An efficient similarity measure for intuitionistic fuzzy sets

Abstract

We introduce a new methodology for measuring the degree of similarity between two intuitionistic fuzzy sets. The new method is developed on the basis of a distance defined on an interval by the use of convex combination of endpoints and also focusing on the property of min and max operators. It is shown that among the existing methods, the proposed method meets all the well-known properties of a similarity measure and has no counter-intuitive examples. The validity and applicability of the proposed similarity measure is illustrated with two examples known as pattern recognition and medical diagnosis.

This is a preview of subscription content, access via your institution.

References

  1. Atanassov KT (1999) Intuitionistic fuzzy sets: theory and application, studies in fuzziness and soft computing, vol 35. Physica, Heidelberg

  2. Bustince H, Burillo P (1996) Vague sets are intuitionistic fuzzy sets. Fuzzy Sets and Syst 79:403–405

    Article  MATH  MathSciNet  Google Scholar 

  3. Goguen JA (1967) L-fuzzy sets. J Math Anal Appl 18:145–174

    Article  MATH  MathSciNet  Google Scholar 

  4. Huang GS, Liu YS, Wang XD (2005) Some new distances between intuitionistic fuzzy sets. In: Proceedings of the international conference on machine learning and cybernetics (ICMLC 05), pp 2478–2482, Guangzhou, China

  5. Hung WL, Yang MS (2007) Similarity measures of intuitionistic fuzzy sets based on Lp metric. Int J Approx Reason 46:120–136

    Article  MATH  MathSciNet  Google Scholar 

  6. Li DF, Cheng CT (2002) New similarity measure of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recog Lett 23:221–225

    Article  MATH  MathSciNet  Google Scholar 

  7. Li YH, Olson DL, Zheng Q (2007) Similarity measures between intuitionistic fuzzy (vague) sets: a comparative analysis. Pattern Recog Lett 28:278–285

    Article  Google Scholar 

  8. Liu HW (2005) New similarity measures between intuitionistic fuzzy sets and between elements. Math Comput Modell 42:61–70

    Article  MATH  Google Scholar 

  9. Pal SK, King RA (1981) Image enhancement using smoothing with fuzzy sets. IEEE Trans Syst Man Cybernet 11:495–501

    Google Scholar 

  10. Pedrycz W. (1997) Fuzzy sets in pattern recognition: accomplishments and challenges. Fuzzy Sets Syst 90:171–176

    Article  MathSciNet  Google Scholar 

  11. Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst 114:505–518

    Article  MATH  MathSciNet  Google Scholar 

  12. Szmidt E, Kacprzyk J (1996) Intuitionistic fuzzy sets in decision making. Notes IFS 2:22–31

    MATH  MathSciNet  Google Scholar 

  13. Szmidt E, Kacprzyk J (2005) A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making. In: Torra V, Narukawa Y, Miyamoto S (eds) Modelling decision for artificial intelligence, LNAI 3558, Springer 272–282

  14. Turksen IB (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 20:191–210

    Article  MATH  MathSciNet  Google Scholar 

  15. Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information-application to pattern recognition. Pattern Recog Lett 28:197–206

    Article  Google Scholar 

  16. Wang W, Xin X (2005) Distance measure between intuitionistic fuzzy sets. Pattern Recog Lett 26:2063–2069

    Article  Google Scholar 

  17. Wei GW, Lan G (2008) Grey relational analysis method for interval-valued intuitionistic fuzzy multiple attribute decision making. In: Fifth international conference on fuzzy systems and knowledge discovery, pp 291–295

  18. Xu ZS, Chen J (2007) Approach to group decision making based on interval-valued intuitionistic judgement matrices. Syst Engi Theory Pract 27:126–133

    Article  Google Scholar 

  19. Yao J, Dash M (2000) Fuzzy clustering and fuzzy modeling. Fuzzy Sets Syst 113:381–388

    Article  MATH  Google Scholar 

  20. Ye J (2011) Cosine similarity measures for intuitionistic fuzzy sets and their applications. Math Comput Modell 53:91–97

    Article  MATH  Google Scholar 

  21. Zadeh LA (1965) Fuzzy sets. Inform Comput 8:338–353

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author thanks the editor-in-chief professor Antonio Di Nola and the referees for their helpful suggestions which improved the presentation of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to B. Farhadinia.

Additional information

Communicated by G. Acampora.

Appendix

Appendix

In this section we prove the main results stated in the last part of Sect. 3. First we prove a key theorem.

Theorem 6

Let \(A_{IFS},B_{IFS} \in IFS(X).\) The parametric distance \({d}_{IFS}:IFS(X)\times IFS(X)\rightarrow [0, 1]\) given by (2) is monotonically decreasing as the parameter m increases.

Proof

Without loss of the generality, we assume that X = {x 1 = x} and \(A_{IFS},B_{IFS} \in IFS(X)\) are respectively represented by the intervals [a 1, a 2] and [b 1, b 2] where a 1 = μ A (x), a 2 = 1 − ν A (x), b 1 = μ B (x), b 2 = 1 − ν B (x). With the latter in mind, we can now restate d IFS (A IFS B IFS ) in parametric form as follows

$$ d^{(m)}_{IFS}(A_{IFS},B_{IFS})=\sqrt{\frac{1}{m+1}\sum_{j=0}^m [ {\chi}_j(A_{IFS}(x))- {\chi}_j(B_{IFS}(x))]^{2} }, $$
$$ \begin{aligned} {\text{where}} \; \chi_j(A_{IFS}(x))&=\left(1-\frac{j}{m}\right)a_1+\frac{j}{m}a_2,\quad j=0,1,...,m,\\ \chi_j(B_{IFS}(x))&=\left(1-\frac{j}{m}\right)b_1+\frac{j}{m}b_2,\quad j=0,1,...,m. \end{aligned} $$

As a first step toward the general case, we first show that

$$ d^{(1)}_{IFS}(A_{IFS},B_{IFS})\geq d^{(2)}_{IFS}(A_{IFS},B_{IFS}), $$

for any \(A_{IFS},B_{IFS} \in IFS(X). \) By the definition of the parametric distance d (m) IFS , one gets

$$ \begin{aligned} (d^{(1)}_{IFS}(A_{IFS},B_{IFS}))^2- (d^{(2)}_{IFS}(A_{IFS},B_{IFS}))^2&= \frac{1}{2} [ (a_1-b_1)^2+(a_2-b_2)^2]-\frac{1}{3} [ (a_1-b_1)^2\\&\quad+((a_1+\frac{a_2-a_1}{2})-(b_1+\frac{b_2-b_1}{2}))^2+(a_2-b_2)^2]. \end{aligned} $$

In this and subsequent results, it is notationally convenient to set

$$ \begin{aligned} &\alpha=a_1-b_1, \\ &\beta=a_2-b_2. \end{aligned} $$

With the use of the above notations, the following result is obtained

$$ \alpha+k\frac{\beta-\alpha}{m}=\left(a_1+k\frac{a_2-a_1}{m}\right)-\left(b_1+k\frac{b_2-b_1}{m}\right), \quad k=0,1,...,m. $$

Thus, with the above setting in mind, we find that

$$ \begin{aligned} (d^{(1)}_{IFS}(A_{IFS},B_{IFS}))^2- (d^{(2)}_{IFS}(A_{IFS},B_{IFS}))^2 & =\frac{1}{2} [ \alpha^2+\beta^2]-\frac{1}{3} \left[ \alpha^2+\left(\alpha+\frac{\beta-\alpha}{2}\right)^2+\beta^2\right]\\&=\frac{1}{6} \left[ 3\alpha^2+3\beta^2-2\alpha^2-2\left(\frac{\alpha+\beta}{2}\right)^2-2\beta^2\right]\\ &=\frac{1}{6} \left[ \alpha^2+\beta^2-2\left(\frac{\alpha+\beta}{2}\right)^2\right]\\&=\frac{1}{12} (\alpha-\beta)^2 \geq 0, \end{aligned} $$

completing the proof of d (1) IFS (A IFS B IFS ) ≥ d (2) IFS (A IFS B IFS ).

We are now ready to prove the general case where the parameter m is a natural number.

For given m and from definition of the parametric distance d (m) IFS , we have

$$ \begin{aligned} &(d^{(m)}_{IFS}(A_{IFS},B_{IFS}))^2-(d^{(m+1)}_{IFS}(A_{IFS},B_{IFS}))^2 \\&\quad=\frac{1}{m+1} \left[ \alpha^2+\left(\alpha+\frac{\beta-\alpha}{m}\right)^2+\cdots+\left(\alpha+(m-1)\frac{\beta-\alpha}{m}\right)^2+\beta^2\right]- \\&\quad\quad \frac{1}{m+2} \left[ \alpha^2+\left(\alpha+\frac{\beta-\alpha}{m+1}\right)^2+\cdots+\left(\alpha+(m-1)\frac{\beta-\alpha}{m+1}\right)^2+\left(\alpha+(m)\frac{\beta-\alpha}{m+1}\right)^2+\beta^2\right] \\&\quad = \frac{1}{(m+1)(m+2)} \left\{(m+2)\left[\alpha^2+\left(\frac{(m-1)\alpha+\beta}{m}\right)^2+\left(\frac{(m-2)\alpha+2\beta}{m}\right)^2+\cdots+\left(\frac{\alpha+(m-1)\beta}{m}\right)^2\right.\right.\\&\left.\left.\quad\quad +\beta^2\right] -(m+1)\left[\alpha^2+\left(\frac{(m)\alpha+\beta}{m+1}\right)^2+\left(\frac{(m-1)\alpha+2\beta}{m+1}\right)^2+\cdots+\left(\frac{2\alpha+(m-1)\beta}{m+1}\right)^2+ \left(\frac{\alpha+(m)\beta}{m+1}\right)^2\right.\right.\\& \left.\left.\quad\quad+\beta^2\right]\right\} \\&\quad= \frac{1}{(m+1)(m+2)} \left\{\alpha^2+\beta^2+\frac{(m+2)}{m^2}[((m-1)\alpha+\beta)^2+((m-2)\alpha+2\beta)^2+\cdots+(\alpha+(m-1)\beta)^2] \right.\\&\left.\quad\quad-\frac{1}{(m+1)}[((m)\alpha+\beta)^2+((m-1)\alpha+2\beta)^2+\cdots+(2\alpha+(m-1)\beta)^2+ (\alpha+(m)\beta)^2]\right\}\\ &\quad= \frac{1}{(m+1)(m+2)} \left\{\frac{(m+2)}{m^2}([(m-1)^2+(m-2)^2+\cdots+1]\alpha^2+[1+2^2+\cdots+(m-1)^2]\beta^2 \right.\\&\left.\quad\quad+2[1(m-1)+2(m-2)+\cdots+(m-1)1]\alpha\beta) \right.\\& \left.\quad\quad-\frac{1}{(m+1)}([m^2+(m-1)^2+\cdots+1]\alpha^2+[1+2^2+\cdots+(m-1)^2+m^2]\beta^2 \right.\\&\left.\quad\quad+2[1(m)+2(m-1)+\cdots+(m)1]\alpha\beta)\right\} \\&\quad=\frac{1}{(m+1)(m+2)} \left\{\frac{m+2}{6m}\alpha^2+\frac{m+2}{6m}\beta^2-\frac{m+2}{3m}\alpha\beta \right\} \\&\quad=\frac{1}{(m+1)6m}(\alpha- \beta)^2 \geq0, \end{aligned} $$

completing the proof of d (m) IFS (A IFS B IFS ) ≥ d (m+1) IFS (A IFS B IFS ). □

Corollary 1

Let \(A_{IFS},B_{IFS} \in IFS(X). \) The parametric similarity measure \({S}^{d(m)}_{IFS}:IFS(X)\times IFS(X)\rightarrow [0,1]\) given by (4) is a monotone increasing function of the parameter m.

Proof

The proof is concluded by taking definition of S d(m) IFS and Theorem 6 into account. □

Theorem 7

Let \(A_{IFS},B_{IFS} \in IFS(X). \) If \(S^{d}_{IFS}:IFS(X)\times IFS(X)\rightarrow [0,1]\) and \(S^{ {mix}}_{IFS}:IFS(X)\times IFS(X)\rightarrow [0,1]\) are the mappings given by (4) and (5), respectively. Then, the sequence of parametric similarity measures \(S^{(m)}_{IFS}:IFS(X)\times IFS(X)\rightarrow [0,1]\) given by (12) which can be restated as

$$ {S^{(m)}_{IFS}(A_{IFS},B_{IFS})=\frac{1}{2}( S^{d(m)}_{IFS}(A_{IFS},B_{IFS})+S^{ {mix}}_{IFS}(A_{IFS},B_{IFS})),} $$

is a convergent sequence on [0,1].

Proof

Since the parametric similarity measure \({S}^{d(m)}_{IFS}:IFS(X)\times IFS(X)\rightarrow [0,1]\) given by (4) is a monotone increasing function of the parameter m (by Corollary 1 and since S mix IFS is not dependant on the choice of m), we deduce that the parametric similarity measure \({S}^{(m)}_{IFS}:IFS(X)\times IFS(X)\rightarrow [0,1]\) is a monotone increasing function of the parameter m, too. This incorporating with the boundedness of S (m) IFS (by the property (P1) where S (m) IFS (A IFS ,B IFS ) ≤ 1 for any \(A_{IFS},B_{IFS} \in IFS(X)\)) will immediately lead to the convergence property of S (m) IFS . □

The earlier result shows that to have a more precise comparison we need to choose m sufficiently large. This finding is confirmed and illustrated by the graph in Fig. 1 where the curves C1–C6 show the behavior of S (m) IFS applied to each pair of IFSs given in columns 1–6 of Table 2, respectively, as the parameter m increases from 1 to 50.

Fig. 1
figure1

Graphical illustration of the convergence property of S (m) IFS applied to IFSs given in Table 2

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Farhadinia, B. An efficient similarity measure for intuitionistic fuzzy sets. Soft Comput 18, 85–94 (2014). https://doi.org/10.1007/s00500-013-1035-5

Download citation

Keywords

  • Intuitionistic fuzzy sets
  • Similarity measures
  • Distance measure
  • Pattern recognition