1. Cluster Analysis Finding Groups in Data
C. Taillie
February 19, 2004

2. Atypical Example of Clustering
Digitized Photographs of Ten Natural Scenes
Can we group them into clusters so that photos within each cluster are similar as images?

3. Dendrogram for Atypical Example
Dendrogram
Dissimilarity
Photo

4. Dendrogram = Many Groupings

5. Splitogram: Choosing Number of Clusters 1
Splitting Node
Dissimilarity
Splitogram 2 3 4 5 6 9 8 7

6. What makes this example atypical?
Atypical Example
What makes this example atypical?
Splitogram levels off at a dissimilarity of about 0.6 (max=1.4) Typically, splitogram levels off to zero
Number of objects to cluster is small (only 10) Typically, number of objects is in hundreds or thousands, sometimes millions
Number of measurements per object is immense (one grey-scale value for each of several million pixels) Typically, number of measurements per object ranges from a few to a few hundred
Complicated spatial structure among the measurements and a specialized dissimilarity measure tailored to that spatial structure Typically, one uses generic dissimilarity measures like Euclidian distance

7. Entomology of “Dendrogram”
Statisticians did not invent dendrograms
dendron = Tree (Greek)
gramma = Diagram (Greek)
dendrogram = Tree Diagram (specifically for classification or grouping of objects)
dendrograms have been around for aeons, especially in taxonomy

8. Taxonomic Dendrogram for Primates
Artistic Issues:
Is the Root at the top or bottom of the tree? (Don Knuth ……)
Ordering of the leaf nodes? Not significant, except that each cluster represents a consecutive sequence
However, ordering of leaf nodes can influence reader’s perception. Try interchanging “human” with the two “chimp” groupings!!!!

9. Classification vs Clustering
A priori set of labels (categories) with meaningful descriptions/interpretations so that, with sufficient effort, an “expert” could assign a given object to its true category
Landcover categories (forest, urban, desert, grassland, etc.)
Taxonomic categories (species of moths)
Set of “training” data whose objects have been expertly classified
Goal in Statistical Classification: Develop rules so that new objects can be classified without the need for an expert
Object X1 X2 X3 X4
Label
Training Objects 1  C 2 B
. . . n A
New Objects ?

10. Classification X2 Simple Classification Rule
Training Data with Three Categories X1 X2
New Object X2
Complex Classification Rule
New Object X1

11. Classification
Essential feature of Classification: Each object has a TRUE category so the performance of any classification rule can be assessed (doing so may be expensive)
Test Data
Inaccuracy
Good Bad
Best Rule
Training Data
Simple Complex
Rules

12. Classification vs Clustering
Collection of objects and a set of measurements on each object
Goal in Statistical Clustering: Divide the objects into groups so that objects in the same group are “similar” and objects in different groups are “dissimilar” Assign an identifying group label to each object. Objects with the same label belong to the same group, but a clustering algorithm attaches no other meaning or interpretation to the labels. Interpretation of the labels is up to the user.
Clustering algorithm itself does not provide a way of assigning new objects to the clusters
Object X1 X2 X3 X4
Label 1  ? 2
. . . n

13. Which of the two groupings is “correct”?
Clustering
Three Groups X1 X2
Original (ungrouped) Data X2
Four Groups X1 X2 X1
Which of the two groupings is “correct”?

14. Clustering Cluster Analysis is an Exploratory Tool
There is no notion of a TRUE cluster
Therefore, no way of evaluating the performance of a particular clustering algorithm
The only criterion is whether it yields anything useful for you

15. Applications of Cluster Analysis
Performance Evaluation (first few slides---unusual application)
Precursor to Classification Assign meaningful interpretations to cluster labels and obtain an initial training set
Reduce/Simplify a Large Database Databases in economics and commerce can be megabytes or gigabytes in size. Even something as simple as computing a mean can consume huge amounts of computer time

16. Get rid of the variables
Database Reduction - 1
May be millions of records (rows) in database and hundreds or thousands of variables
Object X1 X2 X3 X4
Label 1  C 2 A 3 B 4
. . . n E
Get rid of the variables
Object
Label 1 C 2 A 3 B 4
. . . n E

17. Database Reduction - 2 Collapse every cluster to its centroidX2 X2 B A C X1 X1
Create Auxiliary Table – One row for each cluster
Label
Cluster
Size A  11 B 14 C 12

18. Clustering Algorithms

19. Ward’s Method Within- and Between Group Sum of Squares
Ward, J.H. (1963). Hierarchical groupings to optimize an objective function. J. American Statistical Society, 58,
Within- and Between Group Sum of Squares
SSTotal = SSWithin + SSBetween
A “good” clustering should have:
Small within-group sum of squares (objects in a group should be similar)
Large between group sum of squares (objects in different groups should be dissimilar)
Bottom-up (agglomerative) hierarchical approach:
Start with each object in its own separate cluster (bottom of the dendrogram --- SSWithin = 0)
Combine clusters two at a time; at each step combine the pair of clusters that gives the smallest increase in SSWithin

20. Dendrogram for Ward’s Method
SSTotal
Within-group SS
Value of SSWithin just after the fusion

21. Other Agglomerative Hierarchical Methods
Dissimilarity measure D(a,b) between individual objects a and b
Euclidian distance
Manhattan distance
Minkowski distance
Extend D to a measure of dissimilarity D(A,B) between groups of objects A and B. This is called the linkage method:
Single linkage
Complete linkage
Average linkage
Centroid linkage
Ward’s linkage (should be called Wishart’s linkage)

22. Linkage Methods
Link between A and B a b B A
Single linkage D(A,B) is the shortest link between A and B
Complete linkage D(A,B) is the longest link between A and B
Average linkage D(A,B) is the average of all the links between A and B
Centroid linkage
Ward’s linkage (should really be called Wishart’s linkage) Centroid linkage weighted by the cluster sizes

23. Ward’s Linkage
Wishart showed that when groups A and B are fused, the increase in the within group sum of squares is where D2 is squared Euclidean distance. So it is a weighted form of centroid linkage. The weight can be rewritten as

24. Why is Weighting Desirable ?
10 objects
50 objects
10 objects
50 objects
Which pair of group would you prefer to fuse? The pair on the left or on the right?
The “sample sizes” on the right are larger so there is stronger evidence that the groups on the right are really different.
We can achieve this choice by weighting centroidal distance by the average of the group sizes.

25. Why is Harmonic Mean Better than Arithmetic Mean ?
1 object
50 objects
99 objects
50 objects
Which pair of group would you prefer to fuse? The pair on the left or on the right?
The total sample sizes are the same (100) for each pair. But the “sample sizes” on the right are balanced so there is stronger evidence that the groups on the right are really different.
We can achieve this choice by weighting centroidal distance by the harmonic mean of the group sizes instead of the arithmetic mean.

26. Classification of Clustering Methods
Hierarchical
Partitional
Agglomerative
Divisive

27. Agglomerative vs Divisive: Practicalities
How much computer time is required with N objects?
Agglomerative
There are N(N-1)/2 pairs of objects, so computer time is O(N2)
With N=100, O(N2) is about 10,000
With N=1,000, O(N2) is about one million
With N=10,000, O(N2) is about one hundred million
Divisive
There are 2N-1-1 pairs of nonempty subsets, so computer time is O(2N)
With N=10, O(2N) is about 1,000
With N=20, O(2N) is about one million
With N=30, O(2N) is about one billion
Examination of all possible subsets is hopeless unless N is very small.

28. Partitional Methods
The desired number of clusters, k, is specified beforehand.
Three best-known methods:
k-means (moving centroid method)
k-means (Hartigan’s method)
ISODATA (k-means with many embellishments) Available in many of the GIS and image analysis packages, e.g. ENVI

29. k-Means (Moving Centroids)
Specify the value of k
Specify k points (called “seeds”) in measurement space. The algorithm moves the seeds around until they are the centroids of the k desired groups.
Make a pass through the data points. Assign each data point to its closest seed. This determines a partition of the data into k groups, each labeled by its seed. However, a group’s centroid may fail to coincide with the group’s seed.
For each group, compute its centroid. Use these centroids as the seeds in the next iteration.
Keep iterating until centroids and seeds coincide (to a user- specified degree of accuracy).

30. k-Means (Hartigan’s Method)
Specify the value of k
Specify k nonempty starting groups.
Make a pass through the data points.
For each data point, ask if the within-group sum of squares could be reduced by moving that data point to another group.
If yes, move it; otherwise proceed to the next data point
Keep iterating until no data point can be moved.
Hartigan discovered some computationally simple rules for deciding if a data point should be moved and for finding the best group to move it to.

31. ISODATA
Basically the same as the moving centroid version of k-means, except that the user specifies a range of acceptable values for the desired number of clusters.
After each iteration of moving centroids, the current groups are examined to see if any should be split into two subgroups or fused into a larger group. These decisions are reached using a complicated set of rules based on the within-group standard deviations along each coordinate axis.
Caution: The inner workings of ISODATA tend to be very specific to the particular implementation.

32. Every Pathology Exists
No matter what clustering method you propose, someone will manage to come up with a data set (usually artificial) for which your method produces a foolish clustering

33. Example of a Pathology: 3-Dimensional Chain Link
k-means is a disaster. The centroid of each group is close to many members of the other group
Single linkage does quite well. In general, single linkage is good at finding “snake-like” clusters