Data-Informed Fuzzy Measures for Fuzzy Integration of Intervals and Fuzzy Numbers

Havens, Timothy C.; Anderson, Derek T.; Wagner, Christian

doi:10.1109/tfuzz.2014.2382133

Cited by 29 publications

(33 citation statements)

References 33 publications

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“…Note that this alternative form of the Jaccard similarity measure at (2) is relevant for showing its relationship with the Dice and our proposed similarity measures, detailed in Section III. Beyond crisp sets, the Jaccard similarity measure is used to estimate the similarity for intervals or sets of intervals [9], [10]. A closed intervalĪ i is a set of real numbers characterized by two endpoints I − i and I + i with…”

Section: B Jaccard Similarity Measurementioning

confidence: 99%

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Novel similarity measure for interval-valued data based on overlapping ratio

Kabir

Wagner

Havens

et al. 2017

2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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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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Section: B Jaccard Similarity Measurementioning

confidence: 99%

“…We expand on this in Section III. In [9], [10], the Dice similarity is used along with the Jaccard similarity for interval-valued evidence. By following (4), the Dice similarity for two intervalsĪ i andĪ j can be expressed as…”

Section: Dice Similarity Measurementioning

confidence: 99%

Novel similarity measure for interval-valued data based on overlapping ratio

Kabir

Wagner

Havens

et al. 2017

2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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“…Using standard complexity notation, C RAV can be captured as O(n 2 log n), compared to the O(n log n) complexity of the FI. The complexity of the former is driven by its recursive nature, resulting in the factorial term in (6), while that of the latter is driven by the sorting operation.…”

Section: Computational Complexitymentioning

confidence: 99%

“…Examples include the Sugeno λ FM [3], the decomposable FM [4], and the more recently introduced data-driven FMs [5,6]. It is worth noting that some of these algorithms, including the Sugeno λ FM and the decomposable FM, derive the entire FM lattice from the densities alone, with the sole additional constraint being the mathematical correctness of the FM in respect to P1 and P2 above, while others derive the entire FM lattice directly based on some criteria such as the level of agreement over given subsets of sources.…”

Section: Introductionmentioning

confidence: 99%

The arithmetic recursive average as an instance of the recursive weighted power mean

Wagner

Havens

Anderson

2017

2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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Abstract-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.

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“…For example, intuitionist fuzzy numbers (IFNs) and generalized intuitionist fuzzy numbers (GIFNs), which are intuitionistic sets on the real line satisfying certain conditions, were defined and studied in [16][17][18]. Recently, interval type 2 fuzzy numbers (IT2FNs) have been defined in [19][20][21][22], as interval type 2 fuzzy sets (IT2FSs) on the real line fulfilling some properties. Besides, in [19,22], total preorders were given to ranking IT2FNs, but such preorders do not hold the property of antisymmetry, which is a theoretical and practical weakness.…”

Section: Introductionmentioning

confidence: 99%

New Order on Type 2 Fuzzy Numbers

Hernández

Cubillo

Torres-Blanc

et al. 2017

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Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers.

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Data-Informed Fuzzy Measures for Fuzzy Integration of Intervals and Fuzzy Numbers

Cited by 29 publications

References 33 publications

Novel similarity measure for interval-valued data based on overlapping ratio

Novel similarity measure for interval-valued data based on overlapping ratio

The arithmetic recursive average as an instance of the recursive weighted power mean

New Order on Type 2 Fuzzy Numbers

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