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Interpretability constraints for fuzzy information granulation

Information granules are complex entities that arise in the process of abstraction of data and derivation of knowledge. The automatic generation of information granules from data is an important task, since it gives to machines the ability of acquiring knowledge that can be communicated to users. For this purpose, knowledge acquisition should provide for fuzzy information granules that can be naturally labelled by linguistic terms, i.e. symbols that belong to the natural language. Information granules of this type are called interpretable. However, interpretability cannot be guaranteed until a set of constraints is imposed on the granulation process. In literature, several interpretability constraints have been proposed, but, due to the subjective interpretation of interpretability, there is no agreement on which constraints should be adopted. This survey is an attempt to provide for a complete presentation of interpretability constraints adopted in literature with the following objectives: (i) to give a homogeneous description of all interpretability constraints; (ii) to provide for a critical review of such constraints; (iii) to identify potentially different meanings of interpretability. Hopefully, this survey may serve as a guidance for designing interpretable fuzzy models as well as for identifying new methods of interpretable information granulation.

Interpretability assessment of fuzzy knowledge bases

Computing with words (CWW) relies on linguistic representation of knowledge that is processed by operating at the semantical level defined through fuzzy sets. Linguistic representation of knowledge is a major issue when fuzzy rule based models are acquired from data by some form of empirical learning. Indeed, these models are often requested to exhibit interpretability, which is normally evaluated in terms of structural features, such as rule complexity, properties on fuzzy sets and partitions. In this paper we propose a different approach for evaluating interpretability that is based on the notion of cointension. The interpretability of a fuzzy rule-based model is measured in terms of cointension degree between the explicit semantics, defined by the formal parameter settings of the model, and the implicit semantics conveyed to the reader by the linguistic representation of knowledge. Implicit semantics calls for a representation of users knowledge which is difficult to externalise. Nevertheless, we identify a set of properties -- which we call “logical view” -- that is expected to hold in the implicit semantics and is used in our approach to evaluate the cointension between explicit and implicit semantics. In practice, a new fuzzy rule base is obtained by minimising the fuzzy rule base through logical properties. Semantic comparison is made by evaluating the performances of the two rule bases, which are supposed to be similar when the two semantics are almost equivalent. If this is the case, we deduce that the logical view is applicable to the model, which can be tagged as interpretable from the cointension viewpoint. These ideas are then used to define a strategy for assessing interpretability of fuzzy rule-based classifiers (FRBCs). The strategy has been evaluated on a set of pre-existent FRBCs, acquired by different learning processes from a well-known benchmark dataset. Our analysis highlighted that some of them are not cointensive with users knowledge, hence their linguistic representation is not appropriate, even though they can be tagged as interpretable from a structural point of view.

Knowledge discovery by a neuro-fuzzy modeling framework

In this paper a neuro-fuzzy modeling framework is proposed, which is devoted to discover knowledge from data and represent it in the form of fuzzy rules. The core of the framework is a knowledge extraction procedure that is aimed to identify the structure and the parameters of a fuzzy rule base, through a two-phase learning of a neuro-fuzzy network. In order to obtain reliable and readable knowledge, two further stages are integrated with the knowledge extraction procedure: a pre-processing stage, performing variable selection on the available data to obtain simpler and more reliable fuzzy rules, and a post-processing stage, that granulates outputs of the extracted fuzzy rules so as to provide a validity range of estimated outputs. Moreover, the framework can address complex multi-input multi-output problems. In such case, two distinct modeling strategies can be followed with the opportunity of producing both a single MIMO model or a collection of MISO models. The proposed framework is verified on a real-world case study, involving prediction of chemical composition of ashes produced by combustion processes carried out in thermo-electric generators located in Italy.

Distinguishability quantification of fuzzy sets

Distinguishability is a semantic property of fuzzy sets that has a great relevance in the design of interpretable fuzzy models. Distinguishability has been mathematically defined through different measures, which are addressed in this paper. Special emphasis is given to similarity, which exhibits sound theoretical properties but its calculation is usually computationally intensive, and possibility, whose calculation can be very efficient but it does not exhibit the same properties of similarity. It is shown that under mild conditions – usually met in interpretable fuzzy modeling – possibility can be used as a valid measure for assessing distinguishability, thus overcoming the computational inefficiencies of similarity measures. Moreover, procedures that minimize possibility also minimize similarity and, consequently, improve distinguishability. In this sense, the use of possibility is fully justified in interpretable fuzzy modeling.

A neuro-fuzzy network to generate human-understandable knowledge from data

Neuro-fuzzy networks have been successfully applied to extract knowledge from data in the form of fuzzy rules. However, one drawback with the neuro-fuzzy approach is that the fuzzy rules induced by the learning process are not necessarily understandable. The lack of readability is essentially due to the high dimensionality of the parameter space that leads to excessive flexibility in the modification of parameters during learning. In this paper, to obtain readable knowledge from data, we propose a new neuro-fuzzy model and its learning algorithm that works in a parameter space with reduced dimensionality. The dimensionality of the new parameter space is necessary and sufficient to generate human-understandable fuzzy rules, in the sense formally defined by a set of properties. The learning procedure is based on a gradient descent technique and the proposed model is general enough to be applied to other neuro-fuzzy architectures. Simulation studies on a benchmark and a real-life problem are carried out to embody the idea of the paper.

An empirical risk functional to improve learning in a neuro-fuzzy classifier

The paper proposes a new Empirical Risk Functional as cost function for training neuro-fuzzy classifiers. This cost function, called Approximate Differentiable Empirical Risk Functional (ADERF), provides a differentiable approximation of the misclassification rate so that the Empirical Risk Minimization Principle formulated in Vapniks Statistical Learning Theory can be applied. Also, based on the proposed ADERF, a learning algorithm is formulated. Experimental results on a number of benchmark classification tasks are provided and comparison to alternative approaches given.

Similarity-Based Fuzzy Clustering for User Profiling

User profiling is a fundamental task in Web personalization. Fuzzy clustering is a valid approach to derive user profiles by capturing similar user interests from Web usage data available in log files. Often, fuzzy clustering is based on the assumption that data lay on an Euclidean space; however, clustering based on Euclidean distance can lead the clustering process to find user representations that do not capture the semantic information incorporated in the original Web usage data. In this paper, we propose a different approach to express similarity between Web users. The measure is based on the evaluation of similarity between fuzzy sets. The proposed measure is employed in a relational fuzzy clustering algorithm to discover clusters embedded in the Web usage data and derive profiles modeling the real user preferences. An application example on usage data extracted from log files of a sample Web site is reported and a comparison with the results obtained using the cosine measure is shown to demonstrate the effectiveness of the proposed similarity measure.

Interpretability of Fuzzy Systems: Current Research Trends and Prospects

Fuzzy systems are universally acknowledged as valuable tools to model complex phenomena while preserving a readable form of knowledge representation. The resort to natural language for expressing the terms involved in fuzzy rules, in fact, is a key factor to conjugate mathematical formalism and logical inference with human-centered interpretability . That makes fuzzy systems specifically suitable in every real-world context where people are in charge of crucial decisions. This is because the self-explanatory nature of fuzzy rules profitably supports expert assessments. Additionally, as far as interpretability is investigated, it appears that (a) the simple adoption of fuzzy sets in modeling is not enough to ensure interpretability; (b) fuzzy knowledge representation must confront the problem of preserving the overall system accuracy, thus yielding a trade-off which is frequently debated. Such issues have attracted a growing interest in the research community and became to assume a central role in the current literature panorama of computational intelligence. This chapter gives an overview of the topics related to fuzzy system interpretability, facing the ambitious goal of proposing some answers to a number of open challenging questions: What is interpretability? Why interpretability is worth considering? How to ensure interpretability, and how to assess (quantify) it? Finally, how to design interpretable fuzzy models?

A fuzzy clustering approach for mining diagnostic rules

In this paper an approach for automatic discovery of transparent diagnostic rules from data is proposed. The approach is based on a fuzzy clustering technique that is defined by three sequential steps. First, our Crisp Double Clustering algorithm is applied on available symptoms measurements, to provide a set of representative multidimensional prototypes that are further clustered onto each one-dimensional projection. The resulting clusters are used in the second step, where a set of fuzzy relations are defined in terms of transparent fuzzy sets. As a final step, the derived fuzzy relations are employed to define a set of fuzzy rules, which establish the knowledge base of a fuzzy inference system that can be used for fuzzy diagnosis. The approach has been applied to the Aachen Aphasia dataset as a real-world benchmark and compared with related work.

ON THE ROLE OF INTERPRETABILITY IN FUZZY DATA MINING

Data Mining, a central step in the broader overall process of Knowledge Discovery from Databases, concerns with discovering useful properties, called patterns, from data. Understandability is an essential — yet rarely tackled — feature that makes resulting patterns accessible by end users. In this paper we argue that the adoption of Fuzzy Logic for Data Mining can improve understandability of derived patterns. Indeed, Fuzzy Logic is able to represent concepts in a “human-centric” way. Hence, Data Mining methods based on Fuzzy Logic may potentially meet the so-called “Comprehensibility Postulate”, which characterizes the blurry notion of understandability. However, the mere adoption of Fuzzy Logic for Data Mining is not enough to achieve understandability. This paper describes and comments a number of issues that need to be addressed to provide for understandable patterns. A careful consideration of all such issues may end up in a systematic methodology to discover comprehensible knowledge from data.