Similarity measures for general type-2 fuzzy sets based on the -plane representation
Introduction
Zadeh [57] generalized the concept of type-1 fuzzy sets (T1 FS) [25], [26], [39] to type-2 fuzzy sets (T2 FS): A T2 FS is an extension of a T1 FS where the membership function (MF) of a T2 FS is a fuzzy set on the interval . The MF of a T2 FS is three-dimensional, where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its footprint of uncertainty (FOU). A T2 FS lets us incorporate uncertainty about the MF into fuzzy set theory, and, if there is no uncertainty, then a T2 FS reduces to a T1 FS, which is analogous to probability reducing to determinism when unpredictability vanishes [41].
For an interval type-2 fuzzy set (IT2 FS) that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an IT2 FS. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. Most applications of T2 FSs use IT2 FSs (e.g., [3], [7], [8], [9], [12], [15], [16], [17], [18], [20], [21], [22], [27], [29], [30], [32], [34], [38], [50], [55], [56], [61], [62]). IT2 FSs have received the most attention because the mathematics that is needed for such sets–primarily interval arithmetic–is much simpler than the mathematics that is needed for general T2 FSs.
Recently, there has been a growing interest in using general T2 FSs (GT2 FSs) (e.g., [2], [13], [19], [28], [31], [40], [45], [47], [52], [63]). Recall (e.g., [24], [35]) that an IT2 FS can be thought of as a blurred T1 FS that lets one account for MF uncertainties; however, an IT2 FS weights all such uncertainties uniformly, and can therefore be thought of only as a first order fuzzy set uncertainty model. On the other hand, a GT2 FS can also let us account for MF uncertainties, but it weights all such uncertainties non-uniformly, and can therefore be thought of as a second-order fuzzy set uncertainty model. Both IT2 and GT2 FS are parametric models. An IT2 FS is described by more parameters than is a T1 FS and a GT2 FS is described by even more parameters. Because GT2 FSs have more design degrees of freedom then IT2 FSs, and, therefore, have the potential to outperform a system that uses IT2 FSs.
Liu [31] and Mendel et al. [36], [40] proposed a method that represents a GT2 FS, which is called an -plane representation, and they demonstrated that this representation is useful for both theoretical and computational studies for GT2 FSs. Using the -plane representation, set-theoretic operations and centroid computations become very simple, because they can be performed using existing algorithms that are applied to each -plane, which lends themselves to massive parallel processing. Also, in [51], [52], Wagner and Hagras introduced a systematic approach for developing general Type-2 Fuzzy Logic Systems (T2 FLSs) based on “zSlice Representation Theorem (RT)”. Mendel [36] and Mendel and Zhai [65] pointed out that a zSlice is the same as an -plane raised to level .
A similarity measure between fuzzy sets is a very important concept in fuzzy set theory. There have been a lot of different similarity measures proposed in the literature, for both T1 FSs and IT2 FSs [10], [11], [14], [43], [46], [53], [54], [58], [59], [60], [66]. Many applications have been made use of these similarity measures, e.g., Buckley and Hayashi [4] and Turksen and Zhong [49] used a similarity measure between fuzzy sets for rule matching; Turksen and Zhong proposed an approximate analogical reasoning schema based on similarity measures and interval-valued fuzzy sets [48]; Candan et al. [6] applied a similarity measure in multimedia data base query; Meng et al. [42] used a fuzzy set-valued similarity measure for a pattern recognition problem; Mohamed and Abdala [44] applied a new similarity measure to a clustering problem, etc.
In this paper, we propose a new similarity measure between GT2 FSs based on the -plane representation, which is an extension of Jaccard’s similarity measure for IT2 FSs; and, some properties of the new similarity measure are proved. We also provide several numerical examples to demonstrate the use of our proposed similarity measure and discuss the effects of the secondary membership functions (MFs) on the similarity measure.
The rest of this paper is organized as follows: Section 2 reviews the -plane representation and Wu and Mendel’s similarity measure for IT2 FSs; Section 3 explains our new similarity measure for GT2 FSs; Section 4 shows that our new similarity measure satisfies the desired properties for a “good” similarity measure; Section 5 provides two numerical examples of our new similarity measure; Section 6 defines two measures to show how our similarity measure is affected by different secondary MFs; and, Section 7 draws conclusions.
Section snippets
Background
This section reviews some important concepts in T2 FS theory and describes the -plane representation of a GT2 FS developed by Liu [31] and Mendel et al. [36], [40]. It also reviews Jaccard’s [41], [54], [53] similarity measure for IT2 FSs.
A new similarity measure for GT2 FSs
In this section, some desirable properties for a GT2 similarity measure are introduced; then, a new similarity measure for GT2 FSs using the -plane RT is proposed; and, finally, some theorems are proved that demonstrate the new similarity measure satisfies all the desirable properties when trapezoidal secondary MFs are used. Trapezoidal secondary MFs are very widely used [40], [63], [64] and include triangular secondary MFs as a special case; and, in this paper, only this kind of secondary MF
Properties of the new similarity measure for trapezoidal secondary MFs
In this section, two theorems are proved. The first provides a preliminary result that is used in the proof of the second, which in turn demonstrates that our new crisp similarity measure for GT2 FSs, in (23), satisfies all four axiomatic properties defined in Section 3.1. Theorem 2 For two GT2 FSs, and , with trapezoidal secondary MFs, if is a monotonic function of (). Proof A trapezoidal secondary MF for a GT2 FS (at ) is shown in Fig. 4. In this case, and
Numerical examples
In this section, some examples are provided that use the new similarity measure described in the previous sections. We start with an example using two discrete GT2 FSs.
Similarity percentage
As mentioned in the Introduction, GT2 FSs are becoming more and more popular. Of great interest to us is learning whether or not there is much difference between a GT2 FS and an IT2 FS. In this section, we use the similarity results between a GT2 FS with triangular (’s and ’s) or trapezoidal (’s and ’s) secondary MFs and the IT2 FS with the same FOU ( and ) to answer the above question. To be more precise, we use the results in the first row of Table 1, Table 2.
Of course, it is the
Conclusions
In this paper, a new similarity measure for GT2 FSs is proposed using the -plane representation; it is an extension of Wu and Mendel’s similarity measure for IT2 FSs. Some properties of the new similarity measure are proved; numerical examples are given, and the effects of the secondary MFS in different scenarios are discussed. Our overall conclusion about “Is there much of a difference between a GT2 FS and an IT2 FS” can be answered by using our new similarity measure. From our simulations,
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