1. Cluster Analysis

2. 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

3. Hierarchical Clustering
Produces a set of nested clusters organized as a hierarchical tree
Can be visualized as a dendrogram
A tree like diagram that records the sequences of merges or splits

4. Strengths of Hierarchical Clustering
Do not have to assume any particular number of clusters
Any desired number of clusters can be obtained by ‘cutting’ the dendogram at the proper level
They may correspond to meaningful taxonomies
Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

5. Hierarchical Clustering
Two main types of hierarchical clustering
Agglomerative:
Start with the points as individual clusters
At each step, merge the closest pair of clusters until only one cluster (or k clusters) left
Divisive:
Start with one, all-inclusive cluster
At each step, split a cluster until each cluster contains a point (or there are k clusters)
Traditional hierarchical algorithms use a similarity or distance matrix
Merge or split one cluster at a time

6. Agglomerative Clustering Algorithm
More popular hierarchical clustering technique
Basic algorithm is straightforward
Compute the proximity matrix
Let each data point be a cluster
Repeat
Merge the two closest clusters
Update the proximity matrix
Until only a single cluster remains
Key operation is the computation of the proximity of two clusters
Different approaches to defining the distance between clusters distinguish the different algorithms

7. Starting Situation
Start with clusters of individual points and a proximity matrix p1 p3 p5 p4 p2
. . . .
Proximity Matrix

8. Intermediate Situation
After some merging steps, we have some clusters C2 C1 C3 C5 C4 C3 C4
Proximity Matrix C1 C5 C2

9. Intermediate Situation
We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C2 C1 C3 C5 C4 C3 C4
Proximity Matrix C1 C5 C2

10. After Merging The question is “How do we update the proximity matrix?”
C2 U C5 C1 C3 C4 C1 ?
? ? ? ?
C2 U C5 C3 C3 ? C4 ? C4
Proximity Matrix C1
C2 U C5

11. How to Define Inter-Cluster Similarityp1 p3 p5 p4 p2
. . . .
Similarity?
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
Proximity Matrix

12. How to Define Inter-Cluster Similarityp1 p3 p5 p4 p2
. . . .
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
Proximity Matrix

13. How to Define Inter-Cluster Similarityp1 p3 p5 p4 p2
. . . .
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
Proximity Matrix

14. How to Define Inter-Cluster Similarityp1 p3 p5 p4 p2
. . . .
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
Proximity Matrix

15. How to Define Inter-Cluster Similarityp1 p3 p5 p4 p2
. . . .  
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
Proximity Matrix

16. Cluster Similarity: MIN or Single Link
Similarity of two clusters is based on the two most similar (closest) points in the different clusters
Determined by one pair of points, i.e., by one link in the proximity graph. 1 2 3 4 5

17. Hierarchical Clustering: MIN5 1 2 3 4 5 6 4 3 2 1
Nested Clusters
Dendrogram

18. Strength of MIN Two Clusters Original Points
Can handle non-elliptical shapes

19. Limitations of MIN Two Clusters Original Points
Sensitive to noise and outliers

20. Cluster Similarity: MAX or Complete Linkage
Similarity of two clusters is based on the two least similar (most distant) points in the different clusters
Determined by all pairs of points in the two clusters 1 2 3 4 5

21. Hierarchical Clustering: MAX5 4 1 2 3 4 5 6 2 3 1
Nested Clusters
Dendrogram

22. Strength of MAX Two Clusters Original Points
Less susceptible to noise and outliers

23. Limitations of MAX Two Clusters Original Points
Tends to break large clusters
Biased towards globular clusters

24. Cluster Similarity: Group Average
Proximity of two clusters is the average of pairwise proximity between points in the two clusters.
Need to use average connectivity for scalability since total proximity favors large clusters 1 2 3 4 5

25. Hierarchical Clustering: Group Average5 4 1 2 3 4 5 6 2 3 1
Nested Clusters
Dendrogram

26. Hierarchical Clustering: Group Average
Compromise between Single and Complete Link
Strengths
Less susceptible to noise and outliers
Limitations
Biased towards globular clusters

27. Cluster Similarity: Ward’s Method
Similarity of two clusters is based on the increase in squared error when two clusters are merged
Similar to group average if distance between points is distance squared
Less susceptible to noise and outliers
Biased towards globular clusters
Hierarchical analogue of K-means
Can be used to initialize K-means

28. Hierarchical Clustering: Comparison5 5 1 2 3 4 5 6 4 4 1 2 3 4 5 6 3 2 2 1
MIN
MAX 3 1 5 5 1 2 3 4 5 6 1 2 3 4 5 6 4 4 2 2
Ward’s Method 3 3 1
Group Average 1

29. Hierarchical Clustering: Time and Space requirements
O(N2) space since it uses the proximity matrix.
N is the number of points.
O(N3) time in many cases
There are N steps and at each step the size, N2, proximity matrix must be updated and searched
Complexity can be reduced to O(N2 log(N) ) time for some approaches

30. CURE (Clustering Using REpresentatives )
data to be clustered
clusters generated by conventional methods (e.g., k-means, BIRCH)
CURE: proposed by Guha, Rastogi & Shim, 1998
Stops the creation of a cluster hierarchy if a level consists of k clusters
Uses multiple representative points to evaluate the distance between clusters, adjusts well to arbitrary shaped clusters and avoids single-link effect

31. Cure: The Algorithm Draw random sample s.
Partition sample to p partitions with size s/p
Partially cluster partitions into s/pq clusters
Eliminate outliers
By random sampling
If a cluster grows too slow, eliminate it.
Cluster partial clusters.
Label data in disk

32. CURE: cluster representation
Uses a number of points to represent a cluster
Representative points are found by selecting a constant number of points from a cluster and then “shrinking” them toward the center of the cluster
Cluster similarity is the similarity of the closest pair of representative points from different clusters  

33. CURE
Shrinking representative points toward the center helps avoid problems with noise and outliers
CURE is better able to handle clusters of arbitrary shapes and sizes

34. Experimental Results: CURE
Picture from CURE, Guha, Rastogi, Shim.

35. Experimental Results: CURE
(centroid)
(single link)
Picture from CURE, Guha, Rastogi, Shim.

36. CURE Cannot Handle Differing Densities
Original Points

37. ROCK (RObust Clustering using linKs)
Clustering algorithm for data with categorical and Boolean attributes
A pair of points is defined to be neighbors if their similarity is greater than some threshold
Use a hierarchical clustering scheme to cluster the data.
Obtain a sample of points from the data set
Compute the link value for each set of points, i.e., transform the original similarities (computed by Jaccard coefficient) into similarities that reflect the number of shared neighbors between points
Perform an agglomerative hierarchical clustering on the data using the “number of shared neighbors” as similarity measure and maximizing “the shared neighbors” objective function
Assign the remaining points to the clusters that have been found

38. Clustering Categorical Data: The ROCK Algorithm
ROCK: RObust Clustering using linKs
S. Guha, R. Rastogi & K. Shim, ICDE’99
Major ideas
Use links to measure similarity/proximity
Not distance-based
Computational complexity:

39. Similarity Measure in ROCK
Traditional measures for categorical data may not work well, e.g., Jaccard coefficient
Example: Two groups (clusters) of transactions
C1. <a, b, c, d, e>: {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}
C2. <a, b, f, g>: {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g}
Jaccard co-efficient may lead to wrong clustering result
C1: 0.2 ({a, b, c}, {b, d, e}} to 0.5 ({a, b, c}, {a, b, d})
C1 & C2: could be as high as 0.5 ({a, b, c}, {a, b, f})
Jaccard co-efficient-based similarity function:
Ex. Let T1 = {a, b, c}, T2 = {c, d, e}

40. Link Measure in ROCK Links: # of common neighbors
C1 <a, b, c, d, e>: {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}
C2 <a, b, f, g>: {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g}
Let T1 = {a, b, c}, T2 = {c, d, e}, T3 = {a, b, f}
link(T1, T2) = 4, since they have 4 common neighbors
{a, c, d}, {a, c, e}, {b, c, d}, {b, c, e}
link(T1, T3) = 3, since they have 3 common neighbors
{a, b, d}, {a, b, e}, {a, b, g}
Thus link is a better measure than Jaccard coefficient

41. CHAMELEON: Hierarchical Clustering Using Dynamic Modeling (1999)
CHAMELEON: by G. Karypis, E.H. Han, and V. Kumar’99
Measures the similarity based on a dynamic model
Two clusters are merged only if the interconnectivity and closeness (proximity) between two clusters are high relative to the internal interconnectivity of the clusters and closeness of items within the clusters
Cure ignores information about interconnectivity of the objects, Rock ignores information about the closeness of two clusters
A two-phase algorithm
Use a graph partitioning algorithm: cluster objects into a large number of relatively small sub-clusters
Use an agglomerative hierarchical clustering algorithm: find the genuine clusters by repeatedly combining these sub-clusters

42. Overall Framework of CHAMELEON
Construct
Sparse Graph
Partition the Graph
Data Set
Merge Partition
Final Clusters

43. CHAMELEON (Clustering Complex Objects)

44. 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

45. 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)

46. Density-Based Clustering: Background
Neighborhood of point p=all points within distance e from p:
NEps(p)={q | dist(p,q) <= e }
Two parameters:
e : Maximum radius of the neighbourhood
MinPts: Minimum number of points in an e -neighbourhood of that point
If the number of points in the e -neighborhood of p is at least MinPts, then p is called a core object.
Directly density-reachable: A point p is directly density-reachable from a point q wrt. e, MinPts if
1) p belongs to NEps(q)
2) core point condition:
|NEps (q)| >= MinPts p q
MinPts = 5
e = 1 cm

47. 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 p1 q p q o

48. DBSCAN: Density Based Spatial Clustering of Applications with Noise
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
Core
Border
Outlier
Eps = 1cm
MinPts = 5

49. 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.