A Modified Energy Statistic for Unsupervised Anomaly Detection



Published May 12, 2021
Rupam Mukherjee


For prognostics in industrial applications, the degree of anomaly of a test point from a baseline cluster is estimated using a statistical distance metric. Among different statistical distance metrics, energy distance is an interesting concept based on Newton’s Law of Gravitation, promising simpler computation than classical distance metrics. In this paper, we review the state of the art formulations of energy distance and point out several reasons why they are not directly applicable to the anomaly-detection problem. Thereby, we propose a new energy-based metric called the P-statistic which addresses these issues, is applicable to anomaly detection and retains the computational simplicity of the energy distance. We also demonstrate its effectiveness on a real-life data-set.

Abstract 479 | PDF Downloads 373



anomaly detection, Mahalanobis Distance, Statistical Distribution, energy distance

Baringhaus, L., & Franz, C. (2004). On a new multivariate two-sample test. Journal of multivariate analysis, 88(1), 190–206.
Bhattacharyya, A. (1943). On a measure of divergence between two statistical populations defined by their probability distributions. Bulletin of the Calcutta Mathematical Society(35), 99-109.
Dixon, A. (1901). On burmann’s theorem. Proceedings of the London Mathematical Society, 1(1), 151–153.
Dua, D., & Graff, C. (2017). UCI machine learning repository. Retrieved from http://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+\%28Diagnostic\%29
Feuerverger, A. (1993). A consistent test for bivariate dependence. International Statistical Review/Revue Internationale de Statistique, 419–433.
Goldstein, M., & Uchida, S. (2016). A comparative evaluation of unsupervised anomaly detection algorithms for multivariate data. PloS one, 11(4), e0152173.
Harman, R., & Lacko, V. (2010). On decompositional algorithms for uniform sampling from n-spheres and nballs. Journal of Multivariate Analysis, 101(10), 2297–2304.
Kim, A. Y., Marzban, C., Percival, D. B., & Stuetzle, W. (2009). Using labeled data to evaluate change detectors in a multivariate streaming environment. Signal Processing, 89(12), 2529–2536.
Maaten, L. v. d., & Hinton, G. (2008). Visualizing data using t-sne. Journal of machine learning research, 9(Nov), 2579–2605.
Mahalanobis, P. C. (1936). On the generalized distance in statistics.
Matteson, D. S., & James, N. A. (2014). A nonparametric approach for multiple change point analysis of multivariate data. Journal of the American Statistical Association, 109(505), 334–345.
Rizzo, M. L. (2002a). A new rotation invariant goodness-of-fit test (Unpublished doctoral dissertation). Bowling Green University.
Rizzo, M. L. (2002b). A test of homogeneity for two multivariate populations. Proceedings of the American Statistical Association, Physical and Engineering Sciences Section.
Rizzo, M. L. (2009). New goodness-of-fit tests for Pareto distributions. ASTIN Bulletin: The Journal of the IAA, 39(2), 691–715.
Sch¨opf, H., & Supancic, P. (2014). On B¨urmann‘s theorem and its application to problems of linear and nonlinear heat transfer and diffusion. The Mathematica Journal, 16(11).
Scikit-learn breast cancer data-set. (n.d.). Retrieved from https://scikit-learn.org/stable/modules/generated/sklearn.datasets.load breast cancer.html
Statistical distance. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Statistical\ distance
Székely, G. J. (1989). Potential and kinetic energy inétatistics. LBudapest Institue of Technology.
Székely, G. J. (2002). E-statistics: energy of statistical samples (Tech. Rep. No. 02-16). Bowling Green State University, Dep. Math. stat.
Székely, G. J., & Rizzo, M. L. (2005). A new test for multivariate normality. Journal of Multivariate Analysis, 93(1), 58–80.
Székely, G. J., & Rizzo, M. L. (2009). Brownian distance covariance. The annals of applied statistics, 1236–1265.
Székely, G. J., & Rizzo, M. L. (2013). Energy statistics: A class of statistics based on distances. Journal of statistical planning and inference, 143(8), 1249–1272.
Szekely, G. J., & Rizzo, M. L. (2017). The energy of data. Annual Review of Statistics and Its Application, 4, 447–479.
Székely, G. J., Rizzo, M. L., et al. (2004). Testing for equal distributions in high dimension. InterStat, 5(16.10), 1249–1272.
Szekely, G. J., Rizzo, M. L., et al. (2005). Hierarchical clustering via joint between-within distances: Extending ward’s minimum variance method. Journal of classification, 22(2), 151–184.
Yang, G. (2012). The energy goodness-of-fit test for univariate stable distributions (Unpublished doctoral dissertation). Bowling Green State University.
Technical Papers