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 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.

Railway crossing vertical vibration response prediction using a data-driven neuro-fuzzy model – Influence of train factors

2021 ◽  
pp. 095440972098666
Author(s):  
Kaveh Mehrzad ◽  
Shervan Ataei
Keyword(s):  
Model Comparison ◽  
Selection Process ◽  
Fuzzy Model ◽  
Response Prediction ◽  
Vibration Response ◽  
Vibration Monitoring ◽  
Data Driven ◽  
Steady Increase ◽  
Neuro Fuzzy ◽  
Railway Crossing

This paper provides a data-driven model of the vibration response of a railway crossing during vehicle passages. Many of the features of trains passing through instrumented crossing are extracted from measured data. Based on the feature selection process, speed, dynamic axle load and the number of wagons are found proper inputs in the prediction model. Train-crossing interaction response at a crossing due to passing trains is modeled from a data-driven Neuro-Fuzzy soft computing approach. Locally Linear Model Tree (LOLIMOT) is applied to predict the crossing nose acceleration. The model comparison against measurements shows that the ability to predict the extrapolation cases at off-range speeds has satisfactory compatibility. The monitored passing trains are ranked based on the LOLIMOT input space dimension cuts and extrapolation of the model up to higher train speeds. The influence of train factors (i.e. speed, dynamic axle load, number of wagons) on crossing response is demonstrated. Also, based on the analysis results, it is concluded that with a steady increase in train speeds, some trains show a greater amplification in vibration response than others. The results can be applied in data processing in the crossing vibration monitoring and detection of trains with crossing impact sensitive to speed increasing that can lead to proper operation policies to reduce damages and maintenance costs.

A Granular Unified Min-Max Fuzzy-Neuro Framework for Learning Fuzzy Systems

2009 ◽  
Vol 13 (5) ◽  
pp. 520-528 ◽  
Author(s):  
Mokhtar Beldjehem ◽  
Keyword(s):  
Cognitive Processes ◽  
Fuzzy Systems ◽  
Main Idea ◽  
Fuzzy Rules ◽  
Approximation Procedure ◽  
Unified Framework ◽  
Fine Grained ◽  
Neuro Fuzzy ◽  
Fuzzy Partitions ◽  
Hypotheses Generation

We propose a novel computational granular unified framework that is cognitively motivated for learning if-then fuzzy weighted rules by using a hybrid neuro-fuzzy or fuzzy-neuro possibilistic model appropriately crafted as a means to automatically extract or learn fuzzy rules from only input-output examples by integrating some useful concepts from the human cognitive processes and adding some interesting granular functionalities. This learning scheme uses an exhaustive search over the fuzzy partitions of involved variables, automatic fuzzy hypotheses generation, formulation and testing, and approximation procedure of Min-Max relational equations. The main idea is to start learning from coarse fuzzy partitions of the involved variables (both input and output) and proceed progressively toward fine-grained partitions until finding the appropriate partitions that fit the data. According to the complexity of the problem at hand, it learns the whole structure of the fuzzy system, i.e. conjointly appropriate fuzzy partitions, appropriate fuzzy rules, their number and their associated membership functions.

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.

scholarly journals Design of a hierarchical fuzzy model predictive controller

2015 ◽  
Vol 4 (2) ◽  
pp. 342 ◽  
Author(s):  
Zeinab Fallah ◽  
Mojtaba Ahmadieh Khanesar ◽  
Mohammad Teshnehlab
Keyword(s):  
Nonlinear Model ◽  
Fuzzy Systems ◽  
Gradient Descent ◽  
Fuzzy Controller ◽  
Fuzzy Model ◽  
Least Square ◽  
Continuous Stirred Tank Reactor ◽  
Descent Algorithm ◽  
Neuro Fuzzy ◽  
Gradient Descent Algorithm

In order to control a nonlinear system using Nonlinear Model Predictive Control (NMPC), a nonlinear model from system is required. In this paper, a hierarchical neuro-fuzzy model is used for nonlinear identification of the plant. The use of hierarchical neuro-fuzzy systems makes it possible to overcome the curse of dimensionality. In neuro-fuzzy systems, if the input number increases, then the number of rules increases exponentially. One solution to this problem is making use of Hierarchical Fuzzy System Mamdani (HFS) in which the number of the rules increases linearly. Gradient descent and recursive least square algorithm are used simultaneously to train the parameters of the HFS. Gradient Descent Algorithm is utilized to train the parameters, which appear nonlinearly in the output of HFS, and RLS is used to train the parameters of consequent the part, which appears linearly in the output of HFS. Finally, a model predictive fuzzy controller based on a predictive cost function is proposed. Using Gradient Descent Algorithm, the parameters of the controller are optimized. The proposed controller is simulated on the control of continuous stirred tank reactor. It is shown that the proposed method can control the system with high performance.

Rough Set-Based Neuro-Fuzzy System

2011 ◽  
pp. 1396-1403
Author(s):  
Kai Keng Ang ◽  
Chai Quek
Keyword(s):  
Rough Set ◽  
Fuzzy System ◽  
Fuzzy Model ◽  
Fuzzy Rules ◽  
Membership Functions ◽  
Linguistic Terms ◽  
Neural Learning ◽  
Neuro Fuzzy ◽  
Fuzzy Neural ◽  
Modeling Data

Neuro-fuzzy hybridization is the oldest and most popular methodology in soft computing (Mitra & Hayashi, 2000). Neuro-fuzzy hybridization is known as Fuzzy Neural Networks, or Neuro-Fuzzy Systems (NFS) in the literature (Lin & Lee, 1996; Mitra & Hayashi, 2000). NFS is capable of abstracting a fuzzy model from given numerical examples using neural learning techniques to formulate accurate predictions on unseen samples. The fuzzy model incorporates the human-like style of fuzzy reasoning through a linguistic model that comprises of if-then fuzzy rules and linguistic terms described by membership functions. Hence, the main strength of NFS in modeling data is universal approximation (Tikk, Kóczy, & Gedeon, 2003) with the ability to solicit interpretable if-then fuzzy rules (Guillaume, 2001). However, modeling data using NFS involves the contradictory requirements of interpretability versus accuracy. Prevailingly, NFS that focused on accuracy employed optimization which resulted in membership functions that derailed from human-interpretable linguistic terms, or employed large number of if-then fuzzy rules on high-dimensional data that exceeded human level interpretation. This article presents a novel hybrid intelligent Rough set-based Neuro-Fuzzy System (RNFS). RNFS synergizes the sound concept of knowledge reduction from rough set theory with NFS. RNFS reinforces the strength of NFS by employing rough set-based techniques to perform attribute and rule reductions, thereby improving the interpretability without compromising the accuracy of the abstracted fuzzy model.

scholarly journals Training High-Order Takagi-Sugeno Fuzzy Systems Using Batch Least Squares and Particle Swarm Optimization

2019 ◽  
Vol 22 (1) ◽  
pp. 22-34 ◽  
Author(s):  
Krzysztof Wiktorowicz ◽  
Tomasz Krzeszowski
Keyword(s):  
Particle Swarm Optimization ◽  
Least Squares ◽  
Fuzzy Sets ◽  
Fuzzy Systems ◽  
Piecewise Linear ◽  
Particle Swarm ◽  
Fuzzy Model ◽  
Pso Algorithm ◽  
Swarm Optimization ◽  
Takagi Sugeno

AbstractThis paper proposes two methods for training Takagi–Sugeno (T-S) fuzzy systems using batch least squares (BLS) and particle swarm optimization (PSO). The T-S system is considered with triangular and Gaussian membership functions in the antecedents and higher-order polynomials in the consequents of fuzzy rules. In the first method, the BLS determines the polynomials in a system in which the fuzzy sets are known. In the second method, the PSO algorithm determines the fuzzy sets, whereas the BLS determines the polynomials. In this paper, the ridge regression is used to stabilize the solution when the problem is close to the singularity. Thanks to this, the proposed methods can be applied when the number of observations is less than the number of predictors. Moreover, the leave-one-out cross-validation is used to avoid overfitting and this way to choose the structure of a fuzzy model. A method of obtaining piecewise linear regression by means of the zero-order T-S system is also presented.

scholarly journals On classification with missing data using rough-neuro-fuzzy systems

2010 ◽  
Vol 20 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Robert Nowicki
Keyword(s):  
Missing Data ◽  
Fuzzy Sets ◽  
Fuzzy Systems ◽  
Fuzzy Classification ◽  
Fuzzy Classifier ◽  
New Approach ◽  
Logical Type ◽  
Neuro Fuzzy

On classification with missing data using rough-neuro-fuzzy systemsThe paper presents a new approach to fuzzy classification in the case of missing data. Rough-fuzzy sets are incorporated into logical type neuro-fuzzy structures and a rough-neuro-fuzzy classifier is derived. Theorems which allow determining the structure of the rough-neuro-fuzzy classifier are given. Several experiments illustrating the performance of the roughneuro-fuzzy classifier working in the case of missing features are described.

scholarly journals Approximation of two-variable functions using high-order Takagi–Sugeno fuzzy systems, sparse regressions, and metaheuristic optimization

2020 ◽  
Vol 24 (20) ◽  
pp. 15113-15127
Author(s):  
Krzysztof Wiktorowicz ◽  
Tomasz Krzeszowski
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Systems ◽  
Fuzzy Model ◽  
Reference Method ◽  
Ordinary Least Squares ◽  
High Order ◽  
Forward Selection ◽  
Prediction Ability ◽  
Metaheuristic Optimization ◽  
Takagi Sugeno

Abstract This paper proposes a new hybrid method for training high-order Takagi–Sugeno fuzzy systems using sparse regressions and metaheuristic optimization. The fuzzy system is considered with Gaussian fuzzy sets in the antecedents and high-order polynomials in the consequents of fuzzy rules. The fuzzy sets can be chosen manually or determined by a metaheuristic optimization method (particle swarm optimization, genetic algorithm or simulated annealing), while the polynomials are obtained using ordinary least squares, ridge regression or sparse regressions (forward selection, least angle regression, least absolute shrinkage and selection operator, and elastic net regression). A quality criterion is proposed that expresses a compromise between the prediction ability of the fuzzy model and its sparsity. The conducted experiments showed that: (a) the use of sparse regressions and/or metaheuristic optimization can reduce the validation error compared with the reference method, and (b) the use of sparse regressions may simplify the fuzzy model by zeroing some of the coefficients.

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