Using distribution analysis for parameter selection in repstream

Ross Callister; Duc-Son Pham; Mihai Lazarescu

doi:10.3934/mfc.2019015

Article Contents

2019, Volume 2, Issue 3: 215-250. Doi: 10.3934/mfc.2019015

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Using distribution analysis for parameter selection in repstream

Curtin University, Kent Street, Perth, Western Australia 6102, Australia

^* Corresponding author: Ross Callister
^* Corresponding author: Ross Callister

Received: March 31, 2019

Published: September 2019

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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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Mathematics Subject Classification: Primary: 62H30.

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Figure 1. An illustration of the representative and point-level sparse graphs in the RepStream algorithm

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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

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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

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Figure 4. An illustration of relative edge lengths of nearest neighbours in the middle of a cluster versus near the edge of a cluster

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Figure 5. Different cases for the distribution of edge lengths

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Figure 6. Visualisation of the DS1 dataset

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Figure 7. Visualisation of the DS2 dataset

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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

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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

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Figure 10. The evolution of the Closer dataset, showing slices of its 3 sections

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Figure 11. Comparative purity for Tree Cover dataset

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Figure 12. Comparative purity for Tree Cover dataset

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Figure 13. Comparative purity for KDD 99' Cup dataset

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Figure 14. Comparative purity for KDD 99' Cup dataset

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Figure 15. Comparative purity for DS1 dataset

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Figure 16. The $ K $ value selected by our dynamic $ K $ method on the DS1 dataset

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Figure 17. Comparative purity for DS2 dataset

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Figure 18. Comparative purity for SynTest dataset

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Figure 19. Comparative purity for Closer dataset

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Figure 20. Comparative purity for Tree Cover dataset

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Figure 21. Comparative purity for KDD 99' Cup dataset

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Figure 22. The $ K $ value selected by our dynamic $ K $ method on the KDD Dataset

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Figure 23. F-Measure comparison vs RepStream using optimal parameters on DS1 dataset

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Figure 24. F-Measure comparison vs RepStream using optimal parameters on DS2 dataset

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Figure 25. F-Measure comparison vs RepStream using optimal parameters on SynTest dataset

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Figure 26. F-Measure comparison vs RepStream using optimal parameters on Closer dataset

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Figure 27. F-Measure comparison vs RepStream using optimal parameters on Tree Cover dataset

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Figure 28. F-Measure comparison vs RepStream using optimal parameters on KDD 99' Cup dataset

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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 parameters
SWClustering	$ \beta $		$ k $-means			$ \epsilon $, $ N $ Window parameters
DenStream	$ \mu $, $ \beta $				$ \epsilon $
HPStream				$ \lambda $	$ r $	$ l $ Projected Dimensions
D-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 limit
FlockStream	$\beta$, $\mu$			$\lambda$	$\epsilon$
DeBaRa	$dPts$				$f$
DBStream	$\alpha$, $w\_min$			$\lambda$, $t\_gap$	$r$
SNCStream	$\psi$, $\beta$			$\lambda$	$\epsilon$	$K$ Graph connectivity
PASCAL			Evolutionary			Evolutionary EDA hyper-parameters
HASTREAM	$minPts$		Hierarchical	$\lambda$
BEStream	$\Delta$, $\tau$			$\lambda$	$\xi$	$\theta$ Direction parameter
ADStream	$\xi$, $\epsilon$			$\lambda$
PatchWork	$ratio$, $minPoints$, $minCells$	$\epsilon$
RepStream				$\lambda$	$\alpha$	$K$ Graph connectivity

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Table 2. Best and Worst $ K $ values for RepStream, according to $ F $ score

Dataset	Best K	Best $ F $ score	Worst K	Worst $ F $ score
DS1	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

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Table 3. Comparison of our dynamic $ K $ method versus RepStream with optimal and worst $ K $ values

Dataset	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

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