1. RDF: A Density-based Outlier Detection Method Using Vertical Data Representation Dongmei Ren, Baoying Wang, William Perrizo North Dakota State University, U.S.A

2. Introduction  Related Work Breunig et al. [6] first proposed a density-based approach to mining outliers over datasets with different densities. Papadimitriou & Kiragawa [7] introduce local correlation integral (LOCI). Not efficient.  Contributions of this paper 1. a relative density factor (RDF)  RDF expresses the same amount of information as LOF (local outlier factor)[6] and MDEF(multi-granularity deviation factor)[7]  but RDF is easier to compute; 2. RDF-based outlier detection method  it efficiently prunes the data points which are deep in clusters  It detects outliers only within the remaining small subset of the data; 3. a vertical data representation in P-trees  P-Trees improve the efficiency of the method further.

3. Definitions Direct DiskNbr x Indirect DiskNbr Definition 1: Disk Neighborhood --- DiskNbr(x,r)  Given a point x and radius r, the disk neighborhood of x is defined as a set DiskNbr(x, r)={x ’  X | d(x-x ’ )  r}, where d(x-x ’ ) is the distance of x and x ’  Direct & indirect neighbors of x Definition 2: Density of DiskNbr(x, r) --- Dens (x,r), where dim is the number of dimensions

4. Definitions (Continued) Definition 3: Relative Density Factor (RDF) of point x with radius r -- RDF(x,r) RDF is used to measure outlierness. Outliers are points with high RDF values. Direct DiskNbr x Indirect DiskNbr Special case: RDF between DiskNbr(x,r) and {DiskNbr(x,2r)- DiskNbr(x,r)}

5. The Proposed Outlier Detection Method  Given a dataset X, the proposed outlier detection method is processed by: Find Outliers Prune Non-outliers Our method prunes non-outliers (points deep in clusters) efficiently; find outliers over the remaining small subset of the data, which consists of points on cluster boundaries and real outliers.

6. Finding Outliers Three possible distributions with regard to RDF: (a) prune all neighbors, call “ Pruning Non-outliers ” procedure; (b) prune all direct neighbors of x, calculate RDF for each indirect neighbor. (c) x is an outlier, prune indirect neighbors of x. (a) 1/(1+ε) ≤ RDF ≤ (1+ε) (b) RDF (1+ε) x

7. Finding Outliers using P-Trees  P-Tree based direct neighbors --- PDN x r For point x, let X= (x 1,x 2, …,x n ) or X = (x 1,m-1, … x 1,0 ), (x 2,m-1, … x 2,0 ), … (x n,m-1, … x n,0 ), where x i,j is the j th bit value in the i th attribute. For the i th attribute, PDN xi r = P x ’ >xi-r AND P x ’ xi+r For muti-attributes, |DiskNbr(x,r)|= rc(PDN x r )  P-Tree based indirect neighbors --- PIN x r PIN x r = (OR q  Nbr(x,r) PDN q r ) AND PDN x’ r  Pruning is done by P-Trees ANDing based on the above three distributions (a),(c): PU = PU AND PDN x r AND PIN x ’ r (b): PU = PU AND PDN x r ; where PU is a P-tree representing unprocessed data points

8. Pruning Non-outliers  1/(1+ε) ≤ RDF ≤(1+ε)(density stay constant): continue expanding neighborhood by doubling the radius.  RDF < 1/(1+ε) (significant decrease of density): stop expanding, prune DiskNbr(x,kr), and call “ Finding Outliers ” Process;  RDF > (1+ε) (significant increase of density): stop expanding and call “ Pruning Non-outliers ”. The pruning is a neighborhood expanding process. It calculates RDF between {DiskNbr(x,2kr)-DiskNbr(x,kr)} and DiskNbr(x,kr) and prunes based on the value of RDF, where k is an integer.

9. Pruning Non-outliers Using P-Trees  We define ξ- neighbors: it represents the neighbors with ξ bits of dissimilarity with x, e.g. ξ = 1, 2... 8 if x is an 8-bit value  For point x, let X= (x 1,x 2, …,x n ) or X = (x 1,m, … x 1,0 ), (x 2,m, … x 2,0 ), … (x n,m, … x n,0 ), where x i,j is the j th bit value in the i th attribute. For the i th attribute, ξ- neighbors of x is calculated by  The pruning is accomplished by: PU = PU AND P X ξ ’, where P X ξ ’ is the complement set of P X ξ,where

10. RDF-based Outlier Detection Process Algorithm: RDF-based Outlier Detection using P-Trees Input: Dataset X, radius r, distribution parameter ε. Output: An outlier set Ols. // PU — unprocessed points represented by P-Trees; // |PU| — number of points in PU // PO --- outliers; //Build up P-Trees for Dataset X PU  createP-Trees(X); i  1; WHILE |PU| > 0 DO x  PU.first; //pick an arbitrary point x PO  FindOutliers (x, r, ε) ; i  i+1 ENDWHILE

11. “ Find Outliers ” and “ Prune Non-Outliers ” Procedures

12. Experimental Study  NHL data set (1996)  Compare with LOF, aLOCI LOF: Local Outlier Factor Method aLOCI: approximate Local Correlation Integral Method  Run Time Comparison  Scalability Comparison Start from 16,384, outperform in terms of scalability and speed

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14. Thank you!

15. Determination of Parameters  Determination of r Breunig et al. shows choosing miniPt = 10-30 work well in general [6] (miniPt-Neighborhood) Choosing miniPts=20, get the average radius of 20- neighborhood, r average. In our algorithm, r = r average =0.5  Determination of ε Selection of ε is a tradeoff between accuracy and speed. The larger ε is, the faster the algorithm works; the smaller ε is, the more accurate the results are. We chose ε=0.8 experimentally, and get the same result (same outliers) as Breunig ’ s, but much faster. The results shown in the experimental part is based on ε=0.8.