Cluster Analysis

Berlin Chen 2004

References:1. Foundations of Statistical Natural Language Processing, Chapter 142. Data Mining: Concepts, Models, Methods and Algorithms, Chapter 63. Modern Information Retrieval, chapters 5, 7

2

Clustering

• Place similar objects in the same group and assign dissimilar objects to different groups– Word clustering

• Neighbor overlap: words occur with the similar left and right neighbors (such as in and on)

– Document clustering• Documents with the similar topics or concepts are put

together

• But clustering cannot give a comprehensive description of the object– How to label objects shown on the visual display

• Clustering is a way of learning

3

Clustering vs. Classification

• Classification is supervised and requires a set of labeled training instances for each group (class)

• Clustering is unsupervised and learns without a teacher to provide the labeling information of the training data set

– Also called automatic or unsupervised classification

4

Types of Clustering Algorithms

• Two types of structures produced by clustering algorithms– Flat or non-hierarchical clustering– Hierarchical clustering

• Flat clustering– Simply consisting of a certain number of clusters and the relation

between clusters is often undetermined

• Hierarchical clustering– A hierarchy with usual interpretation that each node stands for a

subclass of its mother’s node• The leaves of the tree are the single objects• Each node represents the cluster that contains all the objects

of its descendants

5

Hard Assignment vs. Soft Assignment

• Another important distinction between clustering algorithms is whether they perform soft or hard assignment

• Hard Assignment– Each object is assigned to one and only one cluster

• Soft Assignment– Each object may be assigned to multiple clusters– An object has a probability distribution over

clusters where is the probability that is a member of

– Is somewhat more appropriate in many tasks such as NLP, IR, …

ix ( )ixP ⋅jc

jc( )ji cxP ix

6

Hard Assignment vs. Soft Assignment

• Hierarchical clustering usually adopts hard assignment

• While in flat clustering both types of clustering are common

7

Summarized Attributes of Clustering Algorithms • Hierarchical Clustering

– Preferable for detailed data analysis

– Provide more information than flat clustering

– No single best algorithm (each of the algorithms only optimal for some applications)

– Less efficient than flat clustering (minimally have to compute n x nmatrix of similarity coefficients)

8

Summarized Attributes of Clustering Algorithms

• Flat Clustering– Preferable if efficiency is a consideration or data sets are very

large

– K-means is the conceptually method and should probably be used on a new data because its results are often sufficient

– K-means assumes a simple Euclidean representation space, and so cannot be used for many data sets, e.g., nominal data like colors

– The EM algorithm is the most choice. It can accommodate definition of clusters and allocation of objects based on complex probabilistic models

9

Hierarchical Clustering

10

Hierarchical Clustering

• Can be in either bottom-up or top-down manners– Bottom-up (agglomerative)

• Start with individual objects and grouping the most similar ones

– E.g., with the minimum distance apart

• The procedure terminates when one cluster containing all objects has been formed

– Top-down (divisive)• Start with all objects in a group and divide them into groups

so as to maximize within-group similarity

( ) ( )yxdyxsim ,11,

+=

凝集的

分裂的

distance measures willbe discussed later on

11

Hierarchical Agglomerative Clustering (HAC)

• A bottom-up approach

• Assume a similarity measure for determining the similarity of two objects

• Start with all objects in a separate cluster and then repeatedly joins the two clusters that have the most similarity until there is one only cluster survived

• The history of merging/clustering forms a binary tree or hierarchy

12

Hierarchical Agglomerative Clustering (HAC)

• Algorithm

cluster number

Initialization (for tree leaves):Each object is a cluster

merged as a new cluster

The original two clusters are removed

13

Distance Metrics

• Euclidian Distance (L2 norm)

• L1 Norm

• Cosine Similarity (transform to a distance by subtracting from 1)

2

12 )(),( i

m

ii yxyxL −=∑

=

rr

∑=

−=m

iii yxyxL

11 ),(

rr

yxyxrr

rr

⋅−

•1 ranged between 0 and 1

14

Measures of Cluster Similarity• Especially for the bottom-up approaches

• Single-link clustering– The similarity between two clusters is the similarity of the two

closest objects in the clusters

– Search over all pairs of objects that are from the two differentclusters and select the pair with the greatest similarity

Cu Cv

greatest similarity

( ) ( )y,xsim,ccsimji cy,cx

ji

rrrr∈∈

= max

15

Measures of Cluster Similarity

• Complete-link clustering– The similarity between two clusters is the similarity of their two

most dissimilar members

– Sphere-shaped clusters are achieved

– Preferable for most IR and NLP applications

Cu Cv

least similarity

( ) ( )y,xsim,ccsimji cy,cxji

rrrr∈∈

= min

16

Measures of Cluster Similarity

single link

complete link

17

Measures of Cluster Similarity

• Group-average agglomerative clustering– A compromise between single-link and complete-link clustering

– The similarity between two clusters is the average similarity between members

– If the objects are represented as length-normalized vectors and the similarity measure is the cosine

• There exists an fast algorithm for computing the average similarity

( ) ( ) yxyxyxyxyxsim

rrrr

rrrrrr

⋅=⋅

== ,cos,

length-normalized vectors

18

Measures of Cluster Similarity

• Group-average agglomerative clustering (cont.)

– The average similarity SIM between vectors in a cluster cj is defined as

– The sum of members in a cluster cj :

– Express in terms of

( ) ( ) ( ) ( )∑ ∑∑ ∑ ∈≠∈∈

≠∈

⋅−

=−

=j jj j cx

xycyjjcx

xycyjj

j yxccyxsim

cccSIM

rrr

rrrr

r

rrrr

11,

11

( ) ∑∈

=jcx

j xcs rrr

( )jcSIM ( )jcsr

( ) ( ) ( )( ) ( )( ) ( )

( ) ( ) ( )( )1

1

1

−

−⋅=∴

+−=

⋅+−=

⋅=⋅=⋅

∑

∑ ∑∑

∈

∈ ∈∈

jj

jjj

j

jjjj

cxjjj

cx cyj

cxjj

ccccscs

cSIM

ccSIMcc

xxcSIMcc

yxcsxcscs

j

j jj

rr

rr

rrrrrr

r

r rr

=1

length-normalized vector

19

Measures of Cluster Similarity

• Group-average agglomerative clustering (cont.)

-As merging two clusters ci and cj , the cluster sum vectors and are known in advance

– The average similarity for their union will be

( )icsr ( )jcsr

( )( ) ( )( ) ( ) ( )( ) ( )

( )( )1 −+++−+⋅+

=∪

jiji

jijiji

ji

cccccccscscscs

ccSIMrrrr

( ) ( ) ( ) jiNewjiNew ccccscscs +=+= ,rrr

20

Example: Word Clustering

• Words (objects) are described and clustered using a set of features and values– E.g., the left and right neighbors of tokens of words

“be” has least similarity with the other 21 words !

higher nodes:decreasingof similarity

21

Divisive Clustering

• A top-down approach

• Start with all objects in a single cluster

• At each iteration, select the least coherent cluster and split it

• Continue the iterations until a predefined criterion (e.g., the cluster number) is achieved

• The history of clustering forms a binary tree or hierarchy

22

Divisive Clustering

• To select the least coherent cluster, the measures used in bottom-up clustering can be used again here– Single link measure– Complete-link measure– Group-average measure

• How to split a cluster– Also is a clustering task (finding two sub-clusters)– Any clustering algorithm can be used for the splitting operation,

e.g.,• Bottom-up (agglomerative) algorithms• Non-hierarchical clustering algorithms (e.g., K-means)

23

Divisive Clustering

• Algorithm

:split the least coherent cluster

Generate two new clusters and remove the original one

24

Non-Hierarchical Clustering

25

Non-hierarchical Clustering

• Start out with a partition based on randomly selected seeds (one seed per cluster) and then refine the initial partition– In a multi-pass manner

• Problems associated non-hierarchical clustering– When to stop– What is the right number of clusters

• Algorithms introduced here– The K-means algorithm– The EM algorithm

MI, group average similarity, likelihood

k-1 → k → k+1

Hierarchical clustering also has to face this problem

26

The K-means Algorithm

• A hard clustering algorithm

• Define clusters by the center of mass of their members

• Initialization– A set of initial cluster centers is needed

• Recursion– Assign each object to the cluster whose center is closet – Then, re-compute the center of each cluster as the centroid or

mean (average) of its members• Using the medoid as the cluster center ?(a medoid is one of the objects in the cluster)

27

The K-means Algorithm

• Algorithm

cluster centroid

cluster assignment

calculation of new centroids

28

The K-means Algorithm

• Example 1

29

The K-means Algorithm

• Example 2

governmentfinancesports

research

name

yxyxrr

rr

⋅−

•1

30

The K-means Algorithm

• Choice of initial cluster centers (seeds) is important

– Pick at random– Or use another method such as hierarchical clustering algorithm

on a subset of the objects• E.g., buckshot algorithm uses the group-average

agglomerative clustering to randomly sample of the data that has size square root of the complete set

– Poor seeds will result in sub-optimal clustering

• How to break ties in case there are several centers with the same distance from an object

– Randomly assign the object to one of the candidate clusters– Or, perturb objects slightly

31

The K-means Algorithm

• E.g., the LBG algorithm– By Linde, Buzo, and Gray

Global mean Cluster 1 mean

Cluster 2mean

{µ11,Σ11,ω11}{µ12,Σ12,ω12}

{µ13,Σ13,ω13} {µ14,Σ14,ω14}

M→2M at each iteration

32

The EM Algorithm• A soft version of the K-mean algorithm

– Each object could be the member of multiple clusters– Clustering as estimating a mixture of (continuous) probability

distributions

∑ixr

( )1cxP i( )11 cP=π

( )22 cP=π

( )KK cP=π

( )2cxP i

( )Ki cxP

( ) ( ) ( )∑=

=ΘK

kkkii cP;cxPxP

1

ΘΘ rr

( )( )

( ) ( )⎟⎠⎞

⎜⎝⎛ −Σ−−

Σ= − kik

Tki

kmki

xxcxP µµπ

rrrrr 121exp

2

1Θ;

Continuous case:Likelihood function fordata samples:

( ) ( )

( ) ( ) cP;cxP

xPP

n

i

K

kkki

n

ii

i

ii∏ ∑

∏

= =

=

=

=

1 1

1

ΘΘ

ΘΘ

r

rX

A Mixture Gaussian HMM(or A Mixture of Gaussians)

xxx nr

Krr ,,, 21=X

( ) ( ) ( )( )( ) ( )ΘΘmax

ΘΘmax ,max

:instance new of Prediction

kkik

i

kki

ki

k

cP;cx

xPcP;cx

xcP

r

r

rr

=

Θ=Θ

(i.i.d.) ddistributey identicallt independen are s'ixr

33

The EM Algorithm

34

The EM Algorithm

• E–step (Expectation)– Derive the complete data likelihood function

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( ) ( )[ ]( )[ ]

( )[ ]( )[ ]

( )[ ]∑=∑ ∑∑=

∑ ∑∑=

∑ ∑ ∏∑=

∑ ∑ ∏∑=

++×

×++=

∏ ∑=∏=

= ==

= ==

= = ==

= = ==

= ==

CCX

X

Θ̂,

Θ̂,,,,,,

Θ̂,,,,,,

Θ̂,

Θ̂Θ̂

Θ̂Θ̂..Θ̂Θ̂

Θ̂Θ̂..Θ̂Θ̂

Θ̂Θ̂Θ̂Θ̂

1 121

1

1 121

1

1 1 11

1 1 11

11

1111

1 11

121

2

121

2

1 2

1 2

P

cxcxcxP

cxcxcxP

cxP

cP;cxP

cP;cxPcP;cxP

cP;cxPcP;cxP

cP;cxP xPP

K

k

K

kknkk

K

k

K

k

K

kknkk

K

k

K

k

K

k

n

iki

K

k

K

k

K

k

n

ikki

K

k

KKnn

KK

n

i

K

kkki

n

ii

nn

nn

ni

nii

iii

rL

rrL

rL

rrL

rL

rL

rr

Lrr

rr

nn kkkk

nn

ccccxxxx

121

121

−== −

L

rrL

rr

CX

the complete data likelihood function

likelihood function

)kinds?( ofkindsmany How

nKC

xxx nr

Krr ,,, 21=X

?

iika

( )( )( ) ( )∑ ∑ ∑ ∏=

+++++++++=

∏ ∑

= = = =

= =

Kk

Kk

Kk

Tt tk

TKTTKK

Tt

Kk tk

T t

t t

a

aaaaaaaaa

a

1 1 1 1

212222111211

1 1

1 2...

............

:Note

35

The EM Algorithm

• E–step (Expectation)– Define the auxiliary function as the expectation of the

log complete likelihood function LCM with respective to thehidden/latent variable C conditioned on known data

– Maximize the log likelihood function by maximizing the expectation of the log complete likelihood function

• We have shown this when deriving the HMM-based retrieval model

( )Θ̂Θ,Φ

( )Θ,X

( ) [ ] ( )[ ]( ) ( )( )( ) ( )∑

∑

=

=

==Φ

C

C

XCXC

CXXCX

CXXC

CX

Θ̂,logΘΘ,

Θ̂,logΘ,

Θ̂,loglogΘ̂,ΘΘ,Θ,

PP

P

PP

PELE CM

( )Θ̂log XP( )Θ̂Θ,Φ

36

The EM Algorithm• E–step (Expectation)

– The auxiliary function ( )Θ̂Θ,Φ( ) ( )( ) ( )

( )( ) ( )

( ) ( )

( ) ( )

( )[ ] ( )

( )[ ] ( ){ }( ) ( ){ }( ) ( ) ( )[ ]{ }( ) ( ){ } ( ) ( ){ }∑ ∑ ∑ ∑+=

∑ ∑=

∑ ∑=

∑ ∑=

∑ ∑⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∑ ⎥⎦

⎤⎢⎣

⎡∏=

∑ ∑⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡∏⎥⎦

⎤⎢⎣⎡∑=

∑ ⎥⎦⎤

⎢⎣⎡∑⎥

⎦

⎤⎢⎣

⎡∏=

∑ ⎥⎦

⎤⎢⎣

⎡∏

⎥⎥⎦

⎤

⎢⎢⎣

⎡∏=

∑=Φ

= = = =

= =

= =

= =

= = = =

= = ==

= ==

= ==

K

k

n

i

K

k

n

ikijkkjk

K

k

n

ikkijk

K

k

n

ikijk

K

k

n

iikki

K

k

n

i ccc

n

jjkkkki

ccc

K

k

n

jjkki

n

ikk

cccki

n

i

n

jjk

ccc

n

iki

n

j j

kj

cxPxcPcPxcP

cPcxPxcP

cxPxcP

xcPcxP

xcPcxP

xcPcxP

cxPxcP

cxPxP

cxP

PP

P

jj

j

j

nkkkji

nkkkji

nkkkij

nkkki

j

1 1 1 1

1 1

1 1

1 1

1 1 1,

1 11,

11

11

Θ̂,logΘ,Θ̂logΘ,

Θ̂Θ̂,logΘ,

Θ̂,logΘ,

Θ,Θ̂,log

Θ,Θ̂,log

Θ,Θ̂,log

Θ̂,logΘ,

Θ̂,logΘ

Θ,

Θ̂,logΘΘ,

Θ̂,Θ

21

21

21

21

rrr

rr

rr

rr

rr

rr

rr

rr

r

K

K

K

K

C

C

C

C

CCX

XCX

δ

δ

?

⎩⎨⎧ =

=otherwise 0

if 1,

kk ikk i

δ

data will be aligned to a specific mixture

each mixture collects thecorresponding training data

37

The EM Algorithm

– Note that

( )

( )[ ]

( ) ( )

( )

( )Θ,

Θ,1

Θ,Θ,

Θ,

Θ,

,1

1,

,1 1

1 1 1 1,

1,

1 2

21

ik

ikn

ijj

K

cikkk

n

ijj

K

kjk

K

c

K

c

K

c

n

jjkkk

ccc

n

jjkkk

xcP

xcP

xcPxcP

xcP

xcP

ikii

jj

k k nkji

nkkkji

r

r

rr

rL

r

K

=

⎥⎦

⎤⎢⎣

⎡∏=

⎥⎥⎦

⎤

⎢⎢⎣

⎡∑

⎥⎥⎦

⎤

⎢⎢⎣

⎡∏

⎥⎥⎦

⎤

⎢⎢⎣

⎡∑=

∑ ∑ ∑ ∏=

∑ ⎥⎦

⎤⎢⎣

⎡∏

≠=

=≠= =

= = = =

= =

δ

δ

δC

( )( )( ) ( )∑ ∑ ∑ ∏=

+++++++++=

∏ ∑

= = = =

= =

Kk

Kk

Kk

Tt tk

TKTTKK

Tt

Kk tk

T t

t t

a

aaaaaaaaa

a

1 1 1 1

212222111211

1 1

1 2...

............

:Note

ki cx toaligned beonly can r

⎩⎨⎧ =

=otherwise 0

if 1,

kkikk i

δ

38

The EM Algorithm

• E–step (Expectation)– The auxiliary function can also be divided into two:

( ) ( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )

( )

( ) ( ) ( )( ) ( )( ) ( )

( )∑ ∑∑

∑ ∑

∑ ∑∑

∑ ∑

∑ ∑

= =

=

= =

= =

=

= =

= =

=

=Φ

=

=

=Φ

Φ+Φ=Φ

n

i

K

kkiK

llli

kki

n

i

K

kkiikb

n

i

K

kkK

llli

kki

n

i

K

kk

i

kki

n

i

K

kkika

ba

cxPcPcxP

cPcxP

cxPxcP

cPcPcxP

cPcxP

cPxP

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cPxcP

1 1

1

1 1

1 1

1

1 1

1 1

Θ̂,logΘΘ,

ΘΘ,

Θ̂,logΘ,Θ̂Θ,

Θ̂logΘΘ,

ΘΘ,

Θ̂logΘ

ΘΘ,

Θ̂logΘ,Θ̂Θ,

whereΘ̂Θ,Θ̂Θ,Θ̂Θ,

r

r

r

rr

r

r

r

r

r

auxiliary function for mixture weights

auxiliary function for cluster distributions

39

The EM Algorithm

• M-step (Maximization)– Remember that

• Maximize a function F by applying Lagrange multiplier

∑

∑∑∑

∑ ∑∑

=

===

= ==

=∴

−=⇒−=

∀−=⇒=+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=⇒=

N

jj

jj

N

jj

N

jj

N

jj

j

j

j

j

j

N

j

N

jjjj

N

jjj

w

wy

wwy

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yF

yywFywF

1

111

1 11

0ˆ

1logˆlog that Suppose

Multiplier Lagrange applyingBy

ll

ll

l

l

∂∂

Constraint

jj

j

yyy 1log:Note

=∂

∂

40

The EM Algorithm

• M-step (Maximization)– Maximize ( )Θ̂Θ,aΦ

( ) ( ) ( )( ) ( )( ) ( )

( ) ( ) 1Θ̂Θ̂logΘΘ,

ΘΘ,

1Θ̂Θ̂Θ,Θ̂Θ,

11 1

1

1

⎟⎠

⎞⎜⎝

⎛−+=

⎟⎠

⎞⎜⎝

⎛−+Φ=Φ

∑∑ ∑∑

∑

== =

=

=

K

kk

K

k

n

ikK

llli

kki

K

kkaa

cPlcPcPcxP

cPcxP

cPl

r

r

kw ky

( )

( ) ( )( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )

ΘΘ,

ΘΘ,

ΘΘ,

ΘΘ,

ΘΘ,

ΘΘ,

Θ̂ˆ

1

1

1 1

1

1

1

1

n

cPcxP

cPcxP

cPcxP

cPcxP

cPcxP

cPcxP

w

wcP

n

iK

llli

kki

K

k

n

iK

llli

kki

n

iK

llli

kki

K

kk

kkk

∑∑

∑ ∑∑

∑∑

∑

=

=

= =

=

=

=

=

====⇒

r

r

r

r

r

r

π

auxiliary function for mixture weights (or priors for Gaussians)

41

The EM Algorithm

• M-step (Maximization)– Maximize ( )Θ̂Θ,bΦ

( ) ( ) ( )( ) ( )

( )∑ ∑∑= ==

=Φn

i

K

kkiK

llli

kkib cxP

cPcxP

cPcxP

1 1

1

Θ̂,logΘΘ,

ΘΘ, Θ̂Θ, r

r

r

auxiliary function for Gaussian Means and Variances

( )( )

( ) ( )⎟⎠⎞

⎜⎝⎛ −Σ−−

Σ=Θ − ki

Tki

kmki

xxcxPk

µµπ

rrrrr 121exp

2

1;

( ) ( )( ) ( )

and ΘΘ,

ΘΘ,Let

1

,

∑=

= K

llli

kkiik

cPcxP

cPcxPw

r

r ( )( ) ( ) ( )kiTkik

ki

xxm

cxP

kµµπ rrrr

r

−Σ−−Σ−⋅−

=Θ

−1

21log2

12log2

;log

( ) ( ) ( )∑ ∑= =

− +⎥⎦⎤

⎢⎣⎡ −Σ−+Σ−=Φ

n

i

K

kkik

T

kikikb Dxxw1 1

1,

ˆˆˆ2

1log21 Θ̂Θ, µµ rrrr

constant

42

The EM Algorithm

• M-step (Maximization)– Maximize with respect to

( ) ( ) ( )( )( ) ( )

( ) ( )( ) ( )( ) ( )

∑∑

∑∑

∑

∑

∑

=

=

=

=

=

=

=

−

ΘΘ

ΘΘ

ΘΘ

⋅ΘΘ

=⋅

=⇒

=−−Σ⋅⋅⋅−=∂

Φ∂

n

iK

llli

kki

n

iK

llli

ikki

n

iik

n

iiik

k

n

ikikik

k

b

cPcxP

cPcxP

cPcxP

xcPcxP

w

xw

xw

1

1

1

1

1,

1,

1

1,

,

,

,

,

ˆ

01ˆˆ221 ˆ

Θ̂Θ,

r

r

r

rr

r

r

rrr

µ

µµ

( )Θ̂Θ,bΦ kµr

( ) ( ) ( )∑ ∑= =

− +⎥⎦⎤

⎢⎣⎡ −Σ−+Σ−=Φ

n

i

K

kkik

T

kikikb Dxxw1 1

1,

ˆˆˆ2

1ˆlog21 Θ̂Θ, µµ rrrr

( )here symmetric is and

)(

1−

+=

kΣd

d xCCxCxx TT

43

The EM Algorithm

• M-step (Maximization)– Maximize with respect to( )Θ̂Θ,bΦ

( )[ ]

and

)det(det TXXXX −⋅=

dd

kΣ

( ) ( ) ( )∑ ∑= =

− +⎥⎦⎤

⎢⎣⎡ −Σ−+Σ−=Φ

n

i

K

kkik

T

kikikb Dxxw1 1

1,

ˆˆˆ2

1ˆlog21 Θ̂Θ, µµ rrrr

( ) ( )( )

( )( )

( )( )

( )( )

( )( )( ) ( )( ) ( )

( )( )

( ) ( )( ) ( )

∑∑

∑∑

∑

∑

∑∑

∑∑

∑∑

∑

=

=

=

=

=

=

==

=

−−

=

−

=

−−

=

−

=

−−−−

ΘΘ

ΘΘ

−−⋅ΘΘ

ΘΘ

=−−⋅

=Σ⇒

−−⋅=Σ⋅⇒

ΣΣ−−ΣΣ⋅=ΣΣΣ⋅⇒

Σ−−Σ⋅=Σ⋅⇒

=⎥⎦⎤

⎢⎣⎡ Σ−−Σ−Σ⋅Σ⋅Σ⋅−=

Σ∂Φ∂

n

iK

llli

kki

n

i

T

kikiK

llli

kki

n

iik

n

i

T

kikiik

k

n

i

T

kikiik

n

ikik

k

n

ik

T

kikikkik

n

ikkkik

n

ik

T

kikikik

n

ikik

n

ik

T

kikikkkkikk

b

cPcxP

cPcxP

xxcPcxP

cPcxP

w

xxw

xxww

xxww

xxww

xxw

1

1

1

1

1,

1,

1,

1,

1

11,

1

1,

1

11,

1

1,

1

1111

,

,

,

ˆˆ,

,

ˆˆˆ

ˆˆˆ

ˆˆˆˆˆˆˆˆˆ

ˆˆˆˆˆ

0ˆˆˆˆˆ21 ˆ

Θ̂Θ,

r

r

rrrr

r

r

rrrr

rrrr

rrrr

rrrr

rrrr

µµµµ

µµ

µµ

µµ

µµ

111 )( −−− −= XabX

XbXa TT

dd

44

The EM Algorithm

• The initial cluster distributions can be estimated using the K-means algorithm

• The procedure terminates when the likelihoodfunction is converged or maximum numberof iterations is reached

( )ΘXP

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