10. MLSP intro. (Clustering: K-means, EM, GMM, etc.)
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1 10. MLSP intro. (Clustering: K-means, EM, GMM, etc.) Rahil Mahdian LSV Lab, Saarland University, Germany
2 What is clustering? Clustering is the classification of objects into different groups, or more precisely, the partitioning of a data set into subsets (clusters), so that the data in each subset (ideally) share some common trait - often according to some defined distance measure. 2
3 Types of clustering 1.Hierarchical algorithms: these find successive clusters using previously established clusters. 1. Agglomerative ("bottom-up"): Agglomerative algorithms begin with each element as a separate cluster and merge them into successively larger clusters. 2. Divisive ("top-down"): Divisive algorithms begin with the whole set and proceed to divide it into successively smaller clusters. 2. Partitional clustering: Partitional algorithms determine all clusters at once. They include: K-means and derivatives Fuzzy c-means clustering QT clustering algorithm Self Organizing Maps 3
4 K-means clustering Given a K, find a partition of K clusters to optimize the chosen partitioning criterion (cost function). (Converges to Local Minimum) o For global optimum: exhaustively search all partitions The K-means algorithm: a heuristic method o K-means algorithm (MacQueen 67): each cluster is represented by the center of the cluster and the algorithm converges to stable centroids of clusters. o K-means algorithm is the simplest partitioning method for clustering analysis and widely used in data mining applications. 4
5 K-means clustering-alg. d euc ( x, y) n i 1 ( x i y i 2 ) Choose a number of clusters k Initialize cluster centers 1, k Could pick k data points and set cluster centers to these points Or could randomly assign points to clusters and take means of clusters For each data point, compute the cluster center it is closest to (using some distance measure) and assign the data point to this cluster Re-compute cluster centers (mean of data points in cluster) Stop when there are no new reassignments 5
6 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 6
7 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 7
8 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop 8
9 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 9
10 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 10
11 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 11
12 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 12
13 K-means BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 13
14 K-means STOP! BCS Summer School, Exeter, 2003Christopher M. Bishop BCS Summer School, Exeter, 2003Christopher M. Bishop 14
15 Review of concepts- Likelihood Function Data set Assume observed data points generated independently Viewed as a function of the parameters, this is known as the likelihood function BCS Summer School, Exeter, 2003 Christopher M. Bishop
16 Review of concepts- The Gaussian Distribution Multivariate Gaussian mean covariance Define precision to be the inverse of the covariance In 1-dimension BCS Summer School, Exeter, 2003 Christopher M. Bishop
17 Review of concepts- Maximum Likelihood Set the parameters by maximizing the likelihood function Equivalently maximize the log likelihood BCS Summer School, Exeter, 2003 Christopher M. Bishop
18 Maximum Likelihood Solution Maximizing w.r.t. the mean gives the sample mean Maximizing w.r.t covariance gives the sample covariance BCS Summer School, Exeter, 2003 Christopher M. Bishop
19 Mixture of probabilities (Gaussian)- motivated
20 Contours of Probability Distribution BCS Summer School, Exeter, 2003 Christopher M. Bishop
21 Example: Mixture of 3 Gaussians BCS Summer School, Exeter, 2003 Christopher M. Bishop
22 Gaussian Mixtures Linear super-position of Gaussians Normalization and positivity require Can interpret the mixing coefficients as prior probabilities BCS Summer School, Exeter, 2003 Christopher M. Bishop
23 Problems and Solutions How to maximize the log likelihood solved by expectation-maximization (EM) algorithm How to avoid singularities in the likelihood function solved by a Bayesian treatment How to choose number K of components also solved by a Bayesian treatment BCS Summer School, Exeter, 2003 Christopher M. Bishop
24 Expectation Maximization (EM) Iterative method for parameter estimation where you have missing data Has two steps: Expectation (E) and Maximization (M) Applicable to a wide range of problems Old idea (late 50 s) but formalized by Dempster, Laird and Rubin in 1977 Subject of much investigation. See McLachlan & Krishnan book
25 Example: EM START! BCS Summer School, Exeter, 2003Christopher M. Bishop 25
26 EM BCS Summer School, Exeter, 2003Christopher M. Bishop 26
27 EM BCS Summer School, Exeter, 2003Christopher M. Bishop 27
28 EM BCS Summer School, Exeter, 2003Christopher M. Bishop 28
29 EM STOP! BCS Summer School, Exeter, 2003Christopher M. Bishop 29
30 Gaussian Mixture Model marginalization Mixture proportion Mixture component From: Iain Murray, Edinburgh 30
31 Maximum Likelihood Fitting responsibilities, Indicator funtion r ik : How much cluster k, takes the responsibility of generating the data i? N: # of data. r k: total responsibility of cluster k for all data ݎ. points = ݖ)ܫ = ) Each data weighted by its relation to the cluster k From: Iain Murray, Edinburgh 31
32 EM algorithm for GMM See next slide From: Iain Murray, Edinburgh 32
33 Posterior Probabilities We can think of the mixing coefficients as prior probabilities for the components For a given value of we can evaluate the corresponding posterior probabilities, called responsibilities from Bayes theorem by: BCS Summer School, Exeter, 2003 Christopher M. Bishop
34 EM algorithm EM in another view Details in the next slide From: Iain Murray, Edinburgh 34
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42 Decomposition BCS Summer School, Exeter, 2003Christopher M. Bishop 42
43 E-step BCS Summer School, Exeter, 2003Christopher M. Bishop 43
44 M-step BCS Summer School, Exeter, 2003Christopher M. Bishop 44
45 Optimizing the Bound E-step: maximize with respect to equivalent to minimizing KL divergence sets equal to the posterior distribution M-step: maximize bound with respect to equivalent to maximizing expected complete-data log likelihood Each EM cycle must increase incomplete-data likelihood unless already at a (local) maximum BCS Summer School, Exeter, 2003 Christopher M. Bishop
46 By: Aarti Singh, CMU 46
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