Clustering AnalysisCS 685:

Special Topics in Data MiningJinze Liu

Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Subspace Clustering/Bi-clustering Model-Based Clustering

What is Cluster Analysis? Finding groups of objects such that the objects in a group

will be similar (or related) to one another and different from (or unrelated to) the objects in other groups

Inter-cluster distances are maximized

Intra-cluster distances are

minimized

What is Cluster Analysis?

Cluster: a collection of data objects Similar to one another within the same cluster Dissimilar to the objects in other clusters

Cluster analysis Grouping a set of data objects into clusters

Clustering is unsupervised classification: no predefined classes

Clustering is used: As a stand-alone tool to get insight into data distribution

Visualization of clusters may unveil important information As a preprocessing step for other algorithms

Efficient indexing or compression often relies on clustering

Some Applications of Clustering

Pattern Recognition Image Processing

cluster images based on their visual content Bio-informatics WWW and IR

document classification cluster Weblog data to discover groups of similar access

patterns

What Is Good Clustering?

A good clustering method will produce high quality clusters with high intra-class similarity low inter-class similarity

The quality of a clustering result depends on both the similarity measure used by the method and its implementation.

The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.

Requirements of Clustering in Data Mining Scalability Ability to deal with different types of attributes Discovery of clusters with arbitrary shape Minimal requirements for domain knowledge to

determine input parameters Able to deal with noise and outliers Insensitive to order of input records High dimensionality Incorporation of user-specified constraints Interpretability and usability

Outliers Outliers are objects that do not belong to any

cluster or form clusters of very small cardinality

In some applications we are interested in discovering outliers, not clusters (outlier analysis)

cluster

outliers

Data Structures data matrix

(two modes)

dissimilarity or distancematrix

(one mode)

npx...nfx...n1x...............ipx...ifx...i1x...............1px...1fx...11x

0...)2,()1,(:::

)2,3()

...ndnd

0dd(3,10d(2,1)

0

the “classic” data input

attributes/dimensions

tupl

es/o

bjec

ts

Assuming simmetric distance d(i,j) = d(j, i)

objects

obje

cts

Measuring Similarity in Clustering

Dissimilarity/Similarity metric: The dissimilarity d(i, j) between two objects i and j is

expressed in terms of a distance function, which is typically a metricmetric:

d(i, j)0 (non-negativity) d(i, i)=0 (isolation) d(i, j)= d(j, i) (symmetry) d(i, j) ≤ d(i, h)+d(h, j) (triangular inequality)

The definitions of distance functions are usually different for interval-scaled, boolean, categorical, ordinal and ratio-scaled variables.

Weights may be associated with different variables based on applications and data semantics.

Type of data in cluster analysis

Interval-scaled variables e.g., salary, height

Binary variables e.g., gender (M/F), has_cancer(T/F)

Nominal (categorical) variables e.g., religion (Christian, Muslim, Buddhist, Hindu, etc.)

Ordinal variables e.g., military rank (soldier, sergeant, lutenant, captain, etc.)

Ratio-scaled variables population growth (1,10,100,1000,...)

Variables of mixed types multiple attributes with various types

Similarity and Dissimilarity Between Objects Distance metrics are normally used to measure

the similarity or dissimilarity between two data objects

The most popular conform to Minkowski distance:

where i = (xi1, xi2, …, xin) and j = (xj1, xj2, …, xjn) are two n-dimensional data objects, and p is a positive integer

If p = 1, L1 is the Manhattan (or city block) distance:

ppjnxinxp

jx

ixp

jx

ixjipL

/1||...|

22||

11|),(

||...||||),(1 2211 nn jxixjxixjxixjiL

Similarity and Dissimilarity Between Objects (Cont.) If p = 2, L2 is the Euclidean distance:

Properties d(i,j) 0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j) d(i,k) + d(k,j)

Also one can use weighted distance:

)||...|||(|),( 22

22

2

11 nn jxixjxixjxixjid

)||...||2

||1

(),( 22

22

2

11 nn jxixnwjxixwjxixwjid

Binary Variables A binary variable has two states: 0 absent, 1 present A contingency table for binary data

Simple matching coefficient distance (invariant, if the binary variable is symmetric):

Jaccard coefficient distance (noninvariant if the binary variable is asymmetric):

dcbacb jid

),(

cbacb jid

),(

pdbcasumdcdcbaba

sum

01

01

object i

object j

i= (0011101001)

J=(1001100110)

Binary Variables Another approach is to define the similarity of two

objects and not their distance. In that case we have the following:

Simple matching coefficient similarity:

Jaccard coefficient similarity:dcba

da jis

),(

cbaa jis

),(

Note that: s(i,j) = 1 – d(i,j)

Dissimilarity between Binary Variables

Example (Jaccard coefficient)

all attributes are asymmetric binary 1 denotes presence or positive test 0 denotes absence or negative test

Name Fever Cough Test-1 Test-2 Test-3 Test-4 Jack 1 0 1 0 0 0 Mary 1 0 1 0 1 0 Jim 1 1 0 0 0 0

75.0211

21),(

67.0111

11),(

33.0102

10),(

maryjimd

jimjackd

maryjackd

Each variable is mapped to a bitmap (binary vector)

Jack: 101000 Mary: 101010 Jim: 110000

Simple match distance:

Jaccard coefficient:

A simpler definition

Name Fever Cough Test-1 Test-2 Test-3 Test-4 Jack 1 0 1 0 0 0 Mary 1 0 1 0 1 0 Jim 1 1 0 0 0 0

bits ofnumber totalpositionsbit common -non ofnumber ),( jid

in s1' ofnumber in s1' ofnumber 1),(

jijijid

Variables of Mixed Types

A database may contain all the six types of variables symmetric binary, asymmetric binary, nominal, ordinal,

interval and ratio-scaled. One may use a weighted formula to combine their

effects.

)(1

)()(1),(

fij

pf

fij

fij

pf d

jid

Major Clustering Approaches

Partitioning algorithms: Construct random partitions and then iteratively refine them by some criterion

Hierarchical algorithms: Create a hierarchical decomposition of the set of data (or objects) using some criterion

Density-based: based on connectivity and density functions Grid-based: based on a multiple-level granularity structure Model-based: A model is hypothesized for each of the

clusters and the idea is to find the best fit of that model to each other

Partitioning Algorithms: Basic Concept

Partitioning method: Construct a partition of a database D of n objects into a set of k clusters k-means (MacQueen’67): Each cluster is represented by

the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman &

Rousseeuw’87): Each cluster is represented by one of the objects in the cluster

K-means Clustering Partitional clustering approach Each cluster is associated with a centroid (center point) Each point is assigned to the cluster with the closest centroid Number of clusters, K, must be specified The basic algorithm is very simple

K-means Clustering – Details Initial centroids are often chosen randomly.

Clusters produced vary from one run to another. The centroid is (typically) the mean of the points in

the cluster. ‘Closeness’ is measured by Euclidean distance,

cosine similarity, correlation, etc. Most of the convergence happens in the first few

iterations. Often the stopping condition is changed to ‘Until relatively

few points change clusters’ Complexity is O( n * K * I * d )

n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

Two different K-means Clusterings

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Sub-optimal Clustering-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Optimal Clustering

Original Points

Evaluating K-means Clusters

For each point, the error is the distance to the nearest cluster

To get SSE, we square these errors and sum them.

x is a data point in cluster Ci and mi is the representative point for cluster Ci

can show that mi corresponds to the center (mean) of the cluster

Given two clusters, we can choose the one with the smallest error

K

i Cxi

i

xmdistSSE1

2 ),(

Solutions to Initial Centroids ProblemMultiple runs

Helps, but probability is not on your sideSample and use hierarchical clustering to

determine initial centroidsSelect more than k initial centroids and

then select among these initial centroids Select most widely separated

PostprocessingBisecting K-means

Not as susceptible to initialization issues

Limitations of K-meansK-means has problems when clusters are

of differing Sizes Densities Non-spherical shapes

K-means has problems when the data contains outliers. Why?

The K-Medoids Clustering Method Find representative objects, called medoids, in

clusters PAM (Partitioning Around Medoids, 1987)

starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering

PAM works effectively for small data sets, but does not scale well for large data sets

CLARA (Kaufmann & Rousseeuw, 1990) CLARANS (Ng & Han, 1994): Randomized sampling

PAM (Partitioning Around Medoids) (1987) PAM (Kaufman and Rousseeuw, 1987), built in

statistical package S+ Use a real object to represent the a cluster

1. Select k representative objects arbitrarily2. For each pair of a non-selected object h and a selected

object i, calculate the total swapping cost TCih

3. For each pair of i and h, If TCih < 0, i is replaced by h Then assign each non-selected object to the most

similar representative object4. repeat steps 2-3 until there is no change

PAM Clustering: Total swapping cost TCih=jCjih

i is a current medoid, h is a non-selected object

Assume that i is replaced by h in the set of medoids

TCih = 0; For each non-selected object j ≠ h:

TCih += d(j,new_medj)-d(j,prev_medj): new_medj = the closest medoid to j after i is

replaced by h prev_medj = the closest medoid to j before i is

replaced by h

PAM Clustering: Total swapping cost TCih=jCjih

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

j

ih

t

Cjih = 0

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

t

i hj

Cjih = d(j, h) - d(j, i)

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

h

i t

j

Cjih = d(j, t) - d(j, i)

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

t

ih j

Cjih = d(j, h) - d(j, t)

CLARA (Clustering Large Applications) CLARA (Kaufmann and Rousseeuw in 1990)

Built in statistical analysis packages, such as S+ It draws multiple samples of the data set, applies PAM

on each sample, and gives the best clustering as the output

Strength: deals with larger data sets than PAM Weakness:

Efficiency depends on the sample size A good clustering based on samples will not necessarily

represent a good clustering of the whole data set if the sample is biased

CLARANS (“Randomized” CLARA)

CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han’94)

CLARANS draws sample of neighbors dynamically The clustering process can be presented as searching a

graph where every node is a potential solution, that is, a set of k medoids

If the local optimum is found, CLARANS starts with new randomly selected node in search for a new local optimum

It is more efficient and scalable than both PAM and CLARA Focusing techniques and spatial access structures may

further improve its performance (Ester et al.’95)

Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary

Hierarchical Clustering Use distance matrix as clustering criteria. This

method does not require the number of clusters k as an input, but needs a termination condition

Step 0 Step 1 Step 2 Step 3 Step 4

b

dc

e

a a b

d ec d e

a b c d e

Step 4 Step 3 Step 2 Step 1 Step 0

agglomerative(AGNES)

divisive(DIANA)

AGNES (Agglomerative Nesting)

Implemented in statistical analysis packages, e.g., Splus Use the Single-Link method and the dissimilarity matrix. Merge objects that have the least dissimilarity Go on in a non-descending fashion Eventually all objects belong to the same cluster

Single-Link: each time merge the clusters (C1,C2) which are connected by the shortest single link of objects, i.e., minpC1,qC2dist(p,q)

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Decompose data objects into a several levels of nested partitioning (tree of clusters), called a dendrogram.

A clustering of the data objects is obtained by cutting the dendrogram at the desired level, then each connected component forms a cluster.

E.g., level 1 gives 4 clusters: {a,b},{c},{d},{e},level 2 gives 3 clusters: {a,b},{c},{d,e} level 3 gives 2 clusters: {a,b},{c,d,e}, etc.

A Dendrogram Shows How the Clusters are Merged Hierarchically

a b c d e

ab

de

c

level 1

level 2

level 3

level 4

DIANA (Divisive Analysis)

Implemented in statistical analysis packages, e.g., Splus

Inverse order of AGNES Eventually each node forms a cluster on its own

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More on Hierarchical Clustering Methods Major weakness of agglomerative clustering

methods do not scale well: time complexity of at least O(n2), where

n is the number of total objects can never undo what was done previously

Integration of hierarchical with distance-based clustering BIRCH (1996): uses CF-tree and incrementally adjusts the

quality of sub-clusters CURE (1998): selects well-scattered points from the cluster

and then shrinks them towards the center of the cluster by a specified fraction

CHAMELEON (1999): hierarchical clustering using dynamic modeling

BIRCH (1996) Birch: Balanced Iterative Reducing and Clustering

using Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD’96)

Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering Phase 1: scan DB to build an initial in-memory CF tree (a

multi-level compression of the data that tries to preserve the inherent clustering structure of the data)

Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree

Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans

Clustering Feature Vector

Clustering Feature: CF = (N, LS, SS)

N: Number of data points

LS: Ni=1 Xi

SS: Ni=1 (Xi )2

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0 1 2 3 4 5 6 7 8 9 10

CF = (5, (16,30),244)

(3,4)(2,6)(4,5)(4,7)(3,8)

Some Characteristics of CFVs Two CFVs can be aggregated.

Given CF1=(N1, LS1, SS1), CF2 = (N2, LS2, SS2), If combined into one cluster, CF=(N1+N2, LS1+LS2, SS1+SS2).

The centroid and radius can both be computed from CF. centroid is the center of the cluster radius is the average distance between an object and the

centroid.

Other statistical features as well...

N

N

i ixx 10

NR

N

i i xx 21 0

)(

CF-Tree in BIRCH

A CF tree is a height-balanced tree that stores the clustering features for a hierarchical clustering A nonleaf node in a tree has (at most) B descendants or “children” The nonleaf nodes store sums of the CFs of their children A leaf node contains up to L CF entries

A CF tree has two parameters Branching factor B: specify the maximum number of children. threshold T: max radius of a sub-cluster stored in a leaf node

CF Tree (a multiway tree, like the B-tree)

CF1

child1

CF3

child3

CF2

child2

CF6

child6

CF1

child1

CF3

child3

CF2

child2

CF5

child5

CF1 CF2 CF6prev next CF1 CF2 CF4

prev next

Root

Non-leaf node

Leaf node Leaf node

CF-Tree Construction Scan through the database once. For each object, insert into the CF-tree as follows:

At each level, choose the sub-tree whose centroid is closest. In a leaf page, choose a cluster that can absort it (new radius <

T). If no cluster can absorb it, create a new cluster. Update upper levels.