# Incremental Granular Model Improvement Using Particle Swarm Optimization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Methods

#### 2.1. Incremental Granular Model (IGM)

#### 2.1.1. Global Part: Linear Regression (LR)

#### 2.1.2. Local Part: Granular Model (GM)

#### 2.1.3. Context-Based Fuzzy C-Means (CFCM) Clustering

#### 2.1.4. Granular Model (GM)

#### 2.2. Particle Swarm Optimization-Based Incremental Granular Model (PSO-IGM)

#### 2.2.1. Particle Swarm Optimization (PSO)

#### 2.2.2. Particle Swarm Optimization-Based Incremental Granular Model (PSO-IGM)

## 3. Results

#### 3.1. Boston Housing Dataset

#### 3.2. Experimental Method

#### 3.3. Result Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Difference between fuzzy C-means clustering and context-based fuzzy C-means clustering: (

**a**) fuzzy C-means clustering; (

**b**) context-based fuzzy C-means clustering.

No. of Clusters/Fuzzification Coefficient (m = 1.5) | Training RMSE | Testing RMSE |
---|---|---|

2 | 4.26 | 4.34 |

3 | 3.83 | 4.46 |

4 | 3.68 | 4.84 |

5 | 3.57 | 4.36 |

6 | 3.47 | 4.43 |

7 | 3.44 | 4.12 |

8 | 3.24 | 4.46 |

9 | 3.36 | 4.40 |

10 | 3.30 | 4.33 |

11 | 3.25 | 4.49 |

12 | 3.20 | 4.56 |

13 | 3.36 | 4.65 |

14 | 3.35 | 4.80 |

15 | 3.54 | 4.75 |

16 | 3.69 | 4.51 |

17 | 3.59 | 4.62 |

18 | 3.96 | 4.55 |

19 | 3.94 | 4.50 |

20 | 4.00 | 4.56 |

No. of Clusters/Fuzzification Coefficient (m = 1.5) | Training RMSE | Testing RMSE |
---|---|---|

2 | 4.27 | 4.32 |

3 | 4.35 | 5.34 |

4 | 3.71 | 4.31 |

5 | 3.54 | 4.18 |

6 | 3.61 | 4.15 |

7 | 3.49 | 3.95 |

8 | 3.25 | 4.29 |

9 | 3.56 | 4.29 |

10 | 3.52 | 4.14 |

11 | 3.23 | 4.07 |

12 | 3.69 | 4.23 |

13 | 3.76 | 4.20 |

14 | 3.34 | 4.18 |

15 | 3.68 | 4.22 |

16 | 3.84 | 4.26 |

17 | 3.71 | 4.27 |

18 | 4.12 | 4.39 |

19 | 4.34 | 4.48 |

20 | 4.61 | 4.77 |

No. of Clusters/Fuzzification Coefficient (m = 1.5) | Training RMSE | Testing RMSE |
---|---|---|

2 | 4.28 | 4.27 |

3 | 4.21 | 4.96 |

4 | 3.62 | 4.05 |

5 | 3.55 | 4.10 |

6 | 3.70 | 3.96 |

7 | 3.60 | 3.94 |

8 | 3.39 | 3.76 |

9 | 3.11 | 3.74 |

10 | 3.27 | 3.75 |

11 | 3.45 | 3.88 |

12 | 3.11 | 3.91 |

13 | 3.28 | 3.98 |

14 | 3.54 | 3.98 |

15 | 3.76 | 4.19 |

16 | 3.77 | 4.13 |

17 | 4.00 | 4.15 |

18 | 4.22 | 4.20 |

19 | 4.47 | 4.18 |

20 | 4.49 | 4.13 |

No. of Clusters/Fuzzification Coefficient (m = 1.5) | Training RMSE | Testing RMSE |
---|---|---|

2 | 4.28 | 4.26 |

3 | 4.04 | 4.24 |

4 | 3.60 | 4.07 |

5 | 3.63 | 4.08 |

6 | 3.21 | 3.91 |

7 | 3.56 | 3.83 |

8 | 3.96 | 3.83 |

9 | 3.19 | 3.77 |

10 | 3.16 | 3.92 |

11 | 3.29 | 3.83 |

12 | 3.70 | 3.77 |

13 | 3.37 | 3.84 |

14 | 3.36 | 3.72 |

15 | 3.67 | 3.81 |

16 | 3.89 | 4.11 |

17 | 4.11 | 4.08 |

18 | 4.39 | 4.11 |

19 | 4.55 | 4.11 |

20 | 4.74 | 4.05 |

**Table 5.**Predictive performance of the incremental granular model using particle swarm optimization.

Algorithm | No. of Contexts | No. of Clusters/Fuzzification Coefficient | Training RMSE | Testing RMSE |
---|---|---|---|---|

PSO-IGM | 5 | 5 5 4 3 2/2.3703 | 3.60 | 3.94 |

6 | 4 7 8 4 6 3/1.5740 | 3.17 | 3.56 | |

7 | 6 3 3 6 5 3 7/1.9901 | 3.42 | 3.73 | |

8 | 7 5 3 7 3 4 5 2/1.8734 | 3.24 | 3.55 |

Algorithm | No. of Contexts | No. of Clusters/Fuzzification Coefficient | Training RMSE | Testing RMSE |
---|---|---|---|---|

IGM | 5 | 7/1.5 | 3.44 | 4.12 |

PSO-IGM | 5 5 4 3 2/2.3703 | 3.60 | 3.94 | |

IGM | 6 | 7/1.5 | 3.49 | 3.95 |

PSO-IGM | 4 7 8 4 6 3/1.5740 | 3.17 | 3.56 | |

IGM | 7 | 9/1.5 | 3.11 | 3.74 |

PSO-IGM | 6 3 3 6 5 3 7/1.9901 | 3.42 | 3.73 | |

IGM | 8 | 14/1.5 | 3.36 | 3.72 |

PSO-IGM | 7 5 3 7 3 4 5 2/1.8734 | 3.24 | 3.55 |

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**MDPI and ACS Style**

Yeom, C.-U.; Kwak, K.-C.
Incremental Granular Model Improvement Using Particle Swarm Optimization. *Symmetry* **2019**, *11*, 390.
https://doi.org/10.3390/sym11030390

**AMA Style**

Yeom C-U, Kwak K-C.
Incremental Granular Model Improvement Using Particle Swarm Optimization. *Symmetry*. 2019; 11(3):390.
https://doi.org/10.3390/sym11030390

**Chicago/Turabian Style**

Yeom, Chan-Uk, and Keun-Chang Kwak.
2019. "Incremental Granular Model Improvement Using Particle Swarm Optimization" *Symmetry* 11, no. 3: 390.
https://doi.org/10.3390/sym11030390