1. PARTITIONAL CLUSTERING
ACM Student Chapter,
Heritage Institute of Technology
17th February, 2012
SIGKDD Presentation by
Megha Nangia
J. M. Mansa
Koustav Mullick

2. Why do we cluster? Clustering results are 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

3. What is Cluster Analysis?
Cluster analysis or clustering is the task of assigning a set of objects into groups (called clusters) so that the objects in the same cluster are more “similar” (in some sense or another) to each other than to those in other clusters.
Cluster analysis itself is not one specific algorithm. But the general task to be solved is forming similar clusters. It can be achieved by various algorithms.

4. How do we define “similarity”?
Recall that the goal is to group together “similar” data – but what does this mean?
No single answer – it depends on what we want to find or emphasize in the data; this is one reason why clustering is an “art”
The similarity measure is often more important than the clustering algorithm used – don’t overlook this choice!

5. Clustering: Minimize Intra-cluster distance
Maximize Inter-cluster distance

6. Applications:
Clustering is a main task of explorative data mining to reduce the size of large data sets. Its a common technique for statistical data analysis used in many fields, including :
Machine learning
Pattern recognition
Image analysis
Information retrieval
Bioinformatics.
Web applications such as social network analysis, grouping of shopping items, search result grouping etc.

7. Requirements of Clustering in Data Mining
Scalability
Ability to deal with different types of attributes
Discovery of clusters with arbitrary shape
Able to deal with noise and outliers
Insensitive to order of input records
High dimensionality
Interpretability and usability

8. Notion of clustering: How many clusters? Six Clusters Two Clusters
Four Clusters

9. Clustering Algorithms:
Clustering algorithms can be categorized
Some of the major algorithms are:
Hierarchical or connectivity based clustering
Partitional clustering (K-means or centroid-based clustering)
Density based
Grid based
Model based

10. Mammals

12. Partitional Clustering:
In statistics and data mining, k-means clustering is a method of cluster analysis which aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean. This results into a partitioning of the data space into Voronoi cells.
A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset

13. Partitional Clustering :
A Partitional Clustering
Original Points

14. Hierarchical Clustering:
Connectivity based clustering, also known as hierarchical clustering, is based on the core idea of objects being more related to nearby objects than to objects farther away.
As such, these algorithms connect "objects" to form "clusters" based on their distance. At different distances, different clusters will form, which can be represented using a dendrogram.
These algorithms do not provide a single partitioning of the data set, but instead provide an extensive hierarchy of clusters that merge with each other at certain distances.
A set of nested clusters organized as a hierarchical tree

15. Hierarchical Clustering:
Traditional Hierarchical Clustering
Traditional Dendrogram
Non-traditional Hierarchical Clustering
Non-traditional Dendrogram

16. Hierarchical Clustering.
Partitional Clustering.

17. Partitioning Algorithms:
Partitioning method: Construct a partition of n objects into a set of K clusters
Given: a set of objects and the number K
Find: a partition of K clusters that optimizes the chosen partitionin`g criterion
Effective heuristic methods: K-means and K-medoids algorithms

18. Common choices for Similarity/ Distance measures:
Euclidean distance:
City block or Manhattan distance:
Cosine similarity:
Jaccard similarity:

19. 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

20. K-Means Algorithm: START Choose K Centroids
Select K points as initial Centroids.
Repeat:
Form k clusters by assigning all points to their
respective closest centroid.
Re-compute the centroid for each cluster
5. Until: The centroids don`t change.
Form k clusters.
Recompute centroid
YES
Centroids
change NO
END

21. Time Complexity
Assume computing distance between two instances is O(m) where m is the dimensionality of the vectors.
Reassigning clusters: O(kn) distance computations, or O(knm).
Computing centroids: Each instance vector gets added once to some centroid: O(nm).
Assume these two steps are each done once for I iterations: O(Iknm).

22. K-means Clustering: Step 1
Algorithm: k-means, Distance Metric: Euclidean Distance

23. K-means Clustering: Step 2
Algorithm: k-means, Distance Metric: Euclidean Distance

24. K-means Clustering: Step 3
Algorithm: k-means, Distance Metric: Euclidean Distance

25. K-means Clustering: Step 4
Algorithm: k-means, Distance Metric: Euclidean Distance

26. K-means Clustering: Step 5
Algorithm: k-means, Distance Metric: Euclidean Distance k1 k2 k3

27. K-Means Clustering: Example 2

28. K-Means Clustering: Example 2

29. Importance of Choosing Initial Centroids …

30. Importance of Choosing Initial Centroids …

31. Two different K-means Clusterings
Original Points
Optimal Clustering
Sub-optimal Clustering

32. Solutions to Initial Centroids Problem
Multiple runs
Helps, but probability is not on your side
Sample and use hierarchical clustering to determine initial centroids
Select more than k initial centroids and then select among these initial centroids
Select most widely separated
Postprocessing
Bisecting K-means
Not as susceptible to initialization issues

33. Evaluating K-means Clusters
Most common measure is Sum of Squared Error (SSE)
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
One easy way to reduce SSE is to increase K, the number of clusters
A good clustering with smaller K can have a lower SSE than a poor clustering with higher K

34. Strength
Relatively efficient: O(ikn), where n is # objects, k is # clusters, and i is # iterations. Normally, k, i << 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
Also may give rise to Empty-clusters.

35. Outliers cluster outliers
Outliers are objects that do not belong to any cluster or form clusters of very small cardinality
cluster
outliers

36. Which cluster to be picked for bisection ?
Bisecting K-Means:
A variant of k-means, that can produce a partitional or heirarchical clustering.
Which cluster to be picked for bisection ?
Can pick the largest Cluster , or
The cluster With lowest average similarity, or
Cluster with the largest SSE.

37. Bisecting K-Means Algorithm:
START
Bisecting K-Means Algorithm:
Initialize clusters
Initialize the list of clusters.
Repeat:
Select a cluster from the list of clusters.
For i=1 to number_of_iterations
Bisect the cluster using k-means algorithm
End for
Select two clusters having the lowest SSE
Add the two clusters from the bisection to
the list of clusters
9. Until: The list contains k clusters.
Select a cluster NO
i < no. of iterations
YES
Bisect the cluster.
i++
Add the two bisected clusters, having lowest SSE, to list of clusters NO
K clusters
YES
END

38. Bisecting K-means: Example

39. Why bisecting K-means works better than regular K-means?
–Bisecting K-means tends to produce clusters of relatively uniform size.
–Regular K-means tends to produce clusters of widely different sizes.
–Bisecting K-means beats Regular K-means in Entropy measurement

40. Limitations of K-means:
K-means has problems when clusters are of differing
Sizes
Densities
Non-globular shapes
K-means has problems when the data contains outliers.

41. Limitations of K-means: Differing Sizes
Original Points
K-means (3 Clusters)

42. Limitations of K-means: Differing Density
Original Points
K-means (3 Clusters)

43. Limitations of K-means: Non-globular Shapes
Original Points
K-means (2 Clusters)

44. Overcoming K-means Limitations
Original Points K-means Clusters
One solution is to use many clusters.
Find parts of clusters, but need to put together.

45. Overcoming K-means Limitations
Original Points K-means Clusters

46. Overcoming K-means Limitations
Original Points K-means Clusters

47. K-Medoids Algorithm What is a medoid?
A medoid can be defined as the object of a cluster, whose average dissimilarity to all the objects in the cluster is minimal, i.e, it is a most centrally located point in the cluster.
In contrast to the k-means algorithm, k-medoids chooses datapoints as centers(medoids or exemplars)
The most common realisation of k-medoid clustering is the Partitioning Around Medoids (PAM) algorithm.

48. medoids(PAM) algorithm
Partitioning around
medoids(PAM) algorithm
1. Initialize: randomly select k of the n data points as the medoids.
2. Associate each data point to the closest medoid.
3. For each medoid m
1. For each non-medoid data point o
1. Swap m and o and compute the total cost of the configuration.
4. Select the configuration with the lowest cost.
5. Repeat steps 2 to 5 until there is no change in the medoid.

49. Demonstration of PAM
Cluster the following set of ten objects into two clusters i.e. k=2.
Consider a data set of ten objects as follows:
Point
Cordinate 1
Cordinate2 X1 2 6 X2 3 4 X3 8 X4 7 X5 X6 X7 X8 X9 5
X10

50. Distribution of the data

51. Step 1 Initialize k centres. Let us assume c1=(3,4) and c2=(7,4).
So here c1 and c2 are selected as medoids.
Calculating distance so as to associate each data object to its nearest medoid. c1
Data objects (Xi)
Cost 3 4 2 6 8 7 5 C2
Data objects (Xi)
Cost 7 4 2 6 3 8 1 5

52. Then so the clusters become:
The total cost involved is 20

53. Cluster after step 1
Next, we choose a non-medoid point for each medoid, swap it with the medoid and re-compute the cost. If the cost is optimized, we make it the new medoid and proceed similarly, until there is no change in the medoids.

54. Comments on PAM Algorithm
Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean
Pam works well for small data sets but does not scale well for large data sets.

55. Conclusion:
Partitional clustering is a very efficient and easy to implement clustering method.
It helps us find the global and local optimums.
Some of the heuristic approaches involve the K-means and K-medoid algorithms.
However partitional clustering also suffers from a number of shortcomings:
The performance of the algorithm depends on the initial centroids. So
the algorithm gives no guarantee for an optimal solution.
Choosing poor initial centroids may lead to the generation of empty clusters as well.
The number of clusters need to be determined beforehand.
Does not work well with non-globular clusters.
Some of the above stated drawbacks can be solved using the other popular Clustering approach, such as Hierarchical or density based clustering. Nevertheless the importance of partitional clustering cannot be denied.