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DEMass: a new density estimator for big data

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Abstract

Density estimation is the ubiquitous base modelling mechanism employed for many tasks including clustering, classification, anomaly detection and information retrieval. Commonly used density estimation methods such as kernel density estimator and \(k\)-nearest neighbour density estimator have high time and space complexities which render them inapplicable in problems with big data. This weakness sets the fundamental limit in existing algorithms for all these tasks. We propose the first density estimation method, having average case sub-linear time complexity and constant space complexity in the number of instances, that stretches this fundamental limit to an extent that dealing with millions of data can now be done easily and quickly. We provide an asymptotic analysis of the new density estimator and verify the generality of the method by replacing existing density estimators with the new one in three current density-based algorithms, namely DBSCAN, LOF and Bayesian classifiers, representing three different data mining tasks of clustering, anomaly detection and classification. Our empirical evaluation results show that the new density estimation method significantly improves their time and space complexities, while maintaining or improving their task-specific performances in clustering, anomaly detection and classification. The new method empowers these algorithms, currently limited to small data size only, to process big data—setting a new benchmark for what density-based algorithms can achieve.

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Notes

  1. While there are ways to reduce the computational cost of KDE, \(k\)-NN and \(\epsilon \)-neighbourhood, they are usually limited to low-dimensional problems or incur significant preprocessing cost. See Sect. 7 for a discussion.

  2. The implementation of \(T(\cdot )\) used in this paper is a tree-based nonparametric method. The binomial distribution is required for the error analysis only.

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Acknowledgments

This work is partially supported by the Air Force Research Laboratory, under agreement# FA2386-10-1-4052. The U.S. Government is authorised to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. Takashi Washio is partially supported by JSPS (Japan Society for the Promotion of Science) Grant-in-Aid for Scientific Research (B)# 22300054. Sunil Aryal is supported by Monash University Postgraduate Publications Award to complete the work on DEMass-Bayes. Xiao Yu Ge assisted in experiments using ELKI. Hiroshi Motoda, Zhouyu Fu and the anonymous reviewers had provided many helpful comments to improve this paper.

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Authors and Affiliations

  1. Gippsland School of Information Technology, Monash University, Gippsland Campus, Churchill, VIC, 3842, Australia

    Kai Ming Ting, Jonathan R. Wells, Fei Tony Liu & Sunil Aryal

  2. The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibarakishi, Osaka,  5670047, Japan

    Takashi Washio

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  1. Kai Ming Ting
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  2. Takashi Washio
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  3. Jonathan R. Wells
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  5. Sunil Aryal
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Corresponding author

Correspondence to Sunil Aryal.

Additional information

The source codes of DEMass-DBSCAN and DEMass-Bayes are available at http://sourceforge.net/projects/mass-estimation/.

The preliminary version of this paper appeared in Proceedings of the 2011 IEEE International Conference on Data Mining [33].

Appendix: Data characteristic of the RingCurve-Wave-TriGaussian data set

Appendix: Data characteristic of the RingCurve-Wave-TriGaussian data set

The characteristic of the RingCurve-Wave-TriGaussian data set, used in Sect. 5.1, is shown in Fig. 10. Each of the Ring-Curve, Wave and Triangular-Gaussian is a two-dimensional data set, and together, there is a total of seven clusters. Each cluster has 10,000 instances. When used in the scale-up experiment, the data size in each cluster was scaled by a factor of 0.1, 1, 75, 150 to 1,500.

Fig. 10
figure 10

Scatter plot of the clusters in the RingCurve-Wave-TriGaussian data set

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Ting, K.M., Washio, T., Wells, J.R. et al. DEMass: a new density estimator for big data. Knowl Inf Syst 35, 493–524 (2013). https://doi.org/10.1007/s10115-013-0612-3

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