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.

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 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.

Rule Base Simplification and Constrained Learning for Interpretability in TSK Neuro-Fuzzy Modelling

2020 ◽  
Vol 9 (2) ◽  
pp. 31-58
Author(s):  
Sharifa Rajab
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Systems ◽  
Fuzzy Model ◽  
Fuzzy Rules ◽  
Data Driven ◽  
Similarity Analysis ◽  
Rule Base ◽  
Generalization Capability ◽  
Fuzzy Modelling ◽  
Neuro Fuzzy

Neuro-fuzzy systems based on a fuzzy model proposed by Takagi, Sugeno and Kang known as the TSK fuzzy model provide a powerful method for modelling uncertain and highly complex non-linear systems. The initial fuzzy rule base in TSK neuro-fuzzy systems is usually obtained using data driven approaches. This process induces redundancy into the system by adding redundant fuzzy rules and fuzzy sets. This increases complexity which adversely affects generalization capability and transparency of the fuzzy model being designed. In this article, the authors explore the potential of TSK fuzzy modelling in developing comparatively interpretable neuro-fuzzy systems with better generalization capability in terms of higher approximation accuracy. The approach is based on three phases, the first phase deals with automatic data driven rule base induction followed by rule base simplification phase. Rule base simplification uses similarity analysis to remove similar fuzzy sets and resulting redundant fuzzy rules from the rule base, thereby simplifying the neuro-fuzzy model. During the third phase, the parameters of membership functions are fine-tuned using a constrained hybrid learning technique. The learning process is constrained which prevents unchecked updates to the parameters so that a highly complex rule base does not emerge at the end of model optimization phase. An empirical investigation of this methodology is done by application of this approach to two well-known non-linear benchmark forecasting problems and a real-world stock price forecasting problem. The results indicate that rule base simplification using a similarity analysis effectively removes redundancy from the system which improves interpretability. The removal of redundancy also increased the generalization capability of the system measured in terms of increased forecasting accuracy. For all the three forecasting problems the proposed neuro-fuzzy system demonstrated better accuracy-interpretability tradeoff as compared to two well-known TSK neuro-fuzzy models for function approximation.

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.

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.

The Loss of Dexterity in the Bilateral Lower Extremities in Patients With Stroke

2011 ◽  
Vol 27 (2) ◽  
pp. 122-129 ◽  
Author(s):  
Ryoji Kiyama ◽  
Kiyohiro Fukudome ◽  
Toshiki Hiyoshi ◽  
Akihide Umemoto ◽  
Yoichi Yoshimoto ◽  
...  
Keyword(s):  
Root Mean Square Error ◽  
Mean Square Error ◽  
Root Mean Square ◽  
Knee Extensor ◽  
Lower Extremities ◽  
Response Force ◽  
Mean Square ◽  
Control Subjects ◽  
The Difference ◽  
Mean Square Errors

The aim of this study was to examine the dexterity of both lower extremities in patients with stroke. Twenty patients with stroke and 20 age-matched control subjects participated in this study. To determine the dexterity of the lower extremities, we examined the ability to control muscle force during submaximal contractions in the knee extensor muscles using a force tracking task. The root mean square errors were calculated from the difference between the target and response force. The root mean square error was significantly greater in the affected limb of patients with stroke compared with those of the unaffected limb and the control subjects, and in the unaffected limb compared with that of the control subjects. Furthermore, the root mean square error of the affected limb was related significantly to motor function as determined by Fugl-Myer assessment. These results demonstrate impairment of the dexterity of both the affected and the unaffected lower extremities in patients with stroke.

Equivalent Linearization of a Squeeze Film Damper

1986 ◽  
Vol 108 (4) ◽  
pp. 434-440 ◽  
Author(s):  
Songqi Chen ◽  
Shengpei Liu
Keyword(s):  
Squeeze Film ◽  
Equivalent Linearization ◽  
Cycle Period ◽  
Mean Square ◽  
Damping Force ◽  
Average Force ◽  
Squeeze Film Damper ◽  
Spring Force ◽  
The Difference ◽  
Mean Square Errors

In this paper, the equivalent linearization of an intershaft squeeze film damper in a two shaft engine system is investigated. The two shaft centers at the damper position are assumed to move in different elliptical offset orbits and at synchronous frequency with the unbalanced rotor (e.g., the high pressure rotor). The nonlinear damper force is resolved into two orthogonal components along the absolute coordinate directions and, in turn, each of these force components is supposed to be equivalent to the sum of an average force, a linear spring force, and a linear damping force in the corresponding direction. By using the method of equivalent linearization by harmonic balance, the six parameters of the equivalent forces, including two average forces, two equivalent spring coefficients, and two equivalent damping coefficients, are expressed analytically by the squeeze film forces and the assumed orbital motion of the two shaft centers at the damper position. The analytical expressions of the squeeze film forces are derived from an approximate solution of the basic Reynolds equation. The results obtained are verified by the method of equivalent linearization by minimum mean square errors. It shows that the six obtained parameters make the mean square errors minimum over a cycle period of motion, the errors being the difference between the equivalent forces and the actual nonlinear forces.

scholarly journals Which method is more accurate? or errors have error bars

2017 ◽  
Author(s):  
Jan H Jensen
Keyword(s):  
Root Mean Square ◽  
Mean Square ◽  
Data Set ◽  
Error Bars ◽  
The Difference ◽  
Mean Square Errors

This document is my attempt at distilling some of the information in two papers published by Anthony Nicholls (J. Comput. Aided Mol. Des. 2014, 28, 887; ibid 2016, 30, 103). Anthony also very kindly provided some new equations, not found in the papers, in response to my questions. The paper describes how one determines whether the difference in accuracy of two methods in predicting some properties for the same data set is statistically significant using root-mean-square errors, mean absolute errors, mean errors, and Pearsons r values.

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.

Development of a Compact 1-D Micromanipulator with Flexure Manufactured Using Rapid Prototyping

2012 ◽  
Vol 2 (2) ◽  
pp. 47-57 ◽  
Author(s):  
Su Zhao ◽  
Yan Naing Aye ◽  
Cheng Yap Shee ◽  
Wei Tech Ang
Keyword(s):  
Finite Element ◽  
Rapid Prototyping ◽  
Root Mean Square ◽  
Piezoelectric Actuator ◽  
Trajectory Tracking ◽  
Nonlinear Stiffness ◽  
Mean Square ◽  
Manufacturing Costs ◽  
Simulation Results ◽  
Mean Square Errors

Presented is the design and initial experimental results of a compact 1-D micromanipulator. The presented manipulator is a piezoelectric actuator based complaint mechanism with a compact translational flexure manufactured using rapid prototyping. Rapid prototyping allows complicated designs with low manufacturing costs. Analytical and Finite Element (FE) models for designing the flexure are presented. The simulation results are compared with experiments conducted on a prototype. Nonlinear stiffness is measured and evaluated. The importance of pre-loading force is investigated. Trajectory tracking tests at different frequencies are performed on the manipulator. The maximum and root mean square errors are analyzed.

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