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Using distribution analysis for parameter selection in repstream
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Abstract
One of the most significant challenges in data clustering is the evolution of the data distributions over time. Many clustering algorithms have been introduced to deal specifically with streaming data, but common amongst them is that they require users to set input parameters. These inform the algorithm about the criteria under which data points may be clustered together. Setting the initial parameters for a clustering algorithm is itself a non-trivial task, but the evolution of the data distribution over time could mean even optimally set parameters could become non-optimal as the stream evolves. In this paper we extend the RepStream algorithm, a combination graph and density-based clustering algorithm, in a way which allows the primary input parameter, the $ K $ value, to be automatically adjusted over time. We introduce a feature called the edge distribution score which we compute for data in memory, as well as introducing an incremental method for adjusting the $ K $ parameter over time based on this score. We evaluate our methods against RepStream itself, and other contemporary stream clustering algorithms, and show how our method of automatically adjusting the $ K $ value over time leads to higher quality clustering output even when the initial parameters are set poorly.
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Keywords:
- Cluster analysis,
- data clustering,
- parameter adjustment,
- graph-based clustering,
- KNN graphs.
Mathematics Subject Classification: Primary: 62H30.Citation: 
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Figure 1. An illustration of the representative and point-level sparse graphs in the RepStream algorithm
Figure 2. The representative vertex
$ R_1 $ and$ R_2 $ share a reciprocal connection at the representative level, and are also density related. The density radius of the vertices is shown as$ DR_1 $ and$ DR_2 $ respectively, which is the average distance to neighbours at the point level, multiplied by the alpha scaling factor
Figure 3. Intra and Inter-class edges. The Edge
$ E_1 $ is considered an inter-class edge as it connects two vertices$ R_1 $ and$ R_2 $ that belong to different ground-truth classes. Edge$ E_2 $ connects two vertices belonging to the same class and thus is considered an intra-class edge
Figure 4. An illustration of relative edge lengths of nearest neighbours in the middle of a cluster versus near the edge of a cluster
Figure 5. Different cases for the distribution of edge lengths
Figure 6. Visualisation of the DS1 dataset
Figure 7. Visualisation of the DS2 dataset
Figure 8. Two dimensional representations of the 5 classes in the SynTest dataset. The main class is always present and steadily changes shape, the smaller classes appear at various points through the dataset, as shown in Figure 9
Figure 9. The class presence of the classes in the SynTest dataset. A marker indicates that the class is present in the dataset during the given time window
Figure 10. The evolution of the Closer dataset, showing slices of its 3 sections
Figure 11. Comparative purity for Tree Cover dataset
Figure 12. Comparative purity for Tree Cover dataset
Figure 13. Comparative purity for KDD 99' Cup dataset
Figure 14. Comparative purity for KDD 99' Cup dataset
Figure 15. Comparative purity for DS1 dataset
Figure 16. The
$ K $ value selected by our dynamic$ K $ method on the DS1 dataset
Figure 17. Comparative purity for DS2 dataset
Figure 18. Comparative purity for SynTest dataset
Figure 19. Comparative purity for Closer dataset
Figure 20. Comparative purity for Tree Cover dataset
Figure 21. Comparative purity for KDD 99' Cup dataset
Figure 22. The
$ K $ value selected by our dynamic$ K $ method on the KDD Dataset
Figure 23. F-Measure comparison vs RepStream using optimal parameters on DS1 dataset
Figure 24. F-Measure comparison vs RepStream using optimal parameters on DS2 dataset
Figure 25. F-Measure comparison vs RepStream using optimal parameters on SynTest dataset
Figure 26. F-Measure comparison vs RepStream using optimal parameters on Closer dataset
Figure 27. F-Measure comparison vs RepStream using optimal parameters on Tree Cover dataset
Figure 28. F-Measure comparison vs RepStream using optimal parameters on KDD 99' Cup dataset
Table 1. Table showing the input parameters used for each algorithm
Algorithm name Density Parameters Grid Granularity Reclustering Method Decay Parameter Distance Parameter Other Parameters CluStream $ k $ -means$ \delta $ $ \alpha $ ,$ l $ Snapshot parametersSWClustering $ \beta $ $ k $ -means$ \epsilon $ ,$ N $ Window parametersDenStream $ \mu $ ,$ \beta $ $ \epsilon $ HPStream $ \lambda $ $ r $ $ l $ Projected DimensionsD-Stream $ C_m $ ,$ C_l $ ,$ \beta $ $ len $ $ \lambda $ ExCC Grid Granularity in all dimensions MR-Stream $C_H$ ,$C_L$ ,$\beta$ ,$\mu$ Hierarchical $\lambda$ $\epsilon$ $H$ Heirarchical limitFlockStream $\beta$ ,$\mu$ $\lambda$ $\epsilon$ DeBaRa $dPts$ $f$ DBStream $\alpha$ ,$w\_min$ $\lambda$ ,$t\_gap$ $r$ SNCStream $\psi$ ,$\beta$ $\lambda$ $\epsilon$ $K$ Graph connectivityPASCAL Evolutionary Evolutionary EDA hyper-parameters HASTREAM $minPts$ Hierarchical $\lambda$ BEStream $\Delta$ ,$\tau$ $\lambda$ $\xi$ $\theta$ Direction parameterADStream $\xi$ ,$\epsilon$ $\lambda$ PatchWork $ratio$ ,$minPoints$ ,$minCells$ $\epsilon$ RepStream $\lambda$ $\alpha$ $K$ Graph connectivity| Show Table
DownLoad: CSV
Table 2. Best and Worst
$ K $ values for RepStream, according to$ F $ scoreDataset Best K Best $ F $ scoreWorst K Worst $ F $ scoreDS1 7 0.7208 18 0.2767 DS2 7 0.6371 21 0.2594 SynTest 9 0.8614 5 0.5435 Closer 9 0.7989 5 0.4345 TreeCov 29 0.6108 5 0.2978 KDD99 30 0.7898 5 0.2636 | Show Table
DownLoad: CSV
Table 3. Comparison of our dynamic
$ K $ method versus RepStream with optimal and worst$ K $ valuesDataset Best F-Measure Worst F-Measure Dynamic $ K $ DS1 0.7208 0.2767 0.5153 DS2 0.6371 0.2594 0.4137 SynTest 0.7989 0.5435 0.7091 Closer 0.8614 0.4345 0.8410 TreeCov 0.6108 0.2978 0.5920 KDD99 0.7898 0.2636 0.7882 | Show Table
DownLoad: CSV
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- Figure 1. An illustration of the representative and point-level sparse graphs in the RepStream algorithm
-
Figure 2. The representative vertex
$ R_1 $ and$ R_2 $ share a reciprocal connection at the representative level, and are also density related. The density radius of the vertices is shown as$ DR_1 $ and$ DR_2 $ respectively, which is the average distance to neighbours at the point level, multiplied by the alpha scaling factor -
Figure 3. Intra and Inter-class edges. The Edge
$ E_1 $ is considered an inter-class edge as it connects two vertices$ R_1 $ and$ R_2 $ that belong to different ground-truth classes. Edge$ E_2 $ connects two vertices belonging to the same class and thus is considered an intra-class edge - Figure 4. An illustration of relative edge lengths of nearest neighbours in the middle of a cluster versus near the edge of a cluster
- Figure 5. Different cases for the distribution of edge lengths
- Figure 6. Visualisation of the DS1 dataset
- Figure 7. Visualisation of the DS2 dataset
- Figure 8. Two dimensional representations of the 5 classes in the SynTest dataset. The main class is always present and steadily changes shape, the smaller classes appear at various points through the dataset, as shown in Figure 9
- Figure 9. The class presence of the classes in the SynTest dataset. A marker indicates that the class is present in the dataset during the given time window
- Figure 10. The evolution of the Closer dataset, showing slices of its 3 sections
- Figure 11. Comparative purity for Tree Cover dataset
- Figure 12. Comparative purity for Tree Cover dataset
- Figure 13. Comparative purity for KDD 99' Cup dataset
- Figure 14. Comparative purity for KDD 99' Cup dataset
- Figure 15. Comparative purity for DS1 dataset
-
Figure 16. The
$ K $ value selected by our dynamic$ K $ method on the DS1 dataset - Figure 17. Comparative purity for DS2 dataset
- Figure 18. Comparative purity for SynTest dataset
- Figure 19. Comparative purity for Closer dataset
- Figure 20. Comparative purity for Tree Cover dataset
- Figure 21. Comparative purity for KDD 99' Cup dataset
-
Figure 22. The
$ K $ value selected by our dynamic$ K $ method on the KDD Dataset - Figure 23. F-Measure comparison vs RepStream using optimal parameters on DS1 dataset
- Figure 24. F-Measure comparison vs RepStream using optimal parameters on DS2 dataset
- Figure 25. F-Measure comparison vs RepStream using optimal parameters on SynTest dataset
- Figure 26. F-Measure comparison vs RepStream using optimal parameters on Closer dataset
- Figure 27. F-Measure comparison vs RepStream using optimal parameters on Tree Cover dataset
- Figure 28. F-Measure comparison vs RepStream using optimal parameters on KDD 99' Cup dataset