Thomas Sudkamp

Interpolation, completion, and learning fuzzy rules

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. >

On the representation, measurement, and discovery of fuzzy associations

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.

Similarity, interpolation, and fuzzy rule construction

Abstract A method for the construction of fuzzy concepts and fuzzy if-then rules based on similarity and paradigmatic examples is presented. It is shown that all normal fuzzy sets may be realized as the interpolation of paradigmatic examples by a similarity relation. The class of fuzzy if-then rules that may be obtained by interpolation, however, is a proper subset of the rules definable by disjunctive combination.

On probability-possibility transformations

Abstract A probability-possibility transformation is a purely mechanical transformation of probabilistic support to possibilistic support and vice versa. That is, a conversion of the measure of support of one theory into that of the other that is independent of the problem domain. In this paper it is shown that any bijective mapping between probability distributions and normalized possibility distributions that is symmetric and expansible cannot preserve the properties of projection, noninteraction, or conditionalization.

Examples, counterexamples, and measuring fuzzy associations

This paper examines the measurement of the degree to which tuples in a database support a relation among attributes based on a comparison of the number of examples and counterexamples of the relation. In particular, we are concerned with associations that represent imprecise constraints placed upon the value of one attribute by those of other attributes. Associations of this form may be described by fuzzy rules and the analysis requires an assessment of the degree to which a tuple satisfies the imprecise constraint specified by the rule. Standard measures of rule validity are extended to fuzzy associations based on the degree that the tuples are examples, counterexamples, or irrelevant to imprecise relations. A scaling of the relevance of a tuple is proposed to minimize the impact of the accumulation of small membership values on the confidence-based validity measures.

Patterns of fuzzy rule-based inference

Abstract Processing information in fuzzy rule-based systems generally employs one of two patterns of inference: composition or compatibility modification. Composition originated as a generalization of binary logical deduction to fuzzy logic, while compatibility modification was developed to facilitate the evaluation of rules by separating the evaluation of the input from the generation of the output. The first step in compatibility modification inference is to assess the degree to which the input matches the antecedent of a rule. The result of this assessment is then combined with the consequent of the rule to produce the output. This paper examines the relationships between these two patterns of inference and establishes conditions under which they produce equivalent results. The separation of the evaluation of input from the generation of output permits a flexibility in the methods used to compare the input with the antecedent of a rule with multiple clauses. In this case, the degree to which the input and the rule antecedent match is determined by the application of a compatibility measure and an aggregation operator. The order in which these operations are applied may affect the assessment of the degree of matching, which in turn may cause the production of different results. Separability properties are introduced to define conditions under which compatibility modification inference is independent of the input evaluation strategy.

A Method of Converting a Fuzzy System to a Two-Layered Hierarchical Fuzzy System and Its Run-Time Efficiency

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.

Learning fuzzy rules with their implication operators

Fuzzy predicates have been incorporated into machine learning and data mining to extend the types of data relationships that can be represented, to facilitate the interpretation of rules in linguistic terms, and to avoid unnatural boundaries in partitioning attribute domains. The confidence of an association is classically measured by the co-occurrence of attributes in tuples in the database. The semantics of fuzzy rules, however, is not co-occurrence but rather graduality or certainty and is determined by the implication operator that defines the rule. In this paper we present a learning algorithm, based on inductive logic programming, that simultaneously learns the semantics and evaluates the validity of fuzzy rules. The learning algorithm selects the implication that maximizes rule confidence while trying to be as informative as possible. The use of inductive logic programming increases the expressive power of fuzzy rules while maintaining their linguistic interpretability.

Geometric compatibility modification

Abstract Compatibility modification is a rule-based inference strategy that uses the similarity of the input with the antecedent of a rule to modify the consequent. Existing compatibility modification inference techniques have employed a set theoretic assessment of compatibility. In this paper, a distance-based compatibility measure is derived from a generalization of the dissemblance index for fuzzy sets. This measure is then used to develop an inference technique based on geometric compatibility. This geometric approach is compared with two other distance-based inference techniques: linear rule interpolation and bound dependent linear revision.

Model generation by domain refinement and rule reduction

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.