A Proposal for the Genetic Lateral Tuning of Linguistic Fuzzy Systems and Its Interaction With Rule Selection

Linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings. However, since the accuracy and the interpretability of the obtained model are contradictory properties, the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled. To solve this problem, one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level. In this paper, a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions, with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability. To do so, we consider a new rule representation scheme base on the linguistic 2-tuples representation model which allows the lateral variation of the involved labels. Furthermore, the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper, presenting results on different real applications. The obtained results show the good performance of the proposed approach in high-dimensional problems and its ability to cooperate with methods to remove unnecessary rules.     

Interpretable fuzzy partitioning of classified data with variable granularity

Fuzzy rule-based systems are effective tools for acquiring knowledge from data and represent it in a linguistically interpretable form. To achieve interpretability, input features are granulated in fuzzy partitions. A critical design decision is the selection of the granularity level for each input feature. This paper presents an approach, called DC^* (Double Clustering with A^*), for automatically designing interpretable fuzzy partitions with optimal granularity. DC^* is specific for classification problems and is mainly based on a two-stage process: the first stage identifies clusters of multidimensional samples in order to derive class-labeled prototypes; in the second stage the one-dimensional projections of such prototypes are further clustered along each dimension simultaneously, thus minimizing the number of clusters for each feature. Moreover, the resulting one-dimensional clusters provide information to define fuzzy partitions that satisfy a number of interpretability constraints and exhibit variable granularity levels. The fuzzy sets in each partition can be labeled by meaningful linguistic terms and used to represent knowledge in a natural language form. Experimental results on both synthetic and real data show that the derived fuzzy partitions can be exploited to define very compact fuzzy rule-based systems that exhibit high linguistic interpretability and good classification accuracy.     

Design of fuzzy rule-based classifiers with semantic cointension

In computing with words (CWW), knowledge is linguistically represented and has an explicit semantics defined through fuzzy information granules. The linguistic representation, in turn, naturally bears an implicit semantics that belongs to users reading the knowledge base; hence a necessary condition for achieving interpretability requires that implicit and explicit semantics are cointensive. Interpretability is definitely stringent when knowledge must be acquired from data through inductive learning. Therefore, in this paper we propose a methodology for designing interpretable fuzzy models through semantic cointension. We focus our analysis on fuzzy rule-based classifiers (FRBCs), where we observe that rules resemble logical propositions, thus semantic cointension can be partially regarded as the fulfillment of the “logical view”, i.e. the set of basic logical laws that are required in any logical system. The proposed approach is grounded on the employment of a couple of tools: DCf, which extracts interpretable classification rules from data, and Espresso, that is capable of fast minimization of Boolean propositions. Our research demonstrates that it is possible to design models that exhibit good classification accuracy combined with high interpretability in the sense of semantic cointension. Also, structural parameters that quantify model complexity show that the derived models are also simple enough to be read and understood.     

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

One of the problems associated to linguistic fuzzy modeling is its lack of accuracy when modeling some complex systems. To overcome this problem, many different possibilities of improving the accuracy of linguistic fuzzy modeling have been considered in the specialized literature. We will call these approaches as basic refinement approaches. In this work, we present a short study of how these basic approaches can be combined to obtain new hybrid approaches presenting a better trade-off between interpretability and accuracy. As an example of application of these kinds of systems, we analyze seven hybrid approaches to develop accurate and still interpretable fuzzy rule-based systems, which will be tested considering two real-world problems.This work has been supported by the spanish cicyt project tic2002-04036-c05-01 (keel).     

Learning concurrently data and rule bases of Mamdani fuzzy rule-based systems by exploiting a novel interpretability index

Interpretability of Mamdani fuzzy rule-based systems (MFRBSs) has been widely discussed in the last years, especially in the framework of multi-objective evolutionary fuzzy systems (MOEFSs). Here, multi-objective evolutionary algorithms (MOEAs) are applied to generate a set of MFRBSs with different trade-offs between interpretability and accuracy. In MOEFSs interpretability has often been measured in terms of complexity of the rule base and only recently partition integrity has also been considered. In this paper, we introduce a novel index for evaluating the interpretability of MFRBSs, which takes both the rule base complexity and the data base integrity into account. We discuss the use of this index in MOEFSs, which generate MFRBSs by concurrently learning the rule base, the linguistic partition granularities and the membership function parameters during the evolutionary process. The proposed approach has been experimented on six real world regression problems and the results have been compared with those obtained by applying the same MOEA, with only accuracy and complexity of the rule base as objectives. We show that our approach achieves the best trade-offs between interpretability and accuracy.     

Complexity reduction and interpretability improvement for fuzzy rule systems based on simple interpretability measures and indices by bi-objective evolutionary rule selection

The aim of this paper is to develop a general post-processing methodology to reduce the complexity of data-driven linguistic fuzzy models, in order to reach simpler fuzzy models preserving enough accuracy and better fuzzy linguistic performance with respect to their initial values. This post-processing approach is based on rule selection via the formulation of a bi-objective problem with one objective focusing on accuracy and the other on interpretability. The latter is defined via the aggregation of several interpretability measures, based on the concepts of similarity and complexity of fuzzy systems and rules. In this way, a measure of the fuzzy model interpretability is given. Two neuro-fuzzy systems for providing initial fuzzy models, Fuzzy Adaptive System ART based and Neuro-Fuzzy Function Approximation and several case studies, data sets from KEEL Project Repository, are used to check this approach. Both fuzzy and neuro-fuzzy systems generate Mamdani-type fuzzy rule-based systems, each with its own particularities and complexities from the point of view of the fuzzy sets and the rule generation. Based on these systems and data sets, several fuzzy models are generated to check the performance of the proposal under different restrictions of complexity and fuzziness.     

Fuzzy rule weight modification with particle swarm optimisation

The most challenging problem in developing fuzzy rule-based classification systems is the construction of a fuzzy rule base for the target problem. In many practical applications, fuzzy sets that are of particular linguistic meanings, are often predefined by domain experts and required to be maintained in order to ensure interpretability of any subsequent inference results. However, learning fuzzy rules using fixed fuzzy quantity space without any qualification will restrict the accuracy of the resulting rules. Fortunately, adjusting the weights of fuzzy rules can help improve classification accuracy without degrading the interpretability. There have been different proposals for fuzzy rule weight tuning through the use of various heuristics with limited success. This paper proposes an alternative approach using Particle Swarm Optimisation in the search of a set of optimal rule weights, entailing high classification accuracy. Systematic experimental studies are carried out using common benchmark data sets, in comparison to popular rule based learning classifiers. The results demonstrate that the proposed approach can boost classification performance, especially when the size of the initially built rule base is relatively small, and is competitive to popular rule-based learning classifiers.     

A multi-objective genetic optimization of interpretability-oriented fuzzy rule-based classifiers

The paper presents a multi-objective genetic approach to design interpretability-oriented fuzzy rule-based classifiers from data. The proposed approach allows us to obtain systems with various levels of compromise between their accuracy and interpretability. During the learning process, parameters of the membership functions, as well as the structure of the classifier's fuzzy rule base (i.e., the number of rules, the number of rule antecedents, etc.) evolve simultaneously using a Pittsburgh-type genetic approach. Since there is no particular coding of fuzzy rule structures in a chromosome (it reduces computational complexity of the algorithm), original crossover and mutation operators, as well as chromosome-repairing technique to directly transform the rules are also proposed. To evaluate both the accuracy and interpretability of the system, two measures are used. The first one – an accuracy measure – is based on the root mean square error of the system's response. The second one – an interpretability measure – is based on the arithmetic mean of three components: (a) the average length of rules (the average number of antecedents used in the rules), (b) the number of active fuzzy sets and (c) the number of active inputs of the system (an active fuzzy set or input means a set or input used by at least one fuzzy rule). Both measures are used as objectives in multi-objective (2-objective in our case) genetic optimization approaches such as well-known SPEA2 and NSGA-II algorithms. Moreover, for the purpose of comparison with several alternative approaches, the experiments are carried out both considering the so-called strong fuzzy partitions (SFPs) of attribute domains and without them. SFPs provide more semantically meaningful solutions, usually at the expense of their accuracy. The operation of the proposed technique in various classification problems is tested with the use of 20 benchmark data sets and compared to 11 alternative classification techniques. The experiments show that the proposed approach generates classifiers of significantly improved interpretability, while still characterized by competitive accuracy.     

A parsimony fuzzy rule-based classifier using axiomatic fuzzy set theory and support vector machines

This paper proposes a classification method that is based on easily interpretable fuzzy rules and fully capitalizes on the two key technologies, namely pruning the outliers in the training data by SVMs (support vector machines), i.e., eliminating the influence of outliers on the learning process; finding a fuzzy set with sound linguistic interpretation to describe each class based on AFS (axiomatic fuzzy set) theory. Compared with other fuzzy rule-based methods, the proposed models are usually more compact and easily understandable for the users since each class is described by much fewer rules. The proposed method also comes with two other advantages, namely, each rule obtained from the proposed algorithm is simply a conjunction of some linguistic terms, there are no parameters that are required to be tuned. The proposed classification method is compared with the previously published fuzzy rule-based classifiers by testing them on 16 UCI data sets. The results show that the fuzzy rule-based classifier presented in this paper, offers a compact, understandable and accurate classification scheme. A balance is achieved between the interpretability and the accuracy.     

Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system

ObjectiveTo develop a classifier that tackles the problem of determining the risk of a patient of suffering from a cardiovascular disease within the next 10 years. The system has to provide both a diagnosis and an interpretable model explaining the decision. In this way, doctors are able to analyse the usefulness of the information given by the system.MethodsLinguistic fuzzy rule-based classification systems are used, since they provide a good classification rate and a highly interpretable model. More specifically, a new methodology to combine fuzzy rule-based classification systems with interval-valued fuzzy sets is proposed, which is composed of three steps: (1) the modelling of the linguistic labels of the classifier using interval-valued fuzzy sets; (2) the use of the K_α operator in the inference process and (3) the application of a genetic tuning to find the best ignorance degree that each interval-valued fuzzy set represents as well as the best value for the parameter α of the K_α operator in each rule.ResultsThe suitability of the new proposal to deal with this medical diagnosis classification problem is shown by comparing its performance with respect to the one provided by two classical fuzzy classifiers and a previous interval-valued fuzzy rule-based classification system. The performance of the new method is statistically better than the ones obtained with the methods considered in the comparison. The new proposal enhances both the total number of correctly diagnosed patients, around 3% with respect the classical fuzzy classifiers and around 1% vs. the previous interval-valued fuzzy classifier, and the classifier ability to correctly differentiate patients of the different risk categories.ConclusionThe proposed methodology is a suitable tool to face the medical diagnosis of cardiovascular diseases, since it obtains a good classification rate and it also provides an interpretable model that can be easily understood by the doctors.     

Learning knowledge bases of multi-objective evolutionary fuzzy systems by simultaneously optimizing accuracy, complexity and partition integrity

In the last few years, several papers have exploited multi-objective evolutionary algorithms (MOEAs) to generate Mamdani fuzzy rule-based systems (MFRBSs) with different trade-offs between interpretability and accuracy. In this framework, a common approach is to distinguish between interpretability of the rule base (RB), also known as complexity, and interpretability of fuzzy partitions, also known as integrity of the database (DB). Typically, complexity has been used as one of the objectives of the MOEAs, while partition integrity has been ensured by enforcing constraints on the membership function (MF) parameters. In this paper, we propose to adopt partition integrity as an objective of the evolutionary process. To this aim, we first discuss how partition integrity can be measured by using a purposely defined index based on the similarity between the partitions learned during the evolutionary process and the initial interpretable partitions defined by an expert. Then, we introduce a three-objective evolutionary algorithm which generates a set of MFRBSs with different trade-offs between complexity, accuracy and partition integrity by concurrently learning the RB and the MF parameters of the linguistic variables. Accuracy is assessed in terms of mean squared error between the actual and the predicted values, complexity is calculated as the total number of conditions in the antecedents of the rules and integrity is measured by using the purposely defined index. The proposed approach has been experimented on six real-world regression problems. The results have been compared with those obtained by applying the same MOEA, but with only accuracy and complexity as objectives, both to learn only RBs, and to concurrently learn RBs and MF parameters, with and without constraints on the parameter tuning. We show that our approach achieves the best trade-offs between interpretability and accuracy. Finally, we compare our approach with a similar MOEA recently proposed in the literature.     

Performance evaluation of evolutionary multiobjective optimization algorithms for multiobjective fuzzy genetics-based machine learning

Recently, evolutionary multiobjective optimization (EMO) algorithms have been utilized for the design of accurate and interpretable fuzzy rule-based systems. This research area is often referred to as multiobjective genetic fuzzy systems (MoGFS), where EMO algorithms are used to search for non-dominated fuzzy rule-based systems with respect to their accuracy and interpretability. In this paper, we examine the ability of EMO algorithms to efficiently search for Pareto optimal or near Pareto optimal fuzzy rule-based systems for classification problems. We use NSGA-II (elitist non-dominated sorting genetic algorithm), its variants, and MOEA/D (multiobjective evolutionary algorithm based on decomposition) in our multiobjective fuzzy genetics-based machine learning (MoFGBML) algorithm. Classification performance of obtained fuzzy rule-based systems by each EMO algorithm is evaluated for training data and test data under various settings of the available computation load and the granularity of fuzzy partitions. Experimental results in this paper suggest that reported classification performance of MoGFS in the literature can be further improved using more computation load, more efficient EMO algorithms, and/or more antecedent fuzzy sets from finer fuzzy partitions.     

Predicting corporate investment/non-investment grade by using interval-valued fuzzy rule-based systems—A cross-region analysis

Systems for predicting corporate rating have attracted considerable interest in soft computing research due to the requirements for both accuracy and interpretability. In addition, the high uncertainty associated primarily with linguistic uncertainties and disagreement among experts is another challenging problem. To overcome these problems, this study proposes a hybrid evolutionary interval-valued fuzzy rule-based system, namely IVTURS, combined with evolutionary feature selection component. This model is used to predict the investment/non-investment grades of companies from four regions, namely Emerging countries, the EU, the United States, and other developed countries. To evaluate prediction performance, a yield measure is used that combines the return and default rates of companies. Here, we show that using interval-valued fuzzy sets leads to higher accuracy, particularly with the growing granularity at the fuzzy partition level. The proposed prediction model is then compared with several state-of-the-art evolutionary fuzzy rule-based systems. The obtained results show that the proposed model is especially suitable for high-dimensional problems, without facing rule base interpretability issues. This finding indicates that the model is preferable for investors oriented toward developed markets such as the EU and the United States.     

A proposal for improving the accuracy of linguistic modeling

We propose accurate linguistic modeling, a methodology to design linguistic models that are accurate to a high degree and may be suitably interpreted. This approach is based on two main assumptions related to the interpolative reasoning developed by fuzzy rule-based systems: a small change in the structure of the linguistic model based on allowing the linguistic rule to have two consequents associated; and a different way to obtain the knowledge base based on generating a preliminary fuzzy rule set composed of a large number of rules and then selecting the subset of them best cooperating. Moreover, we introduce two variants of an automatic design method for these kinds of linguistic models based on two well-known inductive fuzzy rule generation processes and a genetic process for selecting rules. The accuracy of the proposed methods is compared with other linguistic modeling techniques with different characteristics when solving of three different applications     

Reducing the number of parameters of a fuzzy system using scaling functions

Learning techniques are tailored for fuzzy systems in order to tune them or even for deriving fuzzy rules from data. However, a compromise between accuracy and interpretability has to be found. Flexible fuzzy systems with a large number of parameters and high degrees of freedom tend to function as black boxes. In this paper, we introduce an interpretation of fuzzy systems that enables us to work with a small number of parameters without loosing flexibility or interpretability. In this way, we can provide a learning algorithm that is efficient and yields accuracy as well as interpretability. Our fuzzy system is based on extremely simple fuzzy sets and transformations using interpretable scaling functions of the input variables.     

Tradeoff search methods between interpretability and accuracy of the identification fuzzy systems based on rules

This paper starts a brief historical overview of occurrence and development of fuzzy systems and their applications. Integration methods are proposed to construct a fuzzy system using other AI methods, achieving synergy effect. Accuracy and interpretability are selected as main properties of rule-based fuzzy systems. The tradeoff between interpretability and accuracy is considered to be the actual problem. The purpose of this paper is the in-depth study of the methods and tools to achieve a tradeoff for accuracy and interpretability in rule-based fuzzy systems and to describe our interpretability indexes. A comparison of the existing ways of interpretability estimation has been made We also propose the new way to construct heuristic interpretability indexes as a quantitative measure of interpretability. In the main part of this paper we describe previously used approaches, the current state and original authors’ methods for achieving tradeoff between accuracy and complexity.     

A multistage genetic fuzzy classifier for land cover classification from satellite imagery

A linguistic boosted genetic fuzzy classifier (LiBGFC) is proposed in this paper for land cover classification from multispectral images. The LiBGFC is a three-stage process, aiming at effectively tackling the interpretability versus accuracy tradeoff problem. The first stage iteratively generates fuzzy rules, as directed by a boosting algorithm that localizes new rules in uncovered subspaces of the feature space, implicitly preserving the cooperation with previously derived ones. Each rule is able to select the required features, further improving the interpretability of the obtained model. Special provision is taken in the formulation of the fitness function to avoid the creation of redundant rules. A simplification stage follows the first one aiming at further improving the interpretability of the initial rule base, providing a more compact and interpretable solution. Finally, a genetic tuning stage fine tunes the fuzzy sets database improving the classification performance of the obtained model. The LiBGFC is tested using an IKONOS multispectral VHR image, in a lake-wetland ecosystem of international importance. The results indicate the effectiveness of the proposed system in handling multidimensional feature spaces, producing easily understandable fuzzy models.     

GP-COACH: Genetic Programming-based learning of COmpact and ACcurate fuzzy rule-based classification systems for High-dimensional problems

In this paper we propose GP-COACH, a Genetic Programming-based method for the learning of COmpact and ACcurate fuzzy rule-based classification systems for High-dimensional problems. GP-COACH learns disjunctive normal form rules (generated by means of a context-free grammar) coded as one rule per tree. The population constitutes the rule base, so it is a genetic cooperative-competitive learning approach. GP-COACH uses a token competition mechanism to maintain the diversity of the population and this obliges the rules to compete and cooperate among themselves and allows the obtaining of a compact set of fuzzy rules. The results obtained have been validated by the use of non-parametric statistical tests, showing a good performance in terms of accuracy and interpretability.     

A hierarchical model of a linguistic variable

In this work a theoretical hierarchical model of dichotomous linguistic variables is presented. The model incorporates certain features of the approximate reasoning approach and others of the Fuzzy Control approach to Fuzzy Linguistic Variables. It allows sharing of the same hierarchical structure between the syntactic definition of a linguistic variable and its semantic definition given by fuzzy sets. This fact makes it easier to build symbolic operations between linguistic terms with a better grounded semantic interpretation. Moreover, the family of fuzzy sets which gives the semantics of each linguistic term constitutes a multiresolution system, and thanks to that any fuzzy set can be represented in terms of the set of linguistic terms. The model can also be considered a general framework for building more interpretable fuzzy linguistic variables with a high capacity of accuracy, which could be used to build more interpretable Fuzzy Rule Based Systems (FRBS).