An Unbiased Distance-based Outlier Detection Approach for High Dimensional Data DASFAA 2011 By Hoang Vu Nguyen, Vivekanand Gopalkrishnan and Ira Assent Presented By Salman Ahmed Shaikh (D1)

Contents Introduction Subspace Outlier Detection Challenges Objectives of Research The Approach – Subspace Outlier Score Function: FS out – HighDOD Algorithm Empirical Results and Analysis Conclusion

Introduction An outlier, is one that appears to deviate markedly from other members of the sample in which it occurs. [1] Popular techniques of outlier detection – Distance based – Density base Since these techniques take full- dimensional space into account, their performance is impacted by noisy or irrelevant features. Recently, researchers have switched to subspace anomaly detection. Anomalous Subsequence o 1, o 2 and o 3 are anomalous instances w.r.t. the data

Subspace Outlier Detection Challenges Unavoidable exploration of all subspaces to mine full result set: – As the monotonicity property does not hold in the case of outliers, one cannot apply apriori-like heuristic for mining outliers. Difficulty in devising an outlier notion: – Full-dimensional outlier detection techniques suffer the issue of dimensionality bias in subspaces. – They assign higher outlier score in high dimensional subspaces than in lower dimensions Exposure to high false alarm rate: – Binary decision on each data point (normal or outlier) in each subspace flag too many points as outliers. – Solution is ranking-based algorithm.

Objectives Build an efficient technique for mining outliers in subspaces, which should – Avoid expensive scan of all subspaces while still yielding high detection accuracy – Eases the task of parameter setting – Facilitates the design of pruning heuristics to speed up the detection process – Provide a ranking of outliers across subspaces.

The Approach The authors have made an assertion and given some definitions to explain their research approach. { Non-monotonicity Property: Consider a data point p in the dataset DS. Even if p is not anomalous in subspace S of DS, it may be an outlier in some projection(s) of S. Even if p is a normal data point in all projections of S, it may be an outlier in S A A is an outlier in full space but not in subspace B B is an outlier in subspace but not in fullspace

(Subspace) Outlier Score Function Outlier Score Function: F out as given by Angiulli et al. for full space [2] The dissimilarity of a point p with respect to its k nearest neighbors is known by its cumulative neighborhood distance. This is defined as the total distance from p to its k nearest neighbors in DS. – In order to ensure that non-monotonicity property is not violated, the outlier score function is redefined by the authors as below. Subspace Outlier Score Function: FS out The dissimilarity of a point p with respect to its k nearest neighbors in a subspace S of dimensionality dim(S), is known by its cumulative neighborhood distance. This is defined as the total distance from p to its k nearest neighbors in DS (projected onto S), normalized by dim(S). – Where p s is the projection of a data point p ∊ DS onto S.

FS out is Dimensionality Unbiased FS out assigns multiple outlier scores to each data point and is dimensionality unbiased. Example: let k=1 and l=2 In Fig.(a), A's outlier score in the 2-dimensional space is 1/(2) 1/2 which is the largest across all subspaces. In Fig.(b), the outlier score of B when projected on the subspace of the x-axis is 1, which is also the largest in all subspaces. Hence, FS out flags A and B as outliers.

FS out is Globally Comparable Range of Distance: In each subspace S of DS, the distance between any arbitrary data points p and q is bounded by (dim(S)) 1/l Range of Outlier Score: For an arbitrary data point p and any subspace S, we have

Subspace Outlier Detection Problem Using FS out for outliers in subspaces, mining problem now can be re-defined as Given two positive integers k and n, mine the top n distinct anomalies whose outlier scores (in any subspace) are largest.

HighDOD-Subspace Outlier Detection Algorithm HighDOD (High dimensional Distance based Outlier Detection) is – A Distance based approach towards detecting outliers in very high- dimensional datasets. – Unbiased w.r.t. the dimensionality of different subspaces. – Capable of producing ranking of outliers HighDOD is composed of following 3 algorithms – OutlierDetection – CandidateExtraction – SubspaceMining Algorithm OutlierDetection examine subspaces of dimensionality up to some threshold m = O(logN) as suggested by Aggarwal and Ailon in [3, 4]

Algorithm 1: Outlier Detection Carry out a bottom-up exploration of all subspaces of up to a dimensionality of m = O(logN)

Estimate the data points’ local densities by using a kernel density estimator and choose βn data points with the lowest estimates as potential candidates. Algorithm 2: CandidateExtraction

Algorithm 3: SubspaceMining This procedure is used to update the set of outliers TopOut with 2n candidate outliers extracted from a subspace S.

Empirical Results and Analysis Authors have compared HighDOD with DenSamp, HighOut, PODM and LOF. Experiments have been performed to compare detection accuracy and scalability. Precision-Recall trade-off curve is used to evaluate the quality of an unordered set of retrieved items. Datasets – 4 Real data sets from UCI Repository have been used.

Comparison of Detection Accuracy Detection accuracy of HighDOD, DenSamp, HighOut, PODM and LOF

Comparison of Scalability Since PODM and LOF yields unsatisfactory accuracy, they are not included in this experiment. Scalability test is done with CorelHistogram (CH) dataset consisting of records in 32-dimensional space. Scalability of HighDOD, DenSamp and HighOut

Conclusion Work proposed a new outlier detection technique which is dimensionality unbiased. Extends distance-based anomaly detection to subspace analysis. Facilitates the design of ranking-based algorithm. Introduced HighDOD, a ranking-based technique for subspace outlier mining.

Precision-Recall Curve Precision and recall are used to evaluate the quality of an unordered set of retrieved items. Recall is the measure of the ability of a system to present all the relevant items. Precision is the measure of the ability of a system to present only relevant items.

References [1] Wikipedia [2] Angiulli, F., Pizzuti, C.: Outlier mining in large high- dimensional data sets. IEEE Trans. Knowl. Data Eng., [3] Aggarwal, C.C., Yu, P.S.: An effective and efficient algorithm for high-dimensional outlier detection. VLDB Journal, [4] Ailon, N., Chazelle, B.: Faster dimension reduction. Commun. CACM, 2010.