Abstract-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 modelled using type-1, interval type-2, or general type-2 FSs based on zSlices. The proposed approach is designed to minimise any loss of information when transferring the intervalbased 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 pre-processing 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 modelled 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 fuzzy sets from intervals, namely the Interval Approach (IA) and the Enhanced Interval Approach (EIA) and highlight the practical applicability of the proposed approach.
Abstract-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.
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