# Feature Selection of Grey Wolf Optimizer Based on Quantum Computing and Uncertain Symmetry Rough Set

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis of Combinatorial Optimization Feature Selection Algorithm

#### 2.1. Uncertain Symmetry Rough Set

_{T}(M), the positive region of partition Z/M relative to T, is a collection of all objects of Z, which can determine partition Z/M by T. M depends on T, and we can use T → kM to represent k (0 ≤ k ≤ 1) in P:

#### 2.2. Quantum Computing

^{2}gives the probability of ‘0′state, |α|

^{2}gives the probability of ‘1′state. The absolute square of the amplitude is the probability of measuring the qubit in the “0” or “1” state, and quantum computing always maintain the conservation of probability. This relationship equation is given as follows:

#### 2.3. Grey Wolf Optimizer

## 3. Grey Wolf Optimizer Based on Quantum Computing and Rough Set

Algorithm 1. QCGWORS process |

Input: An extended Information System: FOutput: optimal feature subset${R}_{min}$1. Initialize n quantum grey wolf individuals $QG{W}_{0}$ using (34); 2. Get the group $BG{W}_{0}$ of n binary grey wolves from $QG{W}_{0}\mathrm{by}\text{}\mathrm{the}\text{}\mathrm{Algorithm}\text{}2.$ 3. Search the minimal feature subset ${R}_{x}$ of each binary Wolf ${\mathrm{BGW}}_{\mathrm{x}}$ by the Algorithm 3; 4. ${\mathrm{Evaluate}\text{}\mathrm{each}\text{}R}_{x}$ corresponding of binary wolf $BG{W}_{x}$ using (26); 5. while (t < maximum iterations)6. for i=1:q (all q binary grey wolf individuals) do7. Evaluate the feature subset ${R}_{x}\text{}\mathrm{using}\text{}\left(26\right)$; 8. Update the best feature subset ${\mathrm{Red}}_{\mathrm{min}}\text{}\mathrm{using}\text{}\left(13\right)$; 9. end for10. end while11. return ${\mathrm{Red}}_{\mathrm{min}}$end while12. return ${\mathrm{Red}}_{\mathrm{min}}$ |

#### 3.1. Rough Set Evaluation Function

_{1}, S

_{2}and the decision feature D and, finally, calculates the dependency of conditional feature subset and decision feature. The dependencies between the decision feature set and S

_{1}and S

_{2}are as follows:

#### 3.2. Quantum Representation of Grey Wolf Individual

_{x}(the xth quantum grey wolf in a quantum group) corresponds to a vector ${\mathsf{\Theta}}_{x}$.${\mathsf{\Theta}}_{x}=({\theta}_{x1},\dots ,{\theta}_{xm})$ of variables ${\theta}_{\mathrm{xy}}$,with ${\theta}_{\mathrm{xy}}$∈ [0, $\frac{\mathsf{\pi}}{2}$] for (1 ≤ y ≤ m). Each quantum grey wolf solution ${\mathrm{QGW}}_{\mathrm{x}}$ is a string of qubits, calculated as follows:

#### 3.3. Quantum Computing in Feature Selection

#### 3.4. Quantum Measurement in the Proposed Algorithm

_{x}) is used to generate a binary grey wolf (BGW

_{x}) solution by qubits’ projection. For a quantum bit, a random number r is generated from the interval [0, 1]. When r > sin

^{2}(${\mathsf{\theta}}_{\mathrm{x},\mathrm{y}}$), it sets to 1 to select the corresponding conditional feature; otherwise, the value is 0 and reject the corresponding conditional feature. Therefore, due to the superposition state of the qubits, a quantum superposition solution contains many binary solutions [37]. However, each qubit determines the probability of selecting or rejecting the corresponding feature. In the quantum measurement step, only certain binary solutions are extracted from quantum solutions, and the selection is guided by the probability of quantized coding.

^{2}(${\mathsf{\theta}}_{x,y}$), the feature is selected to refer to a condition, and the quantum measurement operation is repeated until all features are searched. The following algorithm 2 constructs the feasible solution $BG{W}_{x}\text{}$of feature selection by observing the $QG{W}_{x}\text{}$of the quantum grey wolf. The process of algorithm 2 is as follows.

Algorithm 2. Quantum measurement in the proposed algorithm |

Input:${QGW}_{x}$: Quantum Grey individual, C= {${c}_{1}{,\text{}c}_{2}{,\text{\u2026},c}_{\mathrm{x}}$}: Conditional feature setOutput:$BG{W}_{x}$: Binary Grey Wolf Individual, ${R}_{x}$: Feature Subset1. ${R}_{x}$←$\varphi $ 2. for each qubit y of $QG{W}_{x}$ do3. real value r is generated between [0, 1]; 4. if r > ${sin}^{2}{(\theta}_{x,y})$ then5. ${BGW}_{x,y}\text{}$← 1; 6. R ←${\text{}R}_{x\text{}}$∪${\text{}c}_{x,y}$; 7. else8. ${BGW}_{x,y}\text{}$← 0; 9. end if10. end for11. return ${BGW}_{x}$ |

#### 3.5. Update Position of Binary Grey Wolves

Algorithm 3. Process of updating the binary wolves’ position |

Input: I: Information System Output: minimum condition feature subset ${\mathrm{R}}_{\mathrm{min}}$ 1. Calculate the fitness of $BG{W}_{0}$ 2. Initialize search wolf for Alpha, Beta, and Delta. 3. Initialize parameters a, A and C. 4. while (t <Max iterations)5. for each Omega wolf6. calculate the fitness function (26) value of the $BG{W}_{x\text{}}$ 7. for each search wolf8. if there is a search wolf(Alpha, Beta, Delta) position that needs to be replaced9. Update parameters a, A and C. 10. Update the current search wolf position 11. end if12. end For13. end For14. end while15. return Alpha wolf, ${R}_{min}$ |

## 4. Experiments

#### 4.1. Experimental Setup

#### 4.1.1. Classical Part Preparation

#### 4.1.2. Quantum Part Preparation

- It can initialize the qubits and rearrange the qubits if needed;
- It can easily construct the superposition state of qubits;
- It can give abstraction to the quantum operator to easily format the quantum, such as NOT gate, quantum rotation gate, Hadamard gate, and other gate operation;
- It can simplify quantum measurement operation to obtain the definite state of the qubit;
- It can randomly generate correlation matrices, including unitary matrices.

#### 4.2. Analysis of Experimental Data

#### 4.2.1. QCGWORS Improved Classification Accuracy Experiment

#### 4.2.2. Rough Set Evaluation and Comparison Experiment

#### 4.2.3. Swarm Intelligence Algorithm Comparison Experiment

#### 4.2.4. Experiment on the Effect of Quantum Part in Feature Selection

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- References Davies, S.; Russell, S. NP-completeness of searches for smallest possible feature sets. In Proceedings of the AAAI Symposium on Intelligent Relevance, Berkeley, CA, USA, 4–6 February 1994; pp. 37–39. [Google Scholar]
- Gheyas, I.; Smith, L. Feature subset selection in large dimensionality domains. Pattern Recognit.
**2010**, 43, 5–13. [Google Scholar] [CrossRef] - Vergara, J.; Estévez, P. A review of feature selection methods based on mutual information. Neural Comput. Appl.
**2014**, 24, 175–186. [Google Scholar] - Mao, Y.; Zhou, X.B.; Xia, Z.; Yin, Z.; Sun, Y.X. Survey for study of feature selection algorithms. Pattern Recognit. Artif. Intell.
**2007**, 20, 211–218. [Google Scholar] - Gan, J.; Hasan, B.; Tsui, C. A filter-dominating hybrid sequential forward floating search method for feature subset selection in high-dimensional space. Int. J. Mach. Learn. Cybern.
**2014**, 5, 413–423. [Google Scholar] [CrossRef] - Kohavi, R.; John, G.H. Wrappers for Feature Subset Selection. Artif. Intell.
**1997**, 97, 273–324. [Google Scholar] [CrossRef] - Pawlak, Z.; Skowron, A. Rough sets: Some extensions. Inf. Sci.
**2007**, 177, 28–40. [Google Scholar] [CrossRef] - Pawlak, Z.; Skowron, A. Rudiments of rough sets. Inf. Sci.
**2007**, 177, 3–27. [Google Scholar] [CrossRef] - Kryszkiewicz, M. Rough set approach to incomplete information systems. Inf. Sci.
**1998**, 112, 39–49. [Google Scholar] [CrossRef] - Mi, J.S.; Wu, W.Z.; Zhang, W.X. Approaches to knowledge reduction based on variable precision rough set model. Inf. Sci.
**2004**, 159, 255–272. [Google Scholar] [CrossRef] - Jinming, Q.; Kaiquati, S. F-rough law and the discovery of rough law. J. Syst. Eng. Electron.
**2009**, 20, 81–89. [Google Scholar] - Hu, Q.; Yu, D.; Liu, J.; Wu, C. Neighborhood rough set based heterogeneous feature subset selection. Inf. Sci.
**2008**, 178, 3577–3594. [Google Scholar] [CrossRef] - Stefanowski, J.; Tsoukias, A. Incomplete information tables and rough classification. Comput. Intell.
**2001**, 17, 545–566. [Google Scholar] [CrossRef] - Qian, Y.; Liang, J.; Li, D.; Wang, F.; Ma, N. Approximation reduction in inconsistent incomplete decision tables. Knowl. Based Syst.
**2010**, 23, 427–433. [Google Scholar] [CrossRef] - Yang, X.; Chen, Z.; Dou, H.; Zhang, M.; Yang, J. Neighborhood system based rough set: Models and attribute reductions. Int. J. Uncertain. Fuzziness Knowl. Based Syst.
**2012**, 20, 399–419. [Google Scholar] [CrossRef] - Degang, C.; Changzhong, W.; Qinghua, H. A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf. Sci.
**2007**, 177, 3500–3518. [Google Scholar] [CrossRef] - Teng, S.H.; ZHOU, S.L.; SUN, J.X.; Li, Z.Y. Attribute reduction algorithm based on conditional entropy under incomplete information system. J. Natl. Univ. Def. Technol.
**2010**, 32, 90–94. [Google Scholar] - Ren, Y.G.; Wang, Y.; Yan, D.Q. Rough Set Attribute Reduction Algorithm Based on GA. Comput. Eng. Sci.
**2006**, 47, 134–136. [Google Scholar] - Long, N.C.; Meesad, P.; Unger, H. Attribute reduction based on rough sets and the discrete firefly algorithm. In Recent Advances in Information and Communication Technology; Springer: Berlin, Germany, 2014; pp. 13–22. [Google Scholar]
- Wang, X.; Yang, J.; Teng, X.; Xia, W.; Jensen, R. Feature selection based on rough sets and particle swarm optimization. Pattern Recognit. Lett.
**2007**, 28, 459–471. [Google Scholar] [CrossRef] - Inbarani, H.H.; Azar, A.T.; Jothi, G. Supervised hybrid feature selection based on PSO and rough sets for medical diagnosis. Comput. Methods Programs Biomed.
**2014**, 113, 175–185. [Google Scholar] [CrossRef] - Bae, C.; Yeh, W.C.; Chung, Y.Y.; Liu, S.L. Feature selection with intelligent dynamic swarm and rough set. Expert Syst. Appl.
**2010**, 37, 7026–7032. [Google Scholar] [CrossRef] - Chen, Y.; Miao, D.; Wang, R. A rough set approach to feature selection based on ant colony optimization. Pattern Recognit. Lett.
**2010**, 31, 226–233. [Google Scholar] [CrossRef] - Jensen, R.; Shen, Q. Fuzzy-rough data reduction with ant colony optimization. Fuzzy Sets Syst.
**2005**, 149, 5–20. [Google Scholar] [CrossRef] - Ke, L.; Feng, Z.; Ren, Z. An efficient ant colony optimization approach to attribute reduction in rough set theory. Pattern Recognit. Lett.
**2008**, 29, 1351–1357. [Google Scholar] [CrossRef] - Chen, Y.; Zhu, Q.; Xu, H. Finding rough set reducts with fish swarm algorithm. Knowl. Based Syst.
**2015**, 81, 22–29. [Google Scholar] [CrossRef] - Luan, X.Y.; Li, Z.P.; Liu, T.Z. A novel attribute reduction algorithm based on rough set and improved artificial fish swarm algorithm. Neurocomputing
**2016**, 174, 522–529. [Google Scholar] [CrossRef] - Yamany, W.; Emary, E.; Hassanien, A.E.; Schaefer, G.; Zhu, S.Y. An innovative approach for attribute reduction using rough sets and flower pollination optimisation. Procedia Comput. Sci.
**2016**, 96, 403–409. [Google Scholar] [CrossRef] - Chen, Y.; Zeng, Z.; Lu, J. Neighborhood rough set reduction with fish swarm algorithm. Soft Comput.
**2017**, 21, 6907–6918. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] - Schuld, M.; Sinayskiy, I.; Petruccione, F. An introduction to quantum machine learning. Contemp. Phys.
**2015**, 56, 172–185. [Google Scholar] [CrossRef] - Yu, L.; Liu, H. Efficient feature selection via analysis of relevance and redundancy. J. Mach. Learn. Res.
**2004**, 5, 1205–1224. [Google Scholar] - Swiniarski, R.W.; Skowron, A. Rough set methods in feature selection and recognition. Pattern Recognit. Lett.
**2003**, 24, 833–849. [Google Scholar] [CrossRef] - Benioff, P. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J. Stat. Phys.
**1980**, 22, 563–591. [Google Scholar] [CrossRef] - Nielsen, M.A.; Chuang, I. Quantum computation and quantum information. Am. J. Phys.
**2002**, 70, 558–694. [Google Scholar] [CrossRef] - Manju, A.; Nigam, M.J. Applications of quantum inspired computational intelligence: A survey. Artif. Intell. Rev.
**2014**, 42, 79–156. [Google Scholar] [CrossRef] - Zouache, D.; Nouioua, F.; Moussaoui, A. Quantum-inspired firefly algorithm with particle swarm optimization for discrete optimization problems. Soft Comput.
**2016**, 20, 2781–2799. [Google Scholar] [CrossRef] - Jensen, R.; Shen, Q. Finding Rough Set Reducts with Ant Colony Optimization. J. Fussy Sets Sys.
**2003**, 49, 15–22. [Google Scholar] - Wang, G.; Yu, H.; Yang, D.C. Decision table reduction based on conditional information entropy. Chin. J. Comput. Chin. Ed.
**2002**, 25, 759–766. [Google Scholar] - Tharwat, A.; Gabel, T.; Hassanien, A.E. Classification of toxicity effects of biotransformed hepatic drugs using whale optimized support vector machines. J. Biomed. Inform.
**2017**, 68, 132–149. [Google Scholar] [CrossRef] - Tóth, G. QUBIT4MATLAB V3. 0: A program package for quantum information science and quantum optics for MATLAB. Comput. Phys. Commun.
**2008**, 179, 430–437. [Google Scholar]

${\mathbf{z}}_{\mathbf{i}}\in \mathbf{Z}$ | u | v | w | d |
---|---|---|---|---|

${\mathrm{z}}_{1}$ | 0 | 1 | 1 | 0 |

${\mathrm{z}}_{2}$ | 1 | 1 | 1 | 0 |

${\mathrm{z}}_{3}$ | 1 | 0 | 0 | 1 |

${\mathrm{z}}_{4}$ | 0 | 0 | 0 | 0 |

${\mathrm{z}}_{5}$ | 1 | 0 | 1 | 0 |

${\mathrm{z}}_{6}$ | 0 | 0 | 1 | 1 |

${\mathrm{z}}_{7}$ | 1 | 1 | 0 | 0 |

${\mathrm{z}}_{8}$ | 0 | 0 | 0 | 0 |

No. | Dataset | Features | Samples |
---|---|---|---|

1 | Led | 24 | 2000 |

2 | Exactly | 13 | 1000 |

3 | Exactly2 | 13 | 1000 |

4 | DNA | 57 | 318 |

5 | KC1 | 22 | 2109 |

6 | KC2 | 21 | 522 |

7 | Lung | 56 | 32 |

8 | Vote | 16 | 300 |

9 | Zoo | 16 | 101 |

10 | Lymphography | 18 | 148 |

11 | Mushroom | 22 | 8124 |

12 | Soybean-small | 35 | 47 |

13 | Breast cancer | 9 | 699 |

Dataset | ORIGINAL | QCGWORS | ||||
---|---|---|---|---|---|---|

Features | RF | KNN | Features | RF | KNN | |

Led | 24 | 98.20 ± 0.50 | 77.99 ± 0.8 | 5 | 99.57 ± 0.32 | 99.31 ± 0.25 |

Exactly | 13 | 72.53 ± 2.50 | 72.14 ± 0.7 | 6 | 92.23 ± 0.90 | 93.05 ± 0.30 |

Exactly2 | 13 | 66.51 ± 1.50 | 65.38 ± 0.5 | 10 | 72.56 ± 0.05 | 79.63 ± 0.7 |

DNA | 57 | 37.96 ± 213 | 77.80 ± 0.9 | 5 | 36.25 ± 0.23 | 34.09 ± 0.10 |

KC1 | 22 | 59.23 ± 2.51 | 61.14 ± 2.92 | 8 | 70.42 ± 0.16 | 73.42 ± 1.37 |

KC2 | 21 | 62.25 ± 2.17 | 64.25 ± 1.92 | 5 | 76.61 ± 0.23 | 78.61 ± 0.81 |

Lung | 56 | 80.78 ± 0.27 | 63.28 ± 2.64 | 4 | 86.35 ± 0.16 | 85.53 ± 0.74 |

Vote | 16 | 93.08 ± 0.63 | 91.62 ± 0.65 | 8 | 94.12 ± 0.69 | 92.88 ± 0.42 |

Zoo | 16 | 88.74 ± 1.03 | 96.34 ± 0.43 | 5 | 96.06 ± 0.14 | 94.21 ± 0.40 |

Lymphography | 18 | 78.38 ± 2.20 | 79.07 ± 1.52 | 7 | 83.49 ± 2.01 | 78.75 ± 1.22 |

Mushroom | 22 | 99.00 ± 1.07 | 99.00 ± 0.10 | 4 | 99.03 ± 0.10 | 99.71 ± 0.10 |

Soybean-small | 35 | 98.47 ± 2.53 | 99.00 ± 0.10 | 2 | 99.00 ± 0.10 | 99.00 ± 0.10 |

Breast cancer | 9 | 93.44 ± 6.56 | 94.94 ± 0.51 | 4 | 94.65 ± 0.73 | 96.59 ± 0.34 |

Average | 25 | 79.12 | 77.80 | 6 | 85.17 | 85.59 |

Dataset | ‘RSAR’ | ‘RSAR-Entropy’ | ‘QCGWORS’ |
---|---|---|---|

Led | 6,1,2,4,3,5 | 6,11,24,19,22,8,18,21,9,16,7,1 | 1,2,3,4,5 |

Exactly | 1,2,3,4,5,11,7,9 | 3,5,7,1,4,8,9,11 | 1,3,5,7,9,11 |

Exactly2 | 1,2,3,4,10,9,6,8,7,5 | 2,3,8,6,13,12,5,10,11,4,7 | 1,2,3,4,5,6,7,8,9,10 |

DNA | 1,16,45,24,57,2,3 | 18,42,14,49,9,25 | 5,19,22,26,33 |

KC1 | 4,2,5,8,9,7,10,11,1,6 | 1,5,2,7,3,11,4,21,15,18 | 2,4,5,6,7,8,11,18 |

KC2 | 2,4,5,8,7,18,11,6 | 2,5,6,8,7,4,1,18,5 | 2,4,5,7,8 |

Lung | 1,42,7,4 | 3,9,4,36,13,15 | 3,9,24,42 |

Vote | 1,4,12,16,11,3,13,2,9 | 9,16,8,14,5,10,13,2,15,4,6 | 1,2,3,4,9,11,13,16 |

Zoo | 4,13,12,6,8 | 6,13,1,8,7,5,15,14,12,3 | 3,4,6,8,13 |

Lymphography | 2,18,14,13,16,15 | 1,18,14,5,12,11,16,2 | 8,6,2,13,14,18,15 |

Mushroom | 5,20,8,12,3 | 14,1,9,3,6 | 5,12,20,22 |

Soybean-small | 4,22 | 23,22 | 23,22 |

Breast cancer | 1,7,2,6 | 1,3,4,9 | 1,6,5,8 |

Dataset | Original | RSAR | RSAR-Entropy | QCGWORS |
---|---|---|---|---|

Led | 24 | 6 | 11 | 5 |

Exactly | 13 | 8 | 8 | 6 |

Exactly2 | 13 | 10 | 11 | 10 |

DNA | 57 | 7 | 6 | 5 |

KC1 | 22 | 11 | 10 | 8 |

KC2 | 21 | 7 | 9 | 5 |

Lung | 56 | 4 | 5 | 4 |

Vote | 16 | 9 | 11 | 8 |

Zoo | 16 | 5 | 10 | 5 |

Lymphography | 18 | 6 | 8 | 7 |

Mushroom | 22 | 5 | 5 | 4 |

Soybean-small | 35 | 2 | 2 | 2 |

Breast cancer | 24 | 6 | 11 | 5 |

Algorithm | Parameters |
---|---|

WOARSFS | Population size = 20, $\tau \text{}$= 0.9,$\text{}\mathsf{\mu}$ = 0.1, $\mathrm{a}\in \left(0,2\right),\text{}\mathrm{r}\in \left(0,1\right),\text{}\mathrm{b}=1$ |

FSARSR | Population size = 20, $\tau \text{}$= 0.9,$\text{}\mathsf{\mu}$ = 0.1 |

QCGWORS | Population size = 20, $\tau \text{}$= 0.9,$\text{}\mathsf{\mu}\text{}$= 0.1,$\text{}\mathsf{\theta}\text{}\mathrm{size}\text{}\mathrm{is}\text{}0.025\mathsf{\pi},$ $\mathrm{a}\in \left(0,2\right),\text{}\mathrm{r}\in \left(0,1\right)$ |

**Table 7.**The number of features of the potential best feature subsets found by different feature selection algorithms.

Dataset | Features | WOARSF | FSARSR | QCGWORS |
---|---|---|---|---|

Led | 24 | 6 | 5 | 5 |

Exactly | 13 | 7 | 6 | 6 |

Exactly2 | 13 | 11 | 10 | 10 |

DNA | 57 | 6 | 7 | 5 |

KC1 | 22 | 8 | 9 | 8 |

KC2 | 21 | 5 | 6 | 5 |

Lung | 56 | 4 | 4 | 4 |

Vote | 16 | 9 | 9 | 9 |

Zoo | 16 | 6 | 5 | 5 |

Lymphography | 18 | 7 | 8 | 7 |

Mushroom | 22 | 4 | 5 | 4 |

Soybean-small | 35 | 2 | 2 | 2 |

Breast cancer | 24 | 6 | 5 | 5 |

Dataset | Feature | GWORS | QGWORS | ||||
---|---|---|---|---|---|---|---|

RF | KNN | Feature | RF | KNN | Feature | ||

Lung | 56 | 84.03 ± 0.22 | 85.21 ± 0.81 | 4 | 86.35 ± 0.16 | 85.53 ± 0.74 | 4 |

DNA | 57 | 35.13 ± 0.20 | 33.78 ± 0.10 | 7 | 36.25 ± 0.23 | 34.09± 0.10 | 5 |

Vote | 16 | 91.41 ± 0.29 | 90.64 ± 0.45 | 8 | 93.12 ± 0.69 | 92.88 ± 0.42 | 8 |

Breast cancer | 24 | 92.21 ± 0.28 | 93.91 ± 0.57 | 5 | 94.65 ± 0.73 | 96.59 ± 0.34 | 4 |

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**MDPI and ACS Style**

Zhao, G.; Wang, H.; Jia, D.; Wang, Q.
Feature Selection of Grey Wolf Optimizer Based on Quantum Computing and Uncertain Symmetry Rough Set. *Symmetry* **2019**, *11*, 1470.
https://doi.org/10.3390/sym11121470

**AMA Style**

Zhao G, Wang H, Jia D, Wang Q.
Feature Selection of Grey Wolf Optimizer Based on Quantum Computing and Uncertain Symmetry Rough Set. *Symmetry*. 2019; 11(12):1470.
https://doi.org/10.3390/sym11121470

**Chicago/Turabian Style**

Zhao, Guobao, Haiying Wang, Deli Jia, and Quanbin Wang.
2019. "Feature Selection of Grey Wolf Optimizer Based on Quantum Computing and Uncertain Symmetry Rough Set" *Symmetry* 11, no. 12: 1470.
https://doi.org/10.3390/sym11121470