# Neutrosophic Weighted Support Vector Machines for the Determination of School Administrators Who Attended an Action Learning Course Based on Their Conflict-Handling Styles

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Proposed Neutrosophic Set Support Vector Machines (NS-SVM)

#### 3.1. Support Vector Machine (SVM)

_{i}is a multidimensional feature vector and ${y}_{i}\in \left\{-1,1\right\}$ is the corresponding label, an SVM models a decision boundary between classes of training data as a separating hyperplane. SVM aims to find an optimal solution by maximizing the margin around the separating hyperplane, which is equivalent to minimizing $\left|\left|w\right|\right|$ with the constraint:

#### 3.2. Neutrosophic c-Means Clustering

_{i}is represented as $\left\{\mathrm{T}\left({A}_{i}\right),\text{}\mathrm{I}\left({A}_{i}\right),\mathrm{F}\left({A}_{i}\right)\right\}/{A}_{i}$, where $\mathrm{T}\left({A}_{i}\right)$, $\mathrm{I}\left({A}_{i}\right)$ and $\mathrm{F}\left({A}_{i}\right)$ are the membership values to the true, indeterminate, and false sets. $\mathrm{T}\left({A}_{i}\right)$ is used to measure the belonging degree of the sample to the center of the labeled class, $\mathrm{I}\left({A}_{i}\right)$ for indiscrimination degree between two classes, and $\mathrm{F}\left({A}_{i}\right)$ for the belonging degree to the outliers.

_{ij}. ${\varpi}_{1}$, ${\varpi}_{2}$, and ${\varpi}_{3}$ are constant weights. ${T}_{ij}$ and ${I}_{i}$ are updated at each iteration until $\left|{T}_{ij}^{\left(k+1\right)}-{T}_{ij}^{\left(k\right)}\right|<\epsilon $, where $\epsilon $ is a termination criterion.

#### 3.3. Proposed Neutrosophic Set Support Vector Machine (NS-SVM)

**Input**: Labeled training dataset.**Output**: Predicted class labels.**Step 1**: Calculate the cluster centers according to the labeled dataset and employ NCM algorithm to determine NS memberships T and I for each data point.**Step 2**: Calculate ${g}_{Ni}$ by using T and I components according to Equation (8).**Step 3**: Optimize NS-SVM by minimizing the cost function according to Equation (9).**Step 4**: Calculate the labels of test data.

## 4. Experimental Work and Results

^{−3}, 10

^{2}] at a step size of 10

^{−1}. In addition, for NCM the following parameters are chosen: ε= 10

^{−3}, ${\varpi}_{1}$= 0.75, ${\varpi}_{2}$= 0.125, ${\varpi}_{3}$= 0.125, which were obtained from trial and error. The $\delta $ parameter of NCM method is also searched in the range of $\left\{{2}^{-10},{2}^{-8},\dots ,{2}^{8},{2}^{10}\right\}$. The dataset is normalized with zero mean and unit variance. Table 1 shows the obtained accuracy scores for the first scenario. The obtained results are further compared with FSVM and other SVM types such as Linear, Quadratic, Cubic, Fine Gaussian, Medium Gaussian, and Coarse Gaussian SVMs.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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**Table 1.**Prediction accuracies for the first scenario. The bold case shows the highest accuracy. SVM: Support Vector Machines; FSVM: Fuzzy Support Vector Machines; NS-SVM: Neutrosophic Support Vector Machines.

Classifier Type | Accuracy (%) |
---|---|

Linear SVM | 73.7 |

Quadratic SVM | 68.4 |

Cubic SVM | 68.4 |

Fine Gaussian SVM | 48.7 |

Medium Gaussian SVM | 73.7 |

Coarse Gaussian SVM | 63.2 |

FSVM | 76.9 |

NS-SVM | 81.2 |

**Table 2.**Prediction accuracies for the second scenario. The integrating dimension is used as input. The bold case shows the highest accuracy.

Classifier Type | Accuracy (%) |
---|---|

Linear SVM | 73.7 |

Quadratic SVM | 57.9 |

Cubic SVM | 53.9 |

Fine Gaussian SVM | 60.5 |

Medium Gaussian SVM | 73.7 |

Coarse Gaussian SVM | 67.1 |

FSVM | 76.3 |

NS-SVM | 80.3 |

**Table 3.**Prediction accuracies for the second scenario. The obliging dimension is used as input. The bold case shows the highest accuracy.

Classifier Type | Accuracy (%) |
---|---|

Linear SVM | 61.8 |

Quadratic SVM | 50.0 |

Cubic SVM | 51.3 |

Fine Gaussian SVM | 52.6 |

Medium Gaussian SVM | 61.8 |

Coarse Gaussian SVM | 55.3 |

FSVM | 71.3 |

NS-SVM | 73.8 |

**Table 4.**Prediction accuracies for the second scenario. The dominating dimension is used as input. The bold case shows the highest accuracy.

Classifier Type | Accuracy (%) |
---|---|

Linear SVM | 59.2 |

Quadratic SVM | 57.9 |

Cubic SVM | 52.6 |

Fine Gaussian SVM | 55.3 |

Medium Gaussian SVM | 52.6 |

Coarse Gaussian SVM | 55.3 |

FSVM | 65.0 |

NS-SVM | 70.0 |

**Table 5.**Prediction accuracies for the second scenario. The avoiding dimension is used as input. The bold case shows the highest accuracy.

Classifier Type | Accuracy (%) |
---|---|

Linear SVM | 50.0 |

Quadratic SVM | 43.4 |

Cubic SVM | 53.9 |

Fine Gaussian SVM | 48.7 |

Medium Gaussian SVM | 44.7 |

Coarse Gaussian SVM | 42.1 |

FSVM | 63.8 |

NS-SVM | 66.3 |

**Table 6.**Prediction accuracies for the second scenario. The compromising dimension is used as input. The bold case shows the highest accuracy.

Classifier Type | Accuracy (%) |
---|---|

Linear SVM | 67.1 |

Quadratic SVM | 67.1 |

Cubic SVM | 57.9 |

Fine Gaussian SVM | 65.8 |

Medium Gaussian SVM | 71.1 |

Coarse Gaussian SVM | 68.4 |

FSVM | 73.8 |

NS-SVM | 75.0 |

**Table 7.**Calculated f-measure and running times for the first scenario. The bold cases show the better achievements.

Classifier Type | f-Measure (%) | Time (s) |
---|---|---|

Linear SVM | 73.50 | 0.314 |

Quadratic SVM | 68.50 | 0.129 |

Cubic SVM | 68.50 | 0.122 |

Fine Gaussian SVM | 48.50 | 0.119 |

Medium Gaussian SVM | 71.00 | 0.130 |

Coarse Gaussian SVM | 61.00 | 0.129 |

FSVM | 76.50 | 0.089 |

NS-SVM | 80.00 | 0.065 |

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## Share and Cite

**MDPI and ACS Style**

Turhan, M.; Şengür, D.; Karabatak, S.; Guo, Y.; Smarandache, F.
Neutrosophic Weighted Support Vector Machines for the Determination of School Administrators Who Attended an Action Learning Course Based on Their Conflict-Handling Styles. *Symmetry* **2018**, *10*, 176.
https://doi.org/10.3390/sym10050176

**AMA Style**

Turhan M, Şengür D, Karabatak S, Guo Y, Smarandache F.
Neutrosophic Weighted Support Vector Machines for the Determination of School Administrators Who Attended an Action Learning Course Based on Their Conflict-Handling Styles. *Symmetry*. 2018; 10(5):176.
https://doi.org/10.3390/sym10050176

**Chicago/Turabian Style**

Turhan, Muhammed, Dönüş Şengür, Songül Karabatak, Yanhui Guo, and Florentin Smarandache.
2018. "Neutrosophic Weighted Support Vector Machines for the Determination of School Administrators Who Attended an Action Learning Course Based on Their Conflict-Handling Styles" *Symmetry* 10, no. 5: 176.
https://doi.org/10.3390/sym10050176