In classical single-layer fuzzy systems (FSs), the number of rules and the run-time computational requirements increase exponentially with the number of input domains. In this paper, we present a method for converting a multidimensional FS to a two-layer hierarchical FS that reduces the number of rules and improves the run-time efficiency. The first layer of the two-layer system consists of FSs whose rule bases can be represented as linearly independent vectors. The second layer constructs linear combinations of the rule base vectors. The effectiveness of the hierarchical conversion in reducing the number of rules and improving efficiency is demonstrated on a classic control problem and a simulated higher dimensional FS.

Inductive logic programming (ILP) is a generic tool aiming at learning rules from relational databases. Introducing fuzzy sets arid fuzzy implication connectives in this framework allows us to increase the expressive power of the induced rules while keeping the readability of the rules. Moreover, fuzzy sets facilitate the handling of numerical attributes by avoiding crisp and arbitrary transitions between classes. In this paper, the meaning of a fuzzy rule is encoded by its implication operator, which is to be determined in the learning process. An algorithm is proposed for inducing first order rules having fuzzy predicates, together with the most appropriate implication operator. The benefits of introducing fuzzy logic in ILP and the validation process of what has been learnt are discussed and illustrated on a benchmark

The use of fuzzy sets to describe associations between data extends the types of relationships that may be represented, facilitates the interpretation of rules in linguistic terms, and avoids unnatural boundaries in the partitioning of the attribute domains. In addition, the partial membership values provide a method for incorporating the distribution of the data into the assessment of a rule. This paper investigates techniques to identify and evaluate associations in a relational database that are expressible by fuzzy if-then rules. Extensions of the classical confidence measure based on the /spl alpha/-cut decompositions of the fuzzy sets are proposed to incorporate the distribution of the data into the assessment of a relationship and identify robustness in an association. A rule learning strategy that discovers both the presence and the type of an association is presented.

The granularity and interpretability of a fuzzy model are influenced by the method used to construct the rule base. Models obtained by a heuristic assessment of the underlying system are generally highly granular with interpretable rules, while models algorithmically generated from an analysis of training data consist of a large number of rules with small granularity. This paper presents a method for increasing the granularity of rules while satisfying a prescribed precision bound on the training data. The model is generated by a two-stage process. The first step iteratively refines the partitions of the input domains until a rule base is generated that satisfies the precision bound. In this step, the antecedents of the rules are obtained from decomposable partitions of the input domains and the consequents are generated using proximity techniques. A greedy merging algorithm is then applied to increase the granularity of the rules while preserving the precision bound. To enhance the representational capabilities of a rule and reduce the number of rules required, the rules constructed by the merging procedure have multi-dimensional antecedents. A model defined with rules of this form incorporates advantageous features of both clustering and proximity methods for rule generation. Experimental results demonstrate the ability of the algorithm to reduce the number of rules in a fuzzy model with both precise and imprecise training information.

Fuzzy rule bases have proven to be an effective tool for modeling complex systems and approximating functions. The generation of a fuzzy rule base has generally been accomplished by a heuristic analysis of the relationships of the underlying system or by algorithmic rule generation from training data. Automatic rule generation has utilized clustering algorithms, proximity analysis, and evolutionary techniques to identify approximate relationships between the input and the output. In this research, two general approaches for the evolutionary generation of fuzzy rules are identified and compared: global and local rule generation. Global rule production, which is the standard method of employing evolutionary techniques in fuzzy rule base generation, considers an entire rule base as an element of population. The fitness evaluation of a rule base aggregates the performance of the model over the entire space into a single value. The local approach utilizes the limited scope of a fuzzy rule to evaluate performance in regions of the input space. The local generation of rule bases employs an independent evolutionary search in each region and combines the local results to produce a global model. An experimental suite has been developed to compare the effectiveness of the two strategies for the evolutionary generation of fuzzy models.

A semantics for possibilistic evidential reasoning is presented based on similarity with paradigmatic examples. The acquisition of evidence generates a pseudo-metric on the universe of discourse. It is shown that every possibility distribution can be realized as an embedding in a pseudo-metric space with the possibility values determined by the distance from distinguished elements in that space. Determining support based on similarity captures the fundamental characteristics of possibilistic analysis: optimistic and independent evaluation of the alternatives. The similarity semantics distinguishes possibility from classical approaches to classification and diagnosis problems based on probabilistic techniques.

The characteristics of a fuzzy model are frequently determined by the manner in which the rules are constructed. Rules obtained by a heuristic assessment of a system generally are linguistically interpretable and have large granularity. The generation of rules via learning algorithms that analyse training data produces precise models consisting of multiple rules of small grannularity. In this paper, a greedy algorithm is presented that combines rule learning with a region merging strategy to reduce the number of rules. This approach differs from standard rule reduction techniques in that the latter are employed after the rule base has been completed while the learn-and-merge strategy generates a rule simultaneously with expanding its region of applicability. The objective of the algorithm is to produce fuzzy models with both a small number of interpretable rules and high precision.

This paper presents a framework for studying the effectiveness of evolutionary strategies for generating fuzzy rule bases and function approximations from training data. To facilitate the evolutionary operations that modify the elements of the population, a fuzzy rule base is represented as a real-valued matrix. A comparison of the training data with the function approximation associated with a fuzzy rule base provides a measure of agreement of the rule base with the training data. The analysis of training data provides the ability to generate both global and local fitness assessments. The effectiveness of incorporating local information into the evolutionary search is demonstrated by comparing the generation of rule consequences using the global and local strategies.

Fuzzy models have been designed to represent approximate or imprecise relationships in complex systems and have been successfully employed in control systems, expert systems, and decision analysis. A hierarchical architecture for fuzzy modeling and inference has been developed to learn an initial set of rules from training data and allow adaptation of the rule base via system performance feedback. A general adaptive algorithm is presented and its performance examined for three types of adaptive behavior: continued learning, gradual change, and drastic change. In each of the three types of behavior, the adaptive algorithm has been shown to be able to reconfigure the rule bases to either improve the original approximation or adapt to the new system.

Fuzzy inference systems and neural networks both provide mathematical systems for approximating continuous real-valued functions. Historically, fuzzy rule bases have been constructed by knowledge acquisition from experts while the weights on neural nets have been learned from data. This paper examines algorithms for constructing fuzzy rules from input-output training data. The antecedents of the rules are determined by a fuzzy decomposition of the input domains. The decomposition localizes the learning process, restricting the influence of each training example to a single rule. Fuzzy learning proceeds by determining entries in a fuzzy associative memory using the degree to which the training data matches the rule antecedents. After the training set has been processed, similarity to existing rules and interpolation are used to complete the rule base. Unlike the neural network algorithms, fuzzy learning algorithms require only a single pass through the training set. This produces a computationally efficient method of learning. The effectiveness of the fuzzy learning algorithms is compared with that of a feedforward neural network trained with back-propagation.>

Emitter and system hierarchies are represented by inference nets and propositional relationships. Emitters are the primitive objects of the domain and systems consist of relationships among emitters. Evidence gathered concerning the identification of emitters must be used to classify both emitters and systems. Evidential reasoning and inference nets are used to to combine information at each level. Methods of direct and indirect transfer of evidence between levels are presented.