Anomaly Detection and Classification for Streaming Data using PDEs

Nondominated sorting, also called Pareto Depth Analysis (PDA), is widely used in multi-objective optimization and has recently found important applications in multi-criteria anomaly detection. Recently, a partial differential equation (PDE) continuum limit was discovered for nondominated sorting leading to a very fast approximate sorting algorithm called PDE-based ranking. We propose in this paper a fast real-time streaming version of the PDA algorithm for anomaly detection that exploits the computational advantages of PDE continuum limits. Furthermore, we derive new PDE continuum limits for sorting points within their nondominated layers and show how the new PDEs can be used to classify anomalies based on which criterion was more significantly violated. We also prove statistical convergence rates for PDE-based ranking, and present the results of numerical experiments with both synthetic and real data.

[1]  Alfred O. Hero,et al.  Multicriteria Similarity-Based Anomaly Detection Using Pareto Depth Analysis , 2016, IEEE Transactions on Neural Networks and Learning Systems.

[2]  Alfred O. Hero,et al.  A PDE-based Approach to Nondominated Sorting , 2013, SIAM J. Numer. Anal..

[3]  Barbara Majecka,et al.  Statistical models of pedestrian behaviour in the Forum , 2009 .

[4]  Zhongzhi Shi,et al.  A Fast Nondominated Sorting Algorithm , 2005, 2005 International Conference on Neural Networks and Brain.

[5]  Jeff Calder,et al.  Numerical schemes and rates of convergence for the Hamilton–Jacobi equation continuum limit of nondominated sorting , 2015, Numerische Mathematik.

[6]  Donald Kossmann,et al.  Shooting Stars in the Sky: An Online Algorithm for Skyline Queries , 2002, VLDB.

[7]  Stefan Felsner,et al.  Maximum k-Chains in Planar Point Sets: Combinatorial Structure and Algorithms , 1998, SIAM J. Comput..

[8]  Alfred O. Hero,et al.  A continuum limit for non-dominated sorting , 2014, 2014 Information Theory and Applications Workshop (ITA).

[9]  Marc Parizeau,et al.  Generalizing the improved run-time complexity algorithm for non-dominated sorting , 2013, GECCO '13.

[10]  Satchidananda Dehuri,et al.  Evolutionary Algorithms for Multi-Criterion Optimization: A Survey , 2004 .

[11]  Marco Laumanns,et al.  SPEA2: Improving the strength pareto evolutionary algorithm , 2001 .

[12]  B. Logan,et al.  A Variational Problem for Random Young Tableaux , 1977 .

[13]  Alfred O. Hero,et al.  Pareto-depth for Multiple-query Image Retrieval , 2014, IEEE transactions on image processing : a publication of the IEEE Signal Processing Society.

[14]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[15]  Kalyanmoy Deb,et al.  Muiltiobjective Optimization Using Nondominated Sorting in Genetic Algorithms , 1994, Evolutionary Computation.

[16]  Martin J. Oates,et al.  The Pareto Envelope-Based Selection Algorithm for Multi-objective Optimisation , 2000, PPSN.

[17]  Lothar Thiele,et al.  Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach , 1999, IEEE Trans. Evol. Comput..

[18]  Regina Y. Liu,et al.  Multivariate analysis by data depth: descriptive statistics, graphics and inference, (with discussion and a rejoinder by Liu and Singh) , 1999 .

[19]  Alfred O. Hero,et al.  Multi-criteria Anomaly Detection using Pareto Depth Analysis , 2011, NIPS.

[20]  Vipin Kumar,et al.  Similarity Measures for Categorical Data: A Comparative Evaluation , 2008, SDM.

[21]  VARUN CHANDOLA,et al.  Anomaly detection: A survey , 2009, CSUR.

[22]  B. Bollobás,et al.  The longest chain among random points in Euclidean space , 1988 .

[23]  A. Hero,et al.  Posterior Pareto Front Analysis for Gene Filtering , .

[24]  Victoria J. Hodge,et al.  A Survey of Outlier Detection Methodologies , 2004, Artificial Intelligence Review.

[25]  Mikkel T. Jensen,et al.  Reducing the run-time complexity of multiobjective EAs: The NSGA-II and other algorithms , 2003, IEEE Trans. Evol. Comput..

[26]  Devdatt P. Dubhashi,et al.  Concentration of Measure for the Analysis of Randomized Algorithms: Contents , 2009 .

[27]  J. Hammersley.  A few seedlings of research , 1972 .

[28]  Qian Wang,et al.  An Efficient Non-dominated Sorting Method for Evolutionary Algorithms , 2008, Evolutionary Computation.

[29]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[30]  Jeff Calder,et al.  A direct verification argument for the Hamilton-Jacobi equation continuum limit of nondominated sorting , 2015, 1508.01565.