Clustering offers a general methodology and comes with a remarkably rich conceptual and algorithmic framework for data analysis and data interpretation. As one of the most representative algorithms of fuzzy clustering, fuzzy C-means (FCM) is a widely used objective function-based clustering method exploited in various applications. In this study, a virtual-based fuzzy clustering algorithm is proposed to improve the classification performance coming as a result of using fuzzy clustering. This improvement is achieved by forming a virtual space based on the original data space. First, we construct a piecewise linear transformation function to modify the similarity matrix of the original data and build the so-called virtual similarity matrix (VSM). Considering the VSM, the effect of closeness becomes amplified; in other words, high similarity values (say, larger than α which is a cutoff value of the large and small similarity in this paper) present in the original similarity matrix are made higher, whereas lower similarity levels (say, smaller than α) are further reduced. In addition, data with high similarity (say, larger than a certain threshold value) observed in the original space will overlap (the attributes of the samples are exactly the same) significantly in the virtual space; the overlapping samples can be treated as one sample. This modification makes possible easier to identify clusters. Second, we build a relationship matrix between the original dataset and the determined similarity values and present two closed-form solutions to the problem of building the relationship matrix. Subsequently, a virtual space of the original data space is derived through the modified similarity matrix and the introduced relationship matrix. We offer a thorough analysis behind the developed clustering algorithm. The experimental results are in agreement with the underlying conceptual basis. Furthermore, the resulting classification performance is significantly improved compared with the results produced by the FCM and the kernel-based fuzzy C-means.
The emergence of fuzzy sets makes job-shop scheduling problem (JSSP) become better aligned with the reality. This article addresses the JSSP with fuzzy execution time and fuzzy completion time (FJSSP). We choose the classic differential evolution (DE) algorithm as the basic optimization framework. The advantage of the DE algorithm is that it uses a special evolutionary strategy of difference vector sets to carry out mutation operation. However, DE is not very effective in solving some instances of FJSSP. Therefore, we propose a novel selection mechanism augmenting the generic DE algorithm (NSODE) to achieve better optimization results. The proposed selection operator adopted in this article aims at a temporary retention of all children generated by the parent generation, and then selecting N better solutions as the new individuals from N parents and N children. Various examples of fuzzy shop scheduling problems are experimented with to test the performance of the improved DE algorithm. The NSODE algorithm is compared with a variety of existing algorithms such as ant colony optimization, particle swarm optimization, and cuckoo search. Experimental results show that the NSODE can obtain superior feasible solutions compared with solutions produced by several algorithms reported in the literature.
In this paper, we propose an iterative algorithm for multiple regression with fuzzy independent and dependent variables. While using the standard least squares criterion as a performance index we pose the regression problem as a gradient descent optimisation. Since the differentiation and summation are interchangeable we can calculate the gradient as a sum of separate components thus avoiding undue complication of analytical formulas for multiple regression. We discuss the computational complexity of the proposed algorithm.
The paper introduces a generalization of the fuzzy logic connectives AND and OR. To define the logical connectives different t-norms and t-conorms are used. To generalize the t-norms (t-conorms) the Ordinal Sums are introduced. To learn the parameters of the builded Ordinal Sums and the of weights of the connectives the Genetic Algorithms are applied. Two experiments using both synthetic and benchmark data are made. From one hand, a 2-dimensional classification problem to show the behavior of the approach is considered and on the other hand the Zimmermann-Zysno data set to show the capability of the model is considered.
As an important factor in fine thunderstorm detections, a multi-time scale thunderstorm monitoring, warning and imaging system is proposed in this paper. The first computing phase involves a decomposition, classification, denoising and reconstruction of the atmospheric electric field signals (AEFSs), collected by a self-made three-dimensional AEF apparatus, based on autocorrelation characteristics and Fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C$</tex-math></inline-formula> -Means (FCM). Secondly, FCM classifies the equally divided AEFS components. A scale reconstruction rule is put forward and applied to obtain multi-time scale AEF branch data, according to the component temporal continuity in the same class. A corresponding scale correction strategy is then proposed. Thunderstorm point charge coordinate results are calculated by using branch data, and noise points contained in these results are removed. Finally, the curve fitting of denoised coordinate results is performed to image the point charge moving path. Empirical results confirm that the proposed system effectively warns and images thunderstorms, as well as provides a valid reference for multi-scale thunderstorm monitoring.
In this paper, we discuss an issue of description of highly dimensional data realized in terms of fuzzy sets. The underlying idea is to granulate numeric data using fuzzy sets and afterwards reveal and quantify relationships between these granules. This naturally impacts the dimensionality of any original dataset under discussion and provides with its nonlinear transformation (through the corresponding membership functions). These information granules give rise to the notion of associations-multidimensional information granules. Being fuzzy relations, these constructs are direction free. The directionality arises when one defines inputs and outputs and in this way confines himself to some sort of rules capturing a directional nature of main relationships within the data. Rules arising from associations may be in conflict. The essence of data is then captured via a granular signature regarded as a mixture of associations and rules.
Group decision making situations are part of today’s organizations. It is a type of decision making involving many decision \nmakers which act collectively to choose the best alternative (or alternatives) from a set of feasible alternatives. Usually, numerical \nvalues have been used by the decision makers to express their opinions on the possible alternatives. However, as the standard \nrepresentation of the concepts that humans use for communication is the natural language, words or linguistic terms instead of \nnumerical values should be used by the decision makers to provide their preferences. In such a situation, the linguistic information \nhas to be made operational in order to be fully utilized. In this contribution, assuming that decision makers express their opinions by \nusing linguistic terms, we present an information granulation of such a type of information, which is formulated as an optimization \nproblem in which consistency is maximized by a suitable mapping of the linguistic terms on information granules.
In this study, we are concerned with a two-objective development of information granules completed on a basis of numeric data. The first goal of this design concerns revealing and representing a structure in a data set. As such it is very much oriented towards coping with the underlying relational aspects of the experimental data. The second goal deals with a formation of a mapping between information granules constructed in two spaces (thus it concentrates on the directional aspect of information granulation). The quality of the mapping is directly affected by the information granules over which it operates, so in essence we are interested in the granules that not only reflect the data but also contribute to the performance of such a mapping. The optimization of information granules is realized through a collaboration occurring at the level of the data and the mapping between the data sets. The operational facet of the problem is cast in the realm of fuzzy clustering. As the standard techniques of fuzzy clustering (including a well-known approach of FCM) are aimed exclusively at the first objective identified above, we augment them in order to accomplish sound mapping properties between the granules. This leads to a generalized version of the FCM (and any other clustering technique for this matter). We propose a generalized version of the objective function that includes an additional collaboration component to make the formed information granules in rapport with the mapping requirements (that comes with a directional component captured by the information granules). The additive form of the objective function with a modifiable component of collaborative activities makes it possible to express a suitable level of collaboration and to avoid a phenomenon of potential competition in the case
Presented is a model that integrates three data types (numbers, intervals, and linguistic assessments). Data of these three types come from a variety of sensors. One objective of sensor-fusion models is to provide a common framework for data integration, processing, and interpretation. That is what our model does. We use a small set of artificial data to illustrate how problems as diverse as feature analysis, clustering, cluster validity, and prototype classifier design can all be formulated and attacked with standard methods once the data are converted to the generalized coordinates of our model. The effects of reparameterization on computational outputs are discussed. Numerical examples illustrate that the proposed model affords a natural way to approach problems which involve mixed data types.
