# Adaptive Hierarchical Density-Based Spatial Clustering Algorithm for Streaming Applications

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## Abstract

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## 1. Introduction

## 2. Relevant Work

- The number of Gaussian clusters used to describe pixel history is denoted by k;
- ${\omega}_{i,t}$ is the weight factor associated with cluster i and time t;
- ${\mu}_{i,t}$ and ${\Sigma}_{i,t}$ are the mean and covariance matrix of i-th Gaussian cluster.

## 3. Methodology

#### 3.1. Data Preparation

#### 3.2. Adaptive HDBSCAN Implementation

#### 3.3. Pairwise Distance Calculation

#### 3.4. Euclidean to $\lambda $-Space Transformation

#### 3.5. Determining the Cutoff for Algorithm Selection

#### 3.6. Prim’s Algorithm

#### 3.7. Boruvka’s Algorithm

#### 3.8. Cluster Hierarchy Construction

#### 3.9. Cluster Hierarchy Condensation

#### 3.10. Excess of Mass Calculation

## 4. Experiment Results and Discussions

#### 4.1. Experiment Setup to Assess the Execution Time Improvement of Adaptive HDBSCAN

#### 4.2. Experiment Setup to Assess the Accuracy of Clusters Generated by Adaptive HDBSCAN

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**The computation time performance comparison in absolute terms between Adaptive HDBSCAN and HDBSCAN.

**Figure 7.**Boxplot of the accuracies of cluster prediction using the Adaptive HDBSCAN and the HDBSCAN.

**Table 1.**The computation time performance comparison in absolute terms between Adaptive HDBSCAN and HDBSCAN sampled for every 100 data points increment.

Number of Data Points | Execution Time (Seconds) | |
---|---|---|

HDBSCAN | Adaptive HDBSCAN | |

100 | 0.001490 | 0.001397 |

200 | 0.002667 | 0.002337 |

300 | 0.003855 | 0.003550 |

400 | 0.004973 | 0.004591 |

500 | 0.006595 | 0.006137 |

600 | 0.007446 | 0.007131 |

700 | 0.008563 | 0.008794 |

800 | 0.009830 | 0.009840 |

Number of Data Points | Accuracy (%) | |
---|---|---|

Adaptive HDBSCAN | HDBSCAN | |

100 | 99.6 | 99.8 |

200 | 100.0 | 100.0 |

300 | 99.2 | 99.6 |

400 | 98.6 | 99.2 |

500 | 98.6 | 99.0 |

600 | 98.4 | 99.0 |

700 | 98.8 | 98.8 |

800 | 98.4 | 98.6 |

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**MDPI and ACS Style**

Vijayan, D.; Aziz, I.
Adaptive Hierarchical Density-Based Spatial Clustering Algorithm for Streaming Applications. *Telecom* **2023**, *4*, 1-14.
https://doi.org/10.3390/telecom4010001

**AMA Style**

Vijayan D, Aziz I.
Adaptive Hierarchical Density-Based Spatial Clustering Algorithm for Streaming Applications. *Telecom*. 2023; 4(1):1-14.
https://doi.org/10.3390/telecom4010001

**Chicago/Turabian Style**

Vijayan, Darveen, and Izzatdin Aziz.
2023. "Adaptive Hierarchical Density-Based Spatial Clustering Algorithm for Streaming Applications" *Telecom* 4, no. 1: 1-14.
https://doi.org/10.3390/telecom4010001