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2003 | Buch

Interpretability Issues in Fuzzy Modeling

herausgegeben von: Dr. Jorge Casillas, Dr. Oscar Cordón, Dr. Francisco Herrera, Dr. Luis Magdalena

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Fuzziness and Soft Computing

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Über dieses Buch

Fuzzy modeling has become one of the most productive and successful results of fuzzy logic. Among others, it has been applied to knowledge discovery, automatic classification, long-term prediction, or medical and engineering analysis. The research developed in the topic during the last two decades has been mainly focused on exploiting the fuzzy model flexibility to obtain the highest accuracy. This approach usually sets aside the interpretability of the obtained models. However, we should remember the initial philosophy of fuzzy sets theory directed to serve the bridge between the human understanding and the machine processing. In this challenge, the ability of fuzzy models to express the behavior of the real system in a comprehensible manner acquires a great importance. This book collects the works of a group of experts in the field that advocate the interpretability improvements as a mechanism to obtain well balanced fuzzy models.

Inhaltsverzeichnis

Frontmatter

Overview

Frontmatter
Interpretability Improvements to Find the Balance Interpretability-Accuracy in Fuzzy Modeling: An Overview
Abstract
Abstract System modeling with fuzzy rule-based systems (FRBSs), i.e. fuzzy modeling (FM), usually comes with two contradictory requirements in the obtained model: the interpretability, capability to express the behavior of the real system in an understandable way, and the accuracy, capability to faithfully represent the real system. While linguistic FM (mainly developed by linguistic FRBSs) is focused on the interpretability, precise FM (mainly developed by Takagi-Sugeno-Kang FRBSs) is focused on the accuracy. Since both criteria are of vital importance in system modeling, the balance between them has started to pay attention in the fuzzy community in the last few years.
The chapter analyzes mechanisms to find this balance by improving the interpretability in linguistic FM: selecting input variables, reducing the fuzzy rule set, using more descriptive expressions, or performing linguistic approximation; and in precise FM: reducing the fuzzy rule set, reducing the number of fuzzy sets, or exploiting the local description of the rules.
Jorge Casillas, Oscar Cordón, Francisco Herrera, Luis Magdalena

Improving the Interpretability with Flexible Rule Structures

Frontmatter
Regaining Comprehensibility of Approximative Fuzzy Models via the Use of Linguistic Hedges
Abstract
This chapter presents an effective and efficient approach for translating rules that use approximative sets to rules that use descriptive sets and linguistic hedges of predefined meaning. Following this approach, descriptive models can take advantage of any existing approach to approximative modelling which is generally efficient and accurate, whilst employing rules that are comprehensible to human users. This allows the comprehensibility of approximative models to be restored. Although trapezoidal fuzzy sets, including triangular ones, are most commonly used in fuzzy modelling for computational simplicity, applications of conventional linguistic hedges over such sets typically fail to result in significant changes of the sets definition. In particular, the full membership part of a trapezoid membership function does not change at all. This does not help in many modelling tasks as intended. Therefore, this chapter also presents an improved version of more effective hedges specifically devised for trapezoidal fuzzy sets, including three which do not appear in the literature. Simulation results are provided to demonstrate the advantages of utilising the revised and newly introduced hedges for assisting fuzzy modelling, in comparison to the use of conventional ones.
Javier G. Marín-Blázquez, Qiang Shen
Identifying Flexible Structured Premises for Mining Concise Fuzzy Knowledge
Abstract
Data mining attains growing importance to ease the knowledge-acquisition bottleneck. This chapter discusses the issue of extracting, from accumulated data, a compact fuzzy rule base, to reach concise yet highly generalizing knowledge. For this purpose, we establish flexible structured premises of rules, allowing for not only canonical AND combinations of input fuzzy sets but also OR connectives of linguistic terms as well as incomplete compositions of input variables in the premise constitution. The later two forms of premises are beneficial for rule number reduction, as they achieve bigger coverage of the input space compared with the first premise form. A genetic-based search algorithm is utilized to explore optimal premise structure in combination with parameters of fuzzy set membership functions. Simulation results on several data sets are given to demonstrate the merits and characteristics of the presented method.
N. Xiong, L. Litz

Complexity Reduction in Linguistic Fuzzy Models

Frontmatter
A Multiobjective Genetic Learning Process for joint Feature Selection and Granularity and Contexts Learning in Fuzzy Rule-Based Classification Systems
Abstract
In this contribution, we propose a genetic process to select an appropiate set of features in a Fuzzy Rule-Based Classification System (FRBCS) and to automatically learn the whole Data Base definition using a non linear scaling function to adapt the fuzzy partition contexts and determining an appropiate granularity for each of them. An ad-hoc data covering learning method is considered to obtain the Rule Base. The method uses a multiobjective genetic algorithm in order to obtain a good trade-off between accuracy and interpretability.
Oscar Cordón, María José Del Jesus, Francisco Herrera, Luis Magdalena, Pedro Villar
Extracting Linguistic Fuzzy Models from Numerical Data-AFRELI Algorithm
Abstract
This paper discusses the concepts of linguistic integrity and interpretability. The concepts are used as a framework to design an algorithm to construct linguistic fuzzy models from Numerical Data. The constructed model combines prior knowledge (if present) and numerical information. Two algorithms are presented in this chapter. The main algorithm is the Autonomous Fuzzy Rule Extractor with Linguistic Integrity (AFRELI). This algorithm is complemented with the use of the Fu Zion algorithm created to merge consecutive membership functions while guar anteeing the distinguishability between fuzzy sets. Examples of function approximations and modeling of industrial data are presented as application examples.
Jairo Espinosa, Joos Vandewalle
Constrained Optimization of Fuzzy Decision Trees
Abstract
This paper proposes to build and optimize Takagi-Sugeno-like fuzzy regression trees with constraints aiming at preserving the interpretability of the rules. For this purpose, we state five requirements and deduce some conditions such as membership functions shared by the rules and strong fuzzy partitions on input variable domains. The membership functions are automatically placed thanks to an evolutionary strategy. We propose also a heuristics in order to find a sub-optimal structure of the tree.
Pierre-Yves Glorennec
A new method for inducing a set of interpretable fuzzy partitions and fuzzy inference systems from data
Abstract
To improve the interpretability of a fuzzy rule base generated from data, three conditions are necessary: semantic integrity must be respected, the number of rules should be small, and incomplete rules have to be handled. An incomplete rule is a rule defined only by a few variables. The presence of incomplete rules reflects the fact that all the variables do not have the same importance for all rules.
We propose a new method for learning a fuzzy rule base. A hierarchy of fuzzy partitions is generated for each input variable, based on a special metric suitable for fuzzy partitioning, before being used in building fuzzy inference systems of increasing complexity. Then the rule base is simplified. We introduce an intermediate selection level represented by a group of rules. The simplification tolerates some loss of accuracy while being guided by indices complementary to the usual numerical performance index.
The proposed approach is applied to a rice quality evaluation problem.
Serge Guillaume, Brigitte Charnomordic
A Feature Ranking Algorithm for Fuzzy Modelling Problems
Abstract
This paper presents a feature ranking method adapted to fuzzy modelling with output from a continuous range. Existing feature selection/ranking techniques are mostly suitable for classification problems, where the range of the output is discrete. These techniques result in a ranking of the input feature (variables). Our approach exploits an arbitrary fuzzy clustering of the model output data. Using these output clusters, similar feature ranking methods can be used as for classification, where the membership in a cluster (or class) will no longer be crisp, but a fuzzy value determined by the clustering. We propose the application of the Sequential Backward Selection (SBS) search method to determine the feature ranking by means of different criterion functions. We examined the proposed method and the criterion functions through a comparative analysis.
Domonkos Tikk, Tamás D. Gedeon, Kok Wai Wong
Interpretability in Multidimensional Classification
Abstract
Generating rule-based models from data is an efficient way of inferring information from large datasets. In high-dimensional spaces, the complexity of the model itself can undermine the interpretability of this information. This chapter introduces metrics quantifying the information flow between inputs, feature dimensions and output classes. These metrics are used to estimate the contribution of individual input features to a fuzzy classification task without making explicit use of the data underlying the model. Application of these techniques to a speech classification problem shows that significant reduction in the model dimensionality can be achieved with minimal accuracy loss.
Vincent Vanhoucke, Rosaria Silipo

Complexity Reduction in Precise Fuzzy Models

Frontmatter
Interpretable Semi-Mechanistic Fuzzy Models by Clustering, OLS and FIS Model Reduction
Summary
A semi-mechanistic fuzzy modeling technique is proposed to obtain compact and transparent process models based on small data-sets. Semi-mechanistic models are hybrid models that consist of a white box structure based on mechanistic relationships and black-box substructures to model less defined parts. First, it is shown that certain type of white-box models can be efficiently incorporated into a Takagi-Sugeno fuzzy rule structure. Next, the proposed models are identified from learning data and special attention is paid to transparency and accuracy aspects. The approach is based on a combination of (i) prior knowledge-based model structures, (ii) fuzzy clustering, (iii) orthogonal least-squares, and (iv) the modified Fisher’s interclass separability method. For the identification of the semimechanistic fuzzy model, a new fuzzy clustering method is proposed, i.e., clustering is achieved by the simultaneous identification of fuzzy sets defined on some of the scheduling variables and identification of the parameters of the local semimechanistic submodels. Subsequently, model reduction is applied to make the TS models as compact as possible, i.e., the most relevant consequent variables are selected by an orthogonal least squares method, and the modified Fisher’s interclass separability criteria is used for selection of relevant antecedent (scheduling) variables. The overall procedure is demonstrated by the development of a semimechanistic model for a biochemical process. Although the results do not carry over directly to other engineering fields, the main ideas and conclusions, will certainly hold for other application areas as well.
Janos Abonyi, Hans Roubos, Robert Babuska, Ferenc Szeifert
Trade-off between approximation accuracy and complexity: TS controller design via HOSVD based complexity minimization
Abstract
Higher order singular value decomposition (HOSVD) based complexity reduction method is proposed in this paper to tensor product based model approximation methods, especially, to Takagi-Sugeno (TS) fuzzy or polytopic models. The main motivation is that the TS model has exponentially growing computational complexity with the improvement of its approximation property through, as usually practiced, increasing the density of fuzzy partitions. The reduction technique proposed here is capable of pinpointing the contribution of each local linear model consequents of the fuzzy rules, which serves to remove the weakly contributing ones according to a given threshold. Reducing the number of local models leads directly to complexity reduction. The explicit form in this paper for the proposed reduction can also be applied on polytopic model approximation methods. A detailed illustrative example of a non-linear dynamic model is also given.
Péter Baranyi, Yeung Yam, Domonkos Tikk, Ron J. Patton
Simplification and reduction of fuzzy rules
Abstract
This chapter addresses rule base complexity in fuzzy models obtained from data. Data-driven fuzzy modeling is introduced, and two main approaches to complexity reduction in fuzzy rule-based models are presented: Similarity-driven rule base simplification, and rule reduction with orthogonal transforms.
Magne Setnes
Effect of Rule Representation in Rule Base Reduction
Abstract
An objective of merging rules in rule bases designed for system modeling and function approximation is to increase the scope of the rules and enhance their interpretability. The effectiveness of rule merging depends upon the underlying system, the learning algorithm, and the type of rule. In this paper we examine the ability to merge rules using variations of Mamdani and Takagi-Sugeno-Kang style rules. The generation of the rule base is a two part process; initially a uniform partition of the input domain is used to construct a rule base that satisfies a prescribed precision bound on the training data. A greedy algorithm is then employed to merge adjacent regions while preserving the precision bound. The objective of the algorithm is to produce fuzzy models of acceptable precision with a small number of rules. A set of experiments has been performed to compare the effect of the rule representation on the ability to reduce the number of rules and on the precision of the resulting models.
Thomas Sudkamp, Aaron Knapp, Jon Knapp
Singular Value-Based Fuzzy Reduction With Relaxed Normality Condition
Abstract
This work extends the results of a recent reduction method for fuzzy rule bases. The original approach conducts singular value decomposition (SVD) on the rule consequents and eliminates the weak and redundant components according to the magnitudes of the resulting singular values. The number of reduced rules as resulted depends on the number of singular values retained in the process. Conditions of sum normalization (SN), non-negativeness (NN) and Normality (NO) are imposed to ensure properly interpretable membership functions for the reduced rules. In this work, a new concept of relaxed Normality (RNO) condition is presented to enhance the interpretability of membership functions in situations where the NO condition cannot be strictly satisfied. The price to pay is an increase in the number of reduced rules and errors.
Yeung Yam, Chi Tin Yang, Péter Baranyi

Interpretability Constraints in TSK Fuzzy Rule-Based Systems

Frontmatter
Interpretability, Complexity, and Modular Structure of Fuzzy Systems
Abstract
Zadeh’s original motivation for fuzzy logic and the Fuzzy Rule-Based System (FRBS) was linguistic and hence possessed highly interpretable components. But as the complexity of a typical FRBS increases, it often becomes more like an uninterpretable neural network, and the Principle of Incompatibility predicts a degradation in interpretability for the same accuracy. This is particularly true of the so-called Takagi-Sugeno-Kang (TSK), or simply the Sugeno, approximator. We argue that imposing additional structure on the TSK system can significantly improve the tradeoff inherent in the Principle of Incompatibility. A promising structure was proposed recently in which the membership functions are local and sufficiently differentiable, and the consequent polynomials are rule-centered. This structure leads to the general interpretation that the consequent polynomials are Taylor series expansions. On this interpretation, a foundation for an algebra and a calculus of FRBSs can be built. We will illustrate these aspects of the proposed structure and discuss issues of modularity, functionality, and scalability of FRBSs.
Marwan Bikdash
Hierarchical Genetic Fuzzy Systems: Accuracy, Interpretability and Design Autonomy
Abstract
This chapter addresses hierarchical evolutionary rule-based fuzzy modeling, focusing on accuracy, interpretability and design autonomy issues. Special attention is given to interpretability in terms of visibility, simplicity, compactness, and consistency. As a consequence, fuzzy modeling is viewed as a decision making problem where accuracy, interpretability and autonomy are goals. The approach assumes that goals can be handled via corresponding single-objective ε-constrained decision making problems whose solution is produced by a hierarchical evolutionary process based on genetic algorithms, namely, a hierarchical genetic fuzzy system. In addition to performance improvement and interpretability constraints fulfillment, the hierarchical approach allows automatic tuning of a number of critical parameters and increases autonomy by minimizing user intervention. The fitting, generalization, and interpretation characteristics of the resulting fuzzy models are discussed using function approximation and classification problems.
Myriam Regattieri Delgado, Fernando Von Zuben, Fernando Gomide
About the trade-off between accuracy and interpretability of Takagi-Sugeno models in the context of nonlinear time series forecasting
Abstract
The focus of this chapter is related to the question of how to find appropriate Takagi-Sugeno (TS) rules in the framework of (chaotic) time series forecasting. We propose a generalization of the conventional TS system (GTS) allowing to evaluate the importance of any rule in the inference process. The added value of this feature has been put in light on two well-known chaotic time series.
Local (clustering-based) and global (gradient-based) learning strategies for GTS systems are compared in terms of interpretability and accuracy. It appears that there is an unavoidable compromise between these two objectives. Global learning seems to be superior in term of accuracy but cannot achieve the same level of interpretability as the local approach.
We also present an application of the GTS model in the field of forecasts combination with the target of generating interpretable final rules. This has led us to put additional constraints in the model. More precisely, we force the local linear models (rules conclusions) to achieve convex combination of the forecasts (input patterns). The proposed system may be useful for a wide range of applications where a consensus is required, including forecasts synthesis, controllers and patterns classifiers aggregation.
Antonio Fiordaliso
Accurate, transparent and compact fuzzy models by multi-objective evolutionary algorithms
Abstract
Interpretability aspects of fuzzy models have received quite some attention in recent years and may be obtained by using transparent rule-structures and well characterized fuzzy membership functions. Moreover, model compactness is important for the interpretability and is related to the number of rules and fuzzy sets. Besides these two criteria, the model accuracy should always be taken into account. In this way, several criteria appear in fuzzy modeling and then multiobjective evolutionary algorithms are a suitable, because these are able to capture several non-dominated solutions in a single run of the algorithm. For fuzzy modeling, we describe two multi-objective evolutionary algorithms that consider all three objectives. Differences between both algorithms arise in the fuzzy sets considered, trapezoidal and gaussian respectively. The algorithms apply an accuracy criterium and a transparency criterium, based on fuzzy set similarity, while compactness is achieved by a specific technique, incorporated ad hoc within the evolutionary algorithms. Finally, we propose a decision process to find the most satisfactory non-dominated solution. Results are shown for three approximation problems that were studied before by others authors.
Fernando Jiménez, Antonio F. Gómez-Skarmeta, Gracia Sánchez, Hans Roubos, Robert Babuška
Transparent Fuzzy Systems in Modelling and Control
Abstract
This chapter deals with low-level transparency of fuzzy systems that is necessary to ensure reliable interpretation of linguistic information provided by fuzzy systems. It is shown that for different types of fuzzy systems different definitions of transparency apply. Particular attention is paid to transparency protection mechanisms for data-driven optimisation algorithms such as gradient descent and genetic algorithms that otherwise would destroy the semantics of fuzzy systems in the course of optimisation. The need for transparency in fuzzy control is discussed and further illustrated by a control application of truck backer-upper.
Andri Riid, Ennu Rüstern
Uniform Fuzzy Partitions with Cardinal Splines and Wavelets: Getting Interpretable Linguistic Fuzzy Models
Abstract
In an abridged form, two main steps are habitual to build a Fuzzy Rule Model from a data set:
1.
To select the shape and distribution of the linguistic labels set for each variable
 
2.
To tune the set of rules to the training data set.
 
Adolfo R. de Soto

Assessments on the Interpretability Loss

Frontmatter
Relating the theory of partitions in MV-logic to the design of interpretable fuzzy systems
Abstract
The problem of interpretability of a set of membership functions asks whether each fuzzy set can be associated with a linguistic value and how coherent the association is with respect to any other possible association.
In the same way a component of a classical crisp partition is immediately interpreted, the interpretability of a set of membership functions is guaranteed by the act of partitioning the domain. Although many authors have proposed constraints on membership functions in order to guarantee the semantic interpretability of a fuzzy model, formal definitions of these constraints are usually not well connected with the theory of partitions in many-valued and fuzzy logic.
Our aim here is to relate the theory of partitions in MV-logic to the design of interpretable fuzzy systems, and to analyze the concepts of refinement and joint refinement of partitions.
Paolo Amato, Corrado Manara
A Formal Model of Interpretability of Linguistic Variables
Abstract
The present contribution is concerned with the interpretability of fuzzy rule-based systems. While this property is widely considered to be a crucial one in fuzzy rule-based modeling, a more detailed investigation of what “interpretability” actually means is still missing. So far, interpretability has often been associated with heuristic assumptions about shape and mutual overlapping of fuzzy membership functions. In this chapter, we attempt to approach this problem from a more general and formal point of view. First, we clarify what, in our opinion, the different aspects of interpretability are. Following that, we propose an axiomatic framework for the interpretability of linguistic variables (in Zadeh’s sense) which is underlined by examples and application perspectives.
Ulrich Bodenhofer, Peter Bauer
Expressing Relevance Interpretability and Accuracy of Rule-Based Systems
Summary
We discuss a problem of synthesis and analysis of rules based on experimental numeric data and study their interpretation capabilities. Two descriptors of the rules being viewed individually and en block are introduced. The relevance of the rules is quantified in terms of the data being covered by the antecedents and conclusions standing in the rule. While this index describes each rule individually, the consistency of the rule deals with the quality of the rule viewed vis-à-vis other rules. It expresses how much the rule “interacts ” with others in the sense that its conclusion is distorted by the conclusion parts coming from other rules. We show how the rules are formed by means of fuzzy clustering and their quality evaluated by means of the above indexes. We also discuss a construction of a granular mapping (that is a mapping between fuzzy clusters in the input and output spaces) and quantify its performance (approximation capabilities at the numeric level). Global characteristics of a set of rules are also discussed and related to the number of information granules formed in the space of antecedents and conclusions.
Witold Pedrycz
Conciseness of Fuzzy Models
Abstract
Fuzzy models are used to describe input-output relationships of unknown nonlinear systems in an interpretable manner for humans. Interpretability is one of the indispensable features of fuzzy models, which is closely related to their conciseness. The authors introduce the conciseness of fuzzy models, based on observations that humans grasp the input-output relationships by granules. The conciseness measure is then formulated by introducing De Luca and Termini’s fuzzy entropy and a new measure is derived from the analogy of relative entropy. This chapter also discusses the conflicting relationships between the conciseness and the accuracy of fuzzy models. A fuzzy modeling with Pareto optimal solutions is presented. Numerical experiments are done to demonstrate the effects of the conciseness measure.
Toshihiro Suzuki, Takeshi Furuhashi
Exact trade-off between approximation accuracy and interpretability: solving the saturation problem for certain FRBSs
Abstract
Although, in literature various results can be found claiming that fuzzy rule-based systems (FRBSs) possess the universal approximation property, to reach arbitrary accuracy the necessary number of rules are unbounded. Therefore, the inherent property of FRBSs in the original sense of Zadeh, namely that they can be characterized by a semantic relying on linguistic terms is lost. If we restrict the number of rules, universal approximation is not valid anymore as it was shown for, including others, Sugeno and TSK type models [10,19]. Due to this theoretic bound there is recently a great demand among researchers on finding trade-off techniques between a required accuracy and the number of rules, and as such, they attempt to determine the (optimal) number of rules as a function of accuracy. Naturally, to obtain such results one has to restrict somehow the set of continuous functions, usually requiring some smoothness conditions on the approximated function. In terms of approximation theory this is the so-called saturation problem, the determination of optimal order and class of approximation. Hitherto, saturation classes and orders have not been determined for FRBSs and neural networks. In this paper we solve the saturation problem for a special type of fuzzy controller, for the generalized KH-interpolator, being a suitable inference method in sparse rule bases.
Domonkos Tikk, Péter Baranyi

Interpretation of Black-Box Models as Fuzzy Rule-Based Models

Frontmatter
Interpretability improvement of RBF-based neurofuzzy systems using regularized learning
Abstract
Radial-basis-function (RBF) networks are mathematically equivalent to a class of fuzzy systems under mild conditions. Therefore, RBF networks have widely been used in learning of neurofuzzy systems to improve the performance. However, in most cases, the interpretability of fuzzy system will get lost after neural network learning. This chapter proposes a learning method using interpretability based regularization for neurofuzzy systems. This method can either be used in extracting interpretable fuzzy rules from RBF networks or in improving the interpretability of RBF-based neurofuzzy systems. Two simulation examples are presented to show the effectiveness of the proposed method.
Yaochu Jin
Extracting Fuzzy Classification Rules from Fuzzy Clusters on the Basis of Separating Hyperplanes
Abstract
Fuzzy clustering provides a (fuzzy) classification of data into different classes. From the result of a fuzzy cluster analysis fuzzy classification rules can be derived. The most common techniques for this derivation of rules are based on projections of the clusters. The corresponding rules classify only approximately in the same way as the fuzzy clusters themselves, since a certain loss of information has to be tolerated caused by the projections. In this paper, we propose to compute the class or cluster boundaries induced by the fuzzy clusters explicitly and to build up fuzzy rules that reflect exactly these boundaries.
Birka von Schmidt, Frank Klawonn
Metadaten
Titel
Interpretability Issues in Fuzzy Modeling
herausgegeben von
Dr. Jorge Casillas
Dr. Oscar Cordón
Dr. Francisco Herrera
Dr. Luis Magdalena
Copyright-Jahr
2003
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-37057-4
Print ISBN
978-3-642-05702-1
DOI
https://doi.org/10.1007/978-3-540-37057-4