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Turn Waste into Wealth: On Simultaneous Clustering and Cleaning over Dirty Data Shaoxu Song, Chunping Li, Xiaoquan Zhang Tsinghua University Motivation • Dirty data commonly exist – Often a (very) large portion – E.g., GPS readings • Density-based clustering – Such as DBSCAN – Successfully identify noises – Grouping non-noise points in clusters – Discarding noise points KDD 2015 2 Mining Cleaning Useless Guide Make Valuable KDD 2015 Find 3 Mining + Repairing Repair Knowledge • Constraints • Rules Repaired (Dirty) Data • Density Discover KDD 2015 4 Discarding vs. Repairing • Simply discarding a large number of dirty points (as noises) could greatly affect clustering results (a) Clusters in ground truth C1 C2 (b) Clusters in dirty data without repairing C1 C2 C3 Noise (c) Clusters in dirty data with repairing C1 C2 • Propose to repair and utilize noises to support clustering • Basic idea: simultaneously repairing noise points w.r.t. the density of data during the clustering process KDD 2015 5 Density-based Cleaning • Both the clustering and repairing tasks benefit – Clustering: with more supports from repaired noise points – Repairing: under the guide of density information • Already embedded in the data • Rather than manually specified knowledge KDD 2015 6 Basics • DBSCAN: density-based identification of noise points – Distance threshold 𝞮 – Density threshold 𝞰 • 𝞮-neighbor: – if two points have distance less than 𝞮 • Noise point – With the number of 𝞮-neighbors less than 𝞰 – Not in 𝞮-neighbor of some other points that have 𝞮-neighbors no less than 𝞰 (core points) KDD 2015 7 Modification Repair [SIGMOD’05] [ICDT’09] • A repair over a set of points is a mapping λ : P → P • We denote λ(pi) the location of point pi after repairing • The ε-neighbors of λ(pi) after repairing is Cλ(pi) = { pj ∈ P | δ( λ(pi) , λ(pj) ) ≤ ε } KDD 2015 8 Repair Cost • Following the minimum change principle in data cleaning – Intuition: systems or humans always try to minimize mistakes in practice – prefer a repair close to the input • The repair cost ∆(λ) is defined as ∆(λ) = ∑i w( pi , λ(pi) ) – w( pi , λ(pi) ) is the cost of repairing a point pi to the new location λ(pi) – E.g., by counting modified data points KDD 2015 9 Problem Statement • Given a set of data points P, a distance threshold ε and a density threshold η • Density-based Optimal Repairing and Clustering (DORC) problem is to find a repair λ (a mapping λ : P → P ) such that (1) the repairing cost ∆(λ) is minimized, and (2) each repaired λ(pi) is either a core point or a board point – for each repaired λ(pi), either |Cλ(pi)| ≥ η (core points), – or |Cλ(pj)| ≥ η for some pj with δ(λ(pi),λ(pj)) ≤ ε All the points are utilized, no noise remains KDD 2015 10 Technique Concern • Simply repairing only the noise points to the closest clusters is not sufficient – e.g., repairing all the noise points to C1 does not help in identifying the second cluster C2 • Indeed, it should be considered that dirty points may possibly form clusters with repairing (i.e., C2) KDD 2015 11 Problem Solving • No additional parameters are introduced for DORC – besides the density and distance requirements η and ε for clustering • ILP formulation – Efficient solvers can be applied • Quadratic time approximation – via LP relaxation • Trade-off between Effectiveness and Efficiency – By grouping locally data points into several partitions KDD 2015 12 Experimental Results • Answers the following questions – By utilizing dirty data, can it form more accurate clusters? – By simultaneous repairing and clustering, in practice is the repairing accuracy improved compared with the existing data repairing approaches? dirty – How do the approaches scale? • Criteria truth RMS repair – Clustering Accuracy: purity and NMI – Repairing Accuracy: root-mean-square error (RMS) between truth and repair results KDD 2015 13 Artificial Data Set • Compared to existing methods without repairing – DBSCAN and OPTICS • Proposed DORC (ILP/Quadratic-time-approximation) shows 1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8 (a) e=15, h=6 DBSCAN ILP QDORC LDORC(t/e=1/5) LDORC(t/e=1/10) OPTICS 0.03 0.06 0.09 0.12 0.15 0.18 0.21 Dirty rate KDD 2015 Repairing error Clustering purity – Higher clustering purity 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 (b) e=15, h=6 ILP QDORC LDORC(t/e=1/5) LDORC(t/e=1/10) 0.03 0.06 0.09 0.12 0.15 0.18 0.21 Dirty rate 14 Real GPS Data • With errors naturally embedded, and manually labelled • Compared to Median Filter (MF) – A filtering technique for cleaning the noisy data in timespace correlated time-series • DORC is better than MF+DBSCAN 0.014 0.92 0.88 0.84 (a) e=8E-5, h=8 0.012 0.96 0.8 0.02 KDD 2015 (a) e=8E-5, h=8 Repairing error Clustering purity 1 DBSCAN MF+DBSCAN QDORC MF+QDORC OPTICS 0.06 0.1 Dirty rate 0.01 0.008 0.006 MF QDORC MF+QDORC 0.004 0.14 0.18 0.002 0.02 0.06 0.1 Dirty rate 0.14 0.18 15 Restaurant Data • Tabular data, with artificially injected noises – Widely considered in conventional data cleaning • Compared to FD (a) e=0.22, h=16 DBSCAN FD+DBSCAN QDORC FD+QDORC Clustering purity 0.95 0.9 0.85 0.8 0.75 0.7 0.1 KDD 2015 0.14 0.18 0.22 Dirty rate 0.26 0.3 Repairing error – A repairing approach under integrity constraints (Functional Dependencies), [name,address → city] 0.3 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.1 (a) e=0.22, h=16 FD QDORC FD+QDORC 0.14 0.18 0.22 Dirty rate 0.26 0.3 16 More results • Two labeled publicly available benchmark data, – Iris and Ecoli, from UCI • Normalized mutual information (NMI) clustering accuracy – Similar results are observed – DORC shows higher accuracy than DBSCAN and OPTICS (a) GPS 0.9 0.8 0.8 0.6 0.7 0.6 0.5 0.02 KDD 2015 DBSCAN MF+DBSCAN QDORC MF+QDORC OPTICS 0.06 0.1 Dirty rate (b) UCI 1 NMI NMI 1 DBSCAN OPTICS QDORC 0.4 0.2 0 0.14 0.18 Iris Ecoli Dataset 17 Summary • Preliminary density-based clustering can successfully identify noisy data but – without cleaning them • Existing constraint-based repairing relies on external constraint knowledge – without utilizing density information embedded inside the data • With the happy marriage of clustering and repairing advantages – both the clustering and repairing accuracies are significantly improved KDD 2015 18 References (data repairing) • [SIGMOD’05] P. Bohannon, M. Flaster, W. Fan, and R. Rastogi. A cost-based model and effective heuristic for repairing constraints by value modification. In SIGMOD Conference, pages 143–154, 2005. • [TODS’05] J. Wijsen. Database repairing using updates. ACM Trans. Database Syst., TODS, 30(3):722–768, 2005. • [PODS’08] W. Fan. Dependencies revisited for improving data quality. In PODS, pages 159–170, 2008. • [ICDT’09] S. Kolahi and L. V. S. Lakshmanan. On approximating optimum repairs for functional dependency violations. In ICDT, pages 53–62, 2009. KDD 2015 19 Thanks