Introduction to Fuzzy Networks

Alexander Gegov and Nedyalko Petrov – University of Portsmouth, United Kindom / Technical University of Sofia, Bulgaria

The notion of complexity has recently become a serious challenge to scientific research in a multi-disciplinary context. For example, it is quite common to find complex systems in biology, cosmology, engineering, computing, finance and other areas. However, the understanding of complex systems is often a difficult task.There are two main aspects of complexity – quantitative and qualitative. The quantitative aspect is usually associated with a large scale of an entity or a large number of elements within this entity. The qualitative aspect is often characterised by some uncertainty in the data or knowledge about an entity. An obvious way of coping with quantitative complexity is to use the concept of a general network. The latter consists of nodes and connections, whereby the nodes represent the elements of the entity and the connections reflect the interactions among these elements. In this case, the scale is reflected by the overall size of the network, whereas the number of elements is given by the number of nodes.A possible way of dealing with qualitative complexity is to use the concept of a fuzzy network. The latter consists of nodes and connections, whereby the nodes are fuzzy systems and the connections reflect the interactions among these systems. In this case, the uncertainty in the data or knowledge about the entity is reflected by the rule bases of the fuzzy systems and the underlying fuzzy logic.In the context of these considerations, a fuzzy network represents a natural counterpart of a neural network. Both types of networks are computational intelligence based networks with nodes and connections. However, the nodes in a neural network are neurons, whereas the nodes in a fuzzy network are rule bases.This tutorial introduces the novel concept of a fuzzy network within ten sections. The first section discusses complexity as a systemic feature and the ability of fuzzy systems to handle different attributes of complexity. Section 2 reviews several types of fuzzy systems in the context of systemic complexity, including systems with single, multiple and networked rule bases. Section 3 introduces formal models for fuzzy networks such as Boolean matrices, binary relations, block schemes and topological expressions. Section 4 presents basic operations on nodes in fuzzy networks, including merging and splitting in horizontal, vertical and output context. Section 5 discusses some structural properties of basic operations such as associativity of merging and variability of splitting in horizontal, vertical and output context. Section 6 describes advanced operations on nodes in fuzzy networks, including node transformation for input augmentation, output permutation and feedback equivalence, as well as node identification in horizontal, vertical and output merging. Section 7 shows the application of the theoretical results from Sections 3-6 in feedforward fuzzy networks with single or multiple levels and layers. Section 8 illustrates the application of the theoretical results from Sections 3-6 in feedback fuzzy networks with single or multiple local and global feedback. Section 9 evaluates fuzzy networks using structural metrics for comparison with different types of fuzzy systems, composition into standard fuzzy systems, decomposition into hierarchical fuzzy systems, model performance indicators, as well as by demonstrating some examples and case studies in Matlab. The last section highlights the theoretical significance, the application areas and the methodological impact of fuzzy networks in the context of an overall evaluation of the tutorial contents.

Biography

Alexander Gegov is currently with the School of Computing, University of Portsmouth, UK. He holds a PhD in Control Systems and a DSc in Intelligent Systems – both from the Bulgarian Academy of Sciences. He has been Humboldt Guest Researcher at the University of Duisburg in Germany and EU Visiting Researcher at the Delft University of Technology in the Netherlands.Alexander Gegov’s research interests are in the theory of computational intelligence and complex systems as well as their application for modelling and control. He has authored more than 20 journal articles and 40 conference papers. He is also the sole author of two research monographs published by Springer. He has recently introduced and started the development of the novel theory of fuzzy networks. Alexander Gegov has been Referee for a number of international journals including IEEE Transactions on Fuzzy Systems and IEEE Transactions on Neurak Networks. He has recently presented lectures and tutorials at international conferences such as the IEEE Conference on Fuzzy Sytems and the IEEE Confernce on Intelligent Sytems. He is Member of several international organisations including IFAC and EUSFLAT.

Biography

Nedyalko Petrov is currently graduating as an MSc student in Computer Science at the Technical University of Sofia, Bulgaria. He has held fellowships for participation in the EU programmes for cultural and academic exchange Leonardo da Vinchi and Erasmus. As part of these programmes, he has visited the University of Hamburg in Germany and the University of Portsmouth in the UK. Nedyalko Petrov's research interests are in the application of computational intelligence and complex systems for modelling and simulation of processes in finance, industry and other areas. He is coauthor of several research papers and an academic book chapter. He has recently started the implementation in Matlab of the noval theory of fuzzy networks and has validated some of its methods. Nedyalko Petrov has worked as a software developer for a large international company on projects with leading aircraft manufacturers such as Airbus and Boeing. He has recently awarded a prize in Engineering and Technology in the Spirit of Alfred Nobel by the Swedish Institute. He has also received a number of awards from national student competitions in Physics and Mathematics.