# A Dual Hesitant Fuzzy Rough Pattern Recognition Approach Based on Deviation Theories and Its Application in Urban Traffic Modes Recognition

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Mathematical Methodologies

#### 3.1. Assessment Deviation Analysis on DHFRSs

#### 3.2. Novel Findings on DHFRSs

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 4. Main Results

#### 4.1. Novel Pattern Recognition Approach on DHFRs

**Step 1**According to Equations (2) and (3), we calculate the lower and upper approximations $\underline{\mathbb{R}}\left(\mathbb{A}\right)$ and $\overline{\mathbb{R}}\left(\mathbb{A}\right)$ of DHFSs $\mathbb{A}$ with respect to $(U,V,\mathbb{R})$.

**Step 2**By Definition 5, for any ${u}_{k}\in U(k=1,2,\cdots ,m)$, a set ${S}_{\mathbb{A},{u}_{k}}=\{{s}_{\mathbb{A},{u}_{k}}^{1},{s}_{\mathbb{A},{u}_{k}}^{2},{s}_{\mathbb{A},{u}_{k}}^{3},{s}_{\mathbb{A},{u}_{k}}^{4}\}$ is obtained to describe the membership degree of $\mathbb{A}$ to ${u}_{k}$.

**Step 3**By Equation (19) or Equation (20), a series of weighted membership degree sets are obtained as ${s}_{\mathbb{A},{u}_{k}}^{*}(k=$ 1, 2, ⋯, m).

**Step 4**By Equation (21), for any ${k}_{1},{k}_{2}=$ 1, 2, ⋯, m, ${s}_{\mathbb{A},{u}_{{k}_{1}}}^{*}$ and ${s}_{\mathbb{A},{u}_{{k}_{2}}}^{*}$ are compared, and the optimal pattern recognition that $\mathbb{A}$ belongs is obtained.

#### 4.2. Supplement Explanations

- (1)
- In classical intuitionistic fuzzy rough environments [16], Definitions 5 and 6 can be simplified as follows.

**Theorem**

**1.**

- (i)
- (ii)
- Equation (19) is reduced to ${s}_{1}^{*}={\displaystyle \sum _{i=1}^{4}}{e}_{1i}^{-1}\xb7{({\displaystyle \sum _{i=1}^{4}}{e}_{1i}^{-1})}^{-1}\xb7{s}_{i}$;
- (iii)
- Equation (20) is reduced to ${s}_{2}^{*}={\displaystyle \sum _{i=1}^{4}}(2-{e}_{1i})\xb7{({\displaystyle \sum _{i=1}^{4}}(2-{e}_{1i}))}^{-1}\xb7{s}_{i}$.

- (2)
- In classical hesitant fuzzy rough environments [17], Definitions 5 and 6 can be simplified.

**Theorem**

**2.**

- (i)
- (ii)
- Equations (19) and (20) are reduced to$${s}_{1}^{*}=\left({\displaystyle \sum _{i=1}^{2}}({\displaystyle \frac{{e}_{2i}^{-1}}{{\displaystyle \sum _{i=1}^{2}}{e}_{2i}^{-1}}}\xb7{s}_{i}^{\sigma \left(1\right)}),{\displaystyle \sum _{i=1}^{2}}({\displaystyle \frac{{e}_{2i}^{-1}}{{\displaystyle \sum _{i=1}^{2}}{e}_{2i}^{-1}}}\xb7{s}_{i}^{\sigma \left(2\right)}),\cdots ,{\displaystyle \sum _{i=1}^{2}}({\displaystyle \frac{{e}_{2i}^{-1}}{{\displaystyle \sum _{i=1}^{2}}{e}_{2i}^{-1}}}\xb7{s}_{i}^{\sigma \left({l}_{s}\right)})\right),$$$${s}_{2}^{*}=\left({\displaystyle \sum _{i=1}^{2}}({\displaystyle \frac{(1-{e}_{2i})}{{\displaystyle \sum _{i=1}^{2}}(1-{e}_{2i})}}\xb7{s}_{i}^{\sigma \left(1\right)}),{\displaystyle \sum _{i=1}^{2}}({\displaystyle \frac{(1-{e}_{2i})}{{\displaystyle \sum _{i=1}^{2}}(1-{e}_{2i})}}\xb7{s}_{i}^{\sigma \left(2\right)}),\cdots ,{\displaystyle \sum _{i=1}^{2}}({\displaystyle \frac{(1-{e}_{2i})}{{\displaystyle \sum _{i=1}^{2}}(1-{e}_{2i})}}\xb7{s}_{i}^{\sigma \left({l}_{s}\right)})\right),$$

- (3)
- There are some important properties for Equations (19) and (20), which are as follows.

**Theorem**

**3.**

- (i)
- If $\prod _{i=1}^{4}}H(1-{e}_{1i}-{e}_{2i})=1$, then, the variance of ${W}_{1}$ is larger than which of ${W}_{2}$;
- (ii)
- If $\prod _{i=1}^{4}}H({e}_{1i}+{e}_{2i}-1)=1$, then, the variance of ${W}_{1}$ is smaller than which of ${W}_{2}$;
- (iii)
- If $1\u2a7d{\displaystyle \sum _{i=1}^{4}}H({e}_{1i}+{e}_{2i}-1)<4$, then, the comparing between the variances of ${W}_{1}$ and ${W}_{2}$ is inconclusive.
- (iv)

## 5. Illustrative Example

**Step 1**By Definition 4, the lower and upper approximations of each ${\mathbb{A}}_{0}$ with respect to $(M,F,\mathbb{R})$ can be obtained as

**Step 2**By Definition 5, for any ${u}_{k}\in U(k=$ 1, 2, ⋯, 5), a series of membership sets are obtained as follows:

**Step 3**Take $\lambda =2$, by Equation (19), a series of weighted membership degree sets are obtained as follows:

**Step 4**For any ${k}_{1},{k}_{2}\in \{1,2,3,4\}$, ${s}_{\mathbb{A},{u}_{{k}_{1}}}^{*}$ and ${s}_{\mathbb{A},{u}_{{k}_{2}}}^{*}$ are compared. From the comparison results, it can be obtained that ${S}_{{\mathbb{M}}_{0},{M}_{2}}^{*}\succ {S}_{{\mathbb{M}}_{0},{M}_{3}}^{*}\succ {S}_{{\mathbb{M}}_{0},{M}_{1}}^{*}\succ {S}_{{\mathbb{M}}_{0},{M}_{4}}^{*}$. Therefore, when the shared electric bicycles are put into the market, electric bicycle would be affected by the greatest extent among the four kinds of classical traffic modes, followed closely by classical shared bicycle.

## 6. Conclusions

- (i)
- According to the relationship between intuitionistic fuzzy set and vague set, the DHFRS is transferred into a fuzzy set where the membership of any given element to it has multi-grouped values. And then, by the idea of bootstrap sampling, four sets are generated from the DHFRS information, where the elements of any of the sets are considered as observation values of the aforementioned membership degree.
- (ii)
- By differentiated strategy, the four sets on DHFRSS are weighted by their assessment deviations apart from the real-value of the membership degree on DHFRS, where the bigger the assessment deviation, the smaller the weight of its related set on DHFRS; the smaller the assessment deviation, the bigger the weight of its related set on DHFRS.
- (iii)
- The assessment deviations on the four sets are mainly determined by two parameters, where one quantifies the systematic deviations of the four sets, and the other one quantifies the random deviations of the sets. And then, the true-value of the membership degree of an element to the set on DHFRS is estimated by a deviation-based DHFRWA operator.
- (iv)
- A dual hesitant fuzzy rough pattern recognition approach based on the assessment deviation analysis is proposed, and an illustrative example is given to verify the effectiveness of this approach on DHFRSs.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DHFRS | Dual hesitant fuzzy rough set |

HFS | Hesiant fuzzy set |

HFRS | Hesitant fuzzy rough set |

DHFE | Dual hesitant fuzzy element |

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**MDPI and ACS Style**

Zhang, F.; Chen, J.; Zhu, Y.; Li, J.; Li, Q.; Zhuang, Z.
A Dual Hesitant Fuzzy Rough Pattern Recognition Approach Based on Deviation Theories and Its Application in Urban Traffic Modes Recognition. *Symmetry* **2017**, *9*, 262.
https://doi.org/10.3390/sym9110262

**AMA Style**

Zhang F, Chen J, Zhu Y, Li J, Li Q, Zhuang Z.
A Dual Hesitant Fuzzy Rough Pattern Recognition Approach Based on Deviation Theories and Its Application in Urban Traffic Modes Recognition. *Symmetry*. 2017; 9(11):262.
https://doi.org/10.3390/sym9110262

**Chicago/Turabian Style**

Zhang, Fangwei, Jihong Chen, Yuhua Zhu, Jiaru Li, Qiang Li, and Ziyi Zhuang.
2017. "A Dual Hesitant Fuzzy Rough Pattern Recognition Approach Based on Deviation Theories and Its Application in Urban Traffic Modes Recognition" *Symmetry* 9, no. 11: 262.
https://doi.org/10.3390/sym9110262