This study delivers a general overview of the theory and practice of fuzzy relational equations. We discuss various methods leading to the solutions of these equations starting from analytical approaches, moving through semi-analytic methods and finally elaborating on the neural-like style of finding solutions to relational constructs. The paper addresses important aspects of knowledge representation worked out by these equations and proposes a number of structural enhancements of the existing relational architectures.
In the field of fuzzy inference, the universal triple I algorithm integrated the CRI (Compositional Rule of Inference) algorithm with the triple I algorithm. Later the triple I algorithm was generalized to the QIP (quintuple implication principle) algorithm. Whether the QIP algorithm and the CRI algorithm can be unified has become an interesting question. Therefore, in this study, a fuzzy inference scheme referred to as the universal quintuple implicational (UQI) algorithm is proposed. First, we establish a unified granular computing framework with the UQI algorithm, which is a generalization of the QIP algorithm, the CRI algorithm as well as the universal triple I algorithm. The optimal UQI solutions derived from the fundamental principle of determining inference results are obtained for the FMP (fuzzy modus ponens) problem, in which some specific solutions are also given. Second, the reversible property of the UQI algorithm is verified for FMP, while aiming at the metric derived from the biresiduum operation, the robustness of the UQI algorithm is validated. Third, under the environment of multiple rules, two general cases of FITA (First-Inference-Then-Aggregate) and FATI (First-Aggregate-Then-Inference) are constructed for the UQI algorithm. The corresponding equivalence relation between continuity and interpolation is analyzed. Fourth, the fuzzy system is established based on the UQI algorithm, the singleton fuzzier as well as the centroid defuzzier. Its response ability is analyzed and it is shown that such fuzzy system is a universal approximator. Lastly, we compare the results of the UQI algorithm with the QIP algorithm by five examples for FMP. It is found that the UQI algorithm is able to acquire more and better forms of the fuzzy inference in contrast with the QIP algorithm.
A class of clustering problems that is studied here is concerned with the development of a structure of a global nature given a collection of structures (clusters) constructed locally for data that are represented by several collections (blocks) of features. These blocks of features come with a well-defined semantics. For instance, in spatiotemporal data, a certain block of features concerns a spatial component of the data (say, x-y or x-y-z coordinates), while another one deals with the features that describe time series associated with the corresponding locations. The results of clustering that are being produced locally are reconciled by minimizing a distance between the proximity matrices that are formed at the higher conceptual level and induced by the individual partition matrices. The optimization problem is formulated and presented along with its iterative scheme.
In this paper, we develop a robust and fault-tolerant approach to microwave-based sensitive measurements using fuzzy neural network (FNN). Microwave chemic-identification, recently, is employing active planar ring resonators to enhance the resolutions significantly. However, in practice, when the technology of resolution improves, the results become more prone to minor variations in the measurement setup and user error. In order to eliminate these unwanted and uncontrollable deviations from the final allocations, we propose a novel and robust approach that uses more than one parameter out of measurements and incorporates FNN as a machine learning architecture at the post processing stage of sensing to obtain fault-tolerant classification. We have compared different membership functions used in the FNN and shown improvement in assigning accuracy from 49% (single parameter-dependent) up to 81.5% (three parameters-dependent) on an average of four materials, such as isopropanol-2 (IPA), ethanol, acetone, and water.
In this study, we revisit the well-known notion of fuzzy state machines and discuss their development through learning. The systematic development of fuzzy state machines has not been pursued as intensively as it could have been expected from the breadth of the possible usage of them as various modelling platforms. We concentrate on the generalization of the well known architectures exploited in Boolean system synthesis, namely Moore and Mealy machines and show how these can be implemented in terms of generic functional modules such as fuzzy JK flip-flops and fuzzy logic neurons (AND and OR neurons) organized in the form of logic processors. It is shown that the design of the fuzzy state machines can be accomplished through their learning. The detailed learning algorithm is presented and illustrated with a series of numeric examples. The study reveals an interesting option of constructing digital systems through learning: the original problem is solved in the setting of fuzzy state machines and afterwards "binarised" into the two-valued format realized via the standard digital hardware.
When dealing with the curse of dimensionality (small sample size with many dimensions), feature subset selection is an important preprocessing strategy. This issue is particularly germane to the discrimination of class-labeled high-dimensional biomedical spectra as is often acquired from magnetic resonance and infrared spectrometers. A technique is presented that stochastically selects feature subsets with varying cardinality for discrimination by probabilistic neural networks. The results are benchmarked against two classifiers using the entire feature set both with and without feature averaging. The new technique had significantly fewer misclassifications than either of the benchmarks.
