1 Data Mining and Warehousing: Chapter 8 Cluster Analysis Jia-wei Han han

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Data Mining and Warehousing: Chapter 8

Cluster AnalysisJia-wei Han http://www.cs.sfu.ca/~han

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Clustering analysis

What is Clustering Analysis?

Clustering in Data Mining Applications

Handling Different Types of Variables

Major Clustering Techniques

Outlier Discovery

Problems and Challenges

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What Is Clustering ?

Clustering is a process of partitioning a set of data (or objects)

into a set of meaningful sub-classes, called clusters.

Help users understand the natural grouping or structure in

a data set.

Cluster: a collection of data objects that are “similar” to one

another and thus can be treated collectively as one group.

Clustering: unsupervised classification: no predefined classes.

Used either as a stand-alone tool to get insight into data

distribution or as a preprocessing step for other algorithms.

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What Is Good Clustering?

A good clustering method will produce high quality

clusters in which:

the intra-class (that is, intraintra-cluster) similarity is high.

the inter-class similarity is low.

The quality of a clustering result also 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.

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Requirements of Clustering in Data Mining

Scalability

Dealing 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

Interpretability and usability.

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Clustering analysis





Outlier Discovery


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Applications of Clustering

Clustering has wide applications in

Pattern Recognition

Spatial Data Analysis:

– create thematic maps in GIS by clustering feature spaces

– detect spatial clusters and explain them in spatial data mining.

Image Processing

Economic Science (especially market research)

WWW:

– Document classification

– Cluster Weblog data to discover groups of similar access patterns

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Examples of Clustering Applications

Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs.

Land use: Identification of areas of similar land use in an earth observation database.

Insurance: Identifying groups of motor insurance policy holders with a high average claim cost.

City-planning: Identifying groups of houses according to their house type, value, and geographical location.

Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults.

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Clustering analysis





Outlier Discovery


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Data Structures

Data matrix (two mode)

Dissimilarity matrix (one mode)

npx...nfx...n1x

...............ipx...ifx...i1x

...............1px...1fx...11x

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

:::

)2,3()

...ndnd

0dd(3,1

0d(2,1)

0

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Measure the Quality of Clustering

Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically metric: d(i, j)

There is a separate “quality” function that measures the “goodness” of a cluster.

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

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

It is hard to define “similar enough” or “good enough” the answer is typically highly subjective.

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Type of data in clustering analysis

Interval-scaled variables:

Binary variables:

Nominal, ordinal, and ratio variables:

Variables of mixed types:

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Interval-valued variables

Standardize data

Calculate the mean absolute deviation:

where

Calculate the standardized measurement (z-score)

Using mean absolute deviation is more robust than using

standard deviation

.)...21

1nffff

xx(xn m

|)|...|||(|121 fnffffff

mxmxmxns

f

fifif s

mx z

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Similarity and Dissimilarity Between Objects

Distances are normally used to measure the similarity or dissimilarity between two data objects.

Some popular ones include: Minkowski distance:

where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional

data objects, and q is a positive integer.

qq

pp

qq

jx

ix

jx

ix

jx

ixjid )||...|||(|),(

2211

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Similarity and Dissimilarity Between Objects

If q = 1, d is Manhattan distance.

If q = 2, d is Euclidean distance:

Also one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures.

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

22

2

11 pp jx

ix

jx

ix

jx

ixjid

||...||||),(2211 pp jxixjxixjxixjid

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Binary variables

A contingency table for binary data

Simple matching coefficient (invariant, if the binary variable

is symmetric):

Jaccard coefficient (noninvariant if the binary variable is

asymmetric):

dcbacb jid

),(

pdbcasum

dcdc

baba

sum

0

1

01

cbacb jid

),(

Object i

Object j

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Dissimilarity between binary variables

Example

– gender is symmetric attribute

– the remaining attributes are asymmetric binary

– let the values Y and P be set to 1, and the value N be set to 0

Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4

Jack M Y N P N N NMary F Y N P N P NJim M Y P N N N N

75.0211

21),(

67.0111

11),(

33.0102

10),(

maryjimd

jimjackd

maryjackd

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Clustering analysis





Outlier Discovery


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Clustering techniques have been studied extensively in:

Statistics, machine learning, and data mining

with many methods proposed and studied.

Clustering methods can be classified into 5 approaches:

partitioning algorithms

hierarchical algorithms

density-based method

grid-based method

model-based method

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Five Categories of Clustering Methods

Partitioning algorithms: Construct various partitions and

then evaluate them by some criterion.

Hierarchy 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.

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Partitioning Algorithms: Basic Concept

Partitioning method: Construct a partition of a database D

of n objects into a set of k clusters

Given a k, find a partition of k clusters that optimizes the

chosen partitioning criterion. Global optimal: exhaustively enumerate all partitions.

Heuristic methods: k-means and k-medoids algorithms.

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.

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The K-Means Clustering Method

Given k, the k-means algorithm is implemented in 4 steps: Partition objects into k nonempty subsets Compute seed points as the centroids of the clusters of

the current partition. The centroid is the center (mean point) of the cluster.

Assign each object to the cluster with the nearest seed point.

Go back to Step 2, stop when no more new assignment.

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The K-Means Clustering Method

Example

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10

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0

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0 1 2 3 4 5 6 7 8 9 10

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0 1 2 3 4 5 6 7 8 9 10

0

1

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0 1 2 3 4 5 6 7 8 9 10

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Comments on the K-Means Method

Strength of the k-means: Relatively efficient: O(tkn), where n is # of objects, k is # of

clusters, and t is # of iterations. Normally, k, t << n. Often terminates at a local optimum. The global optimum

may be found using techniques such as: deterministic annealing and genetic algorithms.

Weakness of the k-means: Applicable only when mean is defined, then what about

categorical data? Need to specify k, the number of clusters, in advance. Unable to handle noisy data and outliers. Not suitable to discover clusters with non-convex shapes.

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Variations of the K-Means Method

A few variants of the k-means which differ in: Selection of the initial k means. Dissimilarity calculations. Strategies to calculate cluster means.

Handling categorical data: k-modes (Huang’98): Replacing means of clusters with modes. Using new dissimilarity measures to deal with categorical

objects. Using a frequency-based method to update modes of

clusters. A mixture of categorical and numerical data: k-prototype

method.

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The K-Medoids Clustering Method

Find representative objects, called medoids, in clusters To achieve this goal, only the definition of distance from

any two objects is needed. 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. Focusing + spatial data structure (Ester et al., 1995).

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CLARA (Clustering Large Applications) (1990)

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 of CLARA:

deal with larger data sets than PAM. Weaknesse of CLARA:

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.

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CLARANS (“Randomized” CLARA) (1994)

CLARANS (A Clustering Algorithm based on Randomized Search) by Ng and Han.

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).

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CLARANS

A node is represented by the set of k objects {Om1, …, Omk}

Two nodes are neighbors if their sets differ by only one object S1 = {Om1, …, Omk}, S2 = {Ow1, …, Owk} |S1 S2| = k - 1 each node has k(n-k) neighbors each node represent a collection of k medoids, each node

corresponds to a clustering (dynamic) draws a sample of neighbors in each step of a search if a better neighbor is found, moves to the neighbor’s node

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CLARANS

Two parameters: the maximum number of neighbors examined

(maxneighbor) the number of local minima obtained (numlocal)

Complexity:

Enable the detection of outliers

)( 2nO

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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

d

c

e

a a b

d e

c d e

a b c d e

Step 4 Step 3 Step 2 Step 1 Step 0

agglomerative(AGNES)

divisive(DIANA)

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Density-Based Clustering Methods

Clustering based on density (local cluster criterion), such as density-connected points

Major features: Discover clusters of arbitrary shape Handle noise One scan Need density parameters as termination condition

Several interesting studies: DBSCAN: Ester, et al. (KDD’96) OPTICS: Ankerst, et al (SIGMOD’99). DENCLUE: Hinneburg & D. Keim (KDD’98) CLIQUE: Agrawal, et al. (SIGMOD’98)

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DBSCAN: A Density-Based Clustering Method

DBSCAN: Density Based Spatial Clustering of Applications with Noise. Proposed by Ester, Kriegel, Sander, and Xu (KDD’96) Relies on a density-based notion of cluster: A cluster is

defined as a maximal set of density-connected points Discovers clusters of arbitrary shape in spatial

databases with noise

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Handling Complex Shaped Clusters

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Density-Based Clustering: Background

Two parameters:

Eps: Maximum radius of the neighbourhood

MinPts: Minimum number of points in an Eps-neighbourhood of that point

NEps(p): {q belongs to D | dist(p,q) <= Eps}

Directly density-reachable: A point p is directly density-reachable from a point q wrt. Eps, MinPts if

1) p belongs to NEps(q)

2) core point condition:

|NEps (q)| >= MinPts

pq

MinPts = 5

Eps = 1 cm

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Density-Based Clustering: Background (II)

Density-reachable:

A point p is density-reachable from a point q wrt. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi

Density-connected

A point p is density-connected to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts.

p

qp1

p q

o

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DBSCAN: General Ideas

Core

Border

Outlier

Eps = 1cm

MinPts = 5

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DBSCAN: The Algorithm

Arbitrary select a point p

Retrieve all points density-reachable from p wrt Eps and

MinPts.

If p is a core point, a cluster is formed.

If p is a border point, no points are density-reachable

from p and DBSCAN visits the next point of the

database.

Continue the process until all of the points have been

processed.

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OPTICS: A Cluster-Ordering Method (1999)

OPTICS: Ordering Points To Identify the Clustering Structure Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99). Produces a special order of the database wrt its density-

based clustering structure. This cluster-ordering contains info equiv to the density-

based clusterings corresponding to a broad range of parameter settings.

Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure.

Can be represented graphically or using visualization techniques.

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OPTICS: Some Extension from DBSCAN

Index-based: – k = number of dimensions – N = 20– p = 75%– M = N(1-p) = 5

Complexity: O(kN2) Core Distance

Reachability Distance

D

p2

MinPts = 5

= 3 cm

Max (core-distance (o), d (o, p))

r(p1, o) = 2.8cm. r(p2,o) = 4cm

o

o

p1

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Model-Based Clustering Methods

Use certain models for clusters and attempt to optimize the fit between the data and the model.

Neural network approaches: The best known neural network approach to clustering is the SOM

(self-organizing feature map) method, proposed by Kohonen in 1981. It can be viewed as a nonlinear projection from an m-dimensional

input space onto a lower-order (typically 2-dimensional) regular lattice of cells. Such a mapping is used to identify clusters of elements that are similar (in a Euclidean sense) in the original space.

Machine learning: probability density-based approach: Grouping data based on probability density models: based on how

many (possibly weighted) features are the same. COBWEB (Fisher’87) Assumption: The probability distribution on

different attributes are independent of each other --- This is often too strong because correlation may exist between attributes.

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Model-Based Clustering Methods (2)

Statistical approach: Gaussian mixture model (Banfield and Raftery, 1993): A probabilistic variant of k-means method. It starts by choosing k seeds, and regarding the seeds as means of

Gaussian distributions, then iterates over two steps called the estimation step and the maximization step, until the Gaussians are no longer moving.

Estimation: calculating the responsibility that each Gaussian has for each data point.

Maximization: The mean of each Gaussian is moved towards the centroid of the entire data set.

Statistical Approach: AutoClass (Cheeseman and Stutz, 1996): A thorough implementation of a Bayesian clustering procedure based on mixture models. It uses Bayesian statistical analysis to estimate the number of clusters.

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What Is Outlier Discovery?

What are outliers? The set of objects are considerably dissimilar from the

remainder of the data Example: Sports: Michael Jordon, Wayne Gretzky, ...

Problem Given: Data points Find top n outlier points

Applications: Credit card fraud detection Telecom fraud detection Customer segmentation Medical analysis

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Statistical Approaches

Assume a model underlying distribution that generates data set (e.g. normal distribution)

Use discordancy tests depending on data distribution distribution parameter (e.g., mean, variance) number of expected outliers

Drawbacks most tests are for single attribute In many cases, data distribution may not be known

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Considerable progress has been made in scalable clustering methods: Partitioning: k-means, k-medoids, CLARANS Hierarchical: BIRCH, CURE Density-based: DBSCAN, CLIQUE, OPTICS Grid-based: STING, WaveCluster. Model-based: Autoclass, Denclue, Cobweb.

Current clustering techniques do not address all the requirements adequately.

Constraint-based clustering analysis: Constraints exists in data space (bridges and highways) or in user queries.

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References

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. SIGMOD'98

M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973. M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the

clustering structure, SIGMOD’99. P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scietific, 1996 M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering

clusters in large spatial databases. KDD'96. M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases: Focusing

techniques for efficient class identification. SSD'95. D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning,

2:139-172, 1987. D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on

dynamic systems. In Proc. VLDB’98. S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases.

SIGMOD'98. A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.

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References (2)

L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990.

E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB’98.

G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons, 1988.

P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997. R. Ng and J. Han. Efficient and effective clustering method for spatial data mining.

VLDB'94. E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data

sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105. G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution

clustering approach for very large spatial databases. VLDB’98. W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial

Data Mining, VLDB’97. T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method

for very large databases. SIGMOD'96.

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1 Data Mining and Warehousing: Chapter 8 Cluster Analysis Jia-wei Han han