In this paper, we relax the definition of Rényi information dimension. The power law of the Entropy-Layer in the Galton board is discovered and we calculate its information fractal dimension. When the Galton board is extended to bias or three-dimensional space, we get the same fractal features. In addition, according to the connection between Pascal’s triangle and the Poisson distribution, we find constrained Poisson distribution groups with the same information dimension. This is the first time the information entropy is utilized to explore the fractal features of the Galton board and Pascal’s triangle.
This paper presents a new validity index for fuzzy partitions generated by the fuzzy c-means algorithm. The proposed validity index is based on the calculation of factors from the proximity matrix generated from the membership matrix generated by a fuzzy clustering partition algorithm, such as FCM. The experimental results show that the proposed approach is consistent with other well-known metrics and with the dataset structure as observed from Proximity Matrices.
In this study, we present the results of surveys conducted in a group of employees and students of IT faculties presenting the answers to the most important, in our opinion, issues related to software engineering (SE), IT project management, and programming paradigms. The above topics are chosen because of their high relevance to the professional community. The participants taking part in the experiments quantified their input through the process of pairwise comparisons (a so-called Analytic Hierarchy Process, AHP) using an innovative highly interactive approach based on a graphic communication means. The generic AHP method was augmented by the optimization mechanisms delivered by the Particle Swarm Optimization (PSO) in order to deliver the highest possible consistency of responses of the participants. Moreover, we demonstrate a method based on Fuzzy C-Means (FCM) filtering highly inconsistent and unreal experts' assessments. In a series of experiments, we demonstrate the accuracy and stability of the AHP method based on graphical environment. We discuss two variants of aggregation of experts' opinions according to their level of experience in the field of interest. Finally, we show the efficiency of the FCM as the method of preselection of experts' evaluations.
The paper introduces a design methodology for developing fuzzy systems with the aid of fuzzy J K flip-flops. The network representation of the flip-flops expressed with the aid of generic AND and OR neurons is provided. Detailed design algorithms that essentially exploit a parametric learning of the flip-flops are studied. Numerical examples are given as well.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Information granules have been considered to be the fundamental constructs of Granular Computing (GrC). As a useful unsupervised learning technique, Fuzzy C-Means (FCM) is one of the most frequently used methods to construct information granules. The FCM-based granulation-degranulation mechanism plays a pivotal role in GrC. In this paper, to enhance the quality of the degranulation (reconstruction) process, we augment the FCM-based degranulation mechanism by introducing a vector of fuzzification factors (fuzzification factor vector) and setting up an adjustment mechanism to modify the prototypes and the partition matrix. The design is regarded as an optimization problem, which is guided by a reconstruction criterion. In the proposed scheme, the initial partition matrix and prototypes are generated by the FCM. Then a fuzzification factor vector is introduced to form an appropriate fuzzification factor for each cluster to build up an adjustment scheme of modifying the prototypes and the partition matrix. With the supervised learning mode of the granulation-degranulation process, we construct a composite objective function of the fuzzification factor vector, the prototypes and the partition matrix. Subsequently, the particle swarm optimization (PSO) is employed to optimize the fuzzification factor vector to refine the prototypes and develop the optimal partition matrix. Finally, the reconstruction performance of the FCM algorithm is enhanced. We offer a thorough analysis of the developed scheme. In particular, we show that the classical FCM algorithm forms a special case of the proposed scheme. Experiments completed for both synthetic and publicly available datasets show that the proposed approach outperforms the generic data reconstruction approach.
In group decision making (GDM), considering the individuals' satisfaction (i.e., utilities) about the consensus result is important. However, this issue is ignored by the extant minimum cost consensus models (MCCMs). Altruism describes a way of behaving that the individuals unselfishly concern for the utilities of others. Inspired by this, in this article we discuss the minimum cost consensus problem in social network GDM (SNGDM), where the utility constraints based on altruism are first incorporated. In SNGDM, altruism is reflected as the individuals will take both their personal utility and the utilities of others who connected with them into account to form their total utility. Based on this, we first define the individuals' altruistic utility functions in the consensus process of SNGDM, and we propose several interesting properties of the proposed utility function. Afterward, we present a novel MCCM in SNGDM with opinion dynamics and altruistic utility (MCCM-OD-AU). Notably, in the proposed model, the deviation of opinion adjustment in the consensus process is measured as a psychological distance perceived by the individuals. Furthermore, some simulation studies and comparative analysis are conducted to investigate the effects of the altruistic behavior and the psychological distance of opinion adjustment on the consensus reaching results. Finally, two numerical studies, including an illustrative example and an application in energy-saving target formulation with real-world social network data, are provided to justify the performance of our proposal.
