# Pythagorean Fuzzy Partial Correlation Measure and Its Application

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## Abstract

**:**

## 1. Introduction

- enhancement of an existing PFCC approach to be used for the development of PFPCC,
- development of PFPCC using the enhanced PFCC,
- theoretical descriptions of the PFPCC for the sake of validation, and
- the application of the developed PFPCC in pattern recognition.

## 2. Preliminaries

#### 2.1. Pythagorean Fuzzy Sets

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

- (i)
- $C=D$ iff ${\eta}_{C}\left(x\right)={\eta}_{D}\left(x\right)$ ${\theta}_{C}\left(x\right)={\theta}_{D}\left(x\right)$ for every $x\in X$.
- (ii)
- $C\subseteq D$ iff ${\eta}_{C}\left(x\right)\le {\eta}_{D}\left(x\right)$, ${\theta}_{C}\left(x\right)\ge {\theta}_{D}\left(x\right)$ for every $x\in X$.
- (iii)
- $\overline{C}=\left\{\langle x,{\theta}_{C}\left(x\right),{\eta}_{C}\left(x\right)\rangle \right|x\in X\}$, $\overline{D}=\left\{\langle x,{\theta}_{D}\left(x\right),{\eta}_{D}\left(x\right)\rangle \right|x\in X\}$.
- (iv)
- $C\cup D=\left\{\langle x,max\{{\eta}_{C}\left(x\right),{\eta}_{D}\left(x\right)\},min\{{\theta}_{C}\left(x\right),{\theta}_{D}\left(x\right)\}\rangle \right|x\in X\}$.
- (v)
- $C\cap D=\left\{\langle x,min\{{\eta}_{C}\left(x\right),{\eta}_{D}\left(x\right)\},max\{{\theta}_{C}\left(x\right),{\theta}_{D}\left(x\right)\}\rangle \right|x\in X\}$.

**Definition**

**4**

#### 2.2. Simple Correlation Measures in Pythagorean Fuzzy Domain

**Definition**

**5**

- (i)
- $\sigma (C,D)\in [-1,1]$,
- (ii)
- $\sigma (C,D)=\sigma (D,C)$,
- (iii)
- $\sigma (C,D)=1$ iff $C=D$.

#### 2.2.1. Thao’s Technique

#### 2.2.2. Liu et al.’s Technique

#### 2.2.3. Thao et al.’s Technique

#### 2.2.4. Modified Technique

#### 2.2.5. Comparison for the PFCMs

**Example**

**1.**

**Example**

**2.**

## 3. Partial Correlation Coefficient of PFSs

#### 3.1. First-Order PFPCC

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

- (i)
- If $\tilde{\sigma}(C,E)=0$, then$$\tilde{\sigma}(C,D|E)={\displaystyle \frac{\tilde{\sigma}(C,D)}{{\left((1-{\tilde{\sigma}}^{2}(D,E))\right)}^{\frac{1}{2}}}}=\tilde{\sigma}(D,C|E).$$Furthermore, if $\tilde{\sigma}(D,E)=0$, then$$\tilde{\sigma}(C,D|E)={\displaystyle \frac{\tilde{\sigma}(C,D)}{{\left((1-{\tilde{\sigma}}^{2}(C,E))\right)}^{\frac{1}{2}}}}=\tilde{\sigma}(D,C|E).$$
- (ii)
- If $\tilde{\sigma}(C,D)=0$, $\tilde{\sigma}(C,E)\ne 0$ and $\tilde{\sigma}(D,E)\ne 0$, then$$\tilde{\sigma}(C,D|E)={\displaystyle \frac{-\tilde{\sigma}(C,E)\tilde{\sigma}(D,E)}{{\left((1-{\tilde{\sigma}}^{2}(C,E))(1-{\tilde{\sigma}}^{2}(D,E))\right)}^{\frac{1}{2}}}}.$$

**Proof.**

#### 3.2. nth-Order PFPCC

**Definition**

**7.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 4. Applicative Example in Pattern Recognitions and Classifications

#### Case Study

- (i)
- Pattern ${P}_{1}$ has a negative sway on the correlation between patterns $({P}_{2},{P}_{3})$, $({P}_{2},{P}_{4})$ and a positive effect on the correlation of $({P}_{3},{P}_{4})$.
- (ii)
- Pattern ${P}_{2}$ only has a positive sway on the correlation between patterns $({P}_{1},{P}_{3})$, $({P}_{1},{P}_{4})$ and $({P}_{3},{P}_{4})$.
- (iii)
- Pattern ${P}_{3}$ has a negative effect on the correlation between patterns $\left({P}_{1}{P}_{2}\right)$, $({P}_{2},{P}_{4})$ and a positive effect on the correlation of $({P}_{1},{P}_{4})$.
- (iv)
- Pattern ${P}_{4}$ has a negative influence on the correlation between patterns $({P}_{1},{P}_{2})$, $({P}_{2},{P}_{3})$ and a positive effect on the correlation of $({P}_{1},{P}_{3})$.

- (i)
- Pattern ${P}_{1}$ has a negative effect on the first-order partial correlations ${\tilde{\sigma}}_{{P}_{2}{P}_{4}|{P}_{3}}$, ${\tilde{\sigma}}_{{P}_{2}{P}_{3}|{P}_{4}}$ and a positive effect on ${\tilde{\sigma}}_{{P}_{3}{P}_{4}|{P}_{2}}$.
- (ii)
- Pattern ${P}_{2}$ only has a positive effect on the first-order partial correlations ${\tilde{\sigma}}_{{P}_{1}{P}_{3}|{P}_{4}}$, ${\tilde{\sigma}}_{{P}_{1}{P}_{4}|{P}_{3}}$ and ${\tilde{\sigma}}_{{P}_{3}{P}_{4}|{P}_{1}}$.
- (iii)
- Pattern ${P}_{3}$ has a negative impact on the first-order partial correlations ${\tilde{\sigma}}_{{P}_{1}{P}_{2}|{P}_{4}}$, ${\tilde{\sigma}}_{{P}_{2}{P}_{4}|{P}_{1}}$ and a positive effect on ${\tilde{\sigma}}_{{P}_{1}{P}_{4}|{P}_{2}}$.
- (iv)
- Pattern ${P}_{4}$ has a negative effect on the first-order partial correlation coefficients ${\tilde{\sigma}}_{{P}_{1}{P}_{2}|{P}_{3}}$, ${\tilde{\sigma}}_{{P}_{2}{P}_{3}|{P}_{1}}$ and a positive impact on ${\tilde{\sigma}}_{{P}_{1}{P}_{3}|{P}_{2}}$.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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PFCCs | ${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\sigma}}_{3}$ | $\tilde{\mathit{\sigma}}$ |
---|---|---|---|---|

Example 1 | $-0.9080$ | $-0.9898$ | $-0.8800$ | $-0.7021$ |

Example 2 | $-1.0000$ | $-1.0000$ | $0.0000$ | $-0.3235$ |

Feature Space | |||||
---|---|---|---|---|---|

PFS | ${\mathit{s}}_{\mathbf{1}}$ | ${\mathit{s}}_{\mathbf{2}}$ | ${\mathit{s}}_{\mathbf{3}}$ | ${\mathit{s}}_{\mathbf{4}}$ | ${\mathit{s}}_{\mathbf{5}}$ |

${\eta}_{{P}_{1}}$ | 0.8000 | 0.7000 | 0.9000 | 0.6000 | 0.8000 |

${\theta}_{{P}_{1}}$ | 0.1000 | 0.2000 | 0.0000 | 0.3000 | 0.1000 |

${\eta}_{{P}_{2}}$ | 0.9000 | 0.8000 | 0.8000 | 0.5000 | 0.7000 |

${\theta}_{{P}_{2}}$ | 0.1000 | 0.1000 | 0.1000 | 0.3000 | 0.2000 |

${\eta}_{{P}_{3}}$ | 0.5000 | 0.5000 | 0.9000 | 0.5000 | 0.7000 |

${\theta}_{{P}_{3}}$ | 0.3000 | 0.2000 | 0.0000 | 0.4000 | 0.1000 |

${\eta}_{{P}_{4}}$ | 0.7000 | 0.5000 | 0.9000 | 0.6000 | 0.8000 |

${\theta}_{{P}_{4}}$ | 0.2000 | 0.4000 | 0.1000 | 0.3000 | 0.0000 |

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**MDPI and ACS Style**

Yan, D.; Wu, K.; Ejegwa, P.A.; Xie, X.; Feng, Y.
Pythagorean Fuzzy Partial Correlation Measure and Its Application. *Symmetry* **2023**, *15*, 216.
https://doi.org/10.3390/sym15010216

**AMA Style**

Yan D, Wu K, Ejegwa PA, Xie X, Feng Y.
Pythagorean Fuzzy Partial Correlation Measure and Its Application. *Symmetry*. 2023; 15(1):216.
https://doi.org/10.3390/sym15010216

**Chicago/Turabian Style**

Yan, Dongfang, Keke Wu, Paul Augustine Ejegwa, Xianyang Xie, and Yuming Feng.
2023. "Pythagorean Fuzzy Partial Correlation Measure and Its Application" *Symmetry* 15, no. 1: 216.
https://doi.org/10.3390/sym15010216