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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

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**Density Based Clustering**

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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.

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

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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 p belongs to ε -neighborhood of q, q is a core point, MinPts = 4 p is DDR from q. q is not DDR from p! DDR is an asymmetric relation.

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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 P1, …, Pn , P1=q, Pn=p such that Pi +1is directly density-reachable from Pi . 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.

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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.

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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

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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

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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

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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.

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DBSCAN MinPts = 4

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DBSCAN DBSCAN is Sensitive to Parameters. MinPts = 4

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**Original Points Point types: core, border**

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

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**DBSCAN When DBSCAN works well: Resistant to Noise**

Can handle clusters of different shapes and sizes Original Points Clusters

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**DBSCAN When DBSCAN does not work well: Varying densities**

High-dimensional data

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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(n2).

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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

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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

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OPTICS

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OPTICS

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OPTICS

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OPTICS

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OPTICS

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OPTICS

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OPTICS

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OPTICS

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OPTICS

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OPTICS

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OPTICS Color image segmentation using density-Based clustering

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**DENCLUE Major features DENCLUE (DENsity-based CLUstEring)**

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

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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.

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**DENCLUE Technical Essence DENCLUE is based on the following concepts:**

Influence function Density function Density attractors.

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DENCLUE Influence function : The influence function f y(x) for a point 𝑦∈𝐷(data space) at point x is a positive function that decays to zero as x “moves away” from 𝑦 𝑑 𝑥,𝑦 →∞ . Typical examples are: and where σ is a user-defined function.

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DENCLUE Density function :The density function at x based on a data space of N points; i.e. D = {x1,…, xN}; 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, xj*, j=1,…,m of f D(x) Create a cluster Cj for each xj* and assign to Cj all points of D that lie within the “region of attraction” of xj*.

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**DENCLUE Example: Density Computation D={x1,x2,x3,x4}**

f DGaussian (x) = influence(x1)+influence(x2)+influence(x3)+influence(x4) = =0.78 Remark: the density value of y would be larger than the one for x.

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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.

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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 x0, x1, …, xk such that x0 = x , xk = x* and the gradient of xi-1 is in the direction of xi for 0<i<k.

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DENCLUE Center-Defined Cluster :A center-defined cluster (w.r.t. to σ, ε) for a density attractor x* is a subset C ⊆ D, with x ∈ C being density-attracted by x* and f D(x)≥ ε. Outlier: Point x ∈ D is called outlier if it is density-attracted by a local maximum xo* with f D(xo*) < ε.

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

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DENCLUE An arbitrary-shape cluster : An arbitrary-shape cluster (w.r.t. to σ, ε) for a set of density attractors X is a subset C ⊆ D, where ∀𝑥∈𝐶 ∃ 𝑥 ∗ ∈𝑋: 𝑓 𝐷 𝑥 ∗ ≥ε , x is density-attracted to 𝑥 ∗ , and ∀ 𝑥 1 ∗ , 𝑥 2 ∗ ∈𝑋 : ∃ a path P from 𝑥 1 ∗ to 𝑥 2 ∗ with ∀ 𝑝∈𝑃: 𝑓 𝐷 𝑝 ≥ ε

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DENCLUE Note : that the number of clusters found by DENCLUE varies depending on σ, ε.

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**DENCLUE DENCLUE is able to detect arbitrarily shaped clusters.**

The algorithm deals with noise very satisfactory. The worst-case time complexity of DENCLUE is O(N.log2N). Experimental results indicate that the average time complexity is O(log2N). It works efficiently with high-dimensional data. DENCLUE needs at least 3 parameters to be determined, i.e. σ, ε ,εc.

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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

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**Grid-based Basic Grid-based Algorithm Define a set of grid-cells.**

Assign objects to the appropriate grid cell and compute the density of each cell. Eliminate cells, whose density is below a certain threshold τ. Form clusters from contiguous (adjacent) groups of dense cells (usually minimizing a given objective function).

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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.

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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 , 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.

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