Spring 2003 Data Mining by H. Liu, ASU 1

4. Clustering Methods

ConceptsPartitional (k-Means, k-Medoids)

Hierarchical (Agglomerative & Divisive, COBWEB)

Density-based (DBSCAN, CLIQUE)

Large size data (STING, BIRCH, CURE)

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Concepts of Clustering

• Clusters• Different ways of

representing clusters– Division with boundaries

– Venn diagram or spheres

– Probabilistic

– Dendrograms

– Trees

– Rules

1 2 3

I1

I2

…

In

0.5 0.2 0.3

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• About clusters– Inter-clusters distance maximization

– Intra-clusters distance minimization

• Clustering vs. classification– Which one is more difficult? Why?

– Various possible ways of clustering, which way is the best?

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Major Categories

• Partitioning: Divide into k partitions (k fixed); repartition to get better clustering.

• Hierarchical: Divide into different number of partitions in layers - merge (bottom-up) or divide (top-down).

• Density-based: Continue to grow a cluster as long as the density of the cluster exceeds a threshold

• Grid-based: First divide space into grids, then perform clustering on the grids.

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• Algorithm1. Given k

2. Randomly pick k instances as the initial centers

3. Assign the rest instances to closest one of k clusters

4. Recalculate the mean of each cluster

5. Repeat 3 & 4 until means don’t change

• How good the clusters are– Initial and final clusters

– Within-cluster variation diff(x,mean)^2

– Why don’t we consider inter-cluster distance?

k-Means

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Example

• For simplicity, 1 dimensional objects and k=2.• Objects: 1, 2, 5, 6,7• K-means:

– Randomly select 5 and 6 as initial centroids;

– => Two clusters {1,2,5} and {6,7}; meanC1=8/3, meanC2=6.5

– => {1,2}, {5,6,7}; meanC1=1.5, meanC2=6

– => no change.

– Aggregate dissimilarity = 0.5^2 + 0.5^2 + 1^2 + 1^2 = 2.5

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Discussions

• Limitations: – Means cannot be defined for categorical attributes;

– Choice of k;

– Sensitive to outliers;

– Crisp clustering

• Variants of k-means exist:– Using modes to deal with categorical attributes

• How about distance measures

• Is it similar to or different from k-NN?– With and without learning

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k-Medoids

• k-Means algorithm is sensitive to outliers– Is this true? How to prove it?

• Medoid – the most centrally located point in a cluster, as a representative point of the cluster.

• In contrast, a centroid is not necessarily inside a cluster.

• An example

Initial Medoids

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Partition Around Medoids

• PAM:1. Given k2. Randomly pick k instances as initial medoids

3. Assign each instance to the nearest medoid x4. Calculate the objective function

• the sum of dissimilarities of all instances to their nearest medoids

5. Randomly select an instance y6. Swap x by y if the swap reduces the objective

function7. Repeat (3-6) until no change

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k-Means and k-Medoids

• The key difference lies in how they update means or medoids

• Both require distance calculation and reassignment of instances

• Time complexity– Which one is more costly?

• Dealing with outliers

Outlier (100 unit away)

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EM (Expectation and Maximization)

• Moving away from crisp clusters as in k-Means by allowing an instance to belong to several clusters

• Finite mixtures – a statistical clustering model– A mixture is a set of k probability distributions,

representing k clusters– The simplest finite mixture: one feature with a Gaussian– When k=2, we need to estimate 5 parameters: 2 pairs of μ

and σ and pA, where pB = 1- pA

• EM– Estimate using instances– Maximize the overall likelihood that data came from this

data set

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Agglomerative

• Each object is viewed as a cluster (bottom up).• Repeat until the number of clusters is small

enough– Choose a closest pair of clusters– Merge the two into one

• Defining “closest”: Centroid (mean of cluster) distance, (average) sum of pairwise distance, …– Refer to the Evaluation part

• A dendrogram is a tree that shows clustering process.

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Dendrogram

• Cluster 1, 2, 4, 5, 6, 7 into two clusters (centriod distance)

1

2

4

5

6

7

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An example to show different Links

• Single link– Merge the nearest clusters measured by

the shortest edge between the two

– (((A B) (C D)) E)

• Complete link– Merge the nearest clusters measured by

the longest edge between the two

– (((A B) E) (C D))

• Average link– Merge the nearest clusters measured by

the average edge length between the two

– (((A B) (C D)) E)

A B C D E

A 0 1 2 2 3

B 1 0 2 4 3

C 2 2 0 1 5

D 2 4 1 0 3

E 3 3 5 3 0

A B

C

D

E

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Divisive

• All instances belong to one cluster (top-down)• To find an optimal division at each layer (especially

the top one) is computationally prohibitive.• One heuristic method is based on the Minimum

Spanning Tree (MST) algorithm– Connecting all instances with MST (O(N2))

– Repeatedly cut out the longest edges at each iteration until some stopping criterion is met or until one instance remains in each cluster.

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COBWEB

• Building a conceptual hierarchy incrementally• Category Utility:

kijP(fi=vij)P(fi=vij|ck)P(ck|fi=vij)

– All categories ck, all features fi, all feature values vij

• It attempts to maximize both the probability that two objects in the same category have values in common and the probability that objects in different categories will have different property values

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• Processing one instance at a time by evaluating– Placing the instance in the best existing category– Adding a new category containing only the instance– Merging of two existing categories into a new one and

adding the instance to that category– Splitting of an existing category into two and placing

the instance in the best new resulting category

Grandparent Grandparent

Parent

Child 2Child 1 Child 2Child 1Merge

Split

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Density-based

• BBSCAN –Density-Based Clustering of Applications with Noise

• It grows regions with sufficiently high density into clusters and can discover clusters of arbitrary shape in spatial databases with noise.– Many existing clustering algorithms find spherical

shapes of clusters

• DEBSCAN defines a cluster as a maximal set of density-connected points.

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• Defining density and connection -neighborhood of an object x (core object) (M, P, O)

– MinPts of objects within -neighborhood (say, 3)

– directly density-reachable (Q from M, M from P)

– density-reachable (Q from P, P not from Q) [asymmetric]

– density-connected (O, R, S) [symmetric] for border points• What is the relationship between DR and DC?

QM

QS

R

O

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• Clustering with DBSCAN– Search for clusters by checking the -neighborhood of

each instance x– If the -neighborhood of x contains more than MinPts,

create a new cluster with x as a core object– Iteratively collect directly density-reachable objects from

these core object and merge density-reachable clusters– Terminate when no new point can be add to any cluster

• DBSCAN is sensitive to the thresholds of density, but it is many folds faster than CLARANS

• Time complexity O(N log N) if a spatial index is used, O(N2) otherwise

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Dealing with Large Data

• Key ideas– Reducing the number of instances yet to maintain the

distribution

– Identifying relevant subspaces where clusters possibly exist

– Using summarized information to avoid repeated data access

• Sampling– CLARA (Clustering LARge Applications) working on

samples instead of the whole data

– CLARANS (Clustering Large Applications based on RANdomized Search)

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• Grid: STING (STatistical INformation Grid)– Statistical parameters of higher-level cells can easily be

computed from those of lower-level cells• Attribute-independent: count

• Attribute-dependent: mean, standard deviation, min, max

• Type of distribution: normal, uniform, exponential, or unknown

– Irrelevant cells can be removed

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Representatives

• BIRCH using Clustering Feature (CF) and CF tree– A cluster feature is a triplet about sub-clusters of

instances (N, LS, SS)• N - the number of instances, LS – linear sum, SS – square sum

– Two thresholds: branching factor and the max number of children per non-leaf node

– Two phases1. Build an initial in-memory CF tree

2. Apply a clustering algorithm to cluster the leaf nodes in CF tree

• CURE (Clustering Using REpresentitives) is another example

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• Taking advantage of the property of density– If it’s dense in higher dimensional subspaces, it should

be dense in some lower dimensional subspaces

– CLIQUE (CLustering In QUEst)• With high dimensional data, there are many void subspaces

• Using the property identified, we can start with dense lower dimensional data

• CLIQUE is a density-based method that can automatically find subspaces of the highest dimensionality such that high-density clusters exist in those subspaces

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Chameleon

• A hierarchical Clustering Algorithm Using Dynamic Modeling– Observations on the weakness of CURE and ROCK

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Summary

• There are many clustering algorithms • Good clustering algorithms maximize inter-cluster

dissimilarity and intra-cluster similarity• Without prior knowledge, it is difficult to choose

the best clustering algorithm.• Clustering is an important tool for outlier analysis.

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Bibliography

• I.H. Witten and E. Frank. Data Mining – Practical Machine Learning Tools and Techniques with Java Implementations. 2000. Morgan Kaufmann.

• M. Kantardzic. Data Mining – Concepts, Models, Methods, and Algorithms. 2003. IEEE.

• J. Han and M. Kamber. Data Mining – Concepts and Techniques. 2001. Morgan Kaufmann.

• M. H. Dunham. Data Mining – Introductory and Advanced Topics.