This is a website dedicated to the postdoctoral thesis of Dr. Urszula Bentkowska

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About thesis

The main aim of the thesis to consider interval-valued fuzzy methods that improve the classification results and decision processes under incomplete or imprecise information. However, the presented methods may be also appreciated by the entire community, not only fuzzy, but more generally involved in research under uncertainty or imperfect information.

The key part of the book is the description of the original classification algorithms based on interval-valued fuzzy methods. The described algorithms may be applied in decision support systems, for example in medicine or other disciplines where the incomplete or imprecise information may appear (cf. Chapter 4), or for data sets with very large number of objects or attributes (cf. Chapter 5). The presented solutions may cope with the challenges arising from the growth of data and information in our society since they enter the field of large-scale computing. As a result, they may enable efficient data processing.

The presented applications are based on theoretical results connected with the family of comparability relations defined for intervals and other related notions. It is shown the origin, interpretation and properties of the considered concepts deriving from the epistemic interpretation of intervals. Namely, the epistemic uncertainty represents the idea of partial or incomplete information. Since the subject is wide, the book concentrates on theory and applications of new concepts of aggregation functions in interval-valued fuzzy setting. The theory of aggregation functions became an established area of research in the last 30 years. There are many aggregation methods that try, with success, to resolve the challenges of nowadays problems. In this book the so-called possible and necessary aggregation functions are considered. One of the reasons to make a deeper study on this topic is connected with the fact that these notions of aggregation functions were recently introduced and they have not been widely examined before.

The book consists of two parts. Firstly, theoretical background is presented and in the second part application results are analyzed. In theoretical part, in Chapter 1 elements of fuzzy sets theory and its extensions are provided. There are presented the notions of interval-valued fuzzy calculus. Diverse orders applicable for interval-valued comparing, including interval- valued fuzzy settings, are discussed. Furthermore, in Chapter 2 aggregation functions defined on the unit interval [0,1] are recalled and useful notions and properties are provided. Construction methods of interval-valued aggregation functions derive from the real-line settings and interval-valued aggregation functions often inherit the properties of their component functions defined on the unit interval [0,1].

Part II covers two major topics: decision making and classification problems. Chapter 3 is devoted to decision making problems with interval-valued fuzzy methods involved. It is pointed out the usage of new concepts with possible and necessary interpretation involved. Next, the classification problems are discussed. When classifiers are used there is a problem of lowering its performance due to the large number of objects or attributes and in the case of missing values in attribute data. It is shown, that in such situations applying interval-valued fuzzy methods help to retrieve the information and to improve the quality of classification. These issues are discussed in Chapter 4 and Chapter 5 along with the new classification algorithms.

In Chapter 4 there are proposed methods of optimization problem of k-NN classifiers that may be useful in diverse computer support systems facing the problem of missing values in data sets. Missing values appear very often in data sets of computer support systems designed for the medical diagnosis, where the lack of data may be due to financial reasons or the lack of a specific medical equipment in a given medical center.

Chapter 5 presents methods of dealing with large-scale problems such as large number of objects or attributes in data sets. Specifically, there is presented a method of optimization problem of k- NN classifiers in DNA microarray methods for identification of marker genes, where typically there is faced the problem of huge number of attributes. Finally, in Chapter 6 there is presented the performance of the new types of aggregation function for interval-valued fuzzy setting in the computer support system OvaExpert. The book ends with a brief description of the future research plans in the area of presented problems, both in the theoretical and practical aspects.

The book is aimed at practitioners working in the areas of classification and decision making under uncertainty or in large-scale problems, especially in medical diagnosis. It can serve as a brief introduction into the theory of aggregation functions for interval-valued fuzzy settings and application in decision making and classification problems. It can also be used as a supplementary reading for the students of mathematics and computer science. Moreover, the results on aggregation functions may be interesting for computer scientists, system architects, knowledge engineers, programmers, who face a problem of combining various inputs into a single output.