Multi-Level Control of Fuzzy-Constraint Propagation in Inference Based on α-Cuts and Generalized Mean

2013 ◽  
Vol 17 (4) ◽  
pp. 647-662 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Level Set ◽  
Constraint Propagation ◽  
Fuzzy Rules ◽  
Nonlinear Mapping ◽  
Fuzzy Input ◽  
Fuzzy Constraint ◽  
Multi Level ◽  
Generalized Mean ◽  
Α Level

An inference method is proposed, which can perform nonlinear mapping between convex fuzzy sets and present a scheme of various fuzzy-constraint propagation from given facts to deduced consequences. The basis of nonlinear mapping is provided by α-GEMII (α-level-set and generalized-mean-based inference) whereas the control of fuzzy-constraint propagation is based on the compositional rule of inference (CRI). The fuzzy-constraint propagation is controlled at the multi-level of α in its α-cut-based operations. The proposed method is named α-GEMS (α-level-set and generalized-mean-based inference in synergy with composition). Although α-GEMII can perform the nonlinear mapping according to a number of fuzzy rules in parallel, it has limitations in the control of fuzzy-constraint propagation and therefore has difficulty in constructing models of various given systems. In contrast, CRI-based inference can rather easily control fuzzy-constraint propagation with high understandability especially when a single fuzzy rule is used. It is difficult, however, to perform nonlinear mapping between convex fuzzy sets by using a number of fuzzy rules in parallel. α-GEMS can solve both of these problems. Simulation results show that α-GEMS is performed well in the nonlinear mapping and fuzzy-constraint propagation. α-GEMS is expected to be applied to modeling of given systems with various fuzzy input-output relations.

Multi-Level Control of Fuzzy-Constraint Propagation via Evaluations with Linguistic Truth Values in Generalized-Mean-Based Inference

2016 ◽  
Vol 20 (2) ◽  
pp. 355-377 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Level Set ◽  
Constraint Propagation ◽  
Fuzzy Rules ◽  
Truth Values ◽  
Fuzzy Constraints ◽  
Control Scheme ◽  
Fuzzy Constraint ◽  
Generalized Mean ◽  
Α Level

A method is proposed for fuzzy inference which can propagate convex fuzzy-constraints from given facts to consequences in various forms by applying a number of fuzzy rules, particularly when asymmetric fuzzy sets are used for given facts and/or fuzzy rules. The conventionalmethod, α-GEMS (α-level-set and generalized-mean-based inference in synergy with composition), cannot be performed with asymmetric fuzzy sets; it can be conducted only with symmetric fuzzy sets. In order to cope with asymmetric fuzzy sets as well as symmetric ones, a control scheme is proposed for the fuzzy-constraint propagation, which is α-cut based and can be performed independently at each level of α. It suppresses an excessive specificity decrease in consequences, particularly stemming from the asymmetricity. Thereby, the fuzzy constraints of given facts are reflected to those of consequences, to a feasible extent. The theoretical aspects of the control scheme are also presented, wherein the specificity of the support sets of consequences is evaluated via linguistic truth values (LTVs). The proposed method is named α-GEMST (α-level-set and generalized-meanbased inference in synergy with composition via LTV control) in order to differentiate it from α-GEMS. Simulation results show that α-GEMST can be properly performed, particularly with asymmetric fuzzy sets. α-GEMST is expected to be applied to the modeling of given systems with various fuzzy input-output relations.

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

2017 ◽  
Vol 21 (3) ◽  
pp. 425-447 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Rule ◽  
Constraint Propagation ◽  
Nonlinear Mapping ◽  
Level Control ◽  
Inference Method ◽  
Fuzzy Constraints ◽  
Fuzzy Input ◽  
Fuzzy Constraint ◽  
Multi Level ◽  
Α Level

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.

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

2015 ◽  
Vol 19 (1) ◽  
pp. 74-90 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Lower Bounds ◽  
Infinite Number ◽  
Fuzzy Rule ◽  
Fuzzy Rules ◽  
Nonlinear Mapping ◽  
Membership Functions ◽  
Inference Method ◽  
Generalized Mean ◽  
Simulation Results ◽  
Α Level

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.

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

2011 ◽  
Vol 15 (3) ◽  
pp. 264-287 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Shun Sato ◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Rules ◽  
Nonlinear Mapping ◽  
Mean Square ◽  
Inference Method ◽  
Multi Level ◽  
Simulation Results ◽  
Conventional Methods ◽  
Rule Bases ◽  
Mean Square Errors

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.

Inference with Governing Schemes for Propagation of Fuzzy Convex Constraints Based on α-Cuts

2009 ◽  
Vol 13 (3) ◽  
pp. 321-330 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Takumi Koyama ◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Constraint Propagation ◽  
Convex Constraints ◽  
Inference Method ◽  
Self Tuning ◽  
Fuzzy Constraint ◽  
Generalized Mean

A governing scheme is proposed for fuzzy constraint propagation from given facts to consequences in convex forms, which is applied to the inference method based on α-cuts and generalized mean. The governing is performed by self-tuning which can reflect the distribution forms of fuzzy sets in consequent parts to the forms of deduced consequences. Thereby, the proposed scheme can solve the problems in the conventional inference based on the compositional rule of inference that deduces fuzzy sets with excessive fuzziness increase and specificity decrease. In simulations, it is confirmed that the proposed scheme can effectively perform the constraint propagation from given facts to consequences in convex forms while reflecting fuzzy-set distributions in consequent parts. It is also demonstrated that consequences are deduced without excessively large fuzziness and small specificity.

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

2017 ◽  
Vol 21 (4) ◽  
pp. 597-615
Author(s):  
Kiyohiko Uehara ◽  
Kaoru Hirota ◽  
◽  
Keyword(s):  
Fuzzy Inference ◽  
Constraint Propagation ◽  
Inference Engine ◽  
Fuzzy Rules ◽  
Fuzzy Constraint ◽  
Generalized Mean ◽  
Effective Use ◽  
Inference Methods

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.

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

2013 ◽  
Vol 17 (1) ◽  
pp. 44-59 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Feasibility Study ◽  
Infinite Number ◽  
Fuzzy Rule ◽  
Fuzzy Rules ◽  
Nonlinear Mapping ◽  
Fundamental Analysis ◽  
Mapping Accuracy ◽  
Multi Level

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.

Inference Based on α-Cut and Generalized Mean with Fuzzy Tautological Rules

2010 ◽  
Vol 14 (1) ◽  
pp. 76-88 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Takumi Koyama ◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Systems ◽  
Fuzzy Rules ◽  
Mean Square ◽  
Basic Properties ◽  
Generalized Mean ◽  
Conventional Methods ◽  
The Difference ◽  
Mean Square Errors

Theoretical aspects are provided for inference based on α-cuts and generalized mean (α-GEMII). In order to clarify the basic properties of the inference, fuzzy tautological rules (FTRs) are focused on, which are composed by setting fuzzy sets in consequent parts identical to those in antecedent parts of initially given fuzzy rules. It is mathematically proved that the consequences deduced with FTRs are closer to given facts as the number of FTRs increases. The aspects provided in this paper are appropriate from axiomatic viewpoints and can contribute to interpretability in fuzzy systems constructed with α-GEMII. They are not obtained in conventional methods based on the compositional rule of inference. Simulations are performed by evaluating difference (mean square errors) between given facts and deduced consequences under the condition that convex and symmetric fuzzy sets are given as facts. Their results show that the difference becomes smaller as the number of FTRs increases. Thereby, it is confirmed that α-GEMII has an advantage in the interpretability with respect to FTRs over the conventional methods.

Inference Based on α-Cut and Generalized Mean in Representing Fuzzy-Valued Functions

2010 ◽  
Vol 14 (6) ◽  
pp. 581-592 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Takumi Koyama ◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Systems ◽  
Fuzzy Rules ◽  
Rule Base ◽  
Mean Square ◽  
Generalized Mean ◽  
Simulation Results ◽  
Conventional Methods ◽  
The Difference ◽  
Mean Square Errors

It is mathematically proved that inference based on α-cuts and generalized mean (α-GEMII) deduces consequences converging to fuzzy sets mapped by linear fuzzy-valued functions, to be represented with α-GEMII, as the number of fuzzy rules increases. The proof indicates that α-GEMII satisfies axiomatic properties and can contribute to presenting interpretability in designing fuzzy systems in the rule base. Such properties do not hold in conventional methods based on the compositional rule of inference. Simulation results show that the difference between deduced consequences and fuzzy sets mapped by linear fuzzyvalued functions is smaller as the number of fuzzy rules increases, in which the difference is evaluated by mean square errors. The discussions may lead to improvements of the interpretability in representing nonlinear fuzzy-valued functions by using α-GEMII.

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

2019 ◽  
Vol 23 (6) ◽  
pp. 1027-1043
Author(s):  
Kiyohiko Uehara ◽  
Kaoru Hirota ◽  
Keyword(s):  
Noise Reduction ◽  
Level Set ◽  
Fuzzy Inference ◽  
Fuzzy Rule ◽  
Sampled Data ◽  
Input Output ◽  
Generalized Mean ◽  
Α Level ◽  
Rule Optimization ◽  
Learning Data

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.

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