Similar Fuzzy Sets Research Articles

Spherical fuzzy sets based integrated DEMATEL, ANP, VIKOR approach and its application for renewable energy selection in Turkey

Renewable energy continues occupying the global agenda for the low-carbon transition. Identifying suitable sources for renewable energy involves different criteria and alternatives, which can be defined as a Multi-Criteria Decision-Making (MCDM) problem. MCDM methods can be enriched with Group Decision Making (GDM) to merge individual judgments into group opinions, as well as with fuzzy sets to better simulate human opinions. To handle uncertainties of DMs’ preferences, a recently developed fuzzy set extension, the Spherical Fuzzy Sets (SFSs), is combined in this study with different MCDM tools to establish a decision support tool for the energy source selection problem. This article proposes a novel MCDM approach that integrates SF-DEMATEL (Spherical Fuzzy Decision-Making Trial and Evaluation Laboratory), SF-ANP (Spherical Fuzzy Analytic Network Process), and SF-VIKOR (Spherical Fuzzy Vlse Kriterijumska Optimizacija Kompromisno Resenje) algorithms in a GDM environment and presents its application in a case study for selecting renewable energy sources in Turkey. Comparing wind, geothermal, solar, hydropower, and biogas, the proposed method highlighted both wind and solar energy as a compromise solution as the most suitable options for Turkey. It has been found that compared to similar fuzzy sets, SFS can better reflect the hesitation degree to address the lack of knowledge and errors in the membership function definitions. The robustness and plausibility of these findings are demonstrated with sensitivity and comparative analyses, suggesting this MCDM method can also be applied to similar problems in other settings.

An online low-dimension fuzzy modeling method for time-varying processes

Many industrial processes are full of nonlinearity and noise. Fuzzy modeling often leads to issues with dimension disaster and poor robustness for such processes. To address these issues, an online low-dimension fuzzy method (OLDF) is proposed here. First, an explicit mapping-based low-dimension fuzzy method (EMLF) is proposed to handle intricate nonlinear and noisy processes. In this method, similar fuzzy sets are combined into larger ones using a novel similarity criterion to reduce the fuzzy rules. Then, an explicit mapping strategy is developed to represent the local nonlinearities in each enlarged fuzzy set through an explicit mapping function. Subsequently, a robust objective function is constructed to improve the robustness of EMLF model. This objective function minimizes both the mean and variance of the modeling error, ensuring robust performance even in scenarios featuring non-Gaussian noise or outliers. After that, an online update strategy is designed to account for time-varying dynamics. It designs an adaptive detection mechanism to identify changes in dynamic behaviors. Following detection, global and local updating strategies are developed to dynamically update model parameters. Further analysis and proof highlight the effectiveness of the proposed method. Experiments with time-varying processes also confirm its success.

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

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.

A Novel Three-Dimensional Fuzzy Modeling Method for Nonlinear Distributed Parameter Systems

Data-based spatio-temporal modeling has rapidly developed over the past ten years. However, traditional spatio-temporal modeling methods will encounter some uncertainty caused by model reduction. In addition, the established model is complicated and short of linguistic interpretability. In this study, a novel three-dimensional (3-D) fuzzy modeling framework without model reduction is proposed and a new 3-D fuzzy modeling method based on clustering and support vector regression is developed. This method is based on a 3-D fuzzy system that naturally fuses time/space separation and time/space synthesis into a unified framework. Utilizing the machine learning algorithms (clustering and support vector regression), a 3-D fuzzy system is constructed for modeling an unknown nonlinear distributed parameter system. The advantages of the proposed modeling method over the traditional spatio-temporal modeling method are linguistic interpretability and no reliance on model reduction. The proposed modeling method consists of three steps. First, the nearest neighborhood clustering algorithm is used to learn initial structure model of antecedent sets of 3-D fuzzy rules. Then, similarity measure is used to simplify the initial structure via combining similar fuzzy sets and similar fuzzy rules. Finally, a support vector regression algorithm is applied to calculate spatial functions in the consequent sets of 3-D fuzzy rules. The simulation results demonstrate the effectiveness of the proposed modeling method.

Handling interpretability issues in ANFIS using rule base simplification and constrained learning

Adaptive neuro-fuzzy inference system (ANFIS) is a well-known neuro-fuzzy model for approximating highly complex non-linear systems. ANFIS uses precise fuzzy modelling concept that aims at accuracy of a fuzzy model being designed than on interpretability. But interpretability of a fuzzy system is an equally important aspect of fuzzy modelling as accuracy. So far the research based on ANFIS has been mostly application based due to which the various issues related to the interpretability of ANFIS have not been dealt with. The rule base of ANFIS is typically obtained using data driven clustering algorithms. But this process introduces redundancy in the system in terms of similar fuzzy sets and redundant fuzzy rules which unnecessarily increases system complexity. This in turn reduces both the interpretability as well as generalization capability of ANFIS. Additionally in case of ANFIS, unconstrained gradient descent based learning algorithms are used to fine-tune the membership function parameters which usually result in a rule base with inconsistent, excessively overlapping and indistinguishable membership functions for input variables and thus the interpretability of the final optimized system is not guaranteed. This paper is based on addressing the issue of rule base redundancy in ANFIS to reduce complexity and enforcing constraints during learning phase to ensure interpretability of the final optimized system. Rule base redundancy is removed using similarity analysis based rule base simplification in which similar fuzzy sets are merged and subsequently resulting fuzzy rules with equal premises are combined. Hybrid learning technique which is an efficient parameter tuning method for ANFIS is constrained to prevent inconsistency, excessive overlapping and inclusion of membership functions so that the final fuzzy partitions of inputs stay interpretable. The empirical analysis of the impact of rule base simplification and constrained learning on ANFIS is done by application to two well-known benchmark problems and a real world stock price prediction problem. The introduction of rule base simplification and constrained learning in ANFIS modeling has shown better results in terms of obtaining a desired accuracy-interpretability tradeoff than conventional ANFIS.

Rule base simplification by using a similarity measure of fuzzy sets

In fuzzy models, redundancy may be present in the form of similar fuzzy sets, especially in the construction of a fuzzy system from a set of given training examples. In this paper, a simple formula for calculating the degree of similarity of Trapezoidal membership functions is derived in order to merge similar membership functions and hence to obtain a more transparent rule-base with a minimum number of membership functions. A simplification algorithm by using the proposed similarity measure has been applied to fuzzy models of nonlinear functions approximation.

Interpetable Support Vector Machines in Regression and Classification – Application in Process Engineering

Tools from the armoury of soft computing have been in focus of researches recently, since soft computing techniques are used for fault detection (classification techniques), forecasting of time-series data, inference, hypothesis testing, and modelling of causal relationships (regression techniques) in process engineering. These techniques solve two cardinal problems: learning from experimental data by neural networks and support vector based techniques and embedding existing structured human knowledge into fuzzy models. Support vector based models are one of the most commonly used soft computing techniques. Support vector based models are strong in feature selection and to achieve robust models and fuzzy logic helps to improve the interpretability of models. This paper deals with combining these existing soft computing techniques to get interpretable but accurate models for industrial purposes. The paper describes that trained support vector based models can be used for the construction of fuzzy rule-based classifier or regression models. However, the transformed support vector model does not automatically result in an interpretable fuzzy model because the support vector model results in a complex rulebase, where the number of rules is approximately 40-60% of the number of the training data. Hence, reduction of the support model-initialized fuzzy model is an essential task. For this purpose, a three-step reduction algorithm is used on the combination of previously published model reduction techniques. In the first step, the identification of the SV model is followed by the application of the Reduced Set method to decrease the number of kernel functions. The reduced SV model is then transformed into a fuzzy rule-based model. The interpretability of a fuzzy model highly depends on the distribution of the membership functions. Hence, the second reduction step is achieved by merging similar fuzzy sets based on a similarity measure. Finally, in the third step, an orthogonal least-squares method is used to reduce the number of rules and re-estimate the consequent parameters of the fuzzy rule-based model. The proposed approach is applied for classification problems and applied for Hammerstein system identification to illustrate the effectiveness of the technique.

Automatic construction of feedforward/recurrent fuzzy systems by clustering-aided simplex particle swarm optimization

This paper proposes a new approach for automating the structure and parameter learning of fuzzy systems by clustering-aided simplex particle swarm optimization, called CSPSO. Unlike most evolutionary fuzzy systems, where the structure of the fuzzy system is assigned in advance, an on-line fuzzy clustering approach is proposed for system structure learning. This structure learning not only helps determine the number of rules automatically, but also avoids the generation of highly similar fuzzy sets on each input variable. In addition, it improves subsequent parameter learning performance by assigning suitable initial locations of the fuzzy sets on each input variable. Once a new rule is generated, the corresponding parameters are further tuned by the hybrid of the simplex method and particle swarm optimization (PSO). In CSPSO, each fuzzy system corresponds to a particle in PSO, and the idea of the simplex method is incorporated to improve PSO searching performance. To verify the performance of CSPSO, two simulations on feedforward fuzzy systems design are performed. In addition, design of a recurrent fuzzy controller for a practical experiment on water bath temperature control is performed. Comparisons with other design approaches are also made in these examples.

A genetic method for designing TSK models based on objective weighting: application to classification problems.

This paper proposes a genetic-based algorithm for generating simple and well-defined Takagi-Sugeno-Kang (TSK) models. The method handles several attributes simultaneously, such as the input partition, feature selection and estimation of the consequent parameters. The model building process comprises three stages. In stage one, structure learning is formulated as an objective weighting optimization problem. Apart from the mean square error (MSE) and the number of rules, three additional criteria are introduced in the fitness function for measuring the quality of the partitions. Optimization of these measures leads to models with representative rules, small overlapping and efficient data cover. To obtain models with good local interpretation, the consequent parameters are determined using a local MSE function while the overall model is evaluated on the basis of a global MSE function. The initial model is simplified at stage two using a rule base simplification routine. Similar fuzzy sets are merged and the “don’t care” premises are recognized. Finally, the simplified models are fine-tuned at stage three to improve the model performance. The suggested method is used to generate TSK models with crisp and polynomial consequents for two benchmark classification problems, the iris and the wine data. Simulation results reveal the effectiveness of our method. The resulting models exhibit simple structure, interpretability and superior recognition rates compared to other methods of the literature.

Rule-base self-generation and simplification for data-driven fuzzy models

Data-driven fuzzy modeling has been used in a wide variety of applications. However, in fuzzy rule-based models acquired from numerical data, redundancy often exists in the form of redundant rules or similar fuzzy sets. This results in unnecessary structural complexity and decreases the interpretability of the system. In this paper, a rule-base self-extraction and simplification method is proposed to establish interpretable fuzzy models from numerical data. A fuzzy clustering technique associated with the proposed fuzzy partition validity index is used to extract the initial fuzzy rule-base and find out the optimal number of fuzzy rules. To reduce the complexity of fuzzy models while keeping good model accuracy, some approximate similarity measures are presented and a parameter fine-tuning mechanism is introduced to improve the accuracy of the simplified model. Using the proposed similarity measures, the redundant fuzzy rules are removed and similar fuzzy sets are merged to create a common fuzzy set in the rule base. The simplified rule base is computationally efficient and linguistically interpretable. The approach has been successfully applied to fuzzy models of non-linear function approximation, dynamical system modeling and mechanical property prediction for hot-rolled steels.

Learning Fuzzy Rules from Data

We present an algorithm, FUZZEX, for learning fuzzy rules from a corpus of data mapping input antecedents to output consequents. The input and output spaces are first divided into a grid of cells and primitive if % then rules formulated from each occupied input cell in the role of an antecedent The distribution of output cells into which data in the input cell maps, plays the role of the consequent interpreted as a fuzzy set. Those input cells associated with sufficiently similar fuzzy output sets are then combined to form a composite rule. A concise set of rules in Disjunctive Normal Form (DNF) is formed by combining adjacent input cells belonging to the same rule, thereby simplifying the logical expression of the antecedents. Optionally, more succinctness of expression may be obtained by recruiting into a rule, adjacent input cells with (little or) no data, but which happen to simplify rule expression. Preliminary testing on testbed datasets is presented. FUZZEX can be applied effectively to problems of large dimensionality.

Similarity measures in fuzzy rule base simplification

In fuzzy rule-based models acquired from numerical data, redundancy may be present in the form of similar fuzzy sets that represent compatible concepts. This results in an unnecessarily complex and less transparent linguistic description of the system. By using a measure of similarity, a rule base simplification method is proposed that reduces the number of fuzzy sets in the model. Similar fuzzy sets are merged to create a common fuzzy set to replace them in the rule base. If the redundancy in the model is high, merging similar fuzzy sets might result in equal rules that also can be merged, thereby reducing the number of rules as well. The simplified rule base is computationally more efficient and linguistically more tractable. The approach has been successfully applied to fuzzy models of real world systems.

Oxidation of propylene to acrolein over Cu-on-Al 2O 3 catalysts modified by molybdenum oxides

Adaptive neuro-fuzzy inference system (ANFIS) is a well-known neuro-fuzzy model for approximating highly complex non-linear systems. ANFIS uses precise fuzzy modelling concept that aims at accuracy of a fuzzy model being designed than on interpretability. But interpretability of a fuzzy system is an equally important aspect of fuzzy modelling as accuracy. So far the research based on ANFIS has been mostly application based due to which the various issues related to the interpretability of ANFIS have not been dealt with. The rule base of ANFIS is typically obtained using data driven clustering algorithms. But this process introduces redundancy in the system in terms of similar fuzzy sets and redundant fuzzy rules which unnecessarily increases system complexity. This in turn reduces both the interpretability as well as generalization capability of ANFIS. Additionally in case of ANFIS, unconstrained gradient descent based learning algorithms are used to fine-tune the membership function parameters which usually result in a rule base with inconsistent, excessively overlapping and indistinguishable membership functions for input variables and thus the interpretability of the final optimized system is not guaranteed. This paper is based on addressing the issue of rule base redundancy in ANFIS to reduce complexity and enforcing constraints during learning phase to ensure interpretability of the final optimized system. Rule base redundancy is removed using similarity analysis based rule base simplification in which similar fuzzy sets are merged and subsequently resulting fuzzy rules with equal premises are combined. Hybrid learning technique which is an efficient parameter tuning method for ANFIS is constrained to prevent inconsistency, excessive overlapping and inclusion of membership functions so that the final fuzzy partitions of inputs stay interpretable. The empirical analysis of the impact of rule base simplification and constrained learning on ANFIS is done by application to two well-known benchmark problems and a real world stock price prediction problem. The introduction of rule base simplification and constrained learning in ANFIS modeling has shown better results in terms of obtaining a desired accuracy-interpretability tradeoff than conventional ANFIS.