1. Data Mining Algorithms
Clustering

2. M.Vijayalakshmi VESIT BE(IT) Data Mining
Clustering Outline
Goal: Provide an overview of the clustering problem and introduce some of the basic algorithms
Clustering Problem Overview
Clustering Techniques
Hierarchical Algorithms
Partitional Algorithms
Genetic Algorithm
Clustering Large Databases
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3. 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
Typical applications
As a stand-alone tool to get insight into data distribution
As a preprocessing step for other algorithms
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4. General Applications of Clustering
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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5. 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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6. Clustering vs. Classification
No prior knowledge
Number of clusters
Meaning of clusters
Cluster results are dynamic
Unsupervised learning
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7. Classification vs. Clustering
Classification: Supervised learning:
Learns a method for predicting the instance class from pre-labeled (classified) instances
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8. M.Vijayalakshmi VESIT BE(IT) Data Mining
Clustering
Unsupervised learning:
Finds “natural” grouping of instances given un-labeled data
M.Vijayalakshmi VESIT BE(IT) Data Mining

9. Clustering Houses Geographic Distance Based Size Based
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10. M.Vijayalakshmi VESIT BE(IT) Data Mining
Clustering Methods
Many different method and algorithms:
For numeric and/or symbolic data
Deterministic vs. probabilistic
Exclusive vs. overlapping
Hierarchical vs. flat
Top-down vs. bottom-up
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11. M.Vijayalakshmi VESIT BE(IT) Data Mining
Clustering Issues
Outlier handling
Dynamic data
Interpreting results
Evaluating results
Number of clusters
Data to be used
Scalability
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12. Impact of Outliers on Clustering
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13. Clustering Evaluation
Manual inspection
Benchmarking on existing labels
Cluster quality measures
distance measures
high similarity within a cluster, low across clusters
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14. M.Vijayalakshmi VESIT BE(IT) Data Mining
Data Structures
Data matrix
Dissimilarity matrix
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15. 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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16. Type of data in clustering analysis
Interval-scaled variables:
Binary variables:
Nominal, ordinal, and ratio variables:
Variables of mixed types:
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17. 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
If q = 1, d is Manhattan distance
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18. Similarity and Dissimilarity Between Objects (Cont.)
If q = 2, d is 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)
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19. M.Vijayalakshmi VESIT BE(IT) Data Mining
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):
Object j
Object i
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20. Dissimilarity between Binary Variables
Example
gender is a 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
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21. M.Vijayalakshmi VESIT BE(IT) Data Mining
Nominal Variables
A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green
Method 1: Simple matching
m: # of matches, p: total # of variables
Method 2: use a large number of binary variables
creating a new binary variable for each of the M nominal states
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22. M.Vijayalakshmi VESIT BE(IT) Data Mining
Clustering Problem
Given a database D={t1,t2,…,tn} of tuples and an integer value k, the Clustering Problem is to define a mapping f:Dg{1,..,k} where each ti is assigned to one cluster Kj, 1<=j<=k.
A Cluster, Kj, contains precisely those tuples mapped to it.
Unlike classification problem, clusters are not known a priori.
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23. M.Vijayalakshmi VESIT BE(IT) Data Mining
Types of Clustering
Hierarchical – Nested set of clusters created.
Partitional – One set of clusters created.
Incremental – Each element handled one at a time.
Simultaneous – All elements handled together.
Overlapping/Non-overlapping
M.Vijayalakshmi VESIT BE(IT) Data Mining

24. Major Clustering Approaches
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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25. Clustering Approaches
Hierarchical
Partitional
Categorical
Large DB
Agglomerative
Divisive
Sampling
Compression
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26. M.Vijayalakshmi VESIT BE(IT) Data Mining
Cluster Parameters
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27. Distance Between Clusters
Single Link: smallest distance between points
Complete Link: largest distance between points
Average Link: average distance between points
Centroid: distance between centroids
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28. Hierarchical Clustering
Clusters are created in levels actually creating sets of clusters at each level.
Agglomerative
Initially each item in its own cluster
Iteratively clusters are merged together
Bottom Up
Divisive
Initially all items in one cluster
Large clusters are successively divided
Top Down
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29. 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
agglomerative
(AGNES)
divisive
(DIANA)
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30. Hierarchical Algorithms
Single Link
MST Single Link
Complete Link
Average Link
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31. M.Vijayalakshmi VESIT BE(IT) Data Mining
Dendrogram
A tree data structure which illustrates hierarchical clustering techniques.
Each level shows clusters for that level.
Leaf – individual clusters
Root – one cluster
A cluster at level i is the union of its children clusters at level i+1.
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32. M.Vijayalakshmi VESIT BE(IT) Data Mining
Levels of Clustering
M.Vijayalakshmi VESIT BE(IT) Data Mining

33. Agglomerative ExampleB C D E 1 2 3 4 5 A B E C D
Threshold of 1 2 3 4 5 A B C D E
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34. M.Vijayalakshmi VESIT BE(IT) Data Mining
MST Example A B A B C D E 1 2 3 4 5 E C D
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35. Agglomerative Algorithm
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36. M.Vijayalakshmi VESIT BE(IT) Data Mining
Single Link
View all items with links (distances) between them.
Finds maximal connected components in this graph.
Two clusters are merged if there is at least one edge which connects them.
Uses threshold distances at each level.
Could be agglomerative or divisive.
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37. MST Single Link Algorithm
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38. Single Link Clustering
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39. AGNES (Agglomerative Nesting)
Implemented in statistical analysis packages, e.g., Splus
Use the Single-Link method and the dissimilarity matrix.
Merge nodes that have the least dissimilarity
Go on in a non-descending fashion
Eventually all nodes belong to the same cluster
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40. M.Vijayalakshmi VESIT BE(IT) Data Mining
A Dendrogram Shows How the Clusters are Merged Hierarchically
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.
M.Vijayalakshmi VESIT BE(IT) Data Mining

41. 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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42. Partitional Clustering
Nonhierarchical
Creates clusters in one step as opposed to several steps.
Since only one set of clusters is output, the user normally has to input the desired number of clusters, k.
Usually deals with static sets.
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43. Partitioning Algorithms
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: Each cluster is represented by the center of the cluster
k-medoids or PAM (Partition around medoids): Each cluster is represented by one of the objects in the cluster
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44. Partitional Algorithms
MST
Squared Error
K-Means
Nearest Neighbor
PAM
BEA GA
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45. M.Vijayalakshmi VESIT BE(IT) Data Mining
MST Algorithm
M.Vijayalakshmi VESIT BE(IT) Data Mining

46. M.Vijayalakshmi VESIT BE(IT) Data Mining
K-Means
Initial set of clusters randomly chosen.
Iteratively, items are moved among sets of clusters until the desired set is reached.
High degree of similarity among elements in a cluster is obtained.
Given a cluster Ki={ti1,ti2,…,tim}, the cluster mean is mi = (1/m)(ti1 + … + tim)
M.Vijayalakshmi VESIT BE(IT) Data Mining

47. M.Vijayalakshmi VESIT BE(IT) Data Mining
K-Means Example
Given: {2,4,10,12,3,20,30,11,25}, k=2
Randomly assign means: m1=3,m2=4
K1={2,3}, K2={4,10,12,20,30,11,25}, m1=2.5,m2=16
K1={2,3,4},K2={10,12,20,30,11,25}, m1=3,m2=18
K1={2,3,4,10},K2={12,20,30,11,25}, m1=4.75,m2=19.6
K1={2,3,4,10,11,12},K2={20,30,25}, m1=7,m2=25
Stop as the clusters with these means are the same.
M.Vijayalakshmi VESIT BE(IT) Data Mining

48. 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.
M.Vijayalakshmi VESIT BE(IT) Data Mining

49. Comments on the K-Means Method
Strength
Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # 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
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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50. M.Vijayalakshmi VESIT BE(IT) Data Mining
Nearest Neighbor
Items are iteratively merged into the existing clusters that are closest.
Incremental
Threshold, t, used to determine if items are added to existing clusters or a new cluster is created.
M.Vijayalakshmi VESIT BE(IT) Data Mining

51. 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
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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52. The K-Medoids Clustering Method
Find representative objects, called medoids, in clusters
PAM (Partitioning Around Medoids,)
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
CLARANS): Randomized sampling Focusing + spatial data structure
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53. PAM (Partitioning Around Medoids)
PAM - Use real object to represent the cluster
Select k representative objects arbitrarily
For each pair of non-selected object h and selected object i, calculate the total swapping cost TCih
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 object
repeat steps 2-3 until there is no change
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54. M.Vijayalakshmi VESIT BE(IT) Data Mining
PAM
Partitioning Around Medoids (PAM) (K-Medoids)
Handles outliers well.
Ordering of input does not impact results.
Does not scale well.
Each cluster represented by one item, called the medoid.
Initial set of k medoids randomly chosen.
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55. M.Vijayalakshmi VESIT BE(IT) Data Mining
PAM
M.Vijayalakshmi VESIT BE(IT) Data Mining

56. M.Vijayalakshmi VESIT BE(IT) Data Mining
PAM Algorithm
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57. M.Vijayalakshmi VESIT BE(IT) Data Mining
DBSCAN
Density Based Spatial Clustering of Applications with Noise
Outliers will not effect creation of cluster.
Input
MinPts – minimum number of points in cluster
Eps – for each point in cluster there must be another point in it less than this distance away.
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58. M.Vijayalakshmi VESIT BE(IT) Data Mining
Density Concepts
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59. Comparison of Clustering Techniques
M.Vijayalakshmi VESIT BE(IT) Data Mining