The study of aggregation functions (formerly called aggregation operators) is undeniably one of the most important spin-offs of the fuzzy set community, mainly driven by the need for appropriate logical connectives. The generated body of knowledge has also contributed to the fields of multi-criteria decision making (e.g. OWA operators) and dependence modelling (e.g. quasi-copulas and copulas). However, the focus has increasingly narrowed to the aggregation of real numbers in the unit interval (letting aside a small group of researchers focusing on posets and lattices). In this setting of real numbers, Yager’s penalty functions traditionally play an important role.
In this ‘era of aggregation’, data aggregation has become a central problem in many fields of application, with needs extending largely beyond the setting of real numbers, such as the aggregation of multidimensional data, compositional data, rankings, relations and strings. In this talk, we draw attention to the old notion of a betweenness relation and propose to replace the currently-required property of quasi-convexity of a penalty function by the compatibility with a betweenness relation. Several methods for constructing penalty functions using a monometric, which carries a more appropriate semantics than the more commonly used distance function (metric), are provided. In this way, we are able to model aggregation processes on any set of objects equipped with a betweenness relation, thus considerably expanding the scope of the theory of aggregation.
Interestingly, several existing prominent existing data aggregation methods turn out to fit into this new framework for penalty-based data aggregation. To name but a few, we mention the method of Kemeny for the aggregation of rankings (based on the Kendall distance function), the closest string of a given list of strings (based on the Hamming distance function), the median of a given list of binary relations (based on the ‘symmetric difference’ distance function), and even the basic set-theoretical intersection and union of a given list of sets.
Bernard De Baets is a senior full professor in applied mathematics at the Faculty of Bioscience Engineering (Shanghai rank 37 in Life and Agriculture Sciences) of Ghent University, the top-ranked Belgian university (Shanghai rank 61). He is leading the research unit KERMIT and acts as head of the Department of Data Analysis and Mathematical Modelling. Furthermore, he is an affiliated professor at the Anton de Kom Universiteit (Suriname), an Honorary Professor of Budapest Tech (Hungary), a Doctor Honoris Causa of the University of Turku (Finland) and a Profesor Invitado of the Universidad Central “Marta Abreu” de las Villas (Cuba).
As a trained mathematician, computer scientist and knowledge engineer, Bernard has developed a passion for multi- and interdisciplinary research. He is not only deeply involved in fundamental research in three interlaced research threads, namely knowledge-based, predictive and spatio-temporal modelling, but he also aims at innovative applications in the applied biological sciences. At present, over 30 researchers are involved in the activities of KERMIT. Over the past 20 years, 67 PhD students have graduated under his (co-)supervision.
Bernard is a prolific writer, with a bibliography comprising close to 500 peer-reviewed journal papers, 60 book chapters and 300 contributions to conference proceedings, accumulating over 18000 Google Scholar citations (h-index 67). Several of his works have been bestowed upon with a best paper award. Moreover, he is a much-invited speaker, having delivered over 250 lectures world-wide. In 2011, he was elected Fellow of IFSA (International Fuzzy Systems Association) and in 2012, he was a nominee for the Ghent University Prometheus Award for Research.
Bernard actively serves the research community, in particular as co-editor-in-chief of Fuzzy Sets and Systems and as member of the editorial board of several other journals, including the Internat. J. of Approximate Reasoning, Engineering Applications of Artificial Intelligence, and the Iranian J. of Fuzzy Systems. He is a member of the Administrative Board of the Belgian OR Society.