, are explored and, where required, illustrative counterexamples confirm the falsity of certain implications. Finally, we introduce the ( T , N )-Ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this Ferrers property.