1. CSE 5243 Intro. to Data Mining
Cluster Analysis: Basic Concepts and Methods
Huan Sun, Ohio State University
09/28/2017
Slides adapted from UIUC CS412, Fall 2017, by Prof. Jiawei Han

2. Chapter 10. Cluster Analysis: Basic Concepts and Methods
Cluster Analysis: An Introduction
Partitioning Methods
Hierarchical Methods
Density- and Grid-Based Methods
Evaluation of Clustering
Summary

3. Clustering Algorithms
K-means and its variants
Hierarchical clustering
Density-based clustering

4. 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

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
Build a bottom-up hierarchy of clusters
Divisive:
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

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/similarity between clusters distinguish the different algorithms

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

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

9. After Merging How do we update the proximity matrix? Proximity Matrix
C2 U C5 C1 C3 C4 C1 ?
? ? ? ?
Proximity Matrix
C2 U C5 C3 C3 ? C4 ? C4 C1
C2 U C5

10. How to Define Inter-Cluster Similarityp1 p3 p5 p4 p2
. . . .
Similarity?
MIN
MAX
Group Average
Distance Between Centroids
Proximity Matrix

11. 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.

12. 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.
Let us define the distance between two points using Euclidean distance:
Using single link, the distance between two clusters Ci and Cj is then:
The name comes from the observation that if we choose a line with the minimum distance to connect two points in two clusters, typically only a single link would exist.

13. 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.
What if we define the similarity (not distance) between two points?
Using single link, the similarity between two clusters Ci and Cj is then:
Sim(Ci, Cj) = max{sim(x, y) | x in Ci, y in Cj}
The name comes from the observation that if we choose a line with the minimum distance to connect two points in two clusters, typically only a single link would exist.

14. 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
Similarity matrix (e.g., by cosine similairty)

15. 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
Note: if using distance measure, the smaller, the closer
Distance matrix (e.g., by Euclidean distance, Manhattan distance)
See problem III in the sample midterm!

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.
Update similarity matrix with new cluster {I1, I2}

17. 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
Update similarity matrix with new cluster {I1, I2}

18. 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
Update similarity matrix with new cluster {I1, I2} and {I4, I5}

19. 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
Only two clusters are left.

20. Hierarchical Clustering: MIN5
Distance/dissimilarity
between clusters 1 2 3 4 5 6 4 3 2 1
We can see a sequence of nested clusters
Dendrogram
Nested Clusters

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

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

23. Another example: Single Link
Distance matrix
Initialization

24. Another example: Single Link
Distance matrix
First Cluster

25. Another example: Single Link
Update distance matrix
Second Cluster

26. Another example: Single Link
Update distance matrix
Third Cluster

27. Another example: Single Link
Update distance matrix
Fourth Cluster

28. 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
Let us define the distance between two points using Euclidean distance:
Using MAX link, the distance between two clusters Ci and Cj is then:

29. 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
What if we define the similarity (not distance) between two points?
Using MAX link, the similarity between two clusters Ci and Cj is then:
Sim(Ci, Cj) = min{sim(x, y) | x in Ci, y in Cj}

30. 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
Similarity matrix (e.g., by cosine similairty)

31. 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
How do we get this number? How to update the similarity matrix?
Update similarity matrix with new cluster {I1, I2}

32. 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
At each step, you will always choose the most similar clusters to merge, regardless of the measure you choose.

33. 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
After this, which two clusters should be merged next?

34. 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
Merge {3} with {4,5}, why?

35. Hierarchical Clustering: MAX5 4 1 2 3 4 5 6 2 3 1
A sequence of nested clusters
Dendrogram
Nested Clusters

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

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

38. 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

39. Hierarchical Clustering: Group Average5 4 1 2 3 4 5 6 2 3
A sequence of nested clusters 1
Dendrogram
Nested Clusters

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

41. 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

42. Hierarchical Clustering: Problems and Limitations
Once a decision is made to combine two clusters, it cannot be undone
No objective function is directly minimized
Different schemes have problems with one or more of the following:
Sensitivity to noise and outliers
Difficulty handling different sized clusters and convex shapes
Breaking large clusters

43. Divisive Clustering
DIANA (Divisive Analysis) (Kaufmann and Rousseeuw,1990)
Implemented in some statistical analysis packages, e.g., Splus
Inverse order of AGNES: Eventually each node forms a cluster on its own
Divisive clustering is a top-down approach
The process starts at the root with all the points as one cluster
It recursively splits the higher level clusters to build the dendrogram
Can be considered as a global approach
More efficient when compared with agglomerative clustering

44. More on Algorithm Design for Divisive Clustering
Choosing which cluster to split
Check the sums of squared errors of the clusters and choose the one with the largest value
Splitting criterion: Determining how to split
One may use Ward’s criterion to chase for greater reduction in the difference in the SSE criterion as a result of a split
For categorical data, Gini-index can be used
Handling the noise
Use a threshold to determine the termination criterion (do not generate clusters that are too small because they contain mainly noises)

45. Clustering Algorithms
K-means and its variants
Hierarchical clustering
Density-based clustering

46. Density-Based Clustering Methods
Clustering based on density (a local cluster criterion), such as density-connected points
Major features:
Discover clusters of arbitrary shape
Handle noise
One scan (only examine the local region to justify density)
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) (also, grid-based)

47. DBSCAN DBSCAN (M. Ester, H.-P. Kriegel, J. Sander, and X. Xu, KDD’96)
Discovers clusters of arbitrary shape: Density-Based Spatial Clustering of Applications with Noise

48. DBSCAN DBSCAN is a density-based algorithm.
DBSCAN (M. Ester, H.-P. Kriegel, J. Sander, and X. Xu, KDD’96)
Discovers clusters of arbitrary shape: Density-Based Spatial Clustering of Applications with Noise
DBSCAN is a density-based algorithm.
Density = number of points within a specified radius (Eps)
A point is a core point if it has more than a specified number of points (MinPts) within Eps
These are points that are at the interior of a cluster
A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point
A noise point is any point that is not a core point or a border point.

49. DBSCAN DBSCAN is a density-based algorithm.
Density = number of points within a specified radius (Eps)
A point is a core point if it has more than a specified number of points (MinPts) within Eps
These are points that are at the interior of a cluster
A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point
A noise point is any point that is not a core point or a border point.
Two parameters

50. DBSCAN: Core, Border, and Noise Points
1. A point is a core point if it has more than a specified number of points (MinPts) within Eps
2. A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point
3. A noise point is any point that is not a core point or a border point.

51. DBSCAN Algorithm Eliminate noise points
Perform clustering on the remaining points
The Eps-neighborhood of a point q:
NEps(q): {p belongs to D | dist(p, q) ≤ Eps}

52. DBSCAN: Core, Border and Noise Points
Original Points
Point types: core, border and noise
Eps = 10, MinPts = 4

53. When DBSCAN Works Well Original Points Clusters Resistant to Noise
Can handle clusters of different shapes and sizes

54. When DBSCAN Does NOT Work Well
(MinPts=4, Eps=9.75).
Original Points
Varying densities
High-dimensional data
(MinPts=4, Eps=9.92)

55. Cluster Validity
For supervised classification we have a variety of measures to evaluate how good our model is
Accuracy, precision, recall
For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters?

56. Cluster Validity
For supervised classification we have a variety of measures to evaluate how good our model is
Accuracy, precision, recall
For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters?
But “clusters are in the eye of the beholder”!
Then why do we want to evaluate them?
To avoid finding patterns in noise
To compare clustering algorithms
To compare two sets of clusters
To compare two clusters

57. Clusters found in Random Data
DBSCAN
Random Points
Complete Link
K-means

58. Different Aspects of Cluster Validation
Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data.
Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels.
Evaluating how well the results of a cluster analysis fit the data without reference to external information.
- Use only the data
Comparing the results of two different sets of cluster analyses to determine which is better.
Determining the ‘correct’ number of clusters.
For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.

59. Using Similarity Matrix for Cluster Validation
Order the similarity matrix with respect to cluster labels and inspect visually.

60. Using Similarity Matrix for Cluster Validation
Clusters in random data are not so crisp
DBSCAN

61. Intrinsic Measures of Clustering quality

62. Cohesion and Separation
A proximity graph based approach can also be used for cohesion and separation.
Cluster cohesion is the sum of the weight of all links within a cluster.
Cluster separation is the sum of the weights between nodes in the cluster and nodes outside the cluster.
cohesion
separation

63. Silhouette Coefficient
Silhouette Coefficient combine ideas of both cohesion and separation, but for individual points, as well as clusters and clusterings
For an individual point, i
Calculate a = average distance of i to the points in its cluster
Calculate b = min (average distance of i to points in another cluster)
The silhouette coefficient for a point is then given by s = 1 – a/b if a < b, (or s = b/a if a  b, not the usual case)
Typically between 0 and 1.
The closer to 1 the better.
Can calculate the Average Silhouette width for a cluster or a clustering

64. Other Measures of Cluster Validity
Entropy/Gini
If there is a class label – you can use the entropy/gini of the class label – similar to what we did for classification (Check problem III in sample midterm)
If there is no class label – one can compute the entropy w.r.t each attribute (dimension) and sum up or weighted average to compute the disorder within a cluster
Classification Error
If there is a class label one can compute this in a similar manner

65. Extensions: Clustering Large Databases
Most clustering algorithms assume a large data structure which is memory resident.
Clustering may be performed first on a sample of the database then applied to the entire database.
Algorithms
BIRCH
DBSCAN (we have already covered this)
CURE

66. Backup slides

67. MST: Divisive Hierarchical Clustering
Build MST (Minimum Spanning Tree)
Start with a tree that consists of any point
In successive steps, look for the closest pair of points (p, q) such that one point (p) is in the current tree but the other (q) is not
Add q to the tree and put an edge between p and q

68. MST: Divisive Hierarchical Clustering
Use MST for constructing hierarchy of clusters