# Single-Valued Neutrosophic Set Correlation Coefficient and Its Application in Fault Diagnosis

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## Abstract

**:**

## 1. Introduction

#### 1.1. Research Status

#### 1.2. Contribution of This Work

## 2. Preliminaries

#### 2.1. Triangular Fuzzy Numbers

#### 2.2. Single-Valued Neutrosophic Sets

**Definition**

**1.**

**Definition**

**2.**

## 3. The Proposed Method

#### 3.1. Correlation Coefficient between Single-Valued Neutrosophic Sets

- (1)
- According to the structural symmetry of the Formula (5), the condition $W(A,B)=W(B,A)$ is satisfied.
- (2)
- For each element in the Formula (5), they are satisfied $\ge 0$, so obviously $W(A,B)\ge 0$; The proof of inequality $W(A,B)\le 1$ as follows:$$\begin{array}{cc}\hfill C(A,B)& ={\displaystyle \sum _{i=1}^{n}[{T}_{A}({x}_{i})\xb7{T}_{B}({x}_{i})+{F}_{A}({x}_{i})\xb7{F}_{B}({x}_{i})+{I}_{A}({x}_{i})\xb7{I}_{B}({x}_{i})]}\hfill \\ & ={T}_{A}({x}_{1})\xb7{T}_{B}({x}_{1})+{T}_{A}({x}_{2})\xb7{T}_{B}({x}_{2})+\cdots +{T}_{A}({x}_{n})\xb7{T}_{B}({x}_{n})\hfill \\ & +{F}_{A}({x}_{1})\xb7{F}_{B}({x}_{1})+{F}_{A}({x}_{2})\xb7{F}_{B}({x}_{2})+\cdots +{F}_{A}({x}_{n})\xb7{F}_{B}({x}_{n})\hfill \\ & +{I}_{A}({x}_{1})\xb7{I}_{B}({x}_{1})+{I}_{A}({x}_{2})\xb7{I}_{B}({x}_{2})+\cdots +{I}_{A}({x}_{n})\xb7{I}_{B}({x}_{n})\hfill \end{array}$$And because of the inequality:$$ab\le \frac{{a}^{2}+{b}^{2}}{2}$$Therefore, we can get:$$\begin{array}{cc}\hfill C(A,B)& ={\displaystyle \sum _{i=1}^{n}[{T}_{A}({x}_{i})\xb7{T}_{B}({x}_{i})+{F}_{A}({x}_{i})\xb7{F}_{B}({x}_{i})+{I}_{A}({x}_{i})\xb7{I}_{B}({x}_{i})]}\hfill \\ & ={T}_{A}({x}_{1})\xb7{T}_{B}({x}_{1})+{T}_{A}({x}_{2})\xb7{T}_{B}({x}_{2})+\cdots +{T}_{A}({x}_{n})\xb7{T}_{B}({x}_{n})\hfill \\ & +{F}_{A}({x}_{1})\xb7{F}_{B}({x}_{1})+{F}_{A}({x}_{2})\xb7{F}_{B}({x}_{2})+\cdots +{F}_{A}({x}_{n})\xb7{F}_{B}({x}_{n})\hfill \\ & +{I}_{A}({x}_{1})\xb7{I}_{B}({x}_{1})+{I}_{A}({x}_{2})\xb7{I}_{B}({x}_{2})+\cdots +{I}_{A}({x}_{n})\xb7{I}_{B}({x}_{n})\hfill \\ & \le \frac{{{T}_{A}}^{2}({x}_{1})+{{T}_{B}}^{2}({x}_{1})}{2}+\frac{{{T}_{A}}^{2}({x}_{2})+{{T}_{B}}^{2}({x}_{2})}{2}+\cdots +\frac{{{T}_{A}}^{2}({x}_{n})+{{T}_{B}}^{2}({x}_{n})}{2}\hfill \\ & +\frac{{{F}_{A}}^{2}({x}_{1})+{{F}_{B}}^{2}({x}_{1})}{2}+\frac{{{F}_{A}}^{2}({x}_{2})+{{F}_{B}}^{2}({x}_{2})}{2}+\cdots +\frac{{{F}_{A}}^{2}({x}_{n})+{{F}_{B}}^{2}({x}_{n})}{2}\hfill \\ & +\frac{{{I}_{A}}^{2}({x}_{1})+{{I}_{B}}^{2}({x}_{1})}{2}+\frac{{{I}_{A}}^{2}({x}_{2})+{{I}_{B}}^{2}({x}_{2})}{2}+\cdots +\frac{{{I}_{A}}^{2}({x}_{n})+{{I}_{B}}^{2}({x}_{n})}{2}\hfill \\ & =\frac{1}{2}\{{\displaystyle \sum _{i=1}^{n}[{{T}_{A}}^{2}({x}_{i})+{{F}_{A}}^{2}({x}_{i})+{{I}_{A}}^{2}({x}_{i})]}+{\displaystyle \sum _{i=1}^{n}[{{T}_{B}}^{2}({x}_{i})+{{F}_{B}}^{2}({x}_{i})+{{I}_{B}}^{2}({x}_{i})]}\}\hfill \\ & =\frac{1}{2}[C(A,A)+C(B,B)]\hfill \end{array}$$Therefore:$$C(A,B)\le \frac{1}{2}[C(A,A)+C(B,B)]$$There is:$$2\xb7C(A,B)\le C(A,A)+C(B,B)$$Finally, contacting the previous types, there are:$$W(A,B)=\frac{2\xb7C(A,B)}{C(A,A)+C(B,B)}\le 1$$In summary, the condition $0\le W(A,B)\le 1$ is satisfied;
- (3)
- If $A=B$, so for any ${x}_{i}\in X(i=1,2,\cdots ,n)$, all ${T}_{A}({x}_{i})={T}_{B}({x}_{i})$, ${F}_{A}({x}_{i})={F}_{B}({x}_{i})$, ${I}_{A}({x}_{i})={I}_{B}({x}_{i})$, we can see from the structure of Formula (5), $W(A,B)=1$.

#### 3.2. Fault Diagnosis Method

**Step 1**: For fault template set $A{=\{\mathrm{A}}_{1}{,\mathrm{A}}_{2},\cdots ,{\mathrm{A}}_{m}\}$, and test sample set $C{=\{\mathrm{C}}_{1}{,\mathrm{C}}_{2},\cdots ,{C}_{n}\}$.

**Step 2**: By comparing the three fuzzy number of each attribute of the test sample and the three fuzzy number of the same attribute of the fault template, the degree of determinacy-membership ${T}_{{A}_{i}}({C}_{j})$, the degree of non-membership ${F}_{{A}_{i}}({C}_{j})$, and the degree of indeterminacy-membership ${I}_{{A}_{i}}({C}_{j})$ are obtained, as shown in Figure 3, and the calculation method is as follows:

**Step 3**: ${T}_{{A}_{i}}({C}_{j}),{F}_{{A}_{i}}({C}_{j}),{I}_{{A}_{i}}({C}_{j})$ can be expressed as single-valued neutrosophic set ${a}_{ij}=<{t}_{ij},{f}_{ij},{i}_{ij}>$, at this point, a single-valued neutrsophic set decision matrix can be generated as follows:

**Step 4**: After obtaining the single-valued neutrosophic set decision matrix $D$, the ideal single-valued neutrudophic number for attribute $j(j=1,2,\cdots ,n)$ can be generated by column as follows:

**Step 5**: According to Formula (7), generated weighted correlation coefficient based on single-valued neutrsophic set decision matrix $D$ and the ideal single-valued neutrudophic number ${a}^{*}$, the calculation formula is as follows:

**Step 6**: Finally, sorting the $W({A}_{i},B)$ of each analyzed sample, the largest value indicates that the template data belongs to this kind of fault.

## 4. Illustrative Example and Discussion

#### 4.1. Fault Diagnosis

- (i).
- According to the fault template data, the triangular fuzzy numbers under various attributes are obtained, in turn, as shown in Table 1:According to the analyzed sample data, the triangular fuzzy numbers under various attributes are obtained, in turn, as shown in Table 2:For the analyzed sample Xk ($k=1,2,3,4$ represents the $k$ attribute), Xk and ${G}_{k1-k5}$ (where $G=X,Y,Z$ represent $A,B,C$ three kinds of faults) are used for matching, respectively. The neutrosophic numbers $(T,F,I)$ statistics generated by the determined-membership degree T, non-membership degree F, and indeterminacy-membership degree I, are calculated, as shown in Table 3:
- (ii).
- Next, for the same fault template, neutrosophic sets with different attributes under fuzzy sample X, we can get the single-valued neutrosophic decision matrix, as shown in Table 4:
- (iii).
- According to the single-valued neutrosophic set decision matrix and Formula (11) under sample X in Table 4, the ideal neutrosophic set ${B}_{X}$ can be obtained as follows:$$\begin{array}{cc}\hfill {B}_{X}=& [<0.9612,0.0388,0.6747>,<0.7540,0.2460,0.5972>,\hfill \\ & <0.9836,0.0164,0.6451>,<0.9966,0.0034,0.5757>]\end{array}$$
- (iv).
- The weights of attributes $j(j=1,2,3,4)$ are all the same, that is the weight matrix $w$ is as follows:$$w=[0.25,0.25,0.25,0.25]$$Next, according to Table 4, Formula (7), (13), (14), for the fault template type ${A}_{i}$ (${A}_{1}=\mathrm{X}11-\mathrm{X}45,{A}_{2}=\mathrm{Y}11-\mathrm{Y}45,{A}_{3}=\mathrm{Z}11-\mathrm{Z}45$) and the ideal single-valued neutrosophic set ${B}_{X}$, calculate the improved weight correlation coefficient as follows:$$\{\begin{array}{l}W[{A}_{1},{B}_{X}]=0.8126\\ W[{A}_{2},{B}_{X}]=0.4133\\ W[{A}_{3},{B}_{X}]=0.5398\end{array}$$
- (v).
- Finally, according to Formula (15), ${A}_{1}>{A}_{3}>{A}_{2}$, it can be seen that the analyzed samples X1-X4 belong to the first type of fault, namely, the X fault.

#### 4.2. Fault Diagnosis Accuracy

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Min Value | Average Value | Max Value | Area | ||
---|---|---|---|---|---|

X | X11-X15 | 0.0661 | 0.1614605 | 0.2006 | 0.06725 |

X21-X25 | 0.121 | 0.149226 | 0.3468 | 0.1129 | |

X31-X35 | 0.0899 | 0.1123885 | 0.1296 | 0.01985 | |

X41-X45 | 0.357 | 4.3256515 | 4.666 | 2.1545 | |

Y | Y11-Y15 | 0.1567 | 0.181797 | 0.2038 | 0.02355 |

Y21-Y25 | 0.3071 | 0.329311 | 0.351 | 0.02195 | |

Y31-Y35 | 0.1865 | 0.242014 | 0.3218 | 0.06765 | |

Y41-Y45 | 4.094 | 4.715255 | 8.896 | 2.401 | |

Z | Z11-Z15 | 0.3006 | 0.3294004 | 0.3476 | 0.0235 |

Z21-Z25 | 0.2801 | 0.343854 | 0.3647 | 0.0423 | |

Z31-Z35 | 0.1151 | 0.136169 | 0.1864 | 0.03565 | |

Z41-Z45 | 9.385 | 9.810633 | 10.112 | 0.3635 |

Min Value | Average Value | Max Value | Area | ||
---|---|---|---|---|---|

X | X1 | 0.1416 | 0.14265 | 0.144 | 0.0012 |

X2 | 0.1028 | 0.11092 | 0.3058 | 0.1015 | |

X3 | 0.1279 | 0.133655 | 0.1378 | 0.00495 | |

X4 | 4.06 | 4.0938 | 4.18 | 0.06 |

Analyzed Sample | Fault Template | Neutrosophic Number |
---|---|---|

X1 | X11-X15 | (0.9612,0.0388,0.9914) |

Y11-Y15 | (0,1,0.6751) | |

Z11-Z15 | (0,1,0.6747) | |

X2 | X21-X25 | (0.7540,0.2460,0.6610) |

Y21-Y25 | (0,1,0.5972) | |

Z21-Z25 | (0.0126,0.9874,0.6722) | |

X3 | X31-X35 | (0.0127,0.9873,0.6451) |

Y31-Y35 | (0,1,1) | |

Z31-Z35 | (0.9836,0.0164,0.6952) | |

X4 | X41-X45 | (0.9966,0.0034,0.9348) |

Y41-Y45 | (0.0871,0.9129,0.9989) | |

Z41-Z45 | (0,1,0.5757) |

Diagnosis Fault | X1 | X2 | X3 | X4 |
---|---|---|---|---|

X11-X45 | (0.9612,0.0388,0.9914) | (0.7540,0.2460,0.6610) | (0.0127,0.9873,0.6451) | (0.9966,0.0034,0.9348) |

Y11-Y45 | (0,1,0.6751) | (0,1,0.5972) | (0,1,1) | (0.0871,0.9129,0.9989) |

Z11-Z45 | (0,1,0.6747) | (0.0126,0.9874,0.6722) | (0.9836,0.0164,0.6952) | (0,1,0.5757) |

Unknow Fault | SVNPWA | The Proposed Algorithm | ||
---|---|---|---|---|

Times of Right | Times of Error | Times of Right | Times of Error | |

X | 38 | 2 | 40 | 0 |

Y | 40 | 0 | 39 | 1 |

Z | 40 | 0 | 40 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Iryna, S.; Zhong, Y.; Jiang, W.; Deng, X.; Geng, J.
Single-Valued Neutrosophic Set Correlation Coefficient and Its Application in Fault Diagnosis. *Symmetry* **2020**, *12*, 1371.
https://doi.org/10.3390/sym12081371

**AMA Style**

Iryna S, Zhong Y, Jiang W, Deng X, Geng J.
Single-Valued Neutrosophic Set Correlation Coefficient and Its Application in Fault Diagnosis. *Symmetry*. 2020; 12(8):1371.
https://doi.org/10.3390/sym12081371

**Chicago/Turabian Style**

Iryna, Shchur, Yu Zhong, Wen Jiang, Xinyang Deng, and Jie Geng.
2020. "Single-Valued Neutrosophic Set Correlation Coefficient and Its Application in Fault Diagnosis" *Symmetry* 12, no. 8: 1371.
https://doi.org/10.3390/sym12081371