This paper explores the application of the Adaptive Online Non-Singleton (ADONiS) framework for automated decision making for neonatal oxygen support. Maintaining optimal oxygen saturation (SpO2) levels in preterm infants is critical for preventing severe complications such as chronic lung disease and retinopathy of prematurity. Current clinical practice relies on manual adjustment of oxygen support by bedside caregivers, a process complicated by sensor uncertainty caused by contextual factors, such as challenging placement of sensors on, and high mobility of babies. Studies indicate this manual approach results in infants spending only 30–40% of time within target SpO2ranges, highlighting the potential for automated systems. Crucially however, such systems must combine the handling sensor uncertainty with handling clinical interpretability. The ADONiS framework was designed specifically to address systems where input noise is a challenge while established and ideally immutable rule sets and associated well-defined linguistics terms and fuzzy sets, are available which describe the desired and verified behaviour of a given system. In this paper we explore the applicability of ADONiS to the setting of neonatal oxygen support. We collaborated with a neonatology expert at Queen’s Medical Centre (QMC), one of the largest hospitals in Europe, to define the system’s membership functions and rule base, ensuring clinical interpretability remained central to the design. Additionally, we collected a real-world dataset from six neonates at QMC to validate our approach. Using these expert-derived structures, we apply the ADONiS framework to model oxygen support, systematically evaluating its performance on both synthetic scenarios and the collected real patient data. The resulting oxygen support suggestions were then reviewed by a neonatologist, whose assessments on ADONiS’s usage in comparison to the traditional singleton approaches. Quantitive metrics were not feasible as in neonatal care optimal oxygen adjustment inherently lacks a definitive ground truth which leaves clinical judgment to be the most appropriate evaluation metric. While the ADONiS framework itself is established [1], this work represents an initial, exploratory application of the ADONiS framework to neonatal oxygen support decision support.

Local Interpretability Model-agnostic Explanations (LIME) is a well-known post-hoc technique for explaining black-box models. While very useful, recent research highlights challenges around the explanations generated. In particular, there is a potential lack of stability, where the explanations provided vary over repeated runs of the algorithm, casting doubt on their reliability. This paper investigates the stability of LIME when applied to multivariate time series classification. We demonstrate that the traditional methods for generating neighbours used in LIME carry a high risk of creating 'fake' neighbours, which are out-of-distribution in respect to the trained model and far away from the input to be explained. This risk is particularly pronounced for time series data because of their substantial temporal dependencies. We discuss how these out-of-distribution neighbours contribute to unstable explanations. Furthermore, LIME weights neighbours based on user-defined hyperparameters which are problem-dependent and hard to tune. We show how unsuitable hyperparameters can impact the stability of explanations. We propose a two-fold approach to address these issues. First, a generative model is employed to approximate the distribution of the training data set, from which within-distribution samples and thus meaningful neighbours can be created for LIME. Second, an adaptive weighting method is designed in which the hyperparameters are easier to tune than those of the traditional method. Experiments on real-world data sets demonstrate the effectiveness of the proposed method in providing more stable explanations using the LIME framework. In addition, in-depth discussions are provided on the reasons behind these results.

The rules of a rule-based system provide explanations for its behavior by revealing the relationships between the variables captured. However, ideally, we have AI systems which go beyond explainable AI (XAI), that is, systems which not only explain their behavior, but also communicate their “insights” with respect to the real world. This requires rules to capture causal relationships between variables. In this article, we argue that those systems where the rules reflect causal relationships between variables represent an important class of fuzzy rule-based systems with unique benefits. Specifically, such systems benefit from improved performance and robustness; facilitate global explainability and thus cater to a core ambition for AI: the ability to communicate important relationships among a system's real-world variables to the human users of AI. We establish two causal-rule focused approaches to design fuzzy systems, and show the distinctions in their respective application scenarios for the explanations of the rules obtained by these two methods. The results show that rules which reflect causal relationships are more suitable for XAI than rules which “only” reflect correlations, while also confirming that they offer robustness to over-fitting, in turn supporting strong performance.

Machine learning approaches have enabled increasingly powerful time series classifiers. While performance has improved drastically, the resulting classifiers generally suffer from poor explainability, limiting their applicability in critical areas. Saliency-based methods designed to highlight the critical features are one of the most promising approaches to improving this explainability. Here, current techniques commonly rely on artificially perturbing the features, using, for example, random noise or ‘zeroing’ these features. We first demonstrate that an important drawback of these methods is that the perturbations used can result in unrealistic assessments of the classifier, since the perturbations force the data outside their original distribution. We articulate how this can result in poor identification of critical features, and hence misleading explanations. In order to address this issue and identify the most important features for the output of a black-box model, we propose a dual approach through meaningful perturbation and optimisation. First, leveraging a mechanism originally proposed in image analysis, a generative model is trained to create within-distribution perturbations of the input. These are then used to reliably evaluate whether a set of features is critical. Second, a greedy-based segmentation and identification strategy is proposed to search for the smallest set of critical features. Experiments show that the proposed approach addresses the out-of-distribution problem and identifies fewer critical features than existing methods. In combination, both aspects of the proposed approach offer a qualitative advance towards generating meaningful and robust explanations in the context of time series classification.

This paper adds to a growing body of research into the practical utility of using interval-valued (IV) response modes to efficiently capture richer quantitative data from people— e.g., through surveys. Specifically, IV responses offer a cohesive method of capturing uncertainty, vagueness, or range associated with individual quantitative responses. In turn, IV data provide a strong foundation for subsequent fuzzy set based modelling— e.g., using the Interval Agreement Approach. The present paper focuses on the impact of soliciting IV estimates upon accuracy of group perceptual judgements—the ‘Wisdom of the Crowd’. We report results from two empirical studies, examining the utility of IV data in the context of estimating specific (i.e., discrete point) ground truths, and directly comparing IV perceptual judgements (quantity estimates) against more traditional point estimates. There were two main hypotheses concerning the potential impacts of permitting uncertain (i.e., IV) estimates. First, it is possible that when specific predictions are required, permitting deliberately imprecise responses may reduce (prediction) accuracy versus forcing each respondent to provide their single ‘best guess’. Second, that capturing the uncertainty associated with individual predictions should permit improved aggregation of group estimates, through weighting individual estimates according to their certainty. We report findings from two studies designed to investigate these hypotheses, and outline proposals for future research in this area.

With growing significance of interval-valued data, interest in artificial intelligence methods tailored to this data type is similarly increasing across a range of application domains. Here, regression, i.e., the modelling of the association between interval-valued variables has been shown to be both challenging and rewarding. Beyond the mathematical challenges, fundamentals, such as the visualization of regression models, are not similarly available for interval-valued data, limiting both accessibility and utility of resulting models. Recently, the Interval Regression Graph (IRG) was introduced, providing a powerful visualization tool for interval-valued regression models. In this paper, we demonstrate the IRG in a practical data-science application, showing how it can rapidly highlight powerful insights of data. Specifically, we focus on consumer characteristics, analyzing potential relationships between their demographic characteristics and their product purchase intentions. We conclude with a brief outlook on the potential and remaining challenges of leveraging interval-valued data using fuzzy systems and artificial intelligence more broadly.

The compositional representation of data and associated statistical approaches is a powerful framework for modelling and reasoning about quantities which reflect proportions of a whole. Recently, an increasing body of work has started exploring the adoption of a compositional representation for modelling interval-valued data reflecting uncertainty or vagueness, for example interval-valued questionnaire responses. Results have flagged the intriguing potential of this approach, such as the elegant handling of traditional inference challenges, including implicitly ensuring coherence in linear regression for interval data, i.e. ensuring the estimated left bound of intervals is smaller than the right one. Building on these insights, extending the compositional representation via alpha-cut decomposition to fuzzy sets is an intuitive next step. In this paper, we discuss this compositional representation of fuzzy sets, building on prior interval work. We proceed to explore the adoption of compositional regression approaches to conduct linear regression on fuzzy set valued data sets. We demonstrate the approach, discuss results and in particular flag shortcomings and the challenges for next steps.

Mango is the third most crucial fruit product worldwide in terms of value and production volume, after pineapple and banana. However, assessing the quality grading of mangoes in an agricultural environment as a manual task is inefficient, labour demanding, and prone to errors. Thus, this task entails uncertainty in human decision-making, i.e. choice subjectivity due to a diverse perspective, experience, and knowledge. When dealing with ambiguity, fuzzy logic systems (FLSs) can help aid in the systematic automation of the human grading system. However, the underlying problem with FLSs is that they have difficulty dealing with large and complex real-world situations, in which in the curse of dimensionality is an issue. A possible option is to employ a hierarchical fuzzy system, a subtype of an FLS that is particularly effective in reducing the complexity and increasing the interpretability of the overall system. This study proposes an approach to model uncertainty in mango grading decision-making using a hierarchical fuzzy system. We demonstrate the HFS for mango grading application using a FuzzyR toolkit, together with an FLS for comparison. Additionally, this paper explores the importance of uncertainty arising from human knowledge, which will be critical in determining the most suitable system (FLS or HFS) closest to the experts’ opinion. Additionally, we also evaluate both systems’ interpretability.

EXplainable Artificial Intelligence (XAI) is of in-creasing importance as researchers and practitioners seek better transparency and verifiability of AI systems. Mamdani fuzzy systems can provide explanations based on their linguistic rules, and thus a potential pathway to XAI. A factual rule based explanation generally refers to the given set of rules executed, or fired, for a given input. However, research has shown that human explanations are often counterfactual (CF), i.e. rather than explaining why a given output was reached, they show why other potential outputs were not. Although several machine learning-based CF explanation generation methods have been proposed in recent years, quasi none of them focus on fuzzy systems. Also, where they do, they focus on correlation, which limits the interpretive value of any CF explanations obtained as humans expect a causal relationship in rules, i.e. we are cause-effect thinkers. In this paper, we propose a new rule generation framework for Mamdani fuzzy classification systems, which we refer to as CF-MABLAR, building on the MARkov BLAnket Rules (MABLAR) framework. CF-MABLAR approximates the causal links between inputs and output(s) of fuzzy systems and generates CF rules by leveraging them. Uniquely, the CF rules obtained not only provide a basic CF explanation, but can also articulate how the given inputs would need to be changed to generate a different output, crucial for lay-user insight, verification and sensitivity-evaluation of XAI systems, for example in decision support around credit risk, cyber security and medical assistance.

Time series classification is a challenging research area where machine learning techniques such as deep learning perform well, yet lack interpretability. Identifying the most important features for such classifiers provides a pathway to improving their interpretability. Several Feature Importance (FI) identification methods remove the contributions of features, i.e. observations at certain time steps of, from the input and evaluate the change in the classification result to measure the importance of features. As time series features cannot simply be deleted, current techniques generally rely on replacing features with constant or random values. While effective, this approach risks unexpected results in the classification and thus feature importance estimation-as the replacements used may be different to what the classifier encountered in the training phase. This is referred to as the Out-Of-Distribution problem. The OOD problem has been recognised in image and language models but have not received much attention in the context of time series classification. This work addresses the OOD problem in FI identification for time series classifiers. Specifically, we propose a method based on Conditional Variational Autoencoder to generate possible sets of within-distribution inputs, which are used to evaluate feature importance through marginalisation. Experiments on publicly accessible datasets are carried out showing that the method identifies the most important features with higher accuracy than existing methods, providing the basis for improved explainability of time series classifiers.

Data obtained from the real-world tends to be uncertain: Measurement inaccuracies, variability in opinions, and human errors are just some of the reasons that make the information collection process noisy. In recent years, fuzzy sets have been used to capture the uncertainty in data and then build automatic reasoning systems. In some contexts, data on a given subject is gathered from multiple sources and each instance modeled through a fuzzy set. A typical example of this scenario is represented by surveys, in which many participants express their opinions on the same topics. The fuzzy sets representing individual instances can be combined in a new (type-1 or type-2) fuzzy set in order to capture expert or measurement variation. In this article, we propose a novel approach which combines uncertain data modeled through parametric fuzzy sets in an intuitive manner, using the recently introducedconstrained interval type-2 (IT2) fuzzy sets. By intuitive, we mean that each resultant constrained IT2 fuzzy set preserves the shape used to represent a single data instance, while making use of the footprint of uncertainty to represent uncertainty around its parameters. This novelconstrained parametric approachis applied to interval-valued data gathered from real surveys and compared to the other algorithms in the literature, showing how it differs from them, with discussion of the contexts in which it represents a valuable alternative. Finally, it is shown how this novel approach can be used to model not just intervals but data in which individual instances can be modeled through any parametric fuzzy sets (e.g., triangular).

In this work, we generalize the notion of restricted equivalence function for type-2 fuzzy sets, leading to the notion of extended restricted equivalence functions. We also study how under suitable conditions, these new functions recover the standard axioms for restricted equivalence functions in the real setting. Extended restricted equivalence functions allow us to compare any two general type-2 fuzzy sets and to generate a similarity measure for type-2 fuzzy sets. The result of this similarity is a fuzzy set on the same referential set (i.e., domain) as the considered type-2 fuzzy set. The latter is crucial for applications such as explainable AI and decision-making, as it enables an intuitive interpretation of the similarity within the domain-specific context of the fuzzy sets. We show how this measure can be used to compare type-2 fuzzy sets with different membership functions in such a way that the uncertainty linked to type-2 fuzzy sets is not lost. This is achieved by generating a fuzzy set rather than a single numerical value. Furthermore, we also show how to obtain a numerical value for discrete referential sets.

Recent years have seen a surge in interest in non-singleton fuzzy systems. These systems enable the direct modelling of uncertainty affecting systems' inputs using the fuzzification stage. Moreover, recent work has shown how different composition approaches to modelling the interaction between the non-singleton input and the antecedent fuzzy sets enable the efficient handling of uncertainty without requiring changes in a system's rule base, with benefits both in terms of performance and interpretability. As thus far few current software toolkit support non-singleton fuzzy systems, this paper presents an extension of the FuzzyR toolbox, which is a freely available R package on CRAN, for non-singleton fuzzy logic systems. The updated toolbox enables a non-singleton model to be conveniently built from scratch, or for existing singleton fuzzy logic systems built using FuzzyR to be converted easily. Predefined operations include fuzzification of crisp inputs (e.g. into Gaussian membership functions), and a variety of composition approaches for computing rules' firing-strengths, based on the standard, centroid-based, and similarity-based methods. It is also possible to include user-defined options for these abovementioned methods, without the need to modify (or update) the FuzzyR toolbox itself. In this paper, detailed introductions for the new non-singleton features of the toolkit are presented, complete with code samples in R to facilitate adoption both within and beyond the community. Further, the paper presents a series of validation experiments, replicating a recent empirical analysis of non-singleton fuzzy logic systems in the context of time-series prediction with different levels of noise.

Capturing interval-valued, as opposed to more conventional point-valued data, offers a potentially efficient method of obtaining richer information in individual responses. In turn, interval-valued data provide a strong foundation for subsequent fuzzy set based modelling-e.g., using the Interval Agreement Approach. In 2019, open-source software (DECSYS) was released to enable digital administration of interval-valued surveys using an ellipse response mode. This study follows on from an appraisal of this software and demonstration of practical value of the approach, reported last year, in one of many potential real-world applications (consumer preference research). A key ambition of ellipse-based interval elicitation is to maximise response efficiency-i.e., minimising workload and complexity in obtaining this richer information. User experience is therefore a vital consideration regarding potential for broader adoption. The present paper documents a direct empirical comparison between interval-valued response elicitation (using ellipses) and a conventional point-valued counterpart (using a Visual Analogue Scale), in terms of user experience during completion of a simple quantitative estimation task. We examine differences in perceived ease-of-use, unnecessary complexity and effective communication of desired responses, as well as overall liking-with positive outcomes for the interval-valued response mode in each case. We also report results of multiple regression analyses examining how the first three variables contribute to participants' overall liking of each response mode, as well as exploring differences driven by potentially important demographic factors (i.e., gender, age & native English speaking).

Trust estimation of vehicles is vital for the correct functioning of Vehicular Ad Hoc Networks (VANETs) as it enhances their security by identifying reliable vehicles. However, accurate trust estimation still remains distant as existing works do not consider all malicious features of vehicles, such as dropping or delaying packets, altering content, and injecting false information. Moreover, data consistency of messages is not guaranteed here as they pass through multiple paths and can easily be altered by malicious relay vehicles. This leads to difficulty in measuring the effect of content tampering in trust calculation. Further, unreliable wireless communication of VANETs and unpredictable vehicle behavior may introduce uncertainty in the trust estimation and hence its accuracy. In this view, we put forward three trust factors - captured by fuzzy sets to adequately model malicious properties of a vehicle and apply a fuzzy logic-based algorithm to estimate its trust. We also introduce a parameter to evaluate the impact of content modification in trust calculation. Experimental results reveal that the proposed scheme detects malicious vehicles with high precision and recall and makes decisions with higher accuracy compared to the state-of-the-art.

Regression represents one of the most basic building blocks of data analysis and AI. Despite growing interest in interval-valued data across various fields, approaches to establish regression models for interval-valued data which address and handle the specific properties of given data sets are very limited. For broader use and adoption of regression for intervals, this paper conducts a sensitivity analysis of key extant linear regression approaches in respect to important features of interval-valued data sets, such as the mean and associated standard deviation of the range (size) of the intervals within the data set-a measure of overall size and size-diversity, and the dispersion of interval-centers-a measure of diversity in terms of interval position. Experiments with carefully designed synthetic exemplar data sets with these properties suggest that distant placement of intervals as well as higher standard deviation (uncertainty) of ranges increase estimation errors; that is, they result in lower linear regression model fitness for all regression methods as may intuitively be expected. However, these errors are lower for the best suited Parameterized model in comparison to the MinMax and Constrained Center and Range methods. This paper sheds light on the behaviour and `expectable' performance of key linear regression models designed for interval-valued data and adds a further building block to supporting the broader adoption of intervals (and subsequently fuzzy sets) as a fundamental data type in AI.

Time series forecasting is an essential research field that provides significant data to help professionals in several areas. Thus, growing research and development in this area have been conducted, aiming at developing new forecasting methods with higher performance levels, but always also with low processing costs. One of this methods is Fuzzy Time Series - FTS. However, one great problem of FTS prediction is how to properly deal with the uncertainty associated to the time series and to model's design. Thus, in this paper we propose a univariate interval type-2 fuzzy time series model combined with the concept of Self-organised Direction Aware Data Partitioning Algorithm (SODA) for universe of discourse partitioning. All experiments were performed using the TAIEX data set and the results were then compared to other forecasting models from literature. A sliding window methodology was applied and the forecast error metric chosen was the Root Mean Squared Error (RMSE) for all methods. SODA-T2FTS results show that it outperformed other forecasting methods confirming that interval type-2 fuzzy logic can be a reliable tool for time series prediction.

The use of Hierarchical Fuzzy Systems (HFS) has been well acknowledged as a good approach in reducing the complexity and improving the interpretability of fuzzy logic systems (FLS). Over the past years, many fuzzy logic toolkits have been made available for type-1, interval type-2 and general type-2 fuzzy logic systems under different programming languages. However, it is still challenging for people, especially for those who are not expert in fuzzy systems or programming, to build models based on HFSs. The main reason could be the lack of practical tools and examples of using HFSs. This paper presents a step-by-step guide to the implementation of an HFS with the open-source toolbox, FuzzyR, utilising the R Programming Language.

An important property of fuzzy systems is the interpretability provided by their rules. However, if a fuzzy system is derived through machine learning algorithms, its interpretability is often greatly diminished as membership functions and rules are adjusted to minimize error in respect to a data set. To address part of this challenge, we propose a novel two-step fuzzy rule generation framework leveraging the concept of the Markov blanket, i.e., the set of variables which are causally related to a target variable – such as a system’s output. By estimating the Markov blanket for a given application, we restrict rule learning to (only) the variables which are causally linked to the system output, thus minimising the generation of spurious rules (based on spurious correlations of variables). This decreases the complexity of fuzzy systems and maintains a causal link between rules’ antecedents and consequent(s) – as expected by humans when viewing rules. The proposed framework can improve the interpretability of fuzzy rule based systems which are tuned using machine learning techniques, while also providing performance advantages as are commonly associated with feature selection techniques. Experiment results show that even the initial implementation of the framework proposed here can generate more concise and interpretable rule bases without compromising performance.

Constrained interval type-2 (CIT2) fuzzy sets have been introduced to preserve interpretability when moving from type-1 to interval type-2 (IT2) membership functions. Although they can be used to produce type-2 fuzzy systems with enhanced explainability, so far, the latter comes at the expense of high computational cost. Specifically, the exhaustive type-reduction method for CIT2 Mamdani systems has been shown to be too slow to be used in practical applications and even the current approximation procedure is much slower than modern type-reduction algorithms used for IT2 fuzzy sets. In this article, a novel type-reduction procedure for CIT2 sets is presented, based on the concept of switch indices. The algorithm is applied on a real-world classification problem and compared to other type-reduction approaches used in IT2 and CIT2 systems. In the case studies presented, the new algorithm is significantly faster than the exhaustive and sampling CIT2 approaches while keeping the high level of interpretability of the type-reduction operation that characterizes CIT2 fuzzy sets.

Hierarchical fuzzy systems (HFSs) have been shown to have the potential to improve the interpretability of fuzzy logic systems (FLSs). However, challenges remain, such as “How can we measure their interpretability?” “How can we make an informed assessment of how HFSs should be designed to enhance interpretability?” The challenges consist of measuring the interpretability of HFSs include issues such as their topological structure, the number of layers, the meaning of intermediate variables, and so on. In this article, an initial framework to measure the interpretability of HFSs is proposed, combined with a participatory user design process to create a specific instance of the framework for an application context. This approach enables the subjective views of a range of practitioners, experts in the design and creation of FLSs, to be taken into account in shaping the design of a generic framework for measuring interpretability in HFSs. This design process and framework are demonstrated through two classification application examples, showing the ability of the resulting index to appropriately capture interpretability as perceived by system design experts.

Constrained interval type-2 (CIT2) fuzzy sets are a class of type-2 fuzzy sets that has been recently proposed as a way to extend type-1 membership functions to interval type-2 (IT2) while keeping a semantic connection between the IT2 fuzzy set and the concept it models. Recent work has shown how their mathematical properties can be used to design CIT2 fuzzy logic systems that are able to provide explanations for their outputs. Although the CIT2 representation can be a valuable alternative to the IT2 one, no software library for their implementation is available for the research community. The aim of this paper is to introduce a new Java library, Juzzy Constrained, that has been developed as an extension of the popular type-1 and type-2 Java toolkit Juzzy, adding support for CIT2 sets and systems. Throughout the paper, the main classes and the structure of the new library are described, together with a working example that illustrates how to build a CIT2 fuzzy system from scratch and how it can be used to produce explanations for the output.

The capture and analysis of interval-valued data has seen increased interest over recent years. This offers a direct means to capture and reason about uncertainty in data, whether obtained from sensors or from people. Open-source software (DECSYS [1]) was recently released to facilitate the efficient capture of interval-valued survey responses. Potential real-world applications are broad ranging, and this paper documents an initial test-case of the software and its underpinning methodology, in a marketing-centric application. It provides an illustration of the insights offered by interval-valued responses, in this case relating to consumer preferences. We apply two approaches to describe and draw insights from the data: inferential statistics and descriptive visualisation methods. Statistical results indicate that overall purchase intention was well-described by four factors: value, healthiness, taste and brand. The capture of uncertainty information, afforded by intervals, also permitted identification of six factors that contribute to purchase intention uncertainty- relating to taste, ethics and visual appearance. Visualisations of interval-valued responses, using the IAA [2]-[5], also highlighted factors with high degrees of uncertainty-in particular, product ethics. This information could prove valuable for retailers in determining how to focus future marketing campaigns. It may prove equally valuable for market regulators, by informing where to improve product labelling information. More generally, the case study provides an overview of capturing and analysing intervals, highlighting some of the challenges, but also the unique potential to gain additional insights not available using conventional, `crisp', approaches.

Intervals have frequently been used in the literature to represent uncertainty in data, from eliciting uncertain judgements from experts to representing uncertainty in sensor measurements. This widespread use of intervals has led to research on interval statistics to help understand the data. However, even seemingly trivial statistics (such as variance) cannot be calculated on interval-valued data using the same approach as for point data without incurring substantial loss of precision to a level which can make results close to useless. This loss of precision makes it challenging for decision makers to appropriately interpret interval-valued data using familiar statistics. Although there exist several approaches to computing statistics such as variance, these are all developed for specific properties of the data, and there is no general-case method. In addition, there are many statistical measures for which no efficient and accurate method exist. For such cases, we can use a Monte Carlo sampling approach to generate approximate statistics. While sampling does not generally produce exact solutions, it can provide a useful and efficient approximation to a desired degree of accuracy given sufficient computational resources. In this paper, we focus on the application of Monte Carlo sampling to generate statistics for interval-valued data. Specifically, we explore the optimum sample size required to calculate statistics on interval-valued data for a given degree of accuracy desired. We compare different sizes of data and different sampling methods to demonstrate how these affect the choice of an optimum sample size.

Interpretability is an important and necessary topic that needs to be discussed in relation to the fields of Artificial Intelligence and Machine Learning. Within fuzzy logic systems (FLSs), hierarchical fuzzy systems (HFSs) have been suggested as a key component to help improve the interpretability of FLSs. In this context, complexity is a key component in the interpretability of FLSs. In FLSs, the complexity is commonly expressed using in a rule-based manner, considering the number of rules, variables, and fuzzy terms. Several studies have used indicators (for example, the number of rules) to measure the complexity of FLSs. However, this is not a perfect way of assessing complexity in HFSs that have the structure of multiple subsystems, layers and different topologies. Thus far, complexity assessment associated with the structure of HFSs has not been discussed. In this paper, we aim to put forward a new approach in assessing the complexity of HFSs, which will combine rule-based complexity and structural complexity. A detailed measurement of complexity for different HFSs' topologies, namely parallel and serial, will be presented to showcase the features of the new approach. The contribution of this paper is the introduction of a combined rule-based and structural complexities-based approach in order to establish a comprehensive measurement of complexity in HFSs.

With a growing number of areas leveraging interval-valued data-including in the context of modeling human uncertainty (e.g., in cybersecurity), the capacity to accurately and systematically compare intervals for reasoning and computation is increasingly important. In practice, well established set-theoretic similarity measures, such as the Jaccard and Sørensen-Dice measures, are commonly used, whereas axiomatically, a wide breadth of possible measures have been theoretically explored. This article identifies, articulates, and addresses an inherent and so far not discussed limitation of popular measures-their tendency to be subject to aliasing-where they return the same similarity value for very different sets of intervals. The latter risks counter-intuitive results and poor-automated reasoning in real-world applications dependent on systematically comparing interval-valued system variables or states. Given this, we introduce new axioms establishing desirable properties for robust similarity measures, followed by putting forward a novel set-theoretic similarity measure based on the concept of bidirectional subsethood, which satisfies both traditional and new axioms. The proposed measure is designed to be sensitive to the variation in the size of intervals, thus avoiding aliasing. This article provides a detailed theoretical exploration of the new proposed measure, and systematically demonstrates its behavior using an extensive set of synthetic and real-world data. Specifically, the measure is shown to return robust outputs that follow intuition-essential for real-world applications. For example, we show that it is bounded above and below by the Jaccard and Sørensen-Dice similarity measures (when the minimum t-norm is used). Finally, we show that a dissimilarity or distance measure, which satisfies the properties of a metric, can easily be derived from the proposed similarity measure.

Comparing fuzzy sets by computing their similarity is common, with a large set of measures of similarity available. However, while commonplace in the computational intelligence community, the application and results of similarity measures are less common in the wider scientific context, where statistical approaches are the standard for comparing distributions. This is challenging, as it means that developments around similarity measures arising from the fuzzy community are inaccessible to the wider scientific community and that the fuzzy community fails to take advantage of a strong statistical understanding, which may be applicable to comparing (fuzzy membership) functions. In this paper, we commence a body of work on systematically relating the outputs of similarity measures to the notion of statistically significant difference; that is, how (dis)similar do two fuzzy sets need to be for them to be statistically different? We explain that in this context, it is useful to initially focus on dis-similarity, rather than similarity, as the former aligns directly with the widely used concept of statistical difference. We propose two methods of applying statistical tests to the outputs of fuzzy dissimilarity measures to determine significant difference. We show how the proposed work provides deeper insight into the behavior and possible interpretation of degrees of dis-similarity and, consequently, similarity, and how the interpretation differs with respect to context (e.g., the complexity of the fuzzy sets).

Nonsingleton fuzzy logic systems (NSFLSs) have the potential to capture and handle input noise within the design of input fuzzy sets (FSs). In this article, we propose an online learning method that utilizes a sequence of observations to continuously update the input FSs of an NSFLS, thus providing an improved capacity to deal with variations in the level of input-affecting noise, common in real-world applications. The method removes the requirement for both a priori knowledge of noise levels and relying on offline training procedures to define input FS parameters. To the best of our knowledge, the proposed ADaptive, ONline Nonsingleton (ADONiS) fuzzy logic system (FLS) framework represents the first end-to-end framework to adaptively configure nonsingleton input FSs. The latter is achieved through online uncertainty detection applied to a sliding window of observations. Since real-world environments are influenced by a broad range of noise sources, which can vary greatly in magnitude over time, the proposed technique for combining online determination of noise levels with associated adaptation of input FSs provides an efficient and effective solution which elegantly models input uncertainty in the FLS's input FSs, without requiring changes in any other part (e.g., antecedents, rules or consequents) of the FLS. In this article, two common chaotic time series (Mackey-Glass, Lorenz) are used to perform prediction experiments to demonstrate and evaluate the proposed framework. Results indicate that the proposed adaptive NSFLS framework provides significant advantages, particularly in environments that include high variation in noise levels, which are common in real-world applications.

Aggregation operators are mathematical functions that enable the fusion of information from multiple sources. Fuzzy Integrals (FIs) are widely used aggregation operators, which combine information in respect to a Fuzzy Measure (FM) which captures the worth of both the individual sources and all their possible combinations. However, FIs suffer from the potential drawback of not fusing information according to the intuitively interpretable FM, leading to non-intuitive results. The latter is particularly relevant when a FM has been defined using external information (e.g. experts). In order to address this and provide an alternative to the FI, the Recursive Average (RAV) aggregation operator was recently proposed which enables intuitive data fusion in respect to a given FM. With an alternative fusion operator in place, in this paper, we define the concept of `A Priori' FMs which are generated based on external information (e.g. classification accuracy) and thus provide an alternative to the traditional approaches of learning or manually specifying FMs. We proceed to develop one specific instance of such an a priori FM to support the decision level fusion step in ensemble classification. We evaluate the resulting approach by contrasting the performance of the ensemble classifiers for different FMs, including the recently introduced Uriz and the Sugeno λ-measure; as well as by employing both the Choquet FI and the RAV as possible fusion operators. Results are presented for 20 datasets from machine learning repositories and contextualised to the wider literature by comparing them to state-of-the-art ensemble classifiers such as Adaboost, Bagging, Random Forest and Majority Voting.

Data-driven techniques that capture uncertainty through intervals or fuzzy sets can substantially improve systematic reasoning about uncertain information. Recent years have seen renewed interest in the capture of intervals from a variety of sources - including experts and general survey participants. This approach avoids the more cumbersome batteries of questions that are otherwise required to capture individual uncertainty, and which may not obtain the same degree of fidelity. It also enables respondents to effectively communicate any range (e.g. vagueness) inherent in their response, allowing generation of models that represent this additional information. However, manual methods of obtaining and processing interval-valued data - such as through paper-based questionnaires, are labour and time intensive. This has provided a practical barrier to adoption of interval-valued response-formats in the wider community, from research to industry (e.g. marketing). We argue that establishing an effective and accessible method for interval-valued data-capture will greatly encourage research in and application of uncertainty-aware models of data. Thus, we present DECSYS, a newly developed open-source software tool, which enables the creation and administration of digital surveys that elicit both conventional and interval-valued responses. DECSYS incorporates a range of features, and is designed to maximise versatility for experimenters and usability for participants. Surveys can be conducted either locally or online, and results easily exported. We welcome community feedback, including on how to best tailor the tool in the future to maximise value and support multidisciplinary adoption of uncertainty-aware data collection.

Discontinuous intervals (DIs) arise in a wide range of contexts, from real world data capture of human opinion to α-cuts of non-convex fuzzy sets. Commonly, for assessing the similarity of DIs, the latter are converted into their continuous form, followed by the application of a continuous interval (CI) compatible similarity measure. While this conversion is efficient, it involves the loss of discontinuity information and thus limits the accuracy of similarity results. Further, most similarity measures including the most popular ones, such as Jaccard and Dice, suffer from aliasing, that is, they are liable to return the same similarity for very different pairs of CIs. To address both of these challenges, this paper proposes a generalized approach for calculating the similarity of DIs which leverages the recently introduced bidirectional subsethood based similarity measure (which avoids aliasing) while accounting for all pairs of the continuous subintervals within the DIs to be compared. We provide detail of the proposed approach and demonstrate its behaviour when applying bidirectional subsethood, Jaccard and Dice as similarity measures, using different pairs of synthetic DIs. The experimental results show that the similarity outputs of the new generalized approach follow intuition for all three similarity measures; however, it is only the proposed integration with the bidirectional subsethood similarity measure which also avoids aliasing for DIs.

The capture of interval-valued data is becoming an increasingly common approach in data collection (from survey based research to the collation of sensor data) as an efficient method of obtaining information about uncertainty associated with the data in question. To best utilise this data, several methods of aggregating intervals into fuzzy sets have been proposed in the fuzzy set literature, particularly within the field of Computing with Words. Two key examples are the Interval Approach and the Interval Agreement Approach and their respective extensions. Each method takes a fundamentally different approach to constructing fuzzy sets, making different assumptions in respect to the nature and the reliability of the data. The result is noticeably different fuzzy sets that do not share the same statistical properties (such as central-tendency and standard deviation). This begs the question of how these techniques differ in respect to the relationship between the original interval-valued data and the fuzzy sets produced - and thus when and why each of the methods is the most appropriate. This paper compares the results of both methods of constructing fuzzy sets from interval-valued data. Statistical moments of the fuzzy sets are compared against the interval-valued data to evaluate how well key properties of the fuzzy sets match those of the data; for example, does the standard deviation of the fuzzy set represent the standard deviation of the raw interval-valued data? We use comparisons on real-world data to demonstrate how the methods differ and which is more appropriate given the assumptions of the data.

Recently, there has been much research into modelling of uncertainty in human perception through Fuzzy Sets (FSs). Most of this research has focused on allowing respondents to express their (intra) uncertainty using intervals. Here, depending on the technique used and types of uncertainties being modelled different types of FSs can be obtained (e.g., Type-1, Interval Type-2, General Type-2). Arguably, one of the most flexible techniques is the Interval Agreement Approach (IAA) as it allows to model the perception of all respondents without making assumptions such as outlier removal or predefined membership function types (e.g. Gaussian). A key aspect in the analysis of interval-valued data and indeed, IAA based agreement models of said data, is to determine the position and strengths of agreement across all the sources/participants. While previously, the Agreement Ratio was proposed to measure the strength of agreement in fuzzy set based models of interval data, said measure has only been applicable to type-1 fuzzy sets. In this paper, we extend the Agreement Ratio to capture the degree of inter-group agreement modelled by a General Type-2 Fuzzy Set when using the IAA. This measure relies on using a similarity measure to quantitatively express the relation between the different levels of agreement in a given FS. Synthetic examples are provided in order to demonstrate both behaviour and calculation of the measure. Finally, an application to real-world data is provided in order to show the potential of this measure to assess the divergence of opinions for ambiguous concepts when heterogeneous groups of participants are involved.

Most real-world environments are subject to different sources of uncertainty which may vary in magnitude over time. We propose that while Type-1 (T1) Non-Singleton Fuzzy Logic System (NSFLSs) have the potential to tackle uncertainty within the input Fuzzy Sets (FSs), Type-2 (T2) input FSs provide the ability to also capture variation in uncertainty levels by means of their extra degrees of freedom. Specifically, in this paper, we propose a strategy to design Interval Type-2 (IT2) input Membership Functions (MFs) in an online manner to ensure the parameters of input MFs are updated dynamically, thus capturing varying levels of uncertainty affecting systems' inputs. In this strategy, first, uncertainty detection is performed over a given time-frame (the Uncertainty Estimation Time-frame) and Type-1 (T1) input MFs are constructed by utilising the detected uncertainty level. Second, the variation of the uncertainty levels over a sliding window (the Uncertainty Variation Window) is used to capture the degree of variation in the detected uncertainty levels over time, which in turn informs the size of the Footprint of Uncertainty (FOU) of the IT2 MF associated with the T1 principal MF. Using time-series prediction experiments as an initial evaluation and demonstration platform for the proposed architecture, we show that the proposed strategy of designing IT2 input MFs has the potential to deliver performance benefits. Specifically, it allows systems to not only adapt to specific uncertainty levels but also to be more resilient to the variation of said uncertainty levels over time, thus offering a pathway to robust performance in real-world applications.

Hierarchical fuzzy systems (HFSs) have been seen as an effective approach to reduce the complexity of fuzzy logic systems (FLSs), largely as a result of reducing the number of rules. However, it is not clear completely how complexity of HFSs can be measured. In FLSs, complexity is commonly expressed using a multi-factorial approach, taking into consideration the number of rules, variables, and fuzzy terms. However, this may not be the best way to assess complexity in HFSs that have structures involving multiple subsystems, layers and different topologies. Thus far, structural complexity associated with the structure of HFSs has not been discussed. In the field of software engineering (SE), a complexity measure has been proposed to measure program complexity. This measure uses the concept of graph theory complexity, which considers the control structure complexity. The measure can also be applied to assess the complexity of a collection of programs known as a hierarchical nest. In this paper, we present an approach to mapping an SE complexity measure to HFS design. The approach includes several mapping alternatives that are outlined and illustrated using different HFS designs. This study contributes a new approach for the first time to assessing structural complexity in HFSs based on an approach from SE complexity measure.

Fuzzy sets (FSs) may have complex, non-normal, or non-convex membership functions that occur, for example, in the output of a fuzzy logic system or when automatically generating FSs from data. Measuring the distance between such non-standard FSs can be challenging as there is no clear correct method of comparison and only limited research currently exists that systematically compares existing distance measures (DMs) for these FSs. It is useful to know the distance between these sets, which can tell us how much the results of a system change when the inputs differ, or the amount of disagreement between individual's perceptions or opinions on different concepts. In addition, understanding the direction of difference between such FSs further enables us to rank them, learning if one represents a higher output or higher ratings than another. This paper builds on previous functions of measuring directional distance and, for the first time, presents methods of measuring the directional distance between any type-1 and type-2 FSs with both normal/non-normal and convex/non-convex membership functions. In real-world applications, where data-driven, non-convex, non-normal FSs are the norm, the proposed approaches for measuring the distance enables us to systematically reason about the real-world objects captured by the FSs.

Although driving behavior has been largely studied amongst private motor vehicles drivers, the literature addressing heavy goods vehicle (HGV) drivers is scarce. Identifying the existing groups of driving stereotypes and their proportions enables researchers, companies, and policy makers to establish group-specific strategies to improve safety and economy. In addition, insight into driving styles can help predict drivers' reactions and therefore enable the modeling of interactions between vehicles and the possible obstacles encountered on a journey. Consequently, there are also contributions to the research and development of autonomous vehicles and smart roads. In this paper, our interest lies in investigating driving behavior within the HGV community in the United Kingdom (U.K.). We conduct analysis of a telematics dataset containing the incident information on 21 193 HGV drivers across the U.K. We are interested in answering two research questions: 1) What groups of behavior are we able to uncover? 2) How do these groups complement current findings in the literature? To answer these questions, we apply a two-stage data analysis methodology involving consensus clustering and ensemble classification to the dataset. Through the analysis, eight patterns of behavior are uncovered. It is also observed that although our findings have similarities to those from previous work on driving behavior, further knowledge is obtained, such as extra patterns and driving traits arising from vehicle and road characteristics.

Information or data aggregation is an important part of nearly all analysis problems as summarizing inputs from multiple sources is a ubiquitous goal. In this paper we propose a method for non-linear aggregation of data inputs that take the form of non-normal fuzzy sets. The proposed shape-preserving fuzzy integral (SPFI) is designed to overcome a well-known weakness of the previously-proposed sub-normal fuzzy integral (SuFI). The weakness of SuFI is that the output is constrained to have maximum membership equal to the minimum of the maximum memberships of the inputs; hence, if one input has a small height, then the output is constrained to that height. The proposed SPFI does not suffer from this weakness and, furthermore, preserves in the output the shape of the input sets. That is, the output looks like the inputs. The SPFI method is based on the well-known Choquet fuzzy integral with respect to a capacity measure, i.e., fuzzy measure. We demonstrate SPFI on synthetic and real-world data, comparing it to the SuFI and non-direct fuzzy integral (NDFI).

Similarity measures are useful for reasoning about fuzzy sets. Hence, many classical set-theoretic similarity measures have been extended for comparing fuzzy sets. In previous work, a set-theoretic similarity measure considering the bidirectional subsethood for intervals was introduced. The measure addressed specific concerns of many common similarity measures, and it was shown to be bounded above and below by Jaccard and Dice measures respectively. Herein, we extend our prior measure from similarity on intervals to fuzzy sets. Specifically, we propose a vertical-slice extension where two fuzzy sets are compared based on their membership values. We show that the proposed extension maintains all common properties (i.e., reflexivity, symmetry, transitivity, and overlapping) of the original fuzzy similarity measure. We demonstrate and contrast its behaviour along with common fuzzy set-theoretic measures using different types of fuzzy sets (i.e., normal, non-normal, convex, and non-convex) in respect to different discretization levels.

Real world environments face a wide range of sources of noise and uncertainty. Thus, the ability to handle various uncertainties, including noise, becomes an indispensable element of automated decision making. Non-Singleton Fuzzy Logic Systems (NSFLSs) have the potential to tackle uncertainty within the design of fuzzy systems. The firing strength has a significant role in the accuracy of FLSs, being based on the interaction of the input and antecedent fuzzy sets. Recent studies have shown that the standard technique for determining firing strengths risks substantial information loss in terms of the interaction of the input and antecedents. Recently, this issue has been addressed through exploration of alternative approaches which employ the centroid of the intersection (cen-NS) and the similarity (sim-NS) between input and antecedent fuzzy sets. This paper identifies potential shortcomings in respect to the previously introduced similarity-based NSFLSs in which firing strength is defined as the similarity between an input FS and an antecedent. To address these shortcomings, this paper explores the potential of the subsethood measure to generate a more suitable firing level (sub-NS) in NSFLSs featuring various noise levels. In the experiment, the basic waiter tipping fuzzy logic system is used to examine the behaviour ofsub-NS in comparison with the current approaches. Analysis of the results shows that thesub-NS approach can lead to more stable behaviour in real world applications.

Real-world environments face a wide range of noise (uncertainty) sources and gaining insight into the level of noise is a critical part of many applications. While Non-Singleton Fuzzy Logic Systems (NSFLSs), in particular recently introduced advanced variants such as centroid-based NSFLSs have the capacity to handle known quantities of uncertainty, thus far, the actual level of uncertainty has had to be defined a priori - i.e. prior to run time of a system or controller. This paper does not focus on such advances within the architecture of NSFLSs, but focuses on a novel two-stage approach for uncertainty handling in fuzzy logic systems which integrates: (i) estimation of noise levels and (ii) the appropriate handling of the noise based on this estimate, by means of a dynamically configured NSFLS. As initial evaluation of the approach, two chaotic nonlinear time series (Mackey-Glass and Lorenz), as well as a real-world Darwin sea level pressure series prediction fuzzy logic systems are implemented and compared to commonly used procedures. The results indicate that the proposed strategy of integrating uncertainty/noise estimation with the capacity of non-singleton fuzzy logic systems has the potential to deliver performance benefits in real-world applications without requiring a priori information on noise levels and thus delivers a first step towards smart, noise-adaptive non-singleton fuzzy logic systems and controllers.

Previous studies have investigated the experiences and characteristics of holiday decision-making among groups of travelers. This study adds to the knowledge of group trip holiday planning through exploring influential factors (including the individual and group characteristics of travelers), and linking those with their intragroup interactions when planning a group trip. A total of 261 usable questionnaires were collected across two university campuses in the UK and China. The survey employed a retrospective approach, asking participants to recall one of their past group trip planning experiences within the previous 3 months. This study found that intragroup interactions during a group trip planning process are influenced both by tourists’ individual factors, such as age, gender, and nationality, and by group characteristics, such as group size, common interest, group type, and group travel style. This study shows that common interest is the most influential factor in terms of its positive impact on group collaboration, feeling of connectedness, strength of preparation, and flexibility and spontaneity during group trip planning process. Further, in general, Chinese groups tend to spend less time on planning their trips before departure, but focus more on the details of the itinerary. Finally, the implications for technologies that are designed to facilitate the group trip planning process, with a view to enhancing the level of group enjoyment, are discussed based on the findings in this study.

As non-singleton fuzzy logic controllers (NSFLCs) are capable of capturing input uncertainties, they have been effectively used to control and navigate unmanned aerial vehicles (UAVs) recently. To further enhance the capability to handle the input uncertainty for the UAV applications, a novel NSFLC with the recently introduced similarity-based inference engine, i.e., Sim-NSFLC, is developed. In this paper, a comparative study in a 3D trajectory tracking application has been carried out using the aforementioned Sim-NSFLC and the NSFLCs with the standard as well as centroid composition-based inference engines, i.e., Sta-NSFLC and Cen-NSFLC. All the NSFLCs are developed within the robot operating system (ROS) using the C++ programming language. Extensive ROS Gazebo simulation-based experiments show that the Sim-NSFLCs can achieve better control performance for the UAVs in comparison with the Sta-NSFLCs and Cen-NSFLCs under different input noise levels.

Recently, there has been much research into effective representation and analysis of uncertainty in human responses, with applications in cyber-security, forest and wildlife management, and product development, to name a few. Most of this research has focused on representing the response uncertainty as intervals, e.g., “I give the movie between 2 and 4 stars.” In this paper, we extend upon the model-based interval agreement approach (lAA) for combining interval data into fuzzy sets and propose the efficient IAA (eIAA) algorithm, which enables efficient representation of and operation on the fuzzy sets produced by IAA (and other interval-based approaches, for that matter). We develop methods for efficiently modeling, representing, and aggregating both crisp and uncertain interval data (where the interval endpoints are intervals themselves). These intervals are assumed to be collected from individual or multiple survey respondents over single or repeated surveys; although, without loss of generality, the approaches put forth in this paper could be used for any interval-based data where representation and analysis is desired. The proposed method is designed to minimize loss of information when transferring the interval-based data into fuzzy set models and then when projecting onto a compressed set of basis functions. We provide full details of eIAA and demonstrate it on real-world and synthetic data.

Understanding of consumer preferences and perceptions is a vital challenge for the food and beverage industry. Food and beverage product development is a very complex process that deals with highly uncertain factors, including consumer perceptions and manufacturing complexity. Sensory evaluation is widely used in the food industry for product design and defining market segments. Here, we develop a two-step approach to minimize uncertainty in the food and beverage product development, including consumers as co-creators. First, we develop interval-valued questionnaires to capture sensory perceptions of consumers for the corresponding sensory attributes. The data captured is modelled with fuzzy sets in order to then facilitate the design of new consumer-tailored products. Then, we demonstrate the real-world manufacture of a personalized beverage product with a continuous food formulation system. Finally, we highlight consumers” perceptions for the corresponding sensory attributes and their fuzzy set generated agreement models to capture product acceptance for the formulated and commercial orange juice drinks, and consequently to establish that continuous beverage formulator is capable of making similar commercial products for individuals.

In computing the similarity of intervals, current similarity measures such as the commonly used Jaccard and Dice measures are at times not sensitive to changes in the width of intervals, producing equal similarities for substantially different pairs of intervals. To address this, we propose a new similarity measure that uses a bi-directional approach to determine interval similarity. For each direction, the overlapping ratio of the given interval in a pair with the other interval is used as a measure of uni-directional similarity. We show that the proposed measure satisfies all common properties of a similarity measure, while also being invariant in respect to multiplication of the interval endpoints and exhibiting linear growth in respect to linearly increasing overlap. Further, we compare the behavior of the proposed measure with the highly popular Jaccard and Dice similarity measures, highlighting that the proposed approach is more sensitive to changes in interval widths. Finally, we show that the proposed similarity is bounded by the Jaccard and the Dice similarity, thus providing a reliable alternative.

Multi-criteria decision making (MCDM) problems are a well known category of decision making problem that has received much attention in the literature, with a key approach being the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). While TOPSIS has been developed towards the use of Type-2 Fuzzy Sets (T2FS), to date, the additional information provided by T2FSs in TOPSIS has been largely Ignored since the final output, the Closeness Coefficient (CC), has remained a crisp value. In this paper, we develop an alternative approach to T2 fuzzy TOPSIS, where the final CC values adopt an interval-valued form. We show in a series of systematically designed experiments, how increasing uncertainty in the T2 membership functions affects the interval-valued CC outputs. Specifically, we highlight the complex behaviour in terms of the relationship of the uncertainty levels and the outputs, including non-symmetric and non-linear growth in the CC intervals in response to linearly growing levels of uncertainty. As the first TOPSIS approach which provides an interval-valued output to capture output uncertainty, the proposed method is designed to reduce the loss of information and to maximize the benefit of using T2FSs. The initial results indicate substantial potential in the further development and exploration of the proposed and similar approaches and the paper highlights promising next steps.

Hierarchical fuzzy systems (HFSs) have been shown to have the potential to improve interpretability of fuzzy logic systems (FLSs). In recent years, a variety of indices have been proposed to measure the interpretability of FLSs such as the Nauck index and Fuzzy index. However, interpretability indices associated with HFSs have not so far been discussed. The structure of HFSs, with multiple layers, subsystems, and varied topologies, is the main challenge in constructing interpretability indices for HFSs. Thus, the comparison of interpretability between FLSs and HFSs-even at the index level-is still subject to open discussion. This paper begins to address these challenges by introducing extensions to the FLS Nauck and Fuzzy interpretability indices for HFSs. Using the proposed indices, we explore the concept of interpretability in relation to the different structures in FLSs and HFSs. Initial experiments on benchmark datasets show that based on the proposed indices, HFSs with equivalent function to FLSs produce higher indices, i.e. are more interpretable than their corresponding FLSs.

The aggregation of multiple information sources has a long history and ranges from sensor fusion to the aggregation of individual algorithm outputs and human knowledge. A popular approach to achieve such aggregation is the fuzzy integral (FI) which is defined with respect to a fuzzy measure (FM) (i.e. a normal, monotone capacity). In practice, the discrete FI aggregates information contributed by a discrete number of sources through a weighted aggregation (post-sorting), where the weights are captured by a FM that models the typically subjective `worth' of subsets of the overall set of sources. While the combination of FI and FM has been very successful, challenges remain both in regards to the behavior of the resulting aggregation operators - which for example do not produce symmetrically mirrored outputs for symmetrically mirrored inputs - and also in a manifest difference between the intuitive interpretation of a stand-alone FM and its actual role and impact when used as part of information fusion with a FI. This paper elucidates these challenges and introduces a novel family of recursive average (RAV) operators as an alternative to the FI in aggregation with respect to a FM; focusing specifically on the arithmetic recursive average. The RAV is designed to address the above challenges, while also facilitating fine-grained analysis of the resulting aggregation of different combinations of sources. We provide the mathematical foundations of the RAV and include initial experiments and comparisons to the FI for both numeric and interval-valued data.

Non-Singleton Fuzzy Logic Systems (NSFLSs) have the potential to tackle uncertainty within the design of fuzzy systems. The inference process has a major role in determining results, being partly based on the interaction of input and antecedent fuzzy sets (in generating firing levels). Recent studies have shown that the standard technique for determining firing strengths risks substantial information loss in terms of the interaction of the input and antecedents. To address this issue, alternative approaches, which employ the centroid of intersections (cen-NS) and similarity measures (sim-NS), have been developed. More recently, a novel similarity measure for fuzzy sets has been introduced, but as yet this has not been used for NSFLSs. This paper focuses on exploring the potential of this new similarity measure in combination with the sim-NS approach to generate a more suitable firing level for non-singleton input. Experiments are presented for fuzzy systems trained using both noisy and noise-free time series. The prediction results of NSFLSs for the novel similarity measure and the current approaches are compared. Analysis of the results shows that the novel similarity measure, used within the sim-NS approach, can be a more stable and suitable method suitable to be used in real world applications.

This paper investigates if sensory perceptions of orange drinks (e.g., acidity, thickness, wateriness) can be linked to physical measurements (e.g., pH, particle size, density). Using this information, manufactured drinks can be tailored according to consumer' desires by, for example, the consumer providing a sensory description of their preferred drink. Sensory perceptions of different juices are collected in a survey and used to determine 1) if consumers can distinguish between different drinks using the provided sensory descriptors, and 2) if the perceptions match to physical measurements of the drinks. Results show that most of the given sensory descriptors are useful in describing differences in orange drinks. Additionally, the perceived wateriness and thickness of the drinks can be predicted from measurements. However, the perceived acidity could not be reliably predicted. The results show that personally tailored orange beverages can be manufactured according to some of the consumer's desires and there is scope for future developments tailored to a wider range of drink attributes.

Classification of cancer patients into treatment groups is essential for appropriate diagnosis to increase survival. Previously, a series of papers, largely published in the breast cancer domain have leveraged Computational Intelligence (CI) developments and tools, resulting in ground breaking advances such as the classification of cancer into newly identified classes - leading to improved treatment options. However, the current literature on the use of CI to achieve this is fragmented, making further advances challenging. This paper captures developments in this area so far, with the goal to establish a clear, step-by-step pipeline for cancer subtype identification. Based on establishing the pipeline, the paper identifies key potential advances in CI at the individual steps, thus establishing a roadmap for future research. As such, it is the aim of the paper to engage the CI community to address the research challenges and leverage the strong potential of CI in this important area. Finally, we present a small set of recent findings on the Nottingham Tenovus Primary Breast Carcinoma Series enabling the classification of a higher number of patients into one of the identified breast cancer groups, and introduce Classifusion: a combination of results of multiple classifiers.

Most applications of both type-1 and type-2 fuzzy logic systems are employing singleton fuzzification due to its simplicity and reduction in its computational speed. However, using singleton fuzzification assumes that the input data (i.e., measurements) are precise with no uncertainty associated with them. This paper explores the potential of combining the uncertainty modelling capacity of interval type-2 fuzzy sets with the simplicity of type-1 fuzzy logic systems (FLSs) by using interval type-2 fuzzy sets solely as part of the non-singleton input fuzzifier. This paper builds on previous work and uses the methodological design of the footprint of uncertainty (FOU) of interval type-2 fuzzy sets for given levels of uncertainty. We provide a detailed investigation into the ability of both types of fuzzy sets (type-1 and interval type-2) to capture and model different levels of uncertainty/noise through varying the size of the FOU of the underlying input fuzzy sets from type-1 fuzzy sets to very “wide” interval type-2 fuzzy sets as part of type-1 non-singleton FLSs using interval type-2 input fuzzy sets. By applying the study in the context of chaotic time-series prediction, we show how, as uncertainty/noise increases, interval type-2 input fuzzy sets with FOUs of increasing size become more and more viable.

Fuzzy logic controllers (FLCs) have extensively been used for the autonomous control and guidance of unmanned aerial vehicles (UAVs) due to their capability of handling uncertainties and delivering adequate control without the need for a precise, mathematical system model which is often either unavailable or highly costly to develop. Despite the fact that non-singleton FLCs (NSFLCs) have shown more promising performance in several applications when compared to their singleton counterparts (SFLCs), most of UAV applications are still realized by using SFLCs. In this paper, we explore the potential of both standard and the recently introduced centroid based NSFLCs, i.e., Sta-NSFLC and Cen-NSFLC, for the control of a quadcopter UAV under various input noise conditions using different levels of fuzzifier, and a comparative study has been conducted using the three aforementioned FLCs. We present a series of simulation-based experiments, the simulation results show that the control performances of NSFLCs are better than those of SFLC, and the Cen-NSFLC outperforms the Sta-NSFLC especially under highly noisy conditions.

In this paper we establish a link between linguistic descriptors describing food preferences and product manufacturing processes. We show how this is achieved using a model-based methodology that translates consumer preferences into product and process specifications. The ultimate goal is the large scale personalization of formulated food product manufacture where consumers are also the co-creators of the food products they wish to buy. Firstly, we investigate how those sensory attributes for such products can map onto product and process specifications. Fuzzy set modelling is used to capture the preferences and perception for these attributes by different groups of people. Specifically, type-1 fuzzy sets are generated from interval-valued survey data for the linguistic descriptors (i.e., thin, thick, smooth, pulpy) and the sensory indicators (i.e., smoothness, roughness and orange flavor) that describe how consumers perceive and select orange based beverages. Then, the models are employed to establish the links between such product attributes and the actual formulation parameters to make the product. We demonstrate the manufacture of the desired orange beverage that emerged from the modelling approach by deploying the process parameters, which map onto those descriptors, on the controller of a continuous food formulation system which was selected due to its flexibility and its computer controller that provides the ability to redeploy new formulation specifications rapidly. With this overall methodology we demonstrate for the first time the digital, on-demand manufacture of soft beverages with targeted attributes, selected directly by a consumer group.

Multi Criteria Decision Making is a challenging but vital process for organizations. One of the best-known techniques to support Multi-Criteria Decision Making is the `Technique for Order Preference by Similarity to Ideal Solution' (TOPSIS) approach. In recent years, a variety of extensions, including fuzzy extensions of TOPSIS have been proposed. Besides the many variations of standard TOPSIS, one family of extensions employing fuzzy sets is referred to as fuzzy TOPSIS (FTOPSIS). One challenge that has arisen is that it is not straightforward to choose between the multiple variants of TOPSIS existing today. Previously, none of the papers that have compared the key differences between standard and fuzzy TOPSIS have fully explored each of the step-wise stages. In this paper, we now provide a detailed comparison of these key stages in a systematic stepwise manner, clearly highlighting differences. We also identify and discuss the limitations, issues and challenges which exist in the present FTOPSIS method. The crucial and main issues are identified as relating to concepts of reliability, truth and meaning. Having identified these conceptual issues, we then go on to highlight what we argue to be the main issue, that of reliability, to discuss further. We proceed to present a potential solution and propose a framework to address the issue. This study will provide guidelines to researchers in this field and to provide potential pathways to further solutions, which have the capacity to advance the area of FTOPSIS as a whole.

Decision support systems (DSSs) are a convenient tool to aid plant operators in the selection of process set points. Inputs to these systems for variables that are not easily measured online often come from assessments made by experts, with an associated degree of uncertainty. The application of fuzzy sets and systems as part of DSSs provides a systematic approach to addressing the uncertainty in its variables. This paper builds on prior work on DSSs utilising fuzzy cognitive maps and introduces a non-singleton fuzzification stage which directly addresses uncertainty in system inputs. The motivation of the proposed system is grounded in the real world challenges of producing high-quality olive oil and the paper provides promising application and analysis results as part of the Virgin Olive Oil Production Process.

This paper presents a method of measuring the similarity between general type-2 fuzzy sets that may have non-normal secondary membership functions. Such fuzzy sets are increasingly common in applications such as the modelling of the subjective meaning of linguistic terms by groups of people. By building upon existing similarity measures in the literature, which thus far cannot compare such fuzzy sets, we derive an extended similarity measure which can be applied to both normal and non-normal (in terms of the secondary membership functions) general type-2 fuzzy sets. We provide proofs that the proposed method follows all of the common properties of a similarity measure and demonstrations are given to compare the proposed method with others in the literature.

In the context of cancer treatment and surgery, quality of life assessment is a crucial part of determining treatment success and viability. In order to assess it, patient-completed questionnaires which employ words to capture aspects of patients well-being are the norm. As the results of these questionnaires are often used to assess patient progress and to determine future treatment options, it is important to establish that the words used are interpreted in the same way by both patients and medical professionals. In this paper, we capture and model patients perceptions and associated uncertainty about the words used to describe the level of their physical function used in the highly common (in Sarcoma Services) Toronto Extremity Salvage Score (TESS) questionnaire. The paper provides detail about the interval-valued data capture as well as the subsequent modelling of the data using fuzzy sets. Based on an initial sample of participants, we use Jaccard similarity on the resulting words models to show that there may be considerable differences in the interpretation of commonly used questionnaire terms, thus presenting a very real risk of miscommunication between patients and medical professionals as well as within the group of medical professionals.

In non-singleton fuzzy logic systems (NSFLSs) input uncertainties are modelled with input fuzzy sets in order to capture input uncertainty such as sensor noise. The performance of NSFLSs in handling such uncertainties depends both on the actual input fuzzy sets (and their inherent model of uncertainty) and on the way that they affect the inference process. This paper proposes a novel type of NSFLS by replacing the composition-based inference method of type-1 fuzzy relations with a similarity-based inference method that makes NSFLSs more sensitive to changes in the input's uncertainty characteristics. The proposed approach is based on using the Jaccard ratio to measure the similarity between input and antecedent fuzzy sets, then using the measured similarity to determine the firing strength of each individual fuzzy rule. The standard and novel approaches to NSFLSs are experimentally compared for the well-known problem of Mackey-Glass time series predictions, where the NSFLS's inputs have been perturbed with different levels of Gaussian noise. The experiments are repeated for system training under both noisy and noise-free conditions. Analyses of the results show that the new method outperforms the standard approach by substantially reducing the prediction errors.

In nonsingleton fuzzy logic systems (NSFLSs), input uncertainties are modeled with input fuzzy sets in order to capture input uncertainty (e.g., sensor noise). The performance of NSFLSs in handling such uncertainties depends on both the appropriate modeling in the input fuzzy sets of the uncertainties present in the system's inputs and on how the input fuzzy sets (and their inherent model of uncertainty) interact with the antecedent and, thus, affect the inference within the remainder of the NSFLS. This paper proposes a novel development on the latter. Specifically, an alteration to the standard composition method of type-1 fuzzy relations is proposed and applied to build a new type of NSFLS. The proposed approach is based on employing the centroid of the intersection of input and antecedent sets as origin of the firing degree, rather than the traditional maximum of their intersection, thus making the NSFLS more sensitive to changes in the input's uncertainty characteristics. The traditional and novel approach to NSFLSs are experimentally compared for two well-known problems of Mackey-Glass and Lorenz chaotic time-series predictions, where the NSFLSs' inputs have been perturbed with different levels of Gaussian noise. Experiments are repeated for system training under noisy and noise-free conditions. Analyses of the results show that the new method outperforms the traditional approach. Moreover, it is shown that while formally more complex, in practice, the new method has no significant computational overhead compared with the standard approach.

Capturing the uncertainty arising from system noise has been a core feature of fuzzy logic systems (FLSs) for many years. This paper builds on previous work and explores the methodological transition of type-1 (T1) to interval type-2 fuzzy sets (IT2 FSs) for given “levels” of uncertainty. Specifically, we propose to transition from T1 to IT2 FLSs through varying the size of the Footprint Of Uncertainty (FOU) of their respective FSs while maintaining the original FS shape (e.g., triangular) and keeping the size of the FOU over the FS as constant as possible. The latter is important as it enables the systematic relating of FOU size to levels of uncertainty and vice versa, while the former enables an intuitive comparison between the T1 and T2 FSs. The effectiveness of the proposed method is demonstrated through a series of experiments using the well-known Mackey-Glass (MG) time series prediction problem. The results are compared with the results of the IT2 FS creation method introduced in [1] which follows a similar methodology as the proposed approach but does not maintain the membership function (MF) shape.

Consumer and more generally, human preferences are highly complex, depending on a multitude of factors, most of which are not crisp, but uncertain/fuzzy in nature. Thus, user selection amongst a set of items is dependent on the complex comparison of items based on a large number of imprecise item-attributes such as price, size, colour, etc. This paper proposes the mechanisms to underpin the digital replication of such complex preference-based item selection with the view to enabling improved digital item search and recommendation systems. For example, a user may query “I would like a product of similar size but at a cheaper price.” The proposed method involves splitting query-attributes into two categories; those to remain similar (e.g., size) and those to be changed in a specific direction (e.g., price - to be lower). A combination of similarity and distance measures is then used to compare and rank recommendations. Initial results are presented indicating that the proposed method is effective at ranking items according to intuition and expected user preferences.

A major asset of fuzzy logic systems is dealing with uncertainties arising in their various applications, thus it is important to make them achieve this task as effectively and comprehensively as possible. While singleton fuzzy logic systems provide some capacity to deal with such uncertainty aspects, non-singleton fuzzy logic systems (NSFLSs) have further enhanced this capacity, particularly in handling input uncertainties. This paper proposes a novel approach to NSFLSs, which further develops this potential by changing the method of handling input fuzzy sets within the inference engine. While the standard approach is getting the maximum of the intersection between input's and antecedent's fuzzy sets (in the “pre-filtering” stage), it is proposed to employ the centroid of the intersection as the basis of each rule's firing degree. The motivation is to capture the interaction of input and antecedent fuzzy sets with high fidelity, thus making NSFLSs more sensitive to the input's uncertainty information. The testbed is the common problem of Mackey-Glass time series prediction in the presence of input noise. Analyses of the results show that the new method outperforms the standard approach (by reducing the prediction error) and has potential for a more efficient uncertainty handling in NSFLS applications.

The potentials of non-singleton fuzzy logic systems (NSFLSs) in dealing with uncertainties are widely known. However, their utilities and possible challenges in real-world applications, particularly beyond fuzzy controls, are still not widely examined. This paper presents some user-centric design approaches in making NSFLSs usable in a real-world problem of environmental management. In previous work, a singleton FLS was developed based on an established environmental management framework. After further investigation of the users' requirements, it was realized that the effective capture, representation and visualization of the system's inputs and outputs are critical, particularly when there are uncertainties involved in data collection and decision-making processes. For addressing the new requirements, the system has been extended to a NSFLS, so it can make use of non-singleton fuzzification in handling uncertain (e.g., noisy) environmental data. Inspired by the user-centric design of this particular system extension, the contribution of this paper is the development of some practical methods to capture/represent input/output uncertainties in NSFLSs. Subject to further users evaluation, the explained methods have potential to be employed in many similar real-world applications, thus extending the NSFLSs applicability to a wider context than the present.

The main feature of type-2 fuzzy sets is their ability to represent uncertainties within a system. These uncertainties are captured in the Footprint Of Uncertainty (FOU) of a type2 membership function which can be described by the upper and the lower membership function. One of the challenges in modelling a type-2 fuzzy logic system is the problem of defining the membership function parameters and their FOUs, given noisy data or imperfect measurements. This challenge is increased by the complexity which arises from the increase in the number of parameters of IT2 MFs to be tuned. This paper presents a novel method for designing interval type-2 fuzzy logic systems, in which the FOU creation method presented in [1] is adopted, and then the design parameters are tuned through simulated annealing. The novelty of this approach is that it has fewer parameters to be tuned than the conventional approach, as only a single extra parameter is used to define the IT2 MFs. We demonstrate the approach through application to the Mackey-Glass time series prediction problem, using training data sets corrupted with different levels of noise. By doing so, we demonstrate that this approach is an efficient FOU selection mechanism that produces IT2 FLSs with good performance using less computational time than the conventional approach.

Multi Criteria Decision Making methods have been developed to solve complex real-world decision problems. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is currently one of the most popular methods and has been shown to provide helpful outputs in various application areas. In recent years, a variety of extensions, including fuzzy extensions of TOPSIS have been proposed. One challenge that has arisen is that it is not straightforward to differentiate between the multiple variants of TOPSIS existing today. Thus, in this paper, a comparison between the classical Fuzzy TOPSIS method proposed by Chen in 2000 and the recently Fuzzy TOPSIS proposed extension by Yuen in 2014 is made. The purpose of this comparative study is to show the difference between both methods and to provide context for their respective strengths and limitations both in complexity of application, and expressiveness of results. A detailed synthetic numeric example and comparison of both methods are provided.

One of the biggest challenges in the design of Fuzzy Logic Systems (FLSs) is the construction of their rule base. While fuzzy sets capture aspects of a system's variables and associates them with linguistic labels, it is the rules which capture the logical relationships of these labels and underlying fuzzy sets. Further, while fuzzy systems are credited for dealing well with uncertainty in system inputs and outputs, comparatively little research has focused on the capture of uncertainty in their actual inference rules. This paper focusses on the challenge of capturing the knowledge of multiple human experts on the relationships of linguistic labels in a given problem domain. Specifically, it proposes a novel survey-centric methodology which enables the capture of individual, subjective input from domain (not fuzzy logic) experts with minimal prior training and provides mechanisms to aggregate the resulting survey-data into a working and interpretable fuzzy system. The rule base of the resulting system incorporates weights to capture intra- and inter-expert uncertainty during rule specification. The paper follows a practical style to facilitate reproduction of the proposed methodology by peers. Results and initial evaluation based on real world case studies in the context of environmental conservation in Western Australia are provided.

The fuzzy integral (FI) with respect to a fuzzy measure (FM) is a powerful means of aggregating information. The most popular FIs are the Choquet and Sugeno, and most research focuses on these two variants. The arena of the FM is much more populated, including numerically derived FMs such as the Sugeno λ-measure and decomposable measure, expert-defined FMs, and data-informed FMs. The drawback of numerically derived and expert-defined FMs is that one must know something about the relative values of the input sources. However, there are many problems where this information is unavailable, such as crowdsourcing. This paper focuses on data-informed FMs, or those FMs that are computed by an algorithm that analyzes some property of the input data itself, gleaning the importance of each input source by the data they provide. The original instantiation of a data-informed FM is the agreement FM, which assigns high confidence to combinations of sources that numerically agree with one another. This paper extends upon our previous work in datainformed FMs by proposing the uniqueness measure and additive measure of agreement for interval-valued evidence. We then extend data-informed FMs to fuzzy number (FN)-valued inputs. We demonstrate the proposed FMs by aggregating interval and FN evidence with the Choquet and Sugeno FIs for both synthetic and real-world data.

In this paper, a new approach is presented to model interval-based data using fuzzy sets (FSs). Specifically, we show how both crisp and uncertain intervals (where there is uncertainty about the endpoints of intervals) collected from individual or multiple survey participants over single or repeated surveys can be modeled using type-1, interval type-2, or general type-2 FSs based on zSlices. The proposed approach is designed to minimize any loss of information when transferring the interval-based data into FS models, and to avoid, as much as possible, assumptions about the distribution of the data. Furthermore, our approach does not rely on data preprocessing or outlier removal, which can lead to the elimination of important information. Different types of uncertainty contained within the data, namely intra- and inter-source uncertainty, are identified and modeled using the different degrees of freedom of type-2 FSs, thus providing a clear representation and separation of these individual types of uncertainty present in the data. We provide full details of the proposed approach, as well as a series of detailed examples based on both real-world and synthetic data. We perform comparisons with analogue techniques to derive FSs from intervals, namely the interval approach and the enhanced interval approach, and highlight the practical applicability of the proposed approach.

A recurring theme in research employing type-2 fuzzy sets is the question of how much uncertainty in a given context warrants the application of type-2 fuzzy sets and systems over their type-1 counterparts. In this paper we provide insight into this challenging question through a detailed investigation into the ability of both types of Fuzzy Logic Systems (FLSs) to capture and model different levels of uncertainty/noise through varying the size of the Footprint Of Uncertainty (FOU) of the underlying fuzzy sets from type-1 fuzzy sets to very "wide" interval type-2 fuzzy sets. By applying the study in the well-controlled context of chaotic time-series prediction, we show how, as uncertainty/noise increases, type-2 FLSs with fuzzy sets with FOUs of increasing size become more and more viable. While the work in this paper is focused on a specific application, we believe it provides crucial insight into the challenging question of the viability of interval type-2 over type-1 FLSs.

In this paper, we present a complete inferencing framework based on L-fuzzy sets, comprising fuzzification, inferencing itself, and both linguistic and numeric defuzzification strategies. We present the algorithms for each step, and then present a range of worked examples to illustrate the methods. Finally, we compare the results with similar examples which carry out `standard' Mandani-style inference. To the best of our knowledge, this is the first time that practical algorithms for complete L-fuzzy inference have been presented.

The measure of distance between two fuzzy sets is a fundamental tool within fuzzy set theory, however, distance measures currently within the literature use a crisp value to represent the distance between fuzzy sets. A real valued distance measure is developed into a fuzzy distance measure which better reflects the uncertainty inherent in fuzzy sets and a fuzzy directional distance measure is presented, which accounts for the direction of change between fuzzy sets. A multiplicative version is explored as a full maximal assignment is computationally intractable so an intermediate solution is offered.

Reasoning with fuzzy sets can be achieved through measures such as similarity and distance. However, these measures can often give misleading results when considered independently, for example giving the same value for two different pairs of fuzzy sets. This is particularly a problem where many fuzzy sets are generated from real data, and while two different measures may be used to automatically compare such fuzzy sets, it is difficult to interpret two different results. This is especially true where a large number of fuzzy sets are being compared as part of a reasoning system. This paper introduces a method for combining the results of multiple measures into a single measure for the purpose of analysing and comparing fuzzy sets. The combined measure alleviates ambiguous results and aids in the automatic comparison of fuzzy sets. The properties of the combined measure are given, and demonstrations are presented with discussions on the advantages over using a single measure.

In this paper we explore the characteristics of Type-1 Fuzzy Set agreement models based on interval data through contrasting statistical measures of the fuzzy models and the raw data respectively. We create Type-1 Fuzzy Set models using the Interval Agreement Approach, and then extract a preliminary set of attributes that encapsulate aspects of the agreement models. In order to explore what these attributes can tell us, we compare them with a set of traditional statistical measures of consensus which are applied to the raw data. Two interval-valued survey data sets are employed in this study, a synthetic data set consisting of 30 groups of 10 experts rating 25 objects which is used to provide a large example, and a real-world data set consisting of 7 groups of 4-8 cyber-security experts rating 26 security components that was collected during a decision making exercise at GCHQ, Cheltenham, UK. We show that while there are areas in which traditional methods and the attributes extracted from the Type-1 Fuzzy Set agreement models overlap, there are also attributes that do not appear to be replicated, suggesting that these attributes contain additional information about the consensus within the groups. A discussion of the results is provided, along with the conclusions that can be drawn and considerations for future work on this subject.

Environmental planning is complex, and requires careful consideration of a large number of factors, including quantitative ones (e.g., water balance) and qualitative ones (e.g., heterogeneous stakeholder input). To better integrate these factors, value-driven frameworks have been designed in the environmental conservation community. These frameworks are currently largely utilized manually by conservation and policy experts in order to inform policy design. In this paper, we present a fuzzy logic based system, which has been developed to operationalize the existing manual framework while preserving essential qualities, including the capture of uncertainty in the data sources and a consistent interpretability of the underlying automatic reasoning mechanisms. We provide a detailed description of the current implementation which can be applied in the operationalization of policy design and planning tasks in a range of natural resources management cases, followed by a set of concrete, practical outputs for a studied use case in Western Australia. Finally, we highlight remaining limitations and future work.

In this paper we present an online fuzzy logic toolkit for the design, implementation, execution and sharing of type-1 (T1), interval type-2 (T2) and (zSlices based) general T2 fuzzy logic system (FLSs). The motivation to develop the toolkit stems from the desire to provide a free-to-use fuzzy logic toolkit available which is platform-independent, easily accessible and which does not require any background knowledge of programming. This toolkit aims to help expand the accessibility of FLSs, in particular of T2 FLSs, to both research and industrial applications outside of the fuzzy logic community and computer science more generally. We review the features currently available through the JuzzyOnline toolkit (including a complete, previously unseen visualisation of the inference steps for zSlices based general T2 FLSs) and demonstrate a sample Fuzzy Logic System implementation of the toolkit. Finally, we conclude with some future developments and a call for feedback and contributions to aid in further development.

Distance metrics are broadly used in different research areas and applications, such as bio-informatics, data mining and many other fields. However, there are some metrics, like pg-gram and Edit Distance used specifically for data with a hierarchical structure. Other metrics used for non-hierarchical data are the geometric and Hamming metrics. We have applied these metrics to The Health Improvement Network (THIN) database which has some hierarchical data. The THIN data has to be converted into a tree-like structure for the first group of metrics. For the second group of metrics, the data are converted into a frequency table or matrix, then for all metrics, all distances are found and normalised. Based on this particular data set, our research question: which of these metrics is useful for THIN data?. This paper compares the metrics, particularly the pogram metric on finding the similarities of patients' data. It also investigates the similar patients who have the same close distances as well as the metrics suitability for clustering the whole patient population. Our results show that the two groups of metrics perform differently as they represent different structures of the data. Nevertheless, all the metrics could represent some similar data of patients as well as discriminate sufficiently well in clustering the patient population using k-means clustering algorithm.

The fuzzy integral (FI) is an extremely flexible aggregation operator. It is used in numerous applications, such as image processing, multicriteria decision making, skeletal age-at-death estimation, and multisource (e.g., feature, algorithm, sensor, and confidence) fusion. To date, a few works have appeared on the topic of generalizing Sugeno's original real-valued integrand and fuzzy measure (FM) for the case of higher order uncertain information (both integrand and measure). For the most part, these extensions are motivated by, and are consistent with, Zadeh's extension principle (EP). Namely, existing extensions focus on fuzzy number (FN), i.e., convex and normal fuzzy set- (FS) valued integrands. Herein, we put forth a new definition, called the generalized FI (gFI), and efficient algorithm for calculation for FS-valued integrands. In addition, we compare the gFI, numerically and theoretically, with our non-EP-based FI extension called the nondirect FI (NDFI). Examples are investigated in the areas of skeletal age-at-death estimation in forensic anthropology and multisource fusion. These applications help demonstrate the need and benefit of the proposed work. In particular, we show there is not one supreme technique. Instead, multiple extensions are of benefit in different contexts and applications.

Fuzzy integrals are non-linear combinations of a hypothesis support function and the (possibly subjective) worth of subsets of sources of information, realized by a fuzzy measure. They are used in many applications, with data fusion being the most well-known. In most applications, the fuzzy measure is built by some external knowledge about the worth of subsets of the information sources, whether by a subjective expert or objective sensor property, such as signal-to-noise ratio. In this paper, we investigate fuzzy measures for interval-valued evidence that have no intrinsic known worth; hence, the fuzzy measure cannot or should not be built in the conventional ways. Instead, the fuzzy measure is built directly from the data. We examine the previously proposed fuzzy measure of agreement, which builds the fuzzy measure by a computation of the agreement of combinations of sources (sources from which the contributed evidence has a high degree of agreement with evidence from other sources have a high worth). We also propose a new fuzzy measure of generalized accord that addresses a theoretical weakness in the agreement measure. We compare the two fuzzy measures by performing aggregation experiments with the fuzzy Choquet integral. Tests on both synthetic and real data are performed. We also compare the two measures against the aggregation results obtained by a survey of several example data sets.

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures.

The Fuzzy Integral (FI) is a powerful approach for non-linear data aggregation. It has been used in many settings to combine evidence (typically objective) with the known “worth” (typically subjective) of each data source, where the latter is encoded in a Fuzzy Measure (FM). While initially developed for the case of numeric evidence (integrand) and numeric FM, Grabisch et al. extended the FI to the cases of continuous intervals and normal, convex fuzzy sets (i.e., fuzzy numbers). However, in many real-world applications, e.g., explosive hazard detection based on multi-sensor and/or multi-feature fusion, agreement based modeling of survey data, anthropology and forensic science, or computing with respect to linguistic descriptions of spatial relations from sensor data, discontinuous interval and/or non-convex fuzzy set data may arise. The problem is no theory and algorithm currently exists for calculating the FI for such a case. Herein, we propose an extension of the FI to discontinuous interval- and convex normal Interval Fuzzy Set (IFS)-valued integrands (with a numeric FM). Our approach arises naturally from analysis of the Extension Principle. Further, we provide a computationally efficient approach to computing the proposed extension based on the union of the FIs on the combinations of continuous sub-intervals and we demonstrate the approach using examples for both the Choquet FI (CFI) and Sugeno FI (SFI).

In this paper we explore the practical application of the previously introduced approach [1] to generate fuzzy sets from interval-valued data. We demonstrate two specific example applications where we 1) generate type-1 fuzzy sets from interval-valued survey data for both words (e.g., neutral, excellent) and concepts (e.g., ambience, food) and 2) generate zSlices based general type-2 fuzzy set valued data from multiple iterations of a survey. We highlight the need for the simultaneous rating of both concepts and words in order to maintain context (including timeliness) of the resulting models. Further, in both example applications, we demonstrate using the Jaccard similarity measure how similarity measures can be employed to both relate and attribute word models to concept models (e.g., excellent food) and compare different concepts directly for different contexts (e.g., ambience in venue A vs. ambience in venue B). We provide interpretations for the resulting word/concept models and similarity values and highlight their utility, for example, for the data-driven generation of linguistic descriptions of venues. Finally, we highlight remaining questions and challenges both in technical terms and in application terms.

Singleton interval type-2 fuzzy logic systems (FLSs) have been widely applied in several real-world applications, where it was shown that the singleton interval type-2 FLSs outperform their singleton type-1 counterparts in applications with high uncertainty levels. However, one of the main criticisms of singleton interval type-2 FLSs is the fact that they outperform singleton type-1 FLSs solely based on their use of extra degrees of freedom (extra parameters) and that type-1 FLSs with a sufficiently large number of parameters may provide the same performance as interval type-2 FLSs. In addition, most works on type-2 FLSs only compare their results with singleton type-1 FLSs but fail to consider nonsingleton type-1 systems. In this paper, we aim to directly address and investigate this criticism. In order to do so, we will perform a comparative study between optimized singleton type-1, nonsingleton type-1, and singleton interval type-2 FLSs under the presence of noise. We will also present a multiobjective evolutionary algorithm (MOEA) for the optimization of singleton type-1, nonsingleton type-1, and singleton interval type-2 fuzzy systems for function approximation problems. The MOEA will aim to satisfy two objectives to maximize the accuracy of the FLS and minimize the number of rules in the FLS, thus improving its interpretability. Furthermore, we will present a methodology to obtain “optimal” consequents for the FLSs. Hence, this paper has two main contributions: First, it provides a common methodology to learn the three types of FLSs (i.e., singleton type-1, nonsingleton type-1, and singleton interval type-2 FLSs) from data samples. The second contribution is the creation of a common framework for the comparison of type-1 and type-2 FLSs that allows us to address the aforementioned criticism. We provide details of a series of experiments and include statistical analysis showing that the type-2 FLS is able to handle higher levels of noise than its nonsingleton and singleton type-1 counterparts.

In prior work, Grabisch put forth a direct (i.e., result of the Extension Principle) generalization of the Sugeno fuzzy integral (FI) for fuzzy set (FS)-valued normal (height equal to one) integrands and number-based fuzzy measures (FMs). Grabisch's proof is based in large on Dubois and Prade's analysis of functions on intervals, fuzzy numbers (thus normal FSs) and fuzzy arithmetic. However, a case not studied is the extension of the FI for sub-normal FS integrands. In prior work, we described a real-world forensic application in anthropology that requires fusion and has sub-normal FS inputs. We put forth an alternative non-direct approach for calculating FS results from sub-normal FS inputs based on the use of the number-valued integrand and number-valued FM Sugeno FI. In this article, we discuss a direct generalization of the Sugeno FI for sub-normal FS integrands and numeric FMs, called the Sub-normal Fuzzy Integral (SuFI). To no great surprise, it turns out that the SuFI algorithm is a special case of Grabisch's generalization. An algorithm for calculating SuFI and its mathematical properties are compared to our prior method, the Non-Direct Fuzzy Integral (NDFI). It turns out that SuFI and NDFI fuse in very different ways. We assert that in some settings, e.g., skeletal age-at-death estimation, NDFI is preferred to SuFI. Numeric examples are provided to stress important inner workings and differences between the FI generalizations.

Ambient Intelligence (AmI) is a vision that refers to an information technology paradigm where a physical environment is `aware' of its human occupants' presence/context and is sensitive, adaptive and responsive to their needs. Physical environments that are augmented with AmI are called Ambient Intelligent Environments (AIEs) which are deemed to be intelligent because the system should be able to recognise human occupants, reason with context and program itself to meet the occupants' needs by learning from their behaviour [1]. However, there is a need also to deal with real-world scenarios which involve multiple users occupying a given AIE. In order to handle multi-user AIEs and control them, there is a need to have agents that are able to learn the user(s) behaviours and handle the intra and inter-user uncertainties as people have different preferences and profiles which continuously change. In this paper, we present a zSlices based type-2 fuzzy agent which employs zSlices general type-2 fuzzy systems to learn the user(s) preferences and profiles and handle the encountered intra and inter-user uncertainties. The agent will behave according to a learned user profile that is unique to an individual user or a group of users and so the profile-selection problem manifests when the set of users in an AIE changes (i.e. when people enter/ leave an AIE). The proposed agent employs a novel strategy that we call Dynamic Profile-Selection that uses a cloud-based profile repository in order to support the agent activity in multiple AIEs. To demonstrate the proposed approach, we have conducted real-world experiments on two distinct AIEs which are the intelligent apartment (iSpace) and the intelligent Classroom (iClassroom) located at the University of Essex.

In this paper we describe a method of using interval valued survey responses from multiple experts on multiple occassions to produce General Type-2 fuzzy sets. In the method we propose, both the intra- and inter-person variability are modelled, with no loss of information. The resulting sets are completely determined by the data, providing an accurate representation (in terms of being defined solely by the data) of the opinions being modelled. A description of the method is provided, along with synthetic and real-world numeric examples and a comparison to an alternative method proposed in [1].

Fuzzy measures (FMs) have been used to model the (typically subjective) "worth" of subsets of information sources relative to a decision making problem. The fuzzy integral (FI) is a way to fuse the information encoded in a FM with the (typically objective) confidences in the strength of a hypothesis arising from the information sources. In prior work, Yager discussed a set of aggregation functions for general FMs. However, that work is primarily focused on theoretical exploration versus application. Herein, we investigate the direct extraction of different FMs from data, one for specificity and another for agreement, in the context of crowd sourcing. In crowd sourcing, one often has a lack of a ground truth or information regarding the reliability of sources. That is, all sources must be assumed equal (in terms of knowledge, experience level, etc.). Our goal is the intelligent fusion of this data taking into account as much information as possible from the data itself. Once a set of FMs are extracted from the data, we aggregate the FMs (resulting in what we herein refer to as a meta-measure) and use it in fuzzy integration. The novel aspect of this work is the extraction of multiple FMs directly from the original pool of data and the use of the resultant meta-measure and a FI to fuse the data from which the FMs were extracted. Herein, our data is interval-valued, thus we focus on fusion with respect to the generalized interval FI.

Computing with Words (CWW) aims to investigate the possibility of imitating the unique ability of humans for approximate reasoning on the basis of approximately defined classes and concepts in the form of words. Type-2 fuzzy sets have been used to provide an adequate modeling basis for words in a fuzzy logic context. In the context of type-2 fuzzy sets employed as part of CWW, a variety of research efforts have been made to investigate approaches to model the meaning of specific words using type-2 fuzzy sets. In this paper we start by focusing on the interpretation of classical set-theoretical operations (complement, union and intersection) for crisp and type-1 fuzzy sets. We proceed by extending the interpretations to the results of the union and intersection operations of interval type-2 fuzzy sets, specifically indicating their effect on the uncertainty representation in the sets. We note the impact of the choice of t-norms and t-conorms in particular in the context of CWW applications where the interpretation of the resulting sets and its resemblance to the human intuitive meaning of the concept or word is essential. Finally, we provide the interpretation and reasoning behind the Multi Level Agreement (MLA) operation based on zSlices which was previously introduced and discuss the requirement for the selection of the right operations for the amalgamation of individual fuzzy sets and the potential for investigating this choice in particular in a CWW context.

In this paper, we describe the main functionality of an initial version of a new fuzzy logic software toolkit based on the R language. The toolkit supports the implementation of several types of fuzzy logic inference systems and we discuss and present several aspects of its capabilities to allow the straightforward implementation of type-1 and interval type-2 fuzzy systems. We include source code examples and visualizations both of type-1 and type-2 fuzzy sets as well as output surface visualizations generated using the R toolkit. Finally, we describe the significant benefits of relying on the R language as a language which is employed across several research disciplines (thus enabling access to fuzzy logic tools to a variety of researchers), outline future developments and most importantly call for contributions, comments and feedback to/on this open-source software development effort.

In this paper, we present a novel technique to generate zSlices based general type-2 fuzzy sets using a series of interval type-2 fuzzy sets based around the notion of "agreement" of interval type-2 fuzzy sets. We provide details on how this approach can be applied for a series of readily available interval type-2 fuzzy sets as well as how the proposed approach can be employed to generate zSlices based general type-2 fuzzy sets which are continually updated as new interval type-2 fuzzy sets become available over time. We also describe the proposed approach in the context of Ambient Intelligent Environments (AIEs) which illustrate the benefits of a continuously updated general type-2 membership function and its potential advantage over interval type-2 fuzzy logic based approaches. Subsequently, we demonstrate the approach based on triangular interval type-2 fuzzy sets and we highlight the remaining complexities and complications in terms of the implementation of the proposed technique which stem from the potential for the creation of non-convex fuzzy sets and propose solutions for these problems.

Higher order fuzzy logic systems (FLSs), such as interval type-2 FLSs, have been shown to be very well suited to deal with the high levels of uncertainties present in the majority of real-world applications. General type-2 FLSs are expected to further extend this capability. However, the immense computational complexities associated with general type-2 FLSs have, until recently, prevented their application to real-world control problems. This paper aims to address this problem by the introduction of a complete representation framework, which is referred to as zSlices-based general type-2 fuzzy systems. The proposed approach will lead to a significant reduction in both the complexity and the computational requirements for general type-2 FLSs, while it offers the capability to represent complex general type-2 fuzzy sets. As a proof-of-concept application, we have implemented a zSlices-based general type-2 FLS for a two-wheeled mobile robot, which operates in a real-world outdoor environment. We have evaluated the computational performance of the zSlices-based general type-2 FLS, which is suitable for multiprocessor execution. Finally, we have compared the performance of the zSlices-based general type-2 FLS against type-1 and interval type-2 FLSs, and a series of results is presented which is related to the different levels of uncertainty handled by the different types of FLSs.

Fuzzy logic control is generally credited with being an adequate methodology for real world control applications which are subject to large amounts of uncertainties. Recent work has shown that interval type-2 fuzzy logic controllers (FLCs) can outperform type-1 FLCs in applications which encompass large amounts of uncertainty. However, the application of general type-2 FLCs and investigations of their performance have been very limited. This paper employs the recently introduced concept of zSlices based general type-2 fuzzy sets to implement a zSlices based general type-2 FLC (zFLC). We will present an overview of the implementation and operations of the zFLC for a two-wheel mobile robot navigating in real world outdoor environments. Furthermore, we present a performance analysis of the zFLC which is compared to the type-1 and interval type-2 FLCs.

Intelligent congestion control is vital for encoded video streaming of a clip or film, as network traffic volatility and the associated uncertainties require constant adjustment of the bit rate. Existing solutions, including the standard transmission control protocol (TCP) friendly rate control equation-based congestion controller, are prone to fluctuations in their sending rate and may respond only when packet loss has already occurred. This is a major problem, because both fluctuations and packet loss affect the end-user's perception of the delivered video. A type-1 (T1) fuzzy logic congestion controller (FLC) can operate at video display rates and can reduce packet loss and rate fluctuations, despite uncertainties in measurements of delay arising from congestion and network traffic volatility. However, a T1 FLC employing precise T1 fuzzy sets cannot fully cope with the uncertainties associated with such dynamic network environments. A type-2 FLC using type-2 fuzzy sets can handle such uncertainties to produce improved performance. This paper proposes an interval type-2 FLC that achieves a superior delivered video quality compared with existing traditional controllers and a T1 FLC. To show the response in different network scenarios, tests demonstrate the response both in the presence of typical Internet cross-traffic as well as when other video streams occupy a bottleneck on an All-Internet protocol (IP) network. As All-IP networks are intended for multimedia traffic, it is important to develop a form of congestion control that can transfer to them from the mixed traffic environment of the Internet. It was found that the proposed type-2 FLC, although it is specifically designed for Internet conditions, can also successfully react to the network conditions of an All-IP network. When the control inputs were subject to noise, the type-2 FLC resulted in an order of magnitude performance improvement in comparison with the T1 FLC. The type-2 FLC also showed reduced packet loss when compared with the other controllers, again resulting in superior delivered video quality. When judged by established criteria, such as TCP-friendliness and delayed feedback, fuzzy logic congestion control offers a flexible solution to network bottlenecks. These findings offer the type-2 FLC as a way forward for congestion control of video streaming across packet-switched IP networks.

Higher order fuzzy logic systems such as interval type-2 fuzzy logic systems have been shown to be very well suited to dealing with the large amounts of uncertainties present in the majority of real world applications. General type-2 fuzzy logic systems are expected to further extend this capability. However, the complexity as well as the immense computational requirements have generally prevented a foray into general type-2 fuzzy logic research. This paper introduces an alternative approach termed zSlices for representing general type-2 sets based on interval type-2 sets. Thus, this will lead to a smooth transition from interval to general type-2 fuzzy systems. The proposed approach will lead to a significant reduction in both the complexity and the computational requirements for general type-2 fuzzy logic systems. Hence, this will lead to facilitating the application of general type-2 fuzzy logic to many real world applications.

The type-2 Fuzzy Logic Controller (FLC) has started to emerge as a promising control mechanism for autonomous mobile robots navigating in real world environments. This is because such robots need control mechanisms such as type-2 FLCs which can handle the large amounts of uncertainties present in real world environments. However, manually designing and tuning the type-2 Membership Functions (MFs) for an interval type-2 FLC to give a good response is a difficult task. This paper will present a Genetic Algorithm (GA) based architecture to evolve the type-2 MFs of interval type-2 FLCs for mobile robots that will navigate in real world environments. The GA based system converges after a small number of iterations to type-2 MFs which give a very good performance. We have performed a series of real world experiments in which the evolved type-2 FLCs controlled a real robot in an outdoor arena. The evolved type-2 FLCs dealt with the uncertainties present in the real world to give a very good performance that has outperformed their type-1 counterparts as well as the manually designed type-2 FLCs.