Information granules regarded as the key components of knowledge representation help generalize information while the level of granularity of information granules becomes crucial to the problem description and an overall strategy of problem solving in system modeling. The ultimate challenge is to develop a comprehensive model within which information granules serving as an important design asset helps realize problem solving. In this study, we propose a system modeling framework of granular architectures and evaluate its effectiveness. We develop a new approach to build functional rule-based fuzzy models by focusing on the reduction of input space which is realized by genetic algorithms. A concept and practice of a granular fuzzy model is established with interval results of global character by aggregating some collective sources of knowledge (local models). The granular fuzzy model based on interval analysis directly reflects upon the diversity of the local sources of knowledge in which intervals are constructed through the use of the principle of justifiable granularity. Two approaches are proposed to design granular neural networks. The first method is concerned with a formation of a global granular neural network whose architecture is formed as a result of reconciliation of outcomes produced by local neural networks. The second method is aimed at the realization of granular neural networks through the formation of interval (granular) connections around numeric connections of the original neural networks where single- and multiple-objective particle swarm optimization is used. We develop a granular analytic hierarchy process (AHP), which provides decision--makers a significant level of flexibility (expressed by the granular nature of the underling construct) so that their initial preferences can be adjusted within a certain interval to achieve higher level of consensus within the group. Moreover, a granulation of linguistic information used in the AHP model is adopted to elevate the consistency of the obtained solution. Throughout the overall study, particle swarm optimization is used as a comprehensive optimization framework to realize the design of granular constructs.
This study is aimed at a brief, carefully focused retrospective view at the Computational Intelligence – a paradigm supporting the analysis and synthesis of intelligent systems. We stress the reason behind the emergence of this discipline and identify its main features. We highlight the synergistic aspects of Computational Intelligence arising from an interaction and collaboration of fuzzy sets, neural networks, and evolutionary optimization. Some promising directions of future fundamental and applied research are also identified.
The paper introduces a neural network-based model of logical connectives. The network consists of two types of generic OR and AND neurons structured into a three layer topology. The specificity of the logical connectives is captured by the network within its supervised learning. Further analysis of the connections of the network obtained in this way provides a better insight into the nature of the connectives for fuzzy sets; in particular the analysis can look at their non-monotomic and compensative properties. Numerical studies including the Zimmermann-Zysno data set illustrate the performance of the network.
A visible trend in representing knowledge through information granules manifests in the developments of information granules of higher type and higher order, in particular, type-2 fuzzy sets and order-2 fuzzy sets. All these constructs are aimed at the formalization and processing data at a certain level of abstraction. Along the same line, in the recent years, we have seen intensive developments in fuzzy clustering, which are not surprising in light of a growing impact of clustering on fundamentals of fuzzy sets (as supporting ways to elicit membership functions) as well as algorithms (in which clustering and clusters form an integral functional component of various fuzzy models). In this study, we investigate order-2 information granules (fuzzy sets) by analyzing their formal description and properties to cope with structural and hierarchically organized concepts emerging from data. The design of order-2 information granules on a basis of available experimental evidence is discussed and a way of expressing similarity (resemblance) of two order-2 information granules by engaging semantically oriented distance is discussed. In the sequel, the study reported here delivers highly original contributions in the realm of order-2 clustering algorithms. Formally, the clustering problem under discussion is posed as follows: given is a finite collection of reference information granules. Determine a structure in data defined over the space of such granules. Conceptually, this makes a radical shift in comparison with data defined in the p -dimensional space of real numbers Rp. In this situation, expressing distance between two data deserves prudent treatment so that such distance properly captures the semantics and consequently, the closeness between any two information granules to be determined in cluster formation. Following the proposal of the semantically guided distance (and its ensuing design process), we develop an order-2 variant of the fuzzy C-means (FCM), discuss its detailed algorithmic steps, and deliver interpretation of the obtained clustering results. Several relevant applied scenarios of order-2 FCM are identified for spatially and temporally distributed data, which deliver interesting motivating arguments and underline the practical relevance of this category of clustering. Experimental studies are provided to further elicit the performance of the clustering method and discuss essential ways of interpreting results.
Interval-valued intuitionistic multiplicative preference relations (IVIMPRs) form a suitable conceptual framework to represent and process simultaneously uncertain preferred and nonpreferred judgments of decision makers (DMs). The focus of this paper is on group decision-making (GDM) problems realized with IVIMPRs. First, a consistency index is introduced to evaluate the consistency degree for intuitionistic multiplicative preference relations (IMPRs), and a consistency optimization approach is presented to jointly improve the consistency degrees of several IMPRs that do not satisfy the predefined consistency threshold. Then, a consistency definition and an acceptable consistency definition for IVIMPRs are established by splitting an IVIMPR into two IMPRs. For several IVIMPRs with unacceptable consistency, a goal program-based approach is proposed to simultaneously improve their consistency. Subsequently, by minimizing the degree to which the opinions of individual DMs deviate from those of the group, a maximum consensus-based goal program is established to determine the DMs' weights. Furthermore, an aggregation approach is applied to integrate individual IVIMPRs into a collective one. A linear program is then built to determine the interval-valued intuitionistic multiplicative priority weights of alternatives coming from the collective IVIMPR. A consistency-based GDM algorithm is proposed. Finally, a practical example is offered to show the application of the new algorithm, and a comparative analysis is presented to highlight the advantages of the new method.
One of the main challenges is the analysis of large data sets, in particular those containing various types of data, such as time, place, image, and those assuming categorical values. This type of data may contain numerous outliers. Despite the continuous development of data analysis, many methods can be effectively improved, in particular through the use of efficient solutions based on fuzzy set technologies. In this paper, we analyze the improvement of a well-known method, i.e. Isolation Forest, for which we introduce an innovative modification, referred to as the Fuzzy Set-Based Isolation Forest.
Foreword. Preface. 1. Data Mining and Knowledge Discovery. 2. Rough Sets. 3. Fuzzy Sets. 4. Bayesian Methods. 5. Evolutionary Computing. 6. Machine Learning. 7. Neural Networks. 8. Clustering. 9. Preprocessing. Index.
Numerous learning methods for fuzzy cognitive maps (FCMs), such as the Hebbian-based and the population-based learning methods, have been developed for modeling and simulating dynamic systems. However, these methods are faced with several obvious limitations. Most of these models are extremely time consuming when learning the large-scale FCMs with hundreds of nodes. Furthermore, the FCMs learned by those algorithms lack robustness when the experimental data contain noise. In addition, reasonable distribution of the weights is rarely considered in these algorithms, which could result in the reduction of the performance of the resulting FCM. In this article, a straightforward, rapid, and robust learning method is proposed to learn FCMs from noisy data, especially, to learn large-scale FCMs. The crux of the proposed algorithm is to equivalently transform the learning problem of FCMs to a classic-constrained convex optimization problem in which the least-squares term ensures the robustness of the well-learned FCM and the maximum entropy term regularizes the distribution of the weights of the well-learned FCM. A series of experiments covering two frequently used activation functions (the sigmoid and hyperbolic tangent functions) are performed on both synthetic datasets with noise and real-world datasets. The experimental results show that the proposed method is rapid and robust against data containing noise and that the well-learned weights have better distribution. In addition, the FCMs learned by the proposed method also exhibit superior performance in comparison with the existing methods.
