Clustering - example. Given some data x i X Find a partitioning of the data into k disjunctive clusters Example: k-means clustering

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1 Clustering - example Graph Mining and Graph Kernels Given some data x i X Find a partitioning of the data into k disjunctive clusters Example: k-means clustering x!!!!8!!! 8 x 1 1

2 Clustering - example Graph Mining and Graph Kernels Given some data x i X Find a partitioning of the data into k disjunctive clusters Example: k-means clustering x!!! Can we do something better?!8!!! 8 x 1

3 Generative model view of clustering Instead of partitioning the data try to describe the underlying generative process of the data Each cluster can be seen as one distribution For example Gaussian distributions Objects x i are assumed to be independent samples from their cluster distribution => Gaussian mixture model x i N (µ l, Σ l ) univariate Gaussian prbability density function f(x) c 1 =Normal(,1.5); p(c 1 )=.5 c =Normal(3,.5); p(c )=. c 3 =Normal(!,.7); p(c 3 )=.3.!5 5 x 3

4 Gaussian Mixture Model - Introduction Data x i are independent and identically distributed (i.i.d.) samples from a mixture of k distributions c l x i R d,i {1... N} c l,l {1... k} each cluster is a multivariate Gaussian distribution Sufficient statistics of each cluster: Mean (Centroid) Covariance (empirical covariance matrix) Probability density function of a Gaussian distribution P(x i c l ) f l (x i )= x i N (µ l, Σ l ) 1 (π) d det(σ l ) exp µ l R d Σ l R d d ( 1 (x i µ l ) Σ 1 (x i µ l ) )

5 Gaussian Mixture Model - Introduction Mixture of one-dimensional Gaussians c i = N (µ l, σ l ) univariate Gaussian prbability density function c 1 =Normal(,1.5); p(c 1 )=.5 c =Normal(3,.5); p(c )=. c 3 =Normal(!,.7); p(c 3 )=.3.1 f(x) !5 5 x 5

6 Gaussian Mixture Model - Introduction Mixture of multivariate Gaussians x!!!!8!!! 8 x 1

7 Gaussian Mixture Model - Introduction Mixture of multivariate Gaussians No covariance x! &'() Negative covariance!!!8!!! 8 x 1 µ l Σ l Positive covariance 7

8 Gaussian Mixture Model some maths Probability of a cluster c l P(c l )= 1 N N P(c l x i ) i=1 Empirical estimate of the density of the cluster low density => small P(c l ) x!!! high density => large P(c l )!8!!! 8 x 1 8

9 Gaussian Mixture Model some maths Probability of a cluster c l P(c l )= 1 N Empirical estimate of the density of the cluster Probability of observing an object x i P(x i )= k l=1 N P(c l x i ) i=1 P(c l )P(x i c l ) Probability of observing an object x i given its cluster c l P(x i c l ) 1 (π) d det(σ l ) exp ( 1 ) (x i µ l ) Σ 1 (x i µ l ) 9

10 Gaussian Mixture Model likelihood function Quality measure of the model Probability that the data is generated by the GMM L = = N i=1 N i=1 P(x i ) k l=1 P(c l )P(x i c l ) Also possible to use the log-likelihood log (L) 1

11 Gaussian Mixture Model - clustering Question: How can we use the GMM to partition the data? Choose most likely cluster assignment of each object argmax l P(c l x i ) = argmax l P(c l )P(x i c l ) x!!!!8!!! 8 x 1 11

12 Gaussian Mixture Model - clustering Question: How can we use the GMM to partition the data? Choose most likely cluster assignment of each object argmax l P(c l x i ) = argmax l P(c l )P(x i c l ) x!!! Great! but!8!!! 8 x 1 1

13 This is all we that have How to estimate the sufficient statistics of each cluster? Mean (Centroid) Covariance (empirical covariance matrix) µ l R d Σ l R d d => use Expectation Maximization algorithm x!!!!8!!! 8 x 1 13

14 Expectation Maximization algorithm Original algorithm by [Dempster, Laird and Rubin, 1977] General method for finding the maximum-likelihood estimate of a data distribution, when the data is partially missing or hidden. How does this apply? data x i are fully observed Trick: the cluster assignments of an object x i can be seen as hidden variable 1

15 Exepectation Maximization algorithm a short sketch of the EM algorithm: Initialize cluster assignments Two alternating steps: E-step: re-estimate the Expected-values of the hidden data (cluster assignments) under the current estimate of the model M-step: re-estimate the model parameters such that the likelihood according to the current estimate of the complete data is maximized until convergence L new L old < 1+ɛ 15

16 Expectation Maximization algorithm E-step: Re-estimate the Expected-values of the hidden data (cluster assignments) under the current estimate of the model P new (c l x i ) = P(c l )P(x i c l ) 1

17 Expectation Maximization algorithm M-step: re-estimate the model parameters by taking the maximum likelihood estimate according to the current estimate of the complete data Cluster densities P new (c l )= 1 N P new (c l x i ) N Cluster means: µ new l = i=1 N i=1 x ip new (c l x i ) N i=1 Pnew (c l x i ) Cluster covariances: Σ new l = N i=1 (x i µ new l )(x i µ new l ) P new (c l x i ) N i=1 Pnew (c l x i ) 17

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