Abstract-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.

A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract-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 compositionbased inference method of type-1 fuzzy relations with a similaritybased 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.

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 uncertaintyrelating 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.

Abstract-The measure of distance between two fuzzy sets is a fundamental tool within fuzzy set theory. However, current distance measures within the literature do not account for the direction of change between fuzzy sets; a useful concept in a variety of applications, such as Computing With Words. In this paper, we highlight this utility and introduce a distance measure which takes the direction between sets into account. We provide details of its application for normal and non-normal, as well as convex and non-convex fuzzy sets. We demonstrate the new distance measure using real data from the MovieLens dataset and establish the benefits of measuring the direction between fuzzy sets.

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 datacapture 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.