On the Mitchell similarity measure and its application to pattern recognition

doi:10.1016/j.patrec.2012.01.008

Pattern Recognition Letters

Volume 33, Issue 9, 1 July 2012, Pages 1219-1223

https://doi.org/10.1016/j.patrec.2012.01.008 Get rights and content

Abstract

This paper is a response to the similarity measure and pattern recognition problem of Mitchell that was published in Pattern Recognition Letters, 2003. The purpose of this paper is threefold. First, we reviewed and revised her computation for similarity measures. Second, we proved that the similarity values for the one-norm should be larger than that for the two-norm for her pattern recognition problem. Third, we proposed a more scattered similarity measure to help researchers determine patterns. Our findings may shed light on the ongoing debate between Li and Cheng, 2002, Mitchell, 2003.

Highlights

► We revised questionable results in (Mitchell, 2003) that had been cited 47 times. ► We proved that similarity measures based on one-norm is larger than that on two-norm. ► We proposed a new similarity measure for pattern recognition problems.

Introduction

Intuitionistic fuzzy sets (IFSs), first proposed by Atanassov, 1986, Atanassov, 1989, Atanassov, 1999, are a valuable approach to handling the vagueness and uncertainty that arise when applied to various fields. Since their first appearance, many different distance and similarity measures of IFSs have been presented, but there still remain some questionable aspects that have yet to be improved. In this paper, we will discuss the findings of Mitchell (2003). To date, forty-seven papers have cited that work; the aims of all of them can be classified into the following four categories:

(a)
The development of new similarity measures: Hung and Yang, 2004, Huang, 2007, Hung and Yang, 2007, Lv et al., 2007a, Vlachos and Sergiadis, 2007, Zhou and Wu, 2007, Hao et al., 2008, Hung and Yang, 2008, Xu and Chen, 2008, Meng et al., 2009, Xia and Xu, 2010, Zhang et al., 2010, Chen, 2011, Wei et al., 2011, Xu and Xia, 2011, Ye, 2011, Yusoff et al., 2011.
(b)
The construction of new solution methods: Mitchell, 2004b, Mitchell, 2005, Wang and Xin, 2005, Li et al., 2006, Quanming et al., 2006, Pelekis et al., 2008, Xu et al., 2008, Zhang and Jiang, 2008, Chen et al., 2009, Lv et al., 2007b, Lv et al., 2009, Own, 2009, Xu, 2009, Yang et al., 2009, Yue et al., 2009, Zeng et al., 2009, Zhang et al., 2009, Pelekis et al., 2011, Zhang et al., 2011,
(c)
The application to decision making problems: Mitchell, 2004a, Maghrebi et al., 2007, Ning et al., 2008, Baccour and Alimi, 2009, Burduk, 2009, Khatibi and Montazer, 2009a, Khatibi and Montazer, 2009b, Yuan et al., 2009.
(d)
The revision of existing similarity measures: Li et al., 2007, Park et al., 2007, Xu and Yager, 2009.

Papers in the first three categories were so eager to pioneer new territory that they failed to perform an in depth examination of the work in question.

The papers in the last category are the only ones that discuss previous similarity measures. We will provide only a brief summary of their findings here; for a more complete picture, please refer to their original papers. Li et al. (2007) pointed out that with one-norm, the similarity measure of Mitchell (2003) will degenerate to the similarity measure of Hong and Kim (1999). On the other hand, Li et al. (2007) provided two counter examples for the similarity measure of Hong and Kim (1999) to show that it cannot help decision makers solve pattern recognition problems. Park et al. (2007) mentioned that Mitchell (2003) improved the axiom for the similarity measures for IFSs. Four new similarity measures of Park et al. (2007) are all satisfied by the axiom provided by Mitchell (2003). Xu and Yager (2009) claimed that Mitchell (2003) made some modifications to the axioms of Li and Cheng (2002). Xu and Yager (2009) also presented a detailed study on the similarity measure of Szmidt and Kacprzyk (2004). These three papers provided valuable comments on previous similarity measures. However, they did not pay attention to the similarity measure and pattern recognition problems of Mitchell (2003). In this paper, we will first review and examine her similarity measure. Then, we will check the pattern recognition problems in her paper and provide revisions.

In addition, we have followed the valuable suggestion of the anonymous referee to add a hesitancy degree in our future research for the study of similarity measures, as done in Szmidt and Kacprzyk, 2004, Xu, 2007, and Xu and Yager (2009).

Our findings will help future researchers obtain a deeper understanding of the similarity measure in question.

Section snippets

Review the similarity measure of Mitchell (2003)

In Mitchell (2003), for IFSs, $\tilde{A} = {〈 u, μ_{A} (u), v_{A} (u) 〉 | u \in X}$ where X is the disclose of universe, μ_A is the membership function, and v_A is the non-membership function. She defined a new membership function, $ϕ_{A} (x) = 1 - ν_{A},$ where μ_A and ϕ_A denote the “low” and “high” membership functions. Based on the similarity measure of Li and Cheng (2002) for two functions, f and g, under the p-norm $S_{p} (f, g) = 1 - {[\int w (x) | f (x) - g (x) |^{p} dx]}^{1 / p},$ where w(x) is the weight function that satisfies $w (x) ⩾ 0$ and $\int w (x) dx = 1$ with $p ⩾ 1$ .

Mitchell

Our examinations of her computation for similarity values

Owing to the domain, the pattern recognition of Section 2 is restricted to $0 ⩽ x ⩽ 1,$ and so the uniform weight w(x) yields $w (x) = 1 for 0 ⩽ x ⩽ 1 and w (x) = 0 .$ Figure 2 of her paper is shown as Fig. 1 in this paper; it did not give us the exact values of a and b in the definition of membership functions and non-membership functions. Since, from the information given, we can only know that 0 < a < 0.5 < b < 1, we found that $ρ_{μ} (\tilde{A}, \tilde{B}) = S_{2} (μ_{A}, μ_{B}) = 1 - {(\int_{a}^{b} (0.1)^{2} dx)}^{1 / 2} = 1 - 0.1 \sqrt{b - a},$ and $ρ_{ϕ} (\tilde{A}, \tilde{B}) = S_{2} (ϕ_{A}, ϕ_{B}) = 1 - {(\int_{a}^{b} (0.1)^{2} dx)}^{1 / 2} = 1 - 0.1 \sqrt{b - a}$ which

Our proposed similarity measure

We will propose a new similarity measure between $\tilde{A}$ and $\tilde{B}$ , under p-norm $S_{new, p} (\tilde{A}, \tilde{B}) = 1 - {(\int w (x) | μ_{A} (x) - μ_{B} (x) |^{p} dx)}^{1 / p} - {(\int w (x) | ϕ_{A} (x) - ϕ_{B} (x) |^{p} dx)}^{1 / p} .$ With p = 2 and uniform weight w(x), for the above example in Sections 2 Review the similarity measure of, 3 Our examinations of her computation for similarity values, we found that $S_{new, p = 2} (\tilde{A}, \tilde{B}) = 1 - 0.2 \sqrt{b - a} and S_{new, p = 2} (\tilde{A}, \tilde{C}) = 1 - 0.6 \sqrt{b - a} .$ From the Fig. 1, we may assume that a = 0.2 and b = 0.8, and so we derive that $S_{new, p = 2} (\tilde{A}, \tilde{B}) = 1 - 0.1549 = 0.8451 and S_{new, p = 2} (\tilde{A}, \tilde{C}) = 1 - 0.4648 = 0.5352 .$

Further review of pattern recognition of Mitchell (2003)

Mitchell (2003) provided a pattern recognition application for its new similarity measure. However, the exact membership and non-membership functions for three patterns ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ and ${\tilde{P}}_{3}$ , and one sample $\tilde{Q}$ , were absent from the paper, and only graphic expressions were offered in Figure 3 of that paper. For convenience, her Figure 3 is provided as Fig. 2 in this paper.

We assumed that $μ_{1} (x) = \{\begin{matrix} 1.2 - 0.2 x, & for 1 ⩽ x ⩽ 2 \\ 1 - 0.1 x, & for 2 ⩽ x ⩽ 3 \end{matrix}$ and $ϕ_{1} (x) = \{\begin{matrix} 1, & for 1 ⩽ x ⩽ 2 \\ 1.2 - 0.1 x, & for 2 ⩽ x ⩽ 3 \end{matrix}$ for pattern ${\tilde{P}}_{1}$ . We also assumed that $μ_{2} (x)$

Conclusions

In this paper, we first noted that the results computed in Mitchell (2003) do not fit her definition of a similarity measure. Then we provided a new similarity measure. For a pattern recognition problem, we study her results to prepare a new lemma to verify that the similarity value for one-norm is greater than that for two-norm. Moreover, after correcting Mitchell’s similarity value, we pointed out that our proposed similarity measure can offer a more recognizable difference to help

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On the Mitchell similarity measure and its application to pattern recognition

Abstract

Highlights

Introduction

Section snippets

Review the similarity measure of Mitchell (2003)

Our examinations of her computation for similarity values

Our proposed similarity measure

Further review of pattern recognition of Mitchell (2003)

Conclusions

Fuzzy Sets Systems

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Inform. Sci.

Pattern Recognition Lett.

Int. J. Approximate Reasoning

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Pattern Recognition Lett.

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Pattern Recognition Lett.

Pattern Recognition Lett.

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