3.4 density and grid methods
- 1. Clustering Density and Grid Based 1
- 2. 2 Density based methods Clusters – dense regions of objects Low density regions – Noise DBSCAN Density Based Spatial Clustering of Applications with Noise OPTICS Ordering Points To Identify the Clustering Structure DENCLUE DENsity Based CLUstEring
- 3. 3 DBSCAN Cluster – maximal set of density connected points Grows regions with sufficiently high density into clusters ε-neighborhood MinPts and Core object Directly Density Reachable An object p is directly density reachable from object q if p is within the ε-neighborhood of q and q is a core object p q MinPts = 5
- 4. 4 DBSCAN Density Reachable An object p is density reachable from q, if there is a chain of objects p1, …pn, p1=q and pn=p such that pi+1 is directly density reachable from pi p q p1
- 5. 5 DBSCAN Density Connected An object p is density connected to object q if there is an object o such that both p and q are density reachable from o. p q o
- 6. 6 DBSCAN Arbitrarily select a point p Retrieve all points density-reachable from p If p is a core point, a cluster is formed. If p is a border point, no points are density-reachable from p, then DBSCAN visits the next point of the database. Continue the process until all of the points have been processed. Complexity : O(n log n) / O(n2 )
- 7. 7 OPTICS: A Cluster-Ordering Method OPTICS: Ordering Points To Identify the Clustering Structure Produces a special order of the database with respect to its density-based clustering structure Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure Can be represented graphically or using visualization techniques
- 8. OPTICS In DBSCAN, for a constant MinPts value, density based clusters with respect to a higher density (lower value of ε) are completely contained in lower density sets. DBSCAN is extended so that Objects are processed in a specific order. Selects an object that is density-reachable with respect to lowest ε value Core distance of an object p : smallest ε’ value that makes {p} a core object Reachability distance of an object q with respect to p = max (core-distance of p, d(p,q)) 8
- 9. OPTICS Complexity : O(n log n) 9
- 11. 11 DENCLUE: using density functions DENsity-based CLUstEring Major features Solid mathematical foundation Good for data sets with large amounts of noise Allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets Significantly faster than existing algorithm (faster than DBSCAN by a factor of up to 45) But needs a large number of parameters
- 12. 12 Influence function: describes the impact of a data point within its neighborhood. x, y – objects in Fd – d-dimensional input space Influence of object y on x is: Can be determined by distance: Overall density of the data space can be calculated as the sum of the influence function of all data points. Clusters can be determined mathematically by identifying density attractors. Density attractors are local maximal of the overall density function. DENCLUE ),()( yxfxf B y B = otherwise1or),(0),( σ>= yxdifyxfsquare f x y eGaussian d x y ( , ) ( , ) = − 2 2 2σ
- 13. 13 Density attractor – Local maxima of overall density function A point x is said to be density attracted to a density attractor x* if there exists a set of points x0, x1,..xk such that x0 = x and xk =x* and the gradient of xi-1 is in the direction of xi DENCLUE
- 14. DENCLUE Center defined clusters For a density attractor x* - a subset of points that are density attracted by x* and where density function x* is no less than threshold ξ Others are outliers Arbitrary shape cluster Set of density attractors and set of Cs There should be a path from each density attractor to another where density function value for each point is no less that ξ 14
- 15. DENCLUE 15
- 16. 16 Grid Based Methods Uses a Multi-resolution grid data structure Quantizes space into a finite number of cells that form a grid structure Fast processing time STING WaveCluster CLIQUE – CLustering In QUEst
- 17. 17 STING STatistical Information Grid Spatial area is divided into rectangular cells Several levels of cells – at different levels of resolution High level cell is partitioned into several lower level cells Statistical attributes are stored in cell Mean, Maximum, Minimum
- 18. 18 STING
- 19. 19 STING Parameters of higher level cells are computed from those at lower levels To answer queries Identify level Estimate cell’s relevance to query Process relevant cells at lower levels Continue to lowest level
- 20. 20 STING Computation is query independent Parallel processing – supported Data is processed in a single pass Quality depends on granularity
- 21. 21 WaveCluster A multi-resolution clustering approach which applies wavelet transform to the feature space A wavelet transform is a signal processing technique that decomposes a signal into different frequency sub- band. Both grid-based and density-based Input parameters: # of grid cells for each dimension the wavelet, and the # of applications of wavelet transform.
- 22. 22 WaveCluster Using wavelet transform to find clusters Summarises the data by imposing a multidimensional grid structure onto data space These multidimensional spatial data objects are represented in a n-dimensional feature space Apply wavelet transform on feature space to find the dense regions in the feature space Apply wavelet transform multiple times which result in clusters at different scales from fine to coarse
- 23. 23 Quantization
- 25. 25 WaveCluster Reasons for using Wavelet transformation in clustering Unsupervised clustering It uses filters to emphasize region where points cluster, but simultaneously to suppress weaker information in their boundary Effective removal of outliers Multi-resolution Cost efficiency Major features: Complexity O(N) Detect arbitrary shaped clusters at different scales Not sensitive to noise, not sensitive to input order Only applicable to low dimensional data
- 26. 26 CLIQUE (Clustering In QUEst) Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space CLIQUE can be considered as both density-based and grid-based It partitions each dimension into the same number of equal length interval It partitions an m-dimensional data space into non-overlapping rectangular units A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter A cluster is a maximal set of connected dense units within a subspace