Similarity measures for general type-2 fuzzy sets based on the α-plane representation

Extension of Restricted Equivalence Functions and Similarity Measures for Type-2 Fuzzy Sets

“…Note that this set is defined over the referential {0.25, 0.5, 0.75, 1} so it is not possible to directly compare it to the result T 2SM ( Ã, B) obtained in our example. However, if we calculate the centroid of this set to produce a single numerical, representative value (as proposed in [18]), we get a value of 0.7420, which, again, is a high similarity although lower than the numerical value N T 2SM ( Ã, B) obtained in our case. Nevertheless, we maintain that a direct comparison of the similarity scores for such methods,operating on different referential sets-, is not meaningful.…”

“…It is also worth to mention that, if we consider the expression for similarity given in [18], which is done in terms of the αplane or zSlice representation, considering for α the values {0.25, 0.5, 0.75, 1}, we get that the similarity between à and B of Example 4.3 is given by the type-1 fuzzy set: {(0.25, 0.778), (0.5, 0.7560), (0.75, 0.7426), (1, 0.7255)} .…”

“…Nevertheless, we maintain that a direct comparison of the similarity scores for such methods,operating on different referential sets-, is not meaningful. In this particular case, our proposed method is designed to evaluate the similarity of the sets primarily in respect to the primary degree of membership, while [18] focuses on the similarity of the secondary memberships.…”

“…These two directions can be also found in the context of type-2 similarity measures. In the literature we find both, similarity measures in which the result is a numerical value [4], [12], [16] and similarity measures in which the result is a fuzzy set [17], [18]. We opt for the latter case, coinciding with Bandler and Kohout( [15]) which state that "The natural anticipation, it seems to us, is that the fuzziness will not thereby be diminished".…”

Generalized Type-2 Fuzzy Systems for controlling a mobile robot and a performance comparison with Interval Type-2 and Type-1 Fuzzy Systems

“…In comparison with IT2FS, where the uncertainty is represented as an area, in GT2FS the uncertainty is depicted by a volume, and as such, are more capable of handling uncertainty. As GT2FS research is still fairly new, existing research is fairly limited, some examples of advancements are shown in computing the centroid by means of the centroid-flow algorithm (J M Mendel, 2011), similarity measures (Hao & Mendel, 2014), hierarchical collapsing method for direct defuzzification (Doostparast Torshizi & Fazel Zarandi, 2014), definition of footprint of uncertainty (Mo, Wang, Zhou, Li, & Xiao, 2014), a fast method for computing the centroid (H.-J. Wu, Su, & Lee, 2012), enhanced type-reduction (Yeh et al, 2011), monotone centroid flow algorithm for type-reduction (O. , conversion from IT2FS to GT2FS (Wagner, Miller, Garibaldi, Anderson, & Havens, 2014), computing with words for discrete GT2FS (Zhao, Li, & Li, 2013), matching GT2FS by comparing the vertical slices (Rizzi, Livi, Tahayori, & Sadeghian, 2013), and formation of GT2FS based on the information granule numerical evidence (Sanchez, Castro, & Castillo, 2013).…”