1. Cluster Analysis (1)

2. What is Cluster Analysis?
Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized
Intra-cluster distances are minimized

3. Applications of Cluster Analysis
Clustering for Understanding
Group related documents for browsing
Group genes and proteins that have similar functionality
Group stocks with similar price fluctuations
Segment customers into a small number of groups for additional analysis and marketing activities.
Clustering for Summarization
Reduce the size of large data sets
Clustering precipitation in Australia

4. Similarity and Dissimilarity
Numerical measure of how alike two data objects are.
Higher when objects are more alike.
Can be transformed to fall in interval [0,1] by doing:
s’ = (s – min_s)/(max_s – min_s)
Dissimilarity
Numerical measure of how different are two data objects
Lower when objects are more alike
Minimum dissimilarity is often 0
d’ = (d – min_d)/(max_d – min_d)
These proximity measures for objects with a number of attributes are defined by combining the proximities of individual attributes.

5. Similarity/Dissimilarity for Simple Attributes
p and q are the attribute values for two data objects.
Nominal
E.g. province attribute of an address with values:
{BC, AB, ON, QC, …}
Order not important.
Dissimilarity
d=0 if p=q
d=1 if pq
Similarity
s=1 if p=q
s=0 if pq

6. Similarity/Dissimilarity for Simple Attributes
p and q are the attribute values for two data objects.
Ordinal
E.g. quality attribute of a product with values:
{poor, fair, OK, good, wonderful}
Order is important, but exact difference between values is undefined or not important.
Map the values of the attribute to successive integers
{poor=0, fair=1, OK=2, good=3, wonderful=4}
Dissimilarity
d(p,q) = |p – q| / (max_d – min_d)
e.g. d(wonderful, fair) = |4-1| / (4-0) = .75
Similarity
s(p,q) = 1 – d(p,q) e.g. d(wonderful, fair) = .25

7. Similarity/Dissimilarity for Simple Attributes
p and q are the attribute values for two data objects.
Continuous (or Interval)
E.g. weight attribute of a product
Dissimilarity
d(p,q) = |p – q|
Similarity
s(p,q) = – d(p,q)
Of course, we can transform them in the [0,1] scale.

8. Combining Similarities
Sometimes attributes are of many different types, but an overall similarity/dissimilarity is needed.
For the k-th attribute, compute a similarity sk in the range [0,1].
Then,
Similar formula for dissimilarity

9. Euclidean Distance
When all the attributes are continuous we can use the Euclidean Distance
Where n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q.
Standardization is necessary, if scales differ
E.g. weight, salary have different scales

10. Euclidean Distance
Distance Matrix

11. Minkowski Distance
Minkowski Distance is a generalization of Euclidean Distance
Where r is a parameter, n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q.
Examples
r = 1. City block (Manhattan, taxicab, L1 norm) distance.
r = 2. Euclidean distance
r  . “supremum” (Lmax norm, L norm) distance.
This is the maximum difference between any component of the vectors

12. Minkowski Distance
Distance Matrix

13. Similarity Between Binary Vectors
Common situation is that objects, p and q, have only binary attributes
Compute similarities using the following quantities
M01 = the number of attributes where p was 0 and q was 1
M10 = the number of attributes where p was 1 and q was 0
M00 = the number of attributes where p was 0 and q was 0
M11 = the number of attributes where p was 1 and q was 1
Simple Matching and Jaccard Coefficients
SMC = number of matches / number of attributes
= (M11 + M00) / (M01 + M10 + M11 + M00)
J = number of M11 matches / number of not-both-zero attributes values
= (M11) / (M01 + M10 + M11)

14. SMC versus Jaccard: Example
q =
M01 = 2 (the number of attributes where p was 0 and q was 1)
M10 = 1 (the number of attributes where p was 1 and q was 0)
M00 = 7 (the number of attributes where p was 0 and q was 0)
M11 = 0 (the number of attributes where p was 1 and q was 1)
SMC = (M11 + M00)/(M01 + M10 + M11 + M00) = (0+7) / ( ) = 0.7
J = (M11) / (M01 + M10 + M11) = 0 / ( ) = 0

15. Cosine Similarity If D1 and D2 are two document vectors, then
cos( D1, D2 ) = (D1  D2) / ||D1||.||D2|| ,
where  indicates vector dot product and || D || is the length of vector D.
Example:
D1  D2=
.4* *0 + 0* * *.33 = .0561
||D1|| = sqrt(.40^ ^ ^2) = .55
||D2|| = sqrt(.33^2 + 1^ ^2) = 1.1
cos( D1, D2 ) = / (.55 * 1.1) = .093
If the cosine similarity is 1, the angle between D1 and D2 is 0o, and D1 and D2 are the same except for the magnitude.
If the cosine similarity is 0, then the angle between D1 and D2 is 90o, and they don’t share any terms (words).

16. What is Cluster Analysis?
Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized
Intra-cluster distances are minimized

17. Types of Clusters: Well-Separated
Well-Separated Clusters:
Any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster.
3 well-separated clusters

18. Types of Clusters: Center-Based
An object in a cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster
The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster
4 center-based clusters

19. Types of Clusters: Contiguity-Based
Contiguous Cluster (Nearest neighbor or Transitive)
A point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster.
8 contiguous clusters

20. Types of Clusters: Density-Based
A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density.
Used when the clusters are irregular or intertwined, and when noise and outliers are present.
6 density-based clusters

21. K-means Clustering
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
Basic algorithm is very simple

22. Example

23. K-means Clustering – Details
Initial centroids may be chosen randomly.
Clusters produced vary from one run to another.
The centroid is (typically) the mean of the points in the cluster.
‘Closeness’ is measured by Euclidean distance, cosine similarity, etc.
Most of the convergence happens in the first few iterations.
Often the stopping condition is changed to ‘Until relatively few points change clusters’
Complexity is O(I * K* n * d )
n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

24. 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 up.
x is a data point in cluster Ci and
mi is the representative point for cluster Ci

25. Reducing SSE with Post-processing
Obvious way to reduce the SSE is to find more clusters, i.e., to use a larger K.
However, in many cases, we would like to improve the SSE, but don't want to increase the number of clusters.
Various techniques are used to “fix up” the resulting clusters in order to produce a clustering that has lower SSE.
Commonly used approach: Use alternate cluster splitting and merging phases.
Split a cluster:
split the cluster with the largest SSE
Merge two clusters:
merge the two clusters that result in the smallest increase in total SSE.

26. Limitations of K-means
K-means has problems when clusters are of
Differing Sizes
Differing Densities
Non-globular shapes

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

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

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

30. Overcoming K-means Limitations
Original Points K-means Clusters
One solution is to use many clusters.
Find parts of clusters.
Apply merge strategy

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

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

33. Importance of Choosing Initial Centroids
Starting with two initial centroids in one cluster of each pair of clusters

34. Importance of Choosing Initial Centroids
Starting with two initial centroids in one cluster of each pair of clusters

35. Importance of Choosing Initial Centroids
Starting with some pairs of clusters having three initial centroids, while other have only one.

36. Importance of Choosing Initial Centroids
Starting with some pairs of clusters having three initial centroids, while other have only one.

37. Problems with Selecting Initial Points
Of course, the ideal would be to choose initial centroids, one from each true cluster. However, this is very difficult.
If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small.
Chance is relatively small when K is large
If clusters are the same size, n, then
For example, if K = 10, then probability = 10!/1010 =
Sometimes the initial centroids will readjust themselves in the ‘right’ way, and sometimes they don’t.
Consider an example of five pairs of clusters

38. Solutions to Initial Centroids Problem
Multiple runs
Helps, but probability is not on your side
Bisecting K-means
Not as susceptible to initialization issues

39. Bisecting K­means
Straightforward extension of the basic K­means algorithm. Simple idea:
To obtain K clusters, split the set of points into two clusters, select one of these clusters to split, and so on, until K clusters have been produced.
Algorithm
Initialize the list of clusters to contain the cluster consisting of all points.
repeat
Remove a cluster from the list of clusters.
//Perform several “trial” bisections of the chosen cluster.
for i = 1 to number of trials do
Bisect the selected cluster using basic K­means (i.e. 2-means).
end for
Select the two clusters from the bisection with the lowest total SSE.
Add these two clusters to the list of clusters.
until the list of clusters contains K clusters.

40. Bisecting K-means Example