Hybrid learning models to get the interpretability–accuracy trade-off in fuzzy modeling

A New Measure for Determining the Equivalent Symmetry of Decomposed Subsystems from Large Complex Cyber–Physical Systems

In this paper, we propose a new consistency measurement for classification rule sets that is based on the similarity of their classification abilities. The similarity of the classification abilities of the two rule sets is evaluated though the similarity of the corresponding partitions of the feature space using the different rule sets. The proposed consistency measure can be used to measure the equivalent symmetry of subsystems decomposed from a large, complex cyber–physical system (CPS). It can be used to verify whether the same knowledge is obtained by the sensing data in the different subsystems. In the experiments, five decision tree algorithms and eighteen datasets from the UCI machine learning repository are employed to extract the classification rules, and the consistency between the corresponding rule sets is investigated. The classification rule sets extracted from the use of the C4.5 algorithm on the electrical grid stability dataset have a consistency of 0.88, which implies that the different subsystems contain almost equivalent knowledge about the network stability.

CHECKING ORTHOGONAL TRANSFORMATIONS AND GENETIC ALGORITHMS FOR SELECTION OF FUZZY RULES BASED ON INTERPRETABILITY-ACCURACY CONCEPTS

Fuzzy modeling is one of the most known and used techniques in different areas to model the behavior of systems and processes. In most cases, as in data-driven fuzzy modeling, these fuzzy models reach a high performance from the point of view of accuracy, but from other points of view, such as complexity or interpretability, they can present a poor performance.

Several approaches are found in the bibliography to reduce the complexity and improve the interpretability of the fuzzy models. In this paper, a post-processing approach is carried out via rule selection, whose aim is to choose the most relevant rules for working together on the well-known accuracy-interpretability trade-off. The rule relevancy is based on Orthogonal Transformations, such as the SVD-QR rank revealing approach, the P-QR and OLS transformations. Rule selection is carried out using a genetic algorithm that takes into account the information obtained by the Orthogonal Transformations. The main objective is to check the true significance, drawbacks and advantages of the rule selection based on the orthogonal transformations via the rule firing strength matrix.

In order to carry out this aim, a neuro-fuzzy system, FasArt (Fuzzy Adaptive System ART based), and several case studies, data sets from the KEEL Project Repository, are used to tune and check this selection of rules based on orthogonal transformations, genetic selection and accuracy-interpretability trade-off. This neuro-fuzzy system generates Mamdani fuzzy rule based systems (FRBSs), in an approximative way. NSGA-II is the MOEA tool used to tune the proposed rule selection.

A MULTI-OBJECTIVE GENETIC ALGORITHM FOR TUNING AND RULE SELECTION TO OBTAIN ACCURATE AND COMPACT LINGUISTIC FUZZY RULE-BASED SYSTEMS

This work proposes the application of Multi-Objective Genetic Algorithms to obtain Fuzzy Rule-Based Systems with a better trade-off between interpretability and accuracy in linguistic fuzzy modelling problems. To do that, we present a new post-processing method that by considering selection of rules together with tuning of membership functions gets solutions only in the Pareto zone with the highest accuracy, i.e., containing solutions with the least number of possible rules but still presenting high accuracy. This method is based on the well-known SPEA2 algorithm, applying appropriate genetic operators and including some modifications to concentrate the search in the desired Pareto zone.

SYSTEMATIC DESIGN PROCEDURE OF TS-TYPE FUZZY CONTROLLERS

This paper contributes a new alternative for the synthesis of Takagi–Sugeno fuzzy logic controller with reduced rule base. A Genetic Approach to Fuzzy Supervised Learning algorithm called GAFSL based on the Multiobjective Genetic Algorithms (MGAs) is used to construct the proposed robust fuzzy controller. The result controller is similar to nonlinear PI/PD controllers. The tuning algorithm cannot only tune the scaling factors, the shapes of membership functions, and the consequent values, but also optimize the number of rules as possible with guaranteed desired performances: accuracy and robustness. The construction of the chromosomes is based on the mixed binary–real coding system. The genes of chromosome are arranged into two parts, the first part contains the control genes (the certainty factors) and the second part contains the parameters genes that represent the fuzzy knowledge base. The concept of elite strategy is adopted, where the best individuals in a population are regarded as elites. Computer simulation results on two nonlinear problems that are derived to demonstrate the powerful GAFSL algorithm.

MIXED GENETIC ALGORITHM APPROACH FOR FUZZY CLASSIFIER DESIGN

An important issue in the design of FRBS is the formation of fuzzy if-then rules and the membership functions. This paper presents a Mixed Genetic Algorithm (MGA) approach to obtain the optimal rule set and the membership function of the fuzzy classifier. While applying genetic algorithm for fuzzy classifier design, the membership functions are represented as real numbers and the fuzzy rules are represented as binary string. Modified forms of crossover and mutation operators are proposed to deal with the mixed string. The proposed genetic operators help to improve the convergence of GA and accuracy of the classifier. The performance of the proposed approach is evaluated through development of fuzzy classifier for seven standard data sets. From the simulation study it is found that the proposed algorithm produces a fuzzy classifier with minimum number of rules and high classification accuracy. Statistical analysis of the test results shows the superiority of the proposed algorithm over the existing methods.