Fuzzy mappings are fundamental constructs of granular computing and rule-based systems, in particular. The notion of relevance of the fuzzy mappings is instrumental in the quantification of their quality prior to any detailed construction. For the purposes of such quantification, we introduce shadowed sets and discuss their role therein. It is revealed that shadowed sets deliver an interesting three-valued quantification of the relevance of the fuzzy mappings (labeled as acceptable mapping, marginal mapping, and a lack of mapping). The paper includes a number of detailed calculations concerning two commonly exploited classes of triangular and Gaussian fuzzy sets.
Perception is crucial to data mining. When dealing with real-world problems, e.g., expert systems analyzing stock market and financial data, etc., there is a difference between the real world and what the user (or expert system) perceives to be the real world. Often data mining retrieves perceived data and the problem is to reconcile this perceived data with the real world. For example, data mining might retrieve a perceived set of rules learned from long experience by a plant operator in operating a plant (and which can vary significantly from the mathematical model of the plant system). How do we reconcile the two different models? Which is the real one? Information granulation can help with this problem.
This section provides a brief overview of fuzzy set approaches for database management and querying. Incompletely known information as well as flexible query handling capabilities are expected to extend the range of applications for future database management systems. The term fuzzy databases, which is extensively used in the specialized literature, covers several different meanings, which are reviewed. The use of a similarity measure as a fuzzy technique used to extend the relational data model is described, as well as the possibility-theory-based approach, which proves to be useful for representing imperfectly known data and soft constraints. Pattern matching is extended for handling this kind of datum by means of flexible queries. Related database issues, namely, functional dependencies and database security, are also discussed. Alternative fuzzy data models and other uncertainty management problems in databases are briefly mentioned in the conclusion.
In system modeling arises a fundamental question about the level of difficulty one may encounter when designing a model on a basis of some training data. In this study, we advocate that such level of difficulty inherently depends upon the variability of the available function (data). If for a pair of input data which exhibits small differences, the differences of the corresponding outputs are substantial then building a model in the presence of such data becomes more challenging than in cases of data where the differences in the output data are far more limited. Dwelling on this observation, we introduce a variability index quantifying the nature of data in terms of variability observed in input and output data, respectively. The proposed index is model-neutral (model agnostic), namely describes and quantifies the modeling challenge implied by the data irrespectively of the specific model to be constructed. In case of functions, we show that the Lipschitz constant plays a similar role as the variability index computed for experimental data. An original way of reducing values of the variability index through a nonlinear transformation of original data completed by a fuzzy rule-based model is introduced. It is shown that such rule-based architecture gives rise to a piecewise linear transformation (multipoint linear approximation) exhibiting required contraction-dilation characteristics. The optimization of this transformation is carried out with the use of a Particle Swarm Optimization algorithm. We also demonstrate that the index can be used to quantify a concept of adversarial data. Along this line, we introduce a granular characterization of adversarial feature of individual data points. A series of experiments is provided to offer a thorough illustration and detailed insight into the nature and a thorough characterization of publicly available data.
Numeric models (including fuzzy models) produce numeric results. There are no ideal models that deliver a complete match with the data. In this study, we advocate that a way of evaluating the quality of models can be realized at the higher level of abstraction by developing a concept of granular prediction. In this way, modeling results are expressed in the form of information granules, in particular as intervals or fuzzy sets. The study formulates a general conceptual and algorithmically supported statement: a meaningful evaluation framework to assess the quality of numeric models is the one engaging information granules. This general observation comprises a special case commonly investigated in regression analysis, where the quality of numeric results is expressed via granular constructs, namely, confidence or prediction intervals. The original design of prediction information granules is formulated as an optimization problem, in which the criteria of coverage of data and specificity of granular results are considered. In the optimization process, we also engage some nonlinear transformation of the level of information granularity depending upon the value of the numeric result. The proposed development is model agnostic and can support a variety of modeling architectures; the experimental part of the study is focused on rule-based models. Further generalizations of prediction information granules are covered by involving granular parameters in the design process.
Current research on fuzzy set applications has yielded a better understanding of complex design problems involving nondeterministic human factors. This has in turn resulted in a significant number of efficient software case-study oriented realizations. Concurrently, efficient hardware implementations involving integration of a large number of discrete components have become feasible in light of existing technological achievements. Hirota and Ozawa (1989) introduced the notion of the fuzzy JK flip-flop, considering it an essential and basic component for effective processing of fuzzy information. Continuing with this line of reasoning, a discussion is presented of VLSI implementation issues with emphasis on design platforms to be used with the new structures. Next some comparisons are performed pertaining to the usefulness and realization of several logical connectives, followed by an examination of fuzzy sequential system design. The combinational part of the circuit is embodied in a single layer neural network in which weights are adjusted on the basis of training situations. Numerical considerations highlight the performance of the design process.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
The separation of signal-and-noise subspaces is a crucial step in many array signal processing applications, since the performance of most high-resolution methods mainly depends on the accuracy of separated signal-and-noise subspaces. If the resulting signal subspace is inaccurately estimated, high-resolution subspace-based algorithms would most likely fail. In this paper, we propose a high-accuracy method to complete the separation of signal-and-noise subspaces. In the developed scheme, eigenvalues of the covariance matrix of the data received by the uniform linear array are first employed to construct an ideal sample space using the Schur product of matrices. Then, the Gaussian kernel is introduced to map the sample space into a new high-dimensional feature space, where the resulting structure becomes linearly separable. In the sequel, we propose two fuzzy set-based methods to divide the feature space into two subspaces, which correspond to the signal-and-noise subspaces. In this way, the signal subspace becomes separated. Experimental results show that, compared with the results produced by five other commonly used algorithms, the proposed method yields much higher accuracy, especially for low signal-to-noise ratio thresholds and small snapshots.
