Size reduction by interpolation in fuzzy rule bases

On Ordered Weighted Averaging Operator and Monotone Takagi-Sugeno-Kang Fuzzy Inference Systems

The necessary and/or sufficient conditions for a Takagi-Sugeno-Kang Fuzzy Inference System (TSK-FIS) to be monotone has been a key research direction in the last two decades. In this article, we first define fuzzy membership functions (FMFs) with single and continuous support; and consider TSK-FIS with a grid partition strategy for computing its firing strengths with product T-norm (here after denoted as TSK-FIS-product). We also define a more general joint necessary condition, whereby each constituent itself is a necessary condition for the TSK-FIS-product model. The first necessary condition indicates that the normalized firing strength must not be indeterminate (i.e., 0/0), i.e., susceptible to the tomato classification problem. The second necessary condition indicates that all restricted consequents of fuzzy if-then rules must be defined. Based on the principle of the ordered weighted averaging (OWA) operator as well as the concept of increasing orness in OWA and hyperboxes, a general joint sufficient condition for a TSK-FIS-product model to be monotone is derived. Three case studies of the developed methods for undertaking failure mode and effect analysis (FMEA) and image processing tasks are presented. The results are compared, analyzed, and discussed, demonstrating the usefulness of our developed methods.

Approximate reasoning with fuzzy rule interpolation: background and recent advances

Approximate reasoning systems facilitate fuzzy inference through activating fuzzy if–then rules in which attribute values are imprecisely described. Fuzzy rule interpolation (FRI) supports such reasoning with sparse rule bases where certain observations may not match any existing fuzzy rules, through manipulation of rules that bear similarity with an unmatched observation. This differs from classical rule-based inference that requires direct pattern matching between observations and the given rules. FRI techniques have been continuously investigated for decades, resulting in various types of approach. Traditionally, it is typically assumed that all antecedent attributes in the rules are of equal significance in deriving the consequents. Recent studies have shown significant interest in developing enhanced FRI mechanisms where the rule antecedent attributes are associated with relative weights, signifying their different importance levels in influencing the generation of the conclusion, thereby improving the interpolation performance. This survey presents a systematic review of both traditional and recently developed FRI methodologies, categorised accordingly into two major groups: FRI with non-weighted rules and FRI with weighted rules. It introduces, and analyses, a range of commonly used representatives chosen from each of the two categories, offering a comprehensive tutorial for this important soft computing approach to rule-based inference. A comparative analysis of different FRI techniques is provided both within each category and between the two, highlighting the main strengths and limitations while applying such FRI mechanisms to different problems. Furthermore, commonly adopted criteria for FRI algorithm evaluation are outlined, and recent developments on weighted FRI methods are presented in a unified pseudo-code form, easing their understanding and facilitating their comparisons.

Noise Reduction with Fuzzy Inference Based on Generalized Mean and Singleton Input–Output Rules: Toward Fuzzy Rule Learning in a Unified Inference Platform

A method is proposed for reducing noise in learning data based on fuzzy inference methods called α-GEMII (α-level-set and generalized-mean-based inference with the proof of two-sided symmetry of consequences) and α-GEMINAS (α-level-set and generalized-mean-based inference with fuzzy rule interpolation at an infinite number of activating points). It is particularly effective for reducing noise in randomly sampled data given by singleton input–output pairs for fuzzy rule optimization. In the proposed method, α-GEMII and α-GEMINAS are performed with singleton input–output rules and facts defined by fuzzy sets (non-singletons). The rules are initially set by directly using the input–output pairs of the learning data. They are arranged with the facts and consequences deduced by α-GEMII and α-GEMINAS. This process reduces noise to some extent and transforms the randomly sampled data into regularly sampled data for iteratively reducing noise at a later stage. The width of the regular sampling interval can be determined with tolerance so as to satisfy application-specific requirements. Then, the singleton input–output rules are updated with consequences obtained in iteratively performing α-GEMINAS for noise reduction. The noise reduction in each iteration is a deterministic process, and thus the proposed method is expected to improve the noise robustness in fuzzy rule optimization, relying less on trial-and-error-based progress. Simulation results demonstrate that noise is properly reduced in each iteration and the deviation in the learning data is suppressed considerably.

Noise Reduction with Inference Based on Fuzzy Rule Interpolation at an Infinite Number of Activating Points: Toward Fuzzy Rule Learning in a Unified Inference Platform

In order to provide a unified platform for fuzzy inference and fuzzy rule learning with noise-corrupted data, a method is proposed for reducing noise in learning data on the basis of a fuzzy inference method called α-GEMINAS (α-level-set and generalized-mean-based inference with fuzzy rule interpolation at an infinite number of activating points). It is expected to prevent fuzzy rules from overfitting to noise in learning data, especially when there is less learning data available for fuzzy rule optimization. The proposed method is named α-GEMI-ES (α-GEMINAS-based local-evolution toward slight linearity for global smoothness) in this paper. α-GEMI-ES iteratively performs α-GEMINAS and reduces the noise in each iteration. This paper mathematically proves that α-GEMI-ES effectively reduces the noise. The noise-reduction process is decisive and thus relies less on trial-and-error-based progress. The noise is reduced by a large amount in the early iterations and the amount of its reduction is decelerated in the later iterations where the deviation in the learning data is suppressed to a great extent. This property makes it easy to determine the termination conditions for the iterative process. Simulation results demonstrate that α-GEMI-ES properly reduces noise as the mathematical proof suggests. The above-mentioned properties indicate that α-GEMI-ES is feasible in practice for the unified platform.

Fuzzy Inference Based on α-Cuts and Generalized Mean: Relations Between the Methods in its Family and their Unified Platform

This paper clarifies the relations in properties and structures between fuzzy inference methods based on α-cuts and the generalized mean. The group of the inference methods is named the α-GEM (α-cut and generalized-mean-based inference) family. A unified platform is proposed for the inference methods in the α-GEM family by the effective use of the above-mentioned relations. For the unified platform, a criterion is made clear to uniquely determine the value of a parameter in fuzzy-constraint propagation control for facts given by singletons. Moreover, conditions are derived to make the inference methods in the α-GEM family equivalent to singleton-consequent-type fuzzy inference which has been successfully applied to a wide variety of fields. Thereby, the unified platform can contribute to the construction of an inference engine for both the methods in the α-GEM family and singleton-consequent-type fuzzy inference. Such scheme of the inference engine provides an effective way to make these inference methods transformed into each other in learning for selecting the inference methods as well as for optimizing fuzzy rules.

Multi-Level Control of Fuzzy-Constraint Propagation in Inference with Fuzzy Rule Interpolation at an Infinite Number of Activating Points

An inference method is proposed, which can control the degree to which the fuzzy constraints of given facts are propagated to those of consequences via the nonlinear mapping represented by fuzzy rules. The conventional method, α-GEMST (α-level-set and generalized-mean-based inference in synergy with composition via linguistic-truth-value control), has limitations in the control of the propagation degree. In contrast, the proposed method can fully control the fuzzy-constraint propagation to a different degree with each fuzzy rule. After the nonlinear mapping, the proposed method performs fuzzy-logic-based control for further fuzzy-constraint propagation wherein evaluations are conducted via linguistic truth values to suppress the excessive specificity decrease in deduced consequences. Thereby, fuzzy constraints can be propagated in various ways by selecting one pair from the widely available implications and compositional operations. The proposed method controls the fuzzy-constraint propagation at the multi-levels of α in its α-cut-based operations. This scheme contributes to fast computation with parallel processing for each level of α. Simulation results illustrate that the proposed method can properly control the propagation degree. The proposed method is expected to be applied to the modeling of given systems with various fuzzy input-output relations.

Weighted Fuzzy Interpolative Reasoning Based on the Slopes of Fuzzy Sets and Particle Swarm Optimization Techniques

In this paper, we propose a new weighted fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the slopes of fuzzy sets. We also propose a particle swarm optimization (PSO)-based weights-learning algorithm to automatically learn the optimal weights of the antecedent variables of fuzzy rules for weighted fuzzy interpolative reasoning. We apply the proposed weighted fuzzy interpolative reasoning method using the proposed PSO-based weights-learning algorithm to deal with the computer activity prediction problem, the multivariate regression problems, and the time series prediction problems. The experimental results show that the proposed weighted fuzzy interpolative reasoning method using the proposed PSO-based weights-learning algorithm outperforms the existing methods for dealing with the computer activity prediction problem, the multivariate regression problems, and the time series prediction problems.

Inference with Fuzzy Rule Interpolation at an Infinite Number of Activating Points

An inference method for sparse fuzzy rules is proposed which interpolates fuzzy rules at an infinite number of activating points and deduces consequences based on α-GEMII (α-level-set and generalized-mean-based inference). The activating points, proposed in this paper, are determined so as to activate interpolated fuzzy rules by each given fact. The proposed method is named α-GEMINAS (α-GEMII-based inference with fuzzy rule interpolation at an infinite number of activating points). α-GEMINAS solves the problem in infinite-level interpolation where fuzzy rules are interpolated at the least upper and greatest lower bounds of an infinite number of α-cuts of each given fact. The infinite-level interpolation can nonlinearly transform the shapes of given membership functions to those of deduced ones in accordance even with sparse fuzzy rules under some conditions. These conditions are, however, strict from a practical viewpoint. α-GEMINAS can deduce consequences without these conditions and provide nonlinear mapping comparable with infinite-level interpolation. Simulation results demonstrate these properties of α-GEMINAS. Thereby, it is found that α-GEMINAS is practical and applicable to a wide variety of fields.

Multi-Level Interpolation for Inference with Sparse Fuzzy Rules: An Extended Way of Generating Multi-Level Points

As an extended way for inference based on multi-level interpolation, the number of multi-level points, generated with the α-cuts of given facts, is increased by making larger the number of levels of α. The conventional way uses the number of the levels adopted to define each given fact by α-cuts. A basic study is performed with triangular membership functions, where it is examined how the accuracy of mapping with fuzzy rules changes in increasing the number of the levels. It is also examined how deduced consequences behave when the number is increased. Moreover, convergent core sets of consequences are theoretically derived in the increase by the effective use of non-adaptive inference operations for core sets. They are used as references in simulation studies. Increasing the number of the levels provides nonlinear mapping with more precise reflection of distribution forms of sparse fuzzy rules to consequences. The basic study here contributes to improving the reflection accuracy. In simulations for the basic study, it is found that the mapping accuracy improves when the number of levels of α is increased. It is also confirmed that deduced core sets converge to those theoretically derived. Support sets are also found to converge in increasing the number of the levels. The core sets and support sets of deduced consequences do not, however, monotonically converge. This property causes difficulty in determining the optimized number of the levels so as to satisfy required mapping accuracy. In order to solve this problem, further discussions may be possible to theoretically derive convergent consequences and to use them in practical fields, in accordance with the theoretically derived core sets mentioned above.

Infinite-Level Interpolation for Inference with Sparse Fuzzy Rules: Fundamental Analysis Toward Practical Use

Infinite-level interpolation is proposed for inference with sparse fuzzy rules. It is based onmulti-level interpolation where fuzzy rule interpolation is performed at a number of multi-level points. Multi-level points are defined by the bounds of α-cuts of each given fact. As a feasibility study, fundamental analysis is focused on in order to theoretically derive convergent consequences in increasing the number of the levels of α for the α-cuts. By increasing the number of the levels, nonlinear mapping by the inference is made more precise in reflecting the distribution forms of sparse fuzzy rules to consequences. The convergent consequences make it unnecessary to examine the number of the levels for improving the mapping accuracy. It is confirmed that each of the consequences deduced with simulations converges to one theoretically derived with an infinite number of the levels of α. It is thereby proved that the fundamental analysis has its validity. Toward the practical use of the convergent consequences, further discussions may be possible to extend the fundamental analysis, considering practical conditions.

A SIMILARITY-BASED GENERALIZATION OF FUZZY ORDERINGS PRESERVING THE CLASSICAL AXIOMS

Equivalence relations and orderings are key concepts of mathematics. For both types of relations, formulations within the framework of fuzzy relations have been proposed already in the early days of fuzzy set theory. While similarity (indistinguishability) relations have turned out to be very useful tools, e.g. for the interpretation of fuzzy partitions and fuzzy controllers, the utilization of fuzzy orderings is still lagging far behind, although there are a lot of possible applications, for instance, in fuzzy preference modeling and fuzzy control. The present paper is devoted to this missing link. After a brief motivation, we will critically analyze the existing approach to fuzzy orderings. In the main part, an alternative approach to fuzzy orderings, which also takes the strong connection to gradual similarity into account, is proposed and studied in detail, including several constructions and representation results.

Enhancing a Fuzzy Failure Mode and Effect Analysis Methodology with an Analogical Reasoning Technique

In this paper, a fuzzy Failure Mode and Effect Analysis (FMEA) methodology incorporating an analogical reasoning technique is presented. FMEA methodology was introduced as a formal and systematic procedure for evaluation of risk associated with potential failure modes in the 1960s. Bowles and Peláez [1] proposed a Fuzzy Inference System (FIS)-based Risk Priority Number (RPN) model as an alternative to the conventional RPN model. For an FIS-based RPN (a three-input FIS model), a large set of fuzzy rules are required, and it is tedious to collect the full set of rules. With the grid partition strategy, the number of fuzzy rules required increases in an exponential manner, and this phenomenon is known as the “curse of dimensionality” or the combinatorial rule explosion problem. Hence, a rule selection and similarity reasoning technique, i.e., Approximate Analogical Reasoning Schema (AARS) technique are implemented in a fuzzy FMEA in order to solve the problem. The experiment was conducted using a set of data collected from a semiconductor manufacturing line, i.e., underfill dispensing process, and promising results were obtained.

Generalisation of Scale and Move Transformation-Based Fuzzy Interpolation

Fuzzy interpolative reasoning has been extensively studied due to its ability to enhance the robustness of fuzzy systems and reduce system complexity. In particular, the scale and move transformation-based approach is able to handle interpolation with multiple antecedent rules involving triangular, complex polygon, Gaussian and bell-shaped fuzzy membership functions [1]. Also, this approach has been extended to deal with interpolation and extrapolation with multiple multi-antecedent rules [2]. However, the generalised extrapolation approach based on multiple rules may not degenerate back to the basic crisp extrapolation based on two rules. Besides, the approximate function of the extended approach may not be continuous. This paper therefore proposes a new approach to generalising the basic fuzzy interpolation technique of [1] in an effort to eliminate these limitations. Examples are given throughout the paper for illustration and comparative purposes. The result shows that the proposed approach avoids the identified problems, providing more reasonable interpolation and extrapolation.

Inference for Nonlinear Mapping with Sparse Fuzzy Rules Based on Multi-Level Interpolation

An inference method is proposed for sparse fuzzy rules on the basis of interpolations at a number of points determined by α-cuts of given facts. The proposed method can perform nonlinear mapping even with sparse rule bases when each given fact activates a number of fuzzy rules which represent nonlinear relations. The operations for the nonlinear mapping are exactly the same as for the case when given facts activate no fuzzy rules due to the sparseness of rule bases. Such nonlinear mapping cannot be provided by conventional methods for sparse fuzzy rules. In evaluating the proposed method, mean square errors are adopted to indicate difference between deduced consequences and fuzzy sets transformed by nonlinear fuzzy-valued functions to be represented with sparse fuzzy rules. Simulation results show that the proposed method can follow the nonlinear fuzzy-valued functions. The proposed method contributes to both reducing the number of fuzzy rules and providing nonlinear mapping with sparse rule bases.

Fuzzy Rule Interpolation and Extrapolation Techniques: Criteria and Evaluation Guidelines

This paper comprehensively analyzes Fuzzy Rule Interpolation and extrapolation Techniques (FRITs). Because extrapolation techniques are usually extensions of fuzzy rule interpolation, we treat them both as approximation techniques designed to be applied where sparse or incomplete fuzzy rule bases are used, i.e., when classical inference fails. FRITs have been investigated in the literature from aspects such as applicability to control problems, usefulness regarding complexity reduction and logic. Our objectives are to create an overall FRIT standard with a general set of criteria and to set a framework for guiding their classification and comparison. This paper is our initial investigation of FRITs. We plan to analyze details in later papers on how individual techniques satisfy the groups of criteria we propose. For analysis,MATLAB FRI Toolbox provides an easy-to-use testbed, as shown in experiments.

FINs: lattice theoretic tools for improving prediction of sugar production from populations of measurements

This paper presents novel mathematical tools developed during a study of an industrial-yield prediction problem. The set F of fuzzy interval numbers, or FINs for short, is studied in the framework of lattice theory. A FIN is defined as a mapping to a metric lattice of generalized intervals, moreover it is shown analytically that the set F of FINs is a metric lattice. A FIN can be interpreted as a convex fuzzy set, moreover a statistical interpretation is proposed here. Algorithm CALFIN is presented for constructing a FIN from a population of samples. An underlying positive valuation function implies both a metric distance and an inclusion measure function in the set F of FINs. Substantial advantages, both theoretical and practical, are shown. Several examples illustrate geometrically on the plane both the utility and the effectiveness of novel tools. It is outlined comparatively how some of the proposed tools have been employed for improving prediction of sugar production from populations of measurements for Hellenic Sugar Industry, Greece.