A Modified Correlation Coefficient of Probabilistic Hesitant Fuzzy Sets and Its Applications of Decision Making, Medical Diagnosis, Cluster Analysis

Abstract

Correlation coefficients characterize relationships of various fuzzy sets and thus facilitate relevant applications. Regarding correlation coefficients of hesitant fuzzy sets (HFSs), a modified strategy of adjustment factor from hesitation degree is recently proposed, and the corresponding information enrichment can be probabilistically promoted to pursue correlation coefficients of probabilistic hesitant fuzzy sets (PHFSs). In this paper embracing PHFSs, a probabilistic adjustment factor is similarly constructed to modify an existing correlation coefficient, so an improved correlation coefficient is established; this new measure induces practical applications, mainly the decision making, medical diagnosis, and cluster analysis. According to PHFSs, the hesitation degree and adjustment factor are defined to achieve theoretical extensions, and the latter information factor is utilized to modify the existing correlation coefficient; thus, the new correlation coefficient emerges to offer improvements, and it acquires measure properties and example illustrations. Furthermore, the new and old correlation coefficients on PHFSs are comprehensively compared on the basis of a representative correlation coefficient on HFSs; accordingly, related correlation coefficients are applied in decision making, medical diagnosis, and cluster analysis, and our new correlation coefficient is verified to have the effectiveness and applicability.

Article highlights

• A modified correlation coefficient in PHFSs is proposed by fusing an adjustment factor based on hesitation degrees.

• The new correlation coefficient (CC) both extends an existing CC in HFSs and improves a current CC in PHFSs.

• The new CC is applied to decision making, medical diagnosis, cluster analysis to gain effective contrastive performance.

Appendices

Appendix 1: Calculation Details of Example 1

Herein, we complement calculation details of Example 1. By Definition 3, we have

$$\begin{aligned} \begin{aligned}&\bar{h}_{A}(x_{1})=\frac{1}{3}(0.1+0.3+0.6)=0.3333, \\&\bar{h}_{A}(x_{2})=\frac{1}{3}(0.2+0.45+0.75)=0.4667, \\&\bar{h}_{B}(x_{1})=0.6, \bar{h}_{B}(x_{2})=0.6667,\\&C(A,B)=\frac{1}{3}(\vert 0.1-0.3333\vert \vert 0.45-0.6\vert +\vert 0.3-0.3333\vert \vert 0.55-0.6\vert \\&\qquad +\vert 0.6-0.3333\vert \vert 0.8-0.6\vert )\\&\qquad +\frac{1}{3}(\vert 0.2-0.4667\vert \vert 0.5-0.6667\vert +\vert 0.45-0.4667\vert \vert 0.6-0.6667\vert \\&\qquad +\vert 0.75-0.4667\vert \vert 0.9-0.6667\vert )\\&\quad =0.3+0.0372=0.0672,\\&C(A,A)=(\frac{1}{3}(\vert 0.1-0.3333\vert ^{2}+\vert 0.3-0.3333\vert ^{2}+\vert 0.6-0.3333\vert ^{2})\\&\qquad +\frac{1}{3}(\vert 0.2-0.45\vert ^{2}+\vert 0.5-0.6667\vert ^{2}+\vert 0.75-0.45\vert ^{2}))^{\frac{1}{2}}\\&\quad =(0.0422+0.0506)^{\frac{1}{2}}=0.3046,\\&C(B,B)=(0.0217+0.0289)^{\frac{1}{2}}=0.2248, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \rho _{HFS\text{-}X}(A,B)=\frac{0.0672}{0.3046\times 0.2248}=0.9815. \end{aligned} \end{aligned}$$

By Definition 4, we have

$$\begin{aligned}{} & {} \begin{aligned}&\Xi (h_{A}(x_{1}))=\frac{1}{3}(\vert 0.1-0.3\vert +\vert 0.1-0.6\vert +\vert 0.3-0.6\vert )=0.3333,\\&\Xi (h_{A}(x_{2}))=\frac{1}{3}(\vert 0.2-0.45\vert +\vert 0.2-0.75\vert +\vert 0.45-0.75\vert )=0.3667,\\&\Xi (h_{B}(x_{1}))=\frac{1}{3}(\vert 0.45-0.55\vert +\vert 0.45-0.8\vert +\vert 0.55-0.8\vert )=0.2333,\\&\Xi (h_{B}(x_{2}))=\frac{1}{3}(\vert 0.5-0.9\vert +\vert 0.5-0.9\vert +\vert 0.6-0.9\vert )=0.2667,\\&\xi _{\Xi }(A,B)=\frac{0.3333\times 0.2333+0.3667\times 0.2267}{max\{0.3333^{2}+0.3667^{2},0.2333^{2}+0.2667^{2}\}} =0.7149, \end{aligned} \\ {} & {} \begin{aligned}&\rho _{HFS\text{-}X}^{*}(A,B)=\rho _{HFS\text{-}X}(A,B)\xi _{\Xi }(A,B) =0.9815\times 0.7149=0.7017. \end{aligned} \end{aligned}$$

Similarly, we can get the surplus cases:

$$\begin{aligned} \begin{aligned}&\rho _{HFS\text{-}X}^{*}(A,C)=\rho _{HFS\text{-}X}(A,C)\xi _{\Xi }(A,C)=0.0387\times 0.6063=0.0235,\\&\rho _{HFS\text{-}X}^{*}(B,C)=\rho _{HFS\text{-}X}(B,C)\xi _{\Xi }(B,C)=0.0375\times 0.8407=0.0316. \end{aligned} \end{aligned}$$

Appendix 2: A Special Case with Single-Element Universe Related to Example 1

Example 1 demonstrates a current correlation coefficient in HFSs [15] by virtue of a two-element universe. For further illustration, we supplement a special case of single-element universe. Let $X=\{x_{1}\}$, and its three HFSs are

$$\begin{aligned} \begin{aligned} D=&\{\langle x, \{0.2,0.35,0.6\}\rangle \}, E=\{\langle x, \{0.1,0.5,0.7\}\rangle \}, \\&F=\{\langle x, \{0.25,0.55,0.9\}\rangle \}. \end{aligned} \end{aligned}$$

By Definition 3, we have

$$\begin{aligned} \begin{aligned}&\bar{h}_{D}=\frac{1}{3}(0.2+0.35+0.6)=0.3833, \bar{h}_{E}=\frac{1}{3}(0.1+0.5+0.7)=0.4333,\\&C(D,E)\\&\quad =\frac{1}{3}(\vert 0.2-0.3833\vert \vert 0.1-0.4333\vert +\vert 0.35-0.3833\vert \vert 0.5-0.4333\vert \\&\qquad +\vert 0.6-0.3833\vert \vert 0.7-0.4333\vert )\\&\quad =0.0404,\\&C(D,D)=0.0272, C(E,E)=0.0622,\\&\rho _{HFS\text{-}X}(D,E)=\frac{0.0404}{\sqrt{0.0272}\times \sqrt{0.0622}}=0.9809, \end{aligned} \end{aligned}$$

By Definition 4, we have

$$\begin{aligned} \begin{aligned}&\Xi (h_{D})=\frac{1}{3}(\vert 0.2-0.35\vert +\vert 0.2-0.6\vert +\vert 0.35-0.6\vert )=0.2667,\Xi (h_{E})=0.4,\\&\xi _{\Xi }(D,E)=\frac{0.2667\times 0.4}{max\{0.2667^{2},0.4^{2}\}}=0.6667,\\&\rho _{HFS\text{-}X}^{*}(D,E)=\rho _{HFS\text{-}X}(D,E)\xi _{\Xi }(D,E)=0.9809\times 0.6667=0.6539. \end{aligned} \end{aligned}$$

Similarly, we can get two surplus cases:

$$\begin{aligned} \begin{aligned}&\rho _{HFS\text{-}X}^{*}(D,F)=\rho _{HFS\text{-}X}(D,F)\xi _{\Xi }(D,F)=0.9951\times 0.6154=0.6124,\\&\rho _{HFS\text{-}X}^{*}(E,F)=\rho _{HFS\text{-}X}(E,F)\xi _{\Xi }(E,F)=0.9838\times 0.9231=0.9081. \end{aligned} \end{aligned}$$

$\square$