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1 Outline Part 1: of data clustering Non-Supervised Learning and Clustering : Problem formulation cluster analysis : Taxonomies of Clustering Techniques : Data types and Proximity Measures : Difficulties and open problems Part 2: Clustering Algorithms Hierarchical methods : Single-link : Complete-link : Clustering Based on Dissimilarity Increments Criteria -- Ana Fred 1 From Single Clustering to Ensemble Methods - April 2009 Pattern Recognition Decision Making Supervised Learning : training samples, labeled by their category membership, are used to design a classifier. Labeled training patterns. Labels represent true categories of patterns : Based on a collection of samples without being told their categories. Learn the number of classes and the structure of each class using similarity between unlabeled training patterns. Datamining -- Ana Fred 2 From Single Clustering to Ensemble Methods - April

2 / Clustering : Learn the structure of multidimensional patterns. Mixture Densities Gaussian Mixture Decomposition» The probability structure is known with the exception of the values of the parameters Clustering Procedures : Find subclasses. Data description in terms of clusters or groups of data points that possess strong internal similarities Typical applications:. As a stand-alone tool to get insight into data distribution. As a preprocessing step for other algorithms 3 From Single Clustering to Ensemble Methods - April 2009 Cluster Analysis Organize data into sensible groupings (either as a grouping of patterns or a hierarchy of groups) Clustering : The process of grouping a set of objects into classes of similar objects (extracting hidden structure from data) Cluster : A collection of objects that are similar to one another within the same cluster and are dissimilar to the objects in other clusters 4 From Single Clustering to Ensemble Methods - April

3 Shape Clustering Right Ventricle from MR brain images Cistern from MR brain images The main cluster is drawn using multicolor dots, secondary clusters are drawn in red, green and magenta. Duta, Jain and Jolly, Automatic Construction of 2-D Shape Models, IEEE PAMI, May From Single Clustering to Ensemble Methods - April 2009 Shape Clustering Right Ventricle from MR brain images Cistern from MR brain images The main cluster is drawn using multicolor dots, secondary clusters are drawn in red, green and magenta. Duta, Jain and Jolly, Automatic Construction of 2-D Shape Models, IEEE PAMI, May From Single Clustering to Ensemble Methods - April

4 Identification of Writing Styles 122,000 online characters written by 100 writers Lexemes are identified by clustering data within each character class into subclasses: a string matching measure used to calculate distance between 2 characters Connell and Jain, Writer Adaptation for Online Handwriting Recognition, IEEE PAMI, Mar From Single Clustering to Ensemble Methods - April 2009 Segmentation of Natural Scenes Hermes, Zoller, Bumannn, Parametric Distributional Clustering for Image Segmentation, ECCV From Single Clustering to Ensemble Methods - April

5 What is a Cluster? A set of entities which are alike; entities from different clusters are not alike An aggregation of points such that the distance between any two points in a cluster is less than the distance between any point in the cluster and any point not in it. A relatively high density of points, surrounded by a relatively low density of points Ideal cluster: Compact and Isolated -- Ana Fred 9 From Single Clustering to Ensemble Methods - April 2009 Taxonomy of Clustering Approaches Two main strategies: Hierarchical Methods :Propose a sequence of nested data partitions in a hierarchical structure. Single-Link. Complete Link Partitional Methods :Organize patterns into a small number of clusters. K-means. Spectral clustering -- Ana Fred 10 From Single Clustering to Ensemble Methods - April

6 Taxonomy of Clustering Approaches Clustering Principles: Compactness : K-means : Complete-link : Histogram clustering : Pairwise data clustering Connectedness : Single-linkage : Dissimilarity Increments : Mean Shift clustering Separation : Normalized Cut : Spectral clustering 11 From Single Clustering to Ensemble Methods - April 2009 Taxonomy of Clustering Approaches Clustering Principles: Compactness : K-means : Complete-link : Histogram clustering : Pairwise data clustering Connectedness : Single-linkage : Dissimilarity Increments : Mean Shift clustering Separation : Normalized Cut : Spectral clustering 12 From Single Clustering to Ensemble Methods - April

7 Taxonomy of Clustering Approaches Clustering Principles: Compactness : K-means : Complete-link : Histogram clustering : Pairwise data clustering Connectedness : Single-linkage : Dissimilarity Increments : Mean Shift clustering Separation : Normalized Cut : Spectral clustering From Single Clustering to Ensemble Methods - April 2009 Taxonomy of Clustering Approaches Approaches: Model-based : Patterns can be given a simple and compact description in terms of. Parametrical distribution -- Parametric density approaches (Mixture models). A representative element, such as a centroid, median (central clustering, square-error clustering, k-means, k-medoids) or multiple prototypes per cluster (CURE) -- Prototype-based methods. Some geometrical primitives (lines, planes, circles, curves, surfaces) Shape fitting approaches : These approaches assume particular cluster shapes, partitions being in general obtained as a result of an optimization process using a global criterion 14 From Single Clustering to Ensemble Methods - April

8 Taxonomy of Clustering Approaches Graph-theoretical : Mostly explored in hierarchical methods that can be represented graphically as a tree or dendrogram. Agglomerative methods (Single-link, complete-link). Divisive approaches (ex. Based on Minimum Spanning Tree) : View clustering as a graph partitioning problem Non parametric density-based : Attempt to identify high density clusters separated by low density regions (local cluster criterion, such as density-connected points) (valley seeking clustering algorithms). DBSCAN, OPTICS, DENCLUE, CLIQUE. Discover clusters of arbitrary shape 15 From Single Clustering to Ensemble Methods - April 2009 Data Types in Clustering Problems Data representations: Vector data: n vectors in R d Proximity data: n x n pairwise proximity matrix :All types of data may be addressed by choosing adequate proximity measures 16 From Single Clustering to Ensemble Methods - April

9 Similarity and Dissimilarity Between Objects Distances are normally used to measure the similarity or dissimilarity between two data objects Metrics: : Positivity: d(a, b) >0 and d(a, b)=0, a=b : Symmetry property: d(a,b)=d(b,a). : Triangle inequality: d(a,c) d(a,b) + d(b,c). 17 From Single Clustering to Ensemble Methods - April 2009 Metric Models in Feature Spaces Minskowski distance: (Euclidean distance corresponds to r = 2) Maximum Value Metric:. Considers only most distant features 18 From Single Clustering to Ensemble Methods - April

10 Metric Models in Feature Spaces Absolute Value Metric, Manhattan Distance or City-block (r = 1) d ( a, b) d ( a, b) b a M 1 i i i 1 d Constant Manhattan distance curves:. Reduced computational time; does not penalize much the features with higher dissimilarity. In R 2 : dist 1 ((x 1,y 1 ),(x 2,y 2 ))= x 2 -x 1 + y 2 -y 1, city-block: It is not possible to make short-cuts through corners: it counts the number of blocks that is necessary to pass in order to move from one corner to another 19 From Single Clustering to Ensemble Methods - April 2009 Metric Models in Feature Spaces Euclidean Distance: 2 d ( a, b) d ( a, b) b a e 2 i i i 1 d. R2: dist2((x1,y1),(x2,y2))=((x2-x1)2+(y2-y1)2)1/2.. Emphasizes more features with higher dissimilarity. Mahalanobis Distance T 1 d ( x, y) x y x y Mahalanobis 20 From Single Clustering to Ensemble Methods - April

11 Dissimilarity based on String Editing operations.... The Levensthein distance between two strings s 1, s 2 2 *, D L (s 1, s 2 ), is defined as the minimum number of editing operations needed in order to transform s 1 into s From Single Clustering to Ensemble Methods - April 2009 The Weighted Levensthein distance between two strings s 1, s 2 2 *, is defined by where Normalized Weighted Levensthein distance 22 From Single Clustering to Ensemble Methods - April

12 String Editing operations and String Matching (a) String matching using editing operations. (b) Editing path. String matching. In (b), diagonal path elements represent substitutions, vertical segments correspond to insertions, and horizontal segments correspond to deletions. 23 From Single Clustering to Ensemble Methods - April 2009 Normalized Edit Distance Marzal and Vidal, Computation of normalized edit distance and applications, IEEE PAMI, From Single Clustering to Ensemble Methods - April

13 Dissimilarity based on Error-Correcting Parsing [Fu] : distance between strings based on the modelling of string structure by means of grammars and on the concept of error-correcting parsing : the distance between a string and a reference string is given by the error-correcting parser as the weighted Levensthein distance between the string and the nearest (in terms of edit operations) string generated by the grammar inferred from the reference string (thus exhibiting a similar structure): 25 From Single Clustering to Ensemble Methods - April 2009 ECP distance From Single Clustering to Ensemble Methods - April

14 Dissimilarity based on Error-Correcting Parsing [Fu] : distance between strings based on the modelling of string structure by means of grammars and on the concept of error-correcting parsing : the distance between a string and a reference string is given by the error-correcting parser as the weighted Levensthein distance between the string and the nearest (in terms of edit operations) string generated by the grammar inferred from the reference string (thus exhibiting a similar structure): : In order to preserve symmetry 27 From Single Clustering to Ensemble Methods - April 2009 Grammar Complexity-based Similarity The basic idea is that, if two sentences are structurally similar, then their joint description will be more compact than their isolated description due to sharing of rules of symbol composition; the compactness of the representation is quantified by the grammar complexity, and the similarity is measured by the ratio of decrease in grammar complexity where C(G si ) denotes grammar complexity. Fred, Clustering of Sequences using a Minimum Grammar Complexity Criterion, ICGI 1996 Fred. Similarity measures and clustering of string patterns. In Dechang Chen and Xiuzhen Cheng, editors, Pattern Recognition and String Matching, Kluwer Academic, 2002, 28 From Single Clustering to Ensemble Methods - April

15 RDGC Similarity 29 From Single Clustering to Ensemble Methods - April 2009 Grammar Complexity-based Similarity RDGC Let G=(V N,, R, ) be a context-free grammar, where V N, are the sets of nonterminal and terminal symbols, respectively, is the grammar s start symbol and R is the set of productions written in the form: Let 2 (V N ) *, be a grammatical sentence of length n, in which the symbols a 1, a 2,, a m appear k 1, k 2,, k m times, respectively. The complexity of the sentence, C( ), is given by [Fu] The complexity of the grammar G is defined as 30 From Single Clustering to Ensemble Methods - April

16 Minimum Code Length-based Similarity : Based on Solomonoff s code: a string is represented by a triplet where a coded string is obtained in an iterative procedure where, in each step, intermediate codes are produced by defining sequences of two symbols, which are represented by special symbols, and rewriting the sequences using them. Compact codes are produced when sequences exhibit local or distant inter-symbol interactions.. Code length: sum of the lengths of the descriptions of the three part code above : Extension to sets of strings Fred and Leitão, A Minimum Code Length Technique for Clustering of Syntactic Patterns, ICPR 1996 Fred. Similarity measures and clustering of string patterns. In Dechang Chen and Xiuzhen Cheng, editors, Pattern Recognition and String Matching, Kluwer Academic, 2002, 31 From Single Clustering to Ensemble Methods - April 2009 Minimum Code Length-based Similarity : The basic idea is that global compact codes are produced by considering the inter-symbol dependencies on the ensemble of the strings. The quantification of this reduction in code length forms the basis of the similarity measure designated by Normalized Ratio of decrease in code length - NRDCL with 32 From Single Clustering to Ensemble Methods - April

17 Requirements of Clustering in Data Mining Discovery of clusters with arbitrary shape Ability to deal with different types of attributes Scalability Minimal requirements for domain knowledge to determine input parameters Insensitivity to the order of input records Ability to deal with noisy data High dimensionality -- Ana Fred 33 From Single Clustering to Ensemble Methods - April 2009 Issues in Clustering Which similarity measure and features to use? How many clusters? Which is the best clustering method? Are the individual clusters and the partition valid? How to choose algorithmic parameters? K-means clustering of uniform data (k=4) K-means using Euclidean (blue) and Mahalanobis distance (k=2) (red) -- Ana Fred 34 From Single Clustering to Ensemble Methods - April

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