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Outlier removal

Speech and Image Processing Unit School of Computing University of Eastern Finland. Outlier removal. Clustering Methods: Part 7. Pasi Fränti. Outlier detection methods. Distance-based methods Knorr & Ng Density-based methods KDIST: K th nearest distance MeanDIST: Mean distance

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Outlier removal

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  1. Speech and Image Processing UnitSchool of Computing University of Eastern Finland Outlier removal Clustering Methods: Part 7 Pasi Fränti

  2. Outlier detection methods Distance-based methods • Knorr & Ng Density-based methods • KDIST: Kth nearest distance • MeanDIST: Mean distance Graph-based methods • MkNN: Mutual K-nearest neighbor • ODIN: Indegree of nodes in k-NN graph

  3. What is outlier? One definition: Outlier is an observation that deviates from other observations so much that it is expected to be generated by a different mechanism. Outliers

  4. Distance-based method[Knorr and Ng , 1997: Conf. of CASCR ] Definition: Data point x is an outlier if at most k points are within the distance d from x. Example with k=3 Inlier Inlier Outlier

  5. Selection of distance threshold Too small value of d: false detection of outliers Too large value of d:outliers missed

  6. Density-based method: KDIST [Ramaswamy et al., 2000: ACM SIGMOD] • Define k Nearest Neighbourdistance (KDIST) as the distance to the kth nearest vector. • Vectors are sorted by their KDIST distance. The last n vectors in the list are classified as outliers.

  7. Density-based: MeanDist [Hautamäki et al. , 2004: Int. Conf. Pattern Recognition] MeanDIST = the mean of k nearest distances. User parameters: Cutting point k, and local threshold t:

  8. Comparison of KDIST and MeanDIST

  9. Distribution-based method[Aggarwal and Yu , 2001: ACM SIGMOD]

  10. Detection of sparse cells

  11. Mutual k-nearest neighbor[Brito et al., 1997: Statistics & Probability Letters] • Generate directed k-NN graph. • Create undirected graph as follows: • Vectors a and b are “mutual neighbors” if both linksa b and b a exist. • Change all mutual links ab to undirected link a—b. • Remove the rest. • Connected components are clusters. • Isolated vectors as outliers.

  12. 6 5 1 2 4 5 1 8 6 5 1 2 4 5 1 8 k = 2 Mutual k-NN example 1 • Given a data with one outlier. • For each vector find two nearest neighbours and create directed 2-NN graph. • For each pair of vectors, create edge in mutual graph, if there are edges a→b and b→a. 6 5 1 2 4 5 1 8 2 3 Clusters Outlier

  13. Outlier detection using indegree of nodes (ODIN) [Hautamäki et al., 2004: ICPR] Definition: Given kNN graph, classify data point x as an outlier its indegree  T.

  14. 3 3 6 0 6 6 0 4 4 5 5 5 3 3 Outlier 1 1 1 1 1 1 1 2 4 5 1 8 2 4 5 2 1 4 5 1 8 8 3 6 0 4 5 3 Outliers 1 1 1 2 4 5 1 8 Example of ODIN k = 2 Input data Graph and indegrees Threshold value 0 Threshold value 1

  15. 6 5 1 2 4 5 1 8 Example of FA and FR k = 2 Detected as outlier with different threshold values (T) 3 0 3 4 1 1

  16. 6 5 1 2 4 5 1 8

  17. ExperimentsMeasures • False acceptance (FA): • Number of outliers that are not detected. • False rejection (FR): • Number of good vectors wrongly classified as outlier. • Half total error rate: • HTER = (FR+FA) / 2

  18. Comparison of graph-based methods

  19. Difficulty of parameter setup ODIN: MeanDIST: KDD S1 Value of k is not important as long as threshold below 0.1. A clear valley in error surface between 20-50.

  20. Improved k-means using outlier removal Original After 40 iterations After 70 iterations At each step, remove most diverging data objects and construct new clustering.

  21. Example of removal factor Outlier factor:

  22. CERES algorithm [Hautamäki et al., 2005: SCIA]

  23. Experiments Artificial data sets A1 S3 S4 Image data sets Plot of M2 M1 M2 M3

  24. Comparison

  25. Literature • D.M. Hawkins, Identification of Outliers, Chapman and Hall, London, 1980. • W. Jin, A.K.H. Tung, J. Han, "Finding top-n local outliers in large database", In Proc. 7th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining, pp. 293-298, 2001. • E.M. Knorr, R.T. Ng, "Algorithms for mining distance-based outliers in large datasets", In Proc. 24th Int. Conf. Very Large Data Bases, pp. 392-403, New York, USA, 1998. • M.R. Brito, E.L. Chavez, A.J. Quiroz, J.E. Yukich, "Connectivity of the mutual k-nearest-neighbor graph in clustering and outlier detection", Statistics & Probability Letters, 35 (1), 33-42, 1997.

  26. Literature • C.C. Aggarwal and P.S. Yu, "Outlier detection for high dimensional data", Proc. Int. Conf. on Management of data ACM SIGMOD, pp. 37-46, Santa Barbara, California, United States, 2001. • V. Hautamäki, S. Cherednichenko, I. Kärkkäinen, T. Kinnunen and P. Fränti, Improving K-Means by Outlier Removal, In Proc. 14th Scand. Conf. on Image Analysis (SCIA’2005), 978-987, Joensuu, Finland, June, 2005. • V. Hautamäki, I. Kärkkäinen and P. Fränti, "Outlier Detection Using k-Nearest Neighbour Graph", In Proc. 17th Int. Conf. on Pattern Recognition (ICPR’2004), 430-433, Cambridge, UK, August, 2004.

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