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.

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

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