The paper presents a programmable triangular neighborhood function for application in low power transistor level implemented Kohonen self-organized maps (SOMs). Detailed simulations carried out for the software model of such network show that the triangular function forms a good approximation of the Gaussian function, while being implemented in a much easier way in hardware. The proposed circuit is very flexible and allows for easy adjustments of the slope of the function. It enables the asynchronous and fully parallel operation of all neurons in the network thus making it very fast. The proposed mechanism can be used in custom designed networks either in their analog or digital implementation. Due to the simple structure, the energy consumption per a single input pattern is low (120 pJ in case of the map of 16 x 16 neurons).
A self-contained treatment of fuzzy systems engineering, offering conceptual fundamentals, design methodologies, development guidelines, and carefully selected illustrative material Forty years have passed since the birth of fuzzy sets, in which time a wealth of theoretical developments, conceptual pursuits, algorithmic environments, and other applications have emerged. Now, this reader-friendly book presents an up-to-date approach to fuzzy systems engineering, covering concepts, design methodologies, and algorithms coupled with interpretation, analysis, and underlying engineering knowledge. The result is a holistic view of fuzzy sets as a fundamental component of computational intelligence and human-centric systems. Throughout the book, the authors emphasize the direct applicability and limitations of the concepts being discussed, and historical and bibliographical notes are included in each chapter to help readers view the developments of fuzzy sets from a broader perspective. A radical departure from current books on the subject, Fuzzy Systems Engineering presents fuzzy sets as an enabling technology whose impact, contributions, and methodology stretch far beyond any specific discipline, making it applicable to researchers and practitioners in engineering, computer science, business, medicine, bioinformatics, and computational biology. Additionally, three appendices and classroom-ready electronic resources make it an ideal textbook for advanced undergraduate- and graduate-level courses in engineering and science.
A fuzzy cognitive structure is examined, and the problem of numerical and symbolic processing in terms of neurocomputations and fuzzy sets is addressed. The referential neural network architecture previously developed is recalled, and aspects of the VLSI implementation are explored with emphasis on fault tolerance and testability. Preprocessing and postprocessing are used with a feedforward neural network in the hardware implementation, and development of the supporting software environment is also investigated. The result is a very general structure that can be used to address a wide variety of complex problems
Information granules emerging as a result of an abstract and more condensed and global view at numeric data play an essential role in various pattern recognition pursuits. In this study, we investigate an idea of granular prototypes (representatives) and discuss their role in the realization of classification schemes. A two-stage procedure of a formation of information granules is discussed. We show how the commonly used clustering methods are viewed as a prerequisite for the construction of granular prototypes. In this regard, a certain version of the principle of justifiable granularity is investigated. In the sequel, a characterization of information granules expressed in terms of their information (classification) content is provided and its usage in the realization of a classifier is studied. Experimental studies involving both synthetic and publicly available data are reported.
In real-world regression analysis, statistical data may be linguistically imprecise or vague. Given the co-existence of stochastic and fuzzy uncertainty, real data cannot be characterized by using only the formalism of random variables. In order to address regression problems in the presence of such hybrid uncertain data, fuzzy random variables are introduced in this study to serve as an integral component of regression models. A new class of fuzzy regression models that is based on fuzzy random data is built, and is called the confidence-interval-based fuzzy random regression model (CI-FRRM). First, a general fuzzy regression model for fuzzy random data is introduced. Then, using expectations and variances of fuzzy random variables, sigma-confidence intervals are constructed for fuzzy random input-output data. The CI-FRRM is established based on the sigma-confidence intervals. The proposed regression model gives rise to a nonlinear programming problem that consists of fuzzy numbers or interval numbers. Since sign changes in the fuzzy coefficients modify the entire programming structure of the solution process, the inherent dynamic nonlinearity of this optimization makes it difficult to exploit the techniques of linear programming or classical nonlinear programming. Therefore, we resort to some heuristics. Finally, an illustrative example is provided.
We present a model that integrates three data types (numbers, intervals and linguistic assessments). Data of these three types come from a variety of sensors. Problems as diverse as feature analysis, clustering, cluster validity, and prototype classifier design can be formulated and handled with standard methods once the data are converted to the generalized coordinates of our model. This paper describes the model only.
Three-way decisions play an important role in rough sets and decision theory. As a representative model, decision-theoretic rough sets (DTRSs) provide a sound interpretation of thresholds used in three-way decisions. This problem is associated with the determination of the loss function of DTRSs. In this paper, we investigate a novel way of determining the loss functions of DTRSs with relative values. More specifically, with the aid of analytic hierarchy process (AHP) method, the determination of loss functions is realized in the context of DTRSs. First, a hierarchical structure of DTRSs is constructed. Second, along the hierarchical structure, pairwise comparison matrices are analyzed in a top-down fashion. In light of the generic condition imposed on the loss functions of DTRSs, some constraints on relative values between loss functions are introduced. The relative ratios at each level are computed. Considering the consistency ratio (CR) of the reciprocal matrices, two mathematical programming approaches are developed by exploiting the flexibility of information granularity. Then, we design a decision procedure for the determination of loss functions and deduce three-way decisions. The loss functions are calculated by aggregating the relative ratios being available at each level. With regard to the loss functions, we finally compare the existing studies with the AHP method. We demonstrate that the relative value with AHP improves the restriction of the existing studies and exhibits a certain level of tolerance to inconsistency.
Presented here is a problem of fuzzy clustering with partial supervision, i.e., unsupervised learning completed in the presence of some labeled patterns. The classification information is incorporated additively as a part of an objective function utilized in the standard FUZZY ISODATA. The algorithms proposed in the paper embrace two specific learning scenarios of complete and incomplete class assignment of the labeled patterns. Numerical examples including both synthetic and real-world data arising in the realm of software engineering are also provided.
In this paper, we show that the necessity to make crisp decisions in uncertain (fuzzy) situations leads to the necessity to "approximate" fuzzy sets by crisp sets. We show that seemingly natural approximation ideas - such as using alpha-cut for a given alpha - often do not work, and we describe new approximations which not only work, but which are optimal in some reasonable sense
