# Single-Valued Neutrosophic Clustering Algorithm Based on Tsallis Entropy Maximization

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Fuzzy c-Means

#### 2.2. Intuitionistic Fuzzy Clustering

#### 2.3. Picture Fuzzy Clustering

## 3. The Proposed Model and Solutions

**Definition**

**1.**

**Definition**

**2.**

- The first term of the objection function (18) describes the weighted distance sum of each data point ${X}_{i}$ to the cluster center ${V}_{j}$. ${\mu}_{ij}$ being from the positive aspect and $(4-{\xi}_{ij}-{\gamma}_{ij})$ (four is selected in order to guarantee ${\mu}_{ij}\in [0,1]$ in the iterative calculation) from the negative aspect, denoting the membership degree for ${X}_{i}$ to ${V}_{j}$, we use ${\mu}_{ij}(4-{\xi}_{ij}-{\gamma}_{ij})$ to represent the “integrated true” membership of the i-th data point in the j-th cluster. From the maximum entropy principle, the best to represent the current state of knowledge is the one with largest entropy, so the second term of the objection function (18) describes the negative Tsallis entropy of $\mu (4-\gamma -\xi )$, which means that the minimization of (18) is the maximum Tsallis entropy. $\rho $ is the regularization parameter. If $\gamma =\eta =\xi =0$, the proposed model returns the FCM model.
- Formulary (19) guarantees the definition of the SVNS (Definition 1).
- Formulary (20) implies that the “integrated true” membership of a data point ${X}_{i}$ to the cluster center ${V}_{j}$ satisfies the sum-row constraint of memberships. For convenience, we set ${T}_{ij}={\mu}_{ij}(4-{\xi}_{ij}-{\gamma}_{ij})$, and ${X}_{i}$ belongs to class ${C}_{l}$ if ${T}_{il}=max({T}_{i1},{T}_{i2},\cdots ,{T}_{ik})$.
- Equation (21) guarantees the working of the SVNS since at least one of two uncertain factors, namely indeterminacy membership degree and hesitate membership degree, always exists in the model.

**Theorem**

**1.**

**Proof.**

Algorithm 1: SVNC-TEM | |

Input: Dataset $D=\{{X}_{1},{X}_{2},\cdots ,{X}_{n}\}$ (n elements, d dimensions), number of clusters k, maximal number of iterations (Max-Iter), parameters: $m,\u03f5,\rho $ | |

Output: Cluster result | |

1: | $t=0$; |

2: | Initialize $\mu ,\gamma ,\xi ,$ satisfies Constraints (19) and (20); |

3: | Repeat |

4: | $t=t+1$; |

5: | Update ${V}_{j}^{(t)},(j=1,2,\cdots ,k)$ using Equation (22); |

6: | Update ${\mu}_{ij}^{(t)},(i=1,2,\cdots ,n,j=1,2,\cdots ,k)$ using Equation (23); |

7: | Update ${\gamma}_{ij}^{(t)},(i=1,2,\cdots ,n,j=1,2,\cdots ,k)$ using Equation (24); |

8: | Update ${\eta}_{ij}^{(t)},(i=1,2,\cdots ,n,j=1,2,\cdots ,k)$ using Equation (25); |

9: | Update ${\xi}_{ij}^{(t)},(i=1,2,\cdots ,n,j=1,2,\cdots ,k)$ using Equation (26); |

10: | Update ${T}_{ij}^{(t)}={\mu}_{ij}^{(t)}(4-{\gamma}_{ij}^{(t)}-{\xi}_{ij}^{(t)}),(i=1,2,\cdots ,n,j=1,2,\cdots ,k)$; |

11: | Update ${J}^{(t)}$ using Equation (18); |

12: | Until $|{J}^{(t)}-{J}^{(t-1)}|<\u03f5$ or Max-Iter is reached. |

13: | Assign ${X}_{i}(i=1,2,\cdots ,n)$ to the l-th class if ${T}_{il}=max({T}_{i1},{T}_{i2},\cdots ,{T}_{ik})$. |

## 4. Experimental Results

#### 4.1. Artificial Data to Cluster by the SVNC-TEM Algorithm

#### 4.2. Image Segmentation by the SVNC-TEM Algorithm

#### 4.3. Comparison Analysis Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The demonstration figure of the clustering process for artificial data. (

**a**) The original data. (

**b**–

**e**) The clustering figures when the number of iterations $t=$ 1, 5, 10, 20, respectively. (

**f**) The final clustering result.

**Figure 2.**The image segmentation for the Lena image. (

**a**) The original Lena image. (

**b**–

**f**) The clustering images when the number of clusters $k=$ 2, 5, 8, 11 and 20, respectively.

Dataset | No. of Elements | No. of Attributes | No. of Classes | Elements in Each Classes |
---|---|---|---|---|

IRIS | 150 | 4 | 3 | [50, 50, 50] |

CMC | 1473 | 9 | 3 | [629, 333, 511] |

GLASS | 214 | 9 | 6 | [29, 76, 70, 17, 13, 9] |

BALANCE | 625 | 4 | 3 | [49, 288, 288] |

BREAST | 277 | 9 | 2 | [81, 196] |

Dataset | k-Means | FCM | IFC | FC-PFS | SVNC-TEM |
---|---|---|---|---|---|

IRIS | 0.8803 | 0.8933 | 0.9000 | 0.8933 | 0.9000 |

CMC | 0.3965 | 0.3917 | 0.3958 | 0.3917 | 0.3985 |

GLASS | 0.3219 | 0.2570 | 0.3636 | 0.2935 | 0.3681 |

BALANCE | 0.5300 | 0.5260 | 0.5413 | 0.5206 | 0.5149 |

BREAST | 0.6676 | 0.5765 | 0.6595 | 0.6585 | 0.6686 |

Dataset | k-Means | FCM | IFC | FC-PFS | SVNC-TEM |
---|---|---|---|---|---|

IRIS | 0.7514 | 0.7496 | 0.7102 | 0.7501 | 0.7578 |

CMC | 0.0320 | 0.0330 | 0.0322 | 0.0334 | 0.0266 |

GLASS | 0.0488 | 0.0387 | 0.0673 | 0.0419 | 0.0682 |

BALANCE | 0.1356 | 0.1336 | 0.1232 | 0.1213 | 0.1437 |

BREAST | 0.0623 | 0.0309 | 0.0285 | 0.0610 | 0.0797 |

Dataset | k-Means | FCM | IFC | FC-PFS | SVNC-TEM |
---|---|---|---|---|---|

IRIS | 0.8733 | 0.8797 | 0.8827 | 0.8797 | 0.8859 |

CMC | 0.5576 | 0.5582 | 0.5589 | 0.5582 | 0.5605 |

GLASS | 0.5373 | 0.6294 | 0.4617 | 0.5874 | 0.4590 |

BALANCE | 0.5940 | 0.5928 | 0.5899 | 0.5904 | 0.5999 |

BREAST | 0.5708 | 0.5159 | 0.5732 | 0.5656 | 0.5567 |

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**MDPI and ACS Style**

Li, Q.; Ma, Y.; Smarandache, F.; Zhu, S.
Single-Valued Neutrosophic Clustering Algorithm Based on Tsallis Entropy Maximization. *Axioms* **2018**, *7*, 57.
https://doi.org/10.3390/axioms7030057

**AMA Style**

Li Q, Ma Y, Smarandache F, Zhu S.
Single-Valued Neutrosophic Clustering Algorithm Based on Tsallis Entropy Maximization. *Axioms*. 2018; 7(3):57.
https://doi.org/10.3390/axioms7030057

**Chicago/Turabian Style**

Li, Qiaoyan, Yingcang Ma, Florentin Smarandache, and Shuangwu Zhu.
2018. "Single-Valued Neutrosophic Clustering Algorithm Based on Tsallis Entropy Maximization" *Axioms* 7, no. 3: 57.
https://doi.org/10.3390/axioms7030057