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Clustering Methods Professor: Dr. Mansouri Presented by : Muhammad Abouei &Mohsen Ghahremani Manesh

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Clustering Methods Density-Based Clustering Methods DBSCAN (Density Based Spatial Clustering of Applications with Noise) OPTICS (Ordering Points To Identify the Clustering Structure) DENCLUE (DENsity-based CLUstEring) Grid-based Clustering 2

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Density Based Clustering 3

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DBSCAN Concepts ε -neighborhood: Points within ε distance (radius) of a point. MinPts: minimum number of points in cluster (ε-neighborhood of that point). ε-neighborhood of q ε-neighborhood of p MinPts = 5 where ε and MinPts are a user-defined function. 4

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DBSCAN Concepts Density : number of points within a specified radius ( ε ) Density(p)=5 5

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DBSCAN Concepts Core point : A point is a core point if it has more than a specified number of points (MinPts) within ε These are points that are at the interior of a cluster ε-neighborhood of q ε-neighborhood of p p is a core point (MinPts = 5) q is not a core point. 6

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DBSCAN Concepts Directly density-reachable : point p is directly density- reachable from a point q w.r.t. ε, MinPts if 1. p belongs to ε -neighborhood of q, 2. q is a core point, MinPts = 4 p is DDR from q. q is not DDR from p! DDR is an asymmetric relation. 7

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DBSCAN Concepts Density-reachable : A point p is density-reachable from a point q w.r.t. ε, MinPts if there is a chain of points P 1, …, P n, P 1 =q, P n =p such that P i +1is directly density-reachable from P i. Or, point p is density-reachable form q, if there is a path (chain of points) from p to q consisting of only core points. MinPts = 4 p is DR from q. q is not DR from p! p is not core. DR is an asymmetric relation. 8

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DBSCAN Concepts Density-connectivity: point p is density-connected to point q w.r.t. ε, MinPts if there is a point r such that both, p and q are density-reachable from r w.r.t. ε and MinPts. MinPts = 4 p and q are density-connected. DC is an symmetric relation. 9

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DBSCAN Concepts Border point : A border point has fewer than MinPts within ε, but is in the neighborhood of a core point MinPts =5 ε = circle radius 10

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DBSCAN Concepts Noise (outlier) point : is any point that is not a core point nor a border point. MinPts =5 ε = circle radius 11

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DBSCAN Concepts DBSCAN relies on a density-based notion of cluster. Cluster : a cluster C is a non-empty set of density-connected points that is maximal w.r.t. density-reachability. Maximality: For all p, q; if q ∈ C and if p is density-reachable from q w.r.t. ε and MinPts, then also p ∈ C. MinPts = 3 ε = circle radius 12

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DBSCAN Algorithm Arbitrary select a point p Retrieve all points density-reachable from p w.r.t. ε and MinPts. If p is a core point, a cluster is formed. If p is a border point, no points are density-reachable from p and DBSCAN visits the next point of the database. Continue the process until all of the points have been processed. 13

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DBSCAN MinPts = 4 14

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DBSCAN DBSCAN is Sensitive to Parameters. MinPts = 4 15

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DBSCAN Core, Border and Noise Points: MinPts = 4, ε = 10 Original Points Point types: core, border and noise 16

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DBSCAN When DBSCAN works well: Resistant to Noise Can handle clusters of different shapes and sizes Original PointsClusters 17

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DBSCAN When DBSCAN does not work well: Varying densities High-dimensional data 18

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DBSCAN Complexity If a spatial index (ex, kd-tree, R*-tree) is used, the computational complexity of DBSCAN is O(n.logn), where n is the number of database objects. Otherwise, it is O(n 2 ). 19

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OPTICS Core distance: smallest ε that makes it a core object. If p is not core, it is undefined. Core Distance of p or ε′ : distance between p and its 4-thNN. MinPts = 5 ε = 3 cm 20

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OPTICS Reachability distance: of r w.r.t. p is the greater value of the core distance of p and the Euclidean distance between p & r. If p is not a core object, distance reachability between p & q is undefined. reachability-distance ε, MinPts (p, r) = ε′ reachability-distance ε, MinPts (p, r′) = d(p, r′ ) MinPts = 5 ε = 3 cm 21

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OPTICS 22

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OPTICS 23

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OPTICS 24

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OPTICS 25

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OPTICS 26

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OPTICS 27

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OPTICS 28

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OPTICS 29

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OPTICS 30

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OPTICS 31

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OPTICS 32 Color image segmentation using density-Based clustering

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DENCLUE DENCLUE (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 Significant faster than existing algorithm (faster than DBSCAN by a factor of up to 45) But needs a large number of parameters 33

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DENCLUE Technical Essence Uses grid cells but only keeps information about grid cells that do actually contain data points and manages these cells in a tree- based access structure. 34

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DENCLUE Technical Essence DENCLUE is based on the following concepts: Influence function Density function Density attractors. 35

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DENCLUE 36

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DENCLUE Density function :The density function at x based on a data space of N points; i.e. D = {x 1,…, x N }; is defined as the sum of the influence function of all data points at x : The goal of the definition: Identify all “significant” local maxima, x j *, j=1,…,m of f D (x) Create a cluster C j for each x j * and assign to C j all points of D that lie within the “region of attraction” of x j *. 37

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DENCLUE Example: Density Computation D={x1,x2,x3,x4} f DGaussian (x) = influence(x 1 )+influence(x 2 )+influence(x 3 )+influence(x 4 ) =0.04+0.06+0.08+0.6=0.78 Remark: the density value of y would be larger than the one for x. 38

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DENCLUE Density attractors :Density attractors are local maxima of the overall density function f D (x). Clusters can then be determined mathematically by identifying density attractors. A hill-climbing algorithm guided by the gradient can be used to determine the density attractor of a set of data points. 39

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DENCLUE Density-attracted : A point x is density-attracted to a density attractor x*, if there exists a set of points x 0, x 1, …, x k such that x 0 = x, x k = x* and the gradient of x i-1 is in the direction of x i for 0*
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"name": "DENCLUE Density-attracted : A point x is density-attracted to a density attractor x*, if there exists a set of points x 0, x 1, …, x k such that x 0 = x, x k = x* and the gradient of x i-1 is in the direction of x i for 0
*

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DENCLUE 41

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DENCLUE Multicenter defined clusters : Multicenter defined clusters are a set of center-defined clusters linked by a path of significance. 42

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DENCLUE 43

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DENCLUE 44

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DENCLUE 45

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Grid-based Using multi-resolution grid data structure Clustering complexity depends on the number of populated grid cells and not on the number of objects in the dataset Several interesting methods: CS Tree (Clustering Statistical Tree) STING WaveCluster 46

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Grid-based Basic Grid-based Algorithm 1.Define a set of grid-cells. 2.Assign objects to the appropriate grid cell and compute the density of each cell. 3.Eliminate cells, whose density is below a certain threshold τ. 4.Form clusters from contiguous (adjacent) groups of dense cells (usually minimizing a given objective function). 47

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Grid-based Fast: No distance computations, Clustering is performed on summaries and not individual objects; complexity is usually O(no_of_populated_grid_cells) and not O(no_of_objects ), Easy to determine which clusters are neighboring. 48

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References A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Prentice Hall, 1988. A.K. Jain and M. N. Murty and P.J. Flynn, Data Clustering: A Review, ACM Computing Surveys, vol 31. No 3,pp 264-323, 1999. A. L. N. Fred, J. M. N. Leitão, A New Cluster Isolation Criterion Based on Dissimilarity Increments, IEEE “Optimal grid-clustering: Toward breaking the curse of dimensionality in high- dimensional clustering,”in Proc. 25th VLDB Conf.,1999, pp. 506–517. 49

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