Exploring Constrained Type-2 fuzzy sets

Fuzzy logic has been widely used to model human reasoning thanks to its inherent capability of handling uncertainty. In particular, the introduction of Type-2 fuzzy sets added the possibility of expressing uncertainty even on the definition of the membership functions. Type-2 sets, however, don’t pose any restrictions on the continuity or convexity of their embedded sets while these properties may be desirable in certain contexts. To overcome this problem, Constrained Type-2 fuzzy sets have been proposed. In this paper, we focus on Interval Constrained Type-2 sets to see how their unique structure can be exploited to build a new inference process. This will set some ground work for future developments, such as the design of a new defuzzification process for Constrained Type-2 fuzzy systems

Constrained interval type-2 fuzzy sets

In many contexts, type-2 fuzzy sets are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation there is no restriction on the shape of the footprint of uncertainty and the embedded sets that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the type-2 fuzzy set and the concept it models. As a consequence, the interpretability of some of the embedded sets and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 fuzzy sets have been proposed. However, no formal definitions for some of their key components (e.g. acceptable embedded sets) and constrained operations have been given. The goal of this paper is to provide some theoretical underpinning for the definition of constrained type-2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real world cases and briefly compared to the standard interval type-2 inference and defuzzification method

Novel techniques for modelling uncertain human reasoning in explainable artificial intelligence

In recent years, there has been a growing need for intelligent systems that not only are able to provide reliable predictions but can also produce explanations for their outputs. The demand for increased explainability has led to the emergence of explainable artificial intelligence (XAI) as a specific research field. In this context, fuzzy logic systems represent a promising tool thanks to their inherently interpretable structure. The use of a rule-base and linguistic terms, in fact, have allowed researchers to design models with a transparent decision process, from which it is possible to extract human-understandable explanations. The use of interval type-2 fuzzy logic in the XAI field, however, is limited: the improved performances of interval type-2 fuzzy systems and their ability to handle a higher degree of uncertainty comes at the cost of increased complexity that makes the semantic mapping between the input and outputs harder to understand intuitively. The presence of type-reduction, in some contexts fail to preserve the semantic value of the fuzzy sets and rules involved in the decision process. By semantic value, we specifically refer to the capacity of interpreting the output of the fuzzy system in respect to the pre-defined and thus understood linguistic variables used for the antecedents and consequents of the system. An attempt at increasing the explainability of interval type-2 fuzzy logic was first established by Garibaldi and Guadarrama in 2011, with the introduction of constrained type-2 fuzzy sets. However, extensive work needs to be carried out to develop the algorithms necessary for their practical use in fuzzy systems. The aim of this thesis is to extend the initial work on constrained interval type-2 fuzzy sets to develop a framework that preserves the semantic value throughout the modelling and decision process. Achieving this goal would allow the creation of a new class of fuzzy systems that show additional interpretable properties, and could further encourage the use of interval type-2 fuzzy logic in XAI. After the formal definition of the required components and theorems, different approaches are explored to develop inference algorithms that preserve the semantic value of the sets during the input-output mapping, while keeping reasonable run-times on modern computer hardware. The novel frameworks are then tested in a series of practical applications from the real world, in order to assess both their prediction performances and show the quality of the explanations these models can generate. Finally, the original definitions of constrained intervals type-2 fuzzy sets are refined to produce a novel approach which combines uncertain data and represents them using intuitive constrained interval type-2 fuzzy sets. Overall, as a result of the work presented here, it is now possible to design constrained interval type-2 fuzzy systems that preserve the enhanced semantic value provided by constrained interval-type-2 fuzzy sets throughout the inference, type-reduction and defuzzification stages. This characteristic is then used to improve the semantic interpretability of the system outputs, making constrained interval type-2 fuzzy systems a valuable alternative to interval type-2 fuzzy systems in XAI. The research presented here has resulted in three journal articles, two of which have already been published in IEEE Transactions on Fuzzy Systems, and four papers presented at the FUZZ-IEEE international conference between 2018 and 2020

Novel techniques for modelling uncertain human reasoning in explainable artificial intelligence

In recent years, there has been a growing need for intelligent systems that not only are able to provide reliable predictions but can also produce explanations for their outputs. The demand for increased explainability has led to the emergence of explainable artificial intelligence (XAI) as a specific research field. In this context, fuzzy logic systems represent a promising tool thanks to their inherently interpretable structure. The use of a rule-base and linguistic terms, in fact, have allowed researchers to design models with a transparent decision process, from which it is possible to extract human-understandable explanations. The use of interval type-2 fuzzy logic in the XAI field, however, is limited: the improved performances of interval type-2 fuzzy systems and their ability to handle a higher degree of uncertainty comes at the cost of increased complexity that makes the semantic mapping between the input and outputs harder to understand intuitively. The presence of type-reduction, in some contexts fail to preserve the semantic value of the fuzzy sets and rules involved in the decision process. By semantic value, we specifically refer to the capacity of interpreting the output of the fuzzy system in respect to the pre-defined and thus understood linguistic variables used for the antecedents and consequents of the system. An attempt at increasing the explainability of interval type-2 fuzzy logic was first established by Garibaldi and Guadarrama in 2011, with the introduction of constrained type-2 fuzzy sets. However, extensive work needs to be carried out to develop the algorithms necessary for their practical use in fuzzy systems. The aim of this thesis is to extend the initial work on constrained interval type-2 fuzzy sets to develop a framework that preserves the semantic value throughout the modelling and decision process. Achieving this goal would allow the creation of a new class of fuzzy systems that show additional interpretable properties, and could further encourage the use of interval type-2 fuzzy logic in XAI. After the formal definition of the required components and theorems, different approaches are explored to develop inference algorithms that preserve the semantic value of the sets during the input-output mapping, while keeping reasonable run-times on modern computer hardware. The novel frameworks are then tested in a series of practical applications from the real world, in order to assess both their prediction performances and show the quality of the explanations these models can generate. Finally, the original definitions of constrained intervals type-2 fuzzy sets are refined to produce a novel approach which combines uncertain data and represents them using intuitive constrained interval type-2 fuzzy sets. Overall, as a result of the work presented here, it is now possible to design constrained interval type-2 fuzzy systems that preserve the enhanced semantic value provided by constrained interval-type-2 fuzzy sets throughout the inference, type-reduction and defuzzification stages. This characteristic is then used to improve the semantic interpretability of the system outputs, making constrained interval type-2 fuzzy systems a valuable alternative to interval type-2 fuzzy systems in XAI. The research presented here has resulted in three journal articles, two of which have already been published in IEEE Transactions on Fuzzy Systems, and four papers presented at the FUZZ-IEEE international conference between 2018 and 2020

Optimal Fuzzy Model Construction with Statistical Information using Genetic Algorithm

Fuzzy rule based models have a capability to approximate any continuous function to any degree of accuracy on a compact domain. The majority of FLC design process relies on heuristic knowledge of experience operators. In order to make the design process automatic we present a genetic approach to learn fuzzy rules as well as membership function parameters. Moreover, several statistical information criteria such as the Akaike information criterion (AIC), the Bhansali-Downham information criterion (BDIC), and the Schwarz-Rissanen information criterion (SRIC) are used to construct optimal fuzzy models by reducing fuzzy rules. A genetic scheme is used to design Takagi-Sugeno-Kang (TSK) model for identification of the antecedent rule parameters and the identification of the consequent parameters. Computer simulations are presented confirming the performance of the constructed fuzzy logic controller

Fuzzy Least Squares Twin Support Vector Machines

Least Squares Twin Support Vector Machine (LST-SVM) has been shown to be an efficient and fast algorithm for binary classification. It combines the operating principles of Least Squares SVM (LS-SVM) and Twin SVM (T-SVM); it constructs two non-parallel hyperplanes (as in T-SVM) by solving two systems of linear equations (as in LS-SVM). Despite its efficiency, LST-SVM is still unable to cope with two features of real-world problems. First, in many real-world applications, labels of samples are not deterministic; they come naturally with their associated membership degrees. Second, samples in real-world applications may not be equally important and their importance degrees affect the classification. In this paper, we propose Fuzzy LST-SVM (FLST-SVM) to deal with these two characteristics of real-world data. Two models are introduced for FLST-SVM: the first model builds up crisp hyperplanes using training samples and their corresponding membership degrees. The second model, on the other hand, constructs fuzzy hyperplanes using training samples and their membership degrees. Numerical evaluation of the proposed method with synthetic and real datasets demonstrate significant improvement in the classification accuracy of FLST-SVM when compared to well-known existing versions of SVM

Commonsense knowledge representation and reasoning with fuzzy neural networks

This paper highlights the theory of common-sense knowledge in terms of representation and reasoning. A connectionist model is proposed for common-sense knowledge representation and reasoning. A generic fuzzy neuron is employed as a basic element for the connectionist model. The representation and reasoning ability of the model is described through examples

Multi-objective genetic optimisation for self-organising fuzzy logic control

This is the post-print version of the article. The official published version can be accessed from the link below.A multi-objective genetic algorithm is developed for the purpose of optimizing the rule-base of a Self-Organising Fuzzy Logic Control algorithm (SOFLC). The tuning of the SOFLC optimization is based on selection of the best shaped performance index for modifying the rule-base on-line. A comparative study is conducted between various methods of multi-objective genetic optimisation using the SOFLC algorithm on the muscle relaxant anaesthesia system, which includes a severe non-linearity, varying dynamics and time-delay

Designing Software Architectures As a Composition of Specializations of Knowledge Domains

This paper summarizes our experimental research and software development activities in designing robust, adaptable and reusable software architectures. Several years ago, based on our previous experiences in object-oriented software development, we made the following assumption: ‘A software architecture should be a composition of specializations of knowledge domains’. To verify this assumption we carried out three pilot projects. In addition to the application of some popular domain analysis techniques such as use cases, we identified the invariant compositional structures of the software architectures and the related knowledge domains. Knowledge domains define the boundaries of the adaptability and reusability capabilities of software systems. Next, knowledge domains were mapped to object-oriented concepts. We experienced that some aspects of knowledge could not be directly modeled in terms of object-oriented concepts. In this paper we describe our approach, the pilot projects, the experienced problems and the adopted solutions for realizing the software architectures. We conclude the paper with the lessons that we learned from this experience