Michela Antonelli

Learning concurrently partition granularities and rule bases of Mamdani fuzzy systems in a multi-objective evolutionary framework

In this paper we propose a multi-objective evolutionary algorithm to generate Mamdani fuzzy rule-based systems with different good trade-offs between complexity and accuracy. The main novelty of the algorithm is that both rule base and granularity of the uniform partitions defined on the input and output variables are learned concurrently. To this aim, we introduce the concepts of virtual and concrete rule bases: the former is defined on linguistic variables, all partitioned with a fixed maximum number of fuzzy sets, while the latter takes into account, for each variable, a number of fuzzy sets as determined by the specific partition granularity of that variable. We exploit a chromosome composed of two parts, which codify the variables partition granularities, and the virtual rule base, respectively. Genetic operators manage virtual rule bases, whereas fitness evaluation relies on an appropriate mapping strategy between virtual and concrete rule bases. The algorithm has been tested on two real-world regression problems showing very promising results.

Genetic Training Instance Selection in Multiobjective Evolutionary Fuzzy Systems: A Coevolutionary Approach

When dealing with datasets that are characterized by a large number of instances, multiobjective evolutionary learning (MOEL) of fuzzy rule-based systems (FRBSs) suffers from high computational costs, mainly because of the fitness evaluation. The use of a reduced set of representative instances in place of the overall training set (TS) would considerably lessen the computational effort. Even though a large number of papers have proposed instance selection approaches, mainly in classification problems, how this selection should be performed, especially in the context of regression, is still an open issue. In this paper, we tackle the instance selection problem in the framework of MOEL of FRBSs through a coevolutionary approach. In the execution of the MOEL, periodically, a single-objective genetic algorithm (SOGA) evolves a population of reduced TSs. The SOGA aims to maximize a purposely defined index which measures how much the Pareto fronts computed by using, respectively, the reduced TS and the overall TS are close to each other: The closer the fronts, the more the reduced TS is representative of the overall TS. During the execution of the MOEL, the rule base and the membership function parameters of the fuzzy sets are concurrently learned by maximizing the accuracy and minimizing the complexity. We tested our approach on 12 large datasets. We adopted reduced TSs composed of 5%, 10%, and 20% of the overall TS. Using nonparametric statistical tests, we verified that with 10% and 20% of the overall TS, the Pareto front approximations that are generated by our coevolutionary approach are comparable with the ones generated by applying the MOEL with the overall TS, although the coevolution allows us to save up to 86.36% of the execution time. In addition, the analysis of the behavior of three representative solutions on the test set highlights that the use of the reduced TSs does not affect the generalization capabilities of the generated FRBSs.

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.

Multi-objective evolutionary learning of granularity, membership function parameters and rules of Mamdani fuzzy systems

In this paper, we propose a multi-objective evolutionary algorithm (MOEA) to generate Mamdani fuzzy rule-based systems with different trade-offs between accuracy and complexity by learning concurrently granularities of the input and output partitions, membership function (MF) parameters and rules. To this aim, we introduce the concept of virtual and concrete partitions: the former is defined by uniformly partitioning each linguistic variable with a fixed maximum number of fuzzy sets; the latter takes into account, for each variable, the number of fuzzy sets determined by the evolutionary process. Rule bases and MF parameters are defined on the virtual partitions and, whenever a fitness evaluation is required, mapped to the concrete partitions by employing appropriate mapping strategies. The implementation of the MOEA relies on a chromosome composed of three parts, which codify the partition granularities, the virtual rule base and the membership function parameters, respectively, and on purposely-defined genetic operators. The MOEA has been tested on three real-world regression problems achieving very promising results. In particular, we highlight how starting from randomly generated solutions, the MOEA is able to determine different granularities for different variables achieving good trade-offs between complexity and accuracy.

A fast and efficient multi-objective evolutionary learning scheme for fuzzy rule-based classifiers

During the last years, multi-objective evolutionary algorithms (MOEAs) have been extensively used to generate fuzzy rule-based systems characterized by different trade-offs between accuracy and complexity. In this paper, we propose an MOEA-based approach to learn concurrently the rule and data bases of fuzzy rule-based classifiers (FRBCs). In particular, the rule bases are generated by exploiting a rule and condition selection (RCS) strategy, which selects a reduced number of rules from a heuristically generated set of candidate rules and a reduced number of conditions for each selected rule during the evolutionary process. RCS can be considered as a rule learning in a constrained search space. As regards the data base learning, the membership function parameters of each linguistic term used in the rules are learned concurrently to the application of RCS. We tested our approach on twenty-four classification benchmarks and compared our results with the ones obtained by two similar state-of-the-art MOEA-based approaches and by two well-known non-evolutionary classification algorithms, namely FURIA and C4.5. Using non-parametric statistical tests, we show that our approach generates FRBCs with accuracy and complexity statistically comparable to, and sometimes better than, the ones generated by the two MOEA-based approaches, exploiting, however, only the 5% of the number of fitness evaluations used by these approaches. Further, the classifiers generated by our approach result to be more interpretable than the ones generated by the FURIA and C4.5 algorithms, while achieving the same accuracy level.

Segmentation and reconstruction of the lung volume in CT images

The automated extraction of the pulmonary parenchyma in CT images is the most crucial step in a computer-aided diagnosis (CAD) system. Actually, the following step of analysis of the lungs internal structure, aimed at lesion detection and diagnosis, works on the identified pulmonary regions.In this paper we describe a method, consisting of an appropriate combination of image processing techniques, for the automated identification of the pulmonary volume. We present and discuss the results of the method application to computed-tomography (CT) examinations performed in a screening program for early detection of lung cancer.

A Novel Approach Based on Finite-State Machines with Fuzzy Transitions for Nonintrusive Home Appliance Monitoring

Recent studies have highlighted that a significant part of the electrical energy consumption in residential buildings is caused by an improper use of home appliances. The development of low-cost systems for profiling the consumption of electric appliances can play a key role in stimulating the users to adopt adequate policies for energy saving. In this paper, we describe a novel methodology for extracting the power consumption of each appliance deployed in a domestic environment from the aggregate measures collected by a single smart meter. In order to coarsely describe how each type of appliance works, we use finite-state machines (FSMs) based on fuzzy transitions. An ad-hoc disaggregation algorithm exploits a database of these FSMs for, at each meaningful variation in real and reactive aggregate powers, hypothesizing possible configurations of active appliances. This set of configurations is concurrently managed by the algorithm which, whenever requested, outputs the configuration with the highest confidence with respect to the sequence of detected events. We implemented a prototype of a monitoring system based on the proposed methodology and installed it in a real domestic scenario. We discuss an experiment in which 11 appliances were connected to the same circuit and the aggregate power consumption was measured by a smart meter for approximately 12 h. At the end of the experiment, only two possible configurations were outputs from the system, including the correct one.

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.

An experimental study on evolutionary fuzzy classifiers designed for managing imbalanced datasets

In this paper, we show an experimental study on a set of evolutionary fuzzy classifiers (EFCs) purposely designed to manage imbalanced datasets. Three of these EFCs represent the state-of-the-art of the main approaches to the evolutionary generation of fuzzy rule-based systems for imbalanced dataset classification. The fourth EFC is an extension of a multi-objective evolutionary learning (MOEL) scheme we have recently proposed for managing imbalanced datasets: the rule base and the membership function parameters of a set of FRBCs are concurrently learned by optimizing the sensitivity, the specificity and the complexity.By using non-parametric tests, we first compare the results obtained by the four EFCs in terms of area under the ROC curve. We show that our MOEL scheme outperforms two of the comparison algorithms and results to be statistically equivalent to the third. Further, the classifiers generated by our MOEL scheme are characterized by a lower number of rules than the ones generated by the other approaches.To validate the effectiveness of our MOEL scheme in dealing with imbalanced datasets, we also compare our results with the ones achieved, after rebalancing the datasets, by two state-of-the-art algorithms, namely FURIA and FARC-HD, proposed for generating fuzzy rule-based classifiers for balanced datasets. We show that our MOEL scheme is statistically equivalent to FURIA, which is associated with the highest accuracy rank in the statistical tests. However, the rule bases generated by FURIA are characterized by a low interpretability.Finally, we show that the results achieved by our MOEL scheme are statistically equivalent to the ones achieved by four state-of-the-art approaches, based on ensembles of non-fuzzy classifiers, appropriately designed for dealing with imbalanced datasets.

An efficient multi-objective evolutionary fuzzy system for regression problems

During the last years, multi-objective evolutionary algorithms (MOEAs) have been extensively employed as optimization tools for generating fuzzy rule-based systems (FRBSs) with different trade-offs between accuracy and interpretability from data. Since the size of the search space and the computational cost of the fitness evaluation depend on the number of input variables and instances, respectively, managing high-dimensional and large datasets is a critical issue.In this paper, we focus on MOEAs applied to learn concurrently the rule base and the data base of Mamdani FRBSs and propose to tackle the issue by exploiting the synergy between two different techniques. The first technique is based on a novel method which reduces the search space by learning rules not from scratch, but rather from a heuristically generated rule base. The second technique performs an instance selection by exploiting a co-evolutionary approach where cyclically a genetic algorithm evolves a reduced training set which is used in the evolution of the MOEA.The effectiveness of the synergy has been tested on twelve datasets. Using non-parametric statistical tests we show that, although achieving statistically equivalent solutions, the adoption of this synergy allows saving up to 97.38% of the execution time with respect to a state-of-the-art multi-objective evolutionary approach which learns rules from scratch. We propose an efficient multi-objective evolutionary approach for generating fuzzy rule-based systems.We deal with high-dimensional and large regression datasets.We exploit two techniques aimed at reducing the search space and fitness evaluation time, respectively.An extensive experimental analysis supported by statistical tests is shown.