# A New Swarm Intelligence Approach for Clustering Based on Krill Herd with Elitism Strategy

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## Abstract

**:**

## 1. Introduction

## 2. Fuzzy C-Means (FCM) Clustering Algorithm

_{1}, x

_{2}, …, x

_{n}} be n data samples; c (2 ≤ c ≤ n) is the number of the divided categories for these data samples; {A

_{1}, A

_{2}, …, A

_{c}} indicates that the corresponding c categories, and

**U**is their similarity classification matrix, whose cluster centers are {v

_{1}, v

_{2}, …, v

_{c}}; μ

_{k}(x

_{i}) is the membership degree of x

_{i}in the category A

_{k}(abbreviated as μ

_{ik}). The objective function J

_{b}can be expressed as follows:

_{ik}is the Euclidean distance that is used to measure distance between the i-th sample x

_{i}and the center point of the k-th category. It can be calculated as follows:

_{b}. It is required that the sum of the values of membership degree for a sample in terms of each cluster is 1. That is to say, it can be described as

_{ik}is the membership degree of x

_{i}in the category A

_{k}, and it can be updated as

_{i}} are calculated as

_{ik}= 0.

## 3. KHE Method for Clustering Problem

#### 3.1. KH Method

- (i)
- movement induced by other krill individuals;
- (ii)
- foraging action; and
- (iii)
- random diffusion

_{i}is the motion induced by other krill individuals; F

_{i}is the foraging motion, and D

_{i}is the physical diffusion of the i-th krill individuals.

#### 3.1.1. Motion Induced by Other Krill Individuals

_{i}, is approximately evaluated by the target effect, a local effect, and a repulsive effect. For krill i, it can be defined as:

^{max}is the maximum induced speed, ω

_{n}is the inertia weight of the motion induced, ${N}_{i}^{old}$ is the last motion induced, ${\mathsf{\alpha}}_{i}^{local}$ is the local effect provided by the neighbors and ${\mathsf{\alpha}}_{i}^{target}$ is the target direction effect provided by the best krill individual.

#### 3.1.2. Foraging Motion

_{f}is the foraging speed, ω

_{f}is the inertia weight of the foraging, ${F}_{i}^{old}$ is the last foraging motion, ${\mathsf{\beta}}_{i}^{food}$ is the food attractiveness and ${\mathsf{\beta}}_{i}^{best}$ is the effect of the best fitness of the i-th krill so far.

#### 3.1.3. Random Diffusion

_{max}is the maximum diffusion speed, and δ is the random directional vector.

#### 3.2. KH Method with Elitism Strategy (KHE)

#### 3.3. KHE Method for Clustering Problem

- (1)
- Initialize the control parameters. All the parameters used in KHE are firstly initialized.
- (2)
- Randomly initialize c cluster centers, and generate the initial population, calculate membership degree of each cluster center for all samples by Equation (4), and the fitness of each krill individual value f
_{i}, where i = 1, 2, …, NP. Here, NP is the number of population size. - (3)
- Set t = 0.
- (4)
- Save the KEEP best krill individuals as BEST.
- (5)
- Implement three motions and update the positions of krill individuals in population.
- (6)
- Replace the KEEP worst krill individuals with the KEEP best krill individuals saved in BEST.
- (7)
- Calculate c clustering centers, membership degree and fitness for each individual.
- (8)
- If the t < Maxgen, t = t + 1, go to Equation (4); Otherwise, the algorithm is over and finds the final global optimal solution.

## 4. Simulation Results

No. | Name | Definition |
---|---|---|

F01 | Dixon & Price | $f(\overrightarrow{x})={\left({x}_{1}-1\right)}^{2}+{\displaystyle \sum _{i=2}^{n}i{\left(2{x}_{i}^{2}-{x}_{i-1}\right)}^{2}}$ |

F02 | Griewank | $f(\overrightarrow{x})={\displaystyle \sum _{i=1}^{n}\frac{{x}_{i}^{2}}{4000}}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1}$ |

F03 | Holzman 2 function | $f(\overrightarrow{x})={\displaystyle \sum _{i=1}^{n}i{x}_{i}^{4}}$ |

F04 | Powell | $f(\overrightarrow{x})={\displaystyle \sum _{i=1}^{n/4}{({x}_{4i-3}+10{x}_{4i-2})}^{2}}+5{({x}_{4i-1}-{x}_{4i})}^{2}\text{\hspace{0.05em}+}{({x}_{4i-2}-{x}_{4i-1})}^{4}+10{({x}_{4i-3}-{x}_{4i})}^{4}$ |

F05 | Quartic with noise | $f(\overrightarrow{x})={\displaystyle \sum _{i=1}^{n}(i\cdot {x}_{i}^{4}+U(0,1))}$ |

F06 | Rosenbrock | $f(\overrightarrow{x})={\displaystyle \sum _{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]}$ |

F07 | Sphere | $f(\overrightarrow{x})={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}$ |

_{f}= 0.02, D

^{max}= 0.005 and N

^{max}= 0.01. For the parameters used in the other methods, their settings can be referred to in [4,63].

#### 4.1. Convergent Performance Compared KHE with Six Other Methods

Function | ACO | GA | HS | KH | KHE | PSO | SGA | |
---|---|---|---|---|---|---|---|---|

MEAN | F01 | 3.36E5 | 1.25E5 | 8.26E5 | 1.55E5 | 18.90 | 2.77E5 | 8.64E3 |

F02 | 32.55 | 106.40 | 403.60 | 67.46 | 1.13 | 172.70 | 29.50 | |

F03 | 8.72E4 | 3.51E4 | 2.07E5 | 3.74E4 | 1.86 | 8.36E4 | 2.29E3 | |

F04 | 5.92E3 | 1.88E3 | 6.06E3 | 3.70E3 | 36.01 | 2.47E3 | 182.80 | |

F05 | 17.75 | 7.92 | 54.68 | 10.14 | 4.44E−4 | 13.77 | 0.66 | |

F06 | 5.47E3 | 1.86E3 | 3.99E3 | 1.22E3 | 31.60 | 1.38E3 | 313.10 | |

F07 | 85.19 | 21.57 | 119.50 | 20.03 | 0.04 | 50.05 | 11.02 | |

BEST | F01 | 1.14E5 | 1.71E4 | 3.13E5 | 5.31E4 | 3.37 | 2.44E4 | 1.41E3 |

F02 | 14.90 | 33.08 | 266.10 | 35.84 | 1.02 | 91.07 | 9.64 | |

F03 | 2.36E4 | 6.30E3 | 9.39E4 | 1.76E4 | 0.03 | 1.16E4 | 300.60 | |

F04 | 2.43E3 | 388.10 | 2.26E3 | 1.00E3 | 2.41 | 1.01E3 | 52.56 | |

F05 | 6.12 | 1.28 | 25.52 | 5.10 | 5.13E−6 | 4.22 | 0.08 | |

F06 | 3.73E3 | 513.10 | 2.20E3 | 697.30 | 28.19 | 508.00 | 137.50 | |

F07 | 55.53 | 5.80 | 70.20 | 10.44 | 3.84E−3 | 29.45 | 4.16 | |

WORST | F01 | 8.46E5 | 3.63E5 | 1.26E6 | 2.85E5 | 167.60 | 2.39E6 | 4.90E4 |

F02 | 69.45 | 235.90 | 498.70 | 101.70 | 1.63 | 568.30 | 68.65 | |

F03 | 1.76E5 | 1.35E5 | 3.29E5 | 6.26E4 | 20.86 | 6.00E5 | 8.98E3 | |

F04 | 8.89E3 | 4.42E3 | 1.07E4 | 6.35E3 | 218.40 | 4.72E3 | 558.50 | |

F05 | 37.48 | 28.87 | 81.76 | 17.74 | 7.82E−3 | 32.10 | 5.08 | |

F06 | 8.08E3 | 4.14E3 | 5.70E3 | 1.90E3 | 46.15 | 2.88E3 | 688.00 | |

F07 | 126.40 | 42.72 | 143.40 | 33.13 | 0.31 | 65.62 | 21.11 |

#### 4.2. Clustering Problem Compared KHE with Seven Other Methods

ACO | FCM | GA | HS | KH | KHE | PSO | SGA | |
---|---|---|---|---|---|---|---|---|

Mean | 3.303556 | 3.368558 | 3.303527 | 3.303536 | 3.303624 | 3.303510 | 3.303542 | 3.303523 |

Best | 3.303474 | 3.303478 | 3.303466 | 3.303468 | 3.303471 | 3.303462 | 3.303463 | 3.303462 |

Worst | 3.303766 | 3.728121 | 3.303766 | 3.303766 | 3.303766 | 3.303766 | 3.303766 | 3.303766 |

Std | 5.6032E−5 | 0.09555 | 5.9076E−5 | 4.0144E−5 | 1.1470E−4 | 4.6495E−5 | 5.0819E−5 | 5.1780E−5 |

## 5. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Li, Z.-Y.; Yi, J.-H.; Wang, G.-G. A New Swarm Intelligence Approach for Clustering Based on Krill Herd with Elitism Strategy. *Algorithms* **2015**, *8*, 951-964.
https://doi.org/10.3390/a8040951

**AMA Style**

Li Z-Y, Yi J-H, Wang G-G. A New Swarm Intelligence Approach for Clustering Based on Krill Herd with Elitism Strategy. *Algorithms*. 2015; 8(4):951-964.
https://doi.org/10.3390/a8040951

**Chicago/Turabian Style**

Li, Zhi-Yong, Jiao-Hong Yi, and Gai-Ge Wang. 2015. "A New Swarm Intelligence Approach for Clustering Based on Krill Herd with Elitism Strategy" *Algorithms* 8, no. 4: 951-964.
https://doi.org/10.3390/a8040951