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Prénom Nom Document Analysis: Data Analysis and Clustering Prof. Rolf Ingold, University of Fribourg Master course, spring semester 2008

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© Prof. Rolf Ingold 2 Outline Introduction to data clustering Unsupervised learning Background of clustering K-means clustering & Fuzzy k-means Model based clustering Gaussian mixtures Principal Components Analysis

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© Prof. Rolf Ingold 3 Introduction to data clustering Statistically representative datasets may include implicitly valuable semantic information the aim is to group samples into meaningful classes Various terminologies refer to that principle data clustering, data mining unsupervised learning taxonomy analysis knowledge discovery automatic inference In this course we address two aspects Clustering : perform unsupervised classification Principal Components Analysis : reduce feature spaces

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© Prof. Rolf Ingold 4 Application to document analysis Data analysis and clustering can potentially be applied at many levels of document analysis at pixel level for foreground/background separation on connected components for segmentation or character/symbol recognition on blocks for document understanding on entire pages to perform document classification ...

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© Prof. Rolf Ingold 5 Unsupervised classification Unsupervised learning consists of inferring knowledge about classes by using unlabeled training data, i.e. where samples are not assigned to classes There are at least five good reasons for performing unsupervised learning no labeled samples are available or ground-truthing is too costly useful preprocessing to produce ground-truthed data the classes are not known a priori for some problems classes are evolving over time useful for studying relevant features

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© Prof. Rolf Ingold 6 Background of data clustering (1) Clusters are formed following different criteria members of a class share same or closely related properties members of class have small distances or large similarities members of a class are clearly distinguishable from members of other classes Data clustering can be performed on various data types nominal types (categories) discrete types continuous types time series

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© Prof. Rolf Ingold 7 Background of data clustering (2) Data clustering requires similarity measures various similarity and dissimilarity measures (often in [0,1]) distances (with triangular inequality property) Feature transformation and normalization is often required Clusters may be center based : members are close to a representative model chain based : members are close to at least one other member There is a distinction between hard clustering : each sample is member of exactly one class fuzzy clustering : samples have membership functions (probabilities) associated to each class

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© Prof. Rolf Ingold 8 K-means clustering k-means clustering is a popular algorithm for unsupervised classification assuming the following information to be available the number of classes c a set of unlabeled samples x 1 x n Classes are modeled by their centers 1 c and each sample x k is assigned to the classes of the nearest center The algorithm works as follows initialize the vectors 1 c randomly assign each sample x k to the class that minimizes ||x k m || 2 update the centers 1 c using stop when classes do no more change

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© Prof. Rolf Ingold 9 Illustration of k-means algorithm

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© Prof. Rolf Ingold 10 Convergence of k-means algorithm The k-means algorithm always converges to a local minimum depending of the centers' initialization

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© Prof. Rolf Ingold 11 Fuzzy k-means Fuzzy k-means is a generalization taking into account a membership function P * ( i |x k ) normalized as follows The clustering method consists in minimizing the following cost (where b is fixed) the centers are updated using and the membership functions are updated using

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© Prof. Rolf Ingold 12 Model based clustering In this approach, we assume the following information to be available the number of classes c the a priori probability of each class P( i ) the shape of the feature densities p(x| j, j ) with parameter j a dataset of unlabeled samples {x 1,...,x n }, supposed to be drawn by selecting the class i with probability P( i ) then, selecting x k according to p(x| j, j ) The goal is to estimate the parameter vector 1 c t

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© Prof. Rolf Ingold 13 Maximum likelihood estimation The goal is to estimate that maximizes the likelihood of the set D={x 1,...,x n }, that is Equivalently we can also maximize its logarithm, namely

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© Prof. Rolf Ingold 14 Maximum likelihood estimation (cont.) To find the solution, we require the gradient is zero By assuming that i and j are statistically independent, we can state By combining with the Bayesian rule we finally obtain that for i=1,...,c the estimation of i must satisfy the condition

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© Prof. Rolf Ingold 15 Maximum likelihood estimation (cont.) In most cases the equation can not be solved analytically Instead an iterative gradient descending approach can be used to avoid convergence to a local minimum, an approximate initial should be used

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© Prof. Rolf Ingold 16 Application to Gaussian mixture models We consider the case where of a mixture of Gaussians where the parameter i, i et P( i ) have to be determined By applying the gradient method we can estimate i, i iteratively where

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© Prof. Rolf Ingold 17 Problem with local minimums Maximum likelihood estimation by the gradient descending method can converge to local minimums

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© Prof. Rolf Ingold 18 Conclusion about clustering Unsupervised learning allows to extract valuable information from unlabeled training data it is very useful in practice analytical approaches are generally not practicable iterative methods may be used, but they sometimes converge to local minimums sometimes even the number of classes is not known; in such cases clustering can be performed with several hypothesis and the best solution can be selected by information theory Clustering does not work well in high dimensions there is an interest to reduce the dimensionality of the feature space

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© Prof. Rolf Ingold 19 Objective of Principle Component Analysis (PCA) From a Bayesian point of view, the more features are used, more accurate are the classification results But the higher the dimension of the feature space, more difficult it is to get reliable models PCA can be seen as a systematic way to reduce the dimensionality of the feature space by minimizing the loss of information

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© Prof. Rolf Ingold 20 Center of gravity Let us consider a set of samples described by their feature vectors {x 1,x 2,...,x n } The point that best represents the entire set is x 0 minimizing This point corresponds to the center of gravity since

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© Prof. Rolf Ingold 21 Projection on a line The goal is to find the line crossing the center of gravity m that best approximate the sample set {x 1,x 2,...,x n } let vector e be the unit vector of its direction; then the equation of the line is x = m+ae where a is a scalar representing the distance of x from m the optimal solution is given by minimizing the squared error First, the values for a 1,a 2,...,a n minimizing this function are given that is a k = e t (x k - m) corresponding to the orthogonal projection

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© Prof. Rolf Ingold 22 Scatter matrix The scatter matrix of the set {x 1,x 2,...,x n } is defined as it differs from the covariance matrix by a factor n-1

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© Prof. Rolf Ingold 23 Finding the best line The best line (minimizing) can be obtained by where S is the scatter matrix of the set {x 1,x 2,...,x n }

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© Prof. Rolf Ingold 24 Finding the best line (cont.) To minimize J 1 (e) we must maximize e t Se using the method of Lagrange multipliers and by differentiating we obtain Se = e and e t Se = e t e = This means that to maximize e t Se we need to select the eigenvector corresponding to the largest eigenvalue of the scatter matrix

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© Prof. Rolf Ingold 25 Generalization to d dimensions Principal components analysis can be applied for any dimension d up to the dimension of the original feature space each sample is mapped on a hyperplane defined by the objective function to minimize is the solution for e is given by the eigenvectors corresponding to the d highest eigenvalues

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