1. Lecture 13: Clustering (continued) May 12, 2010

2. Announcements
end of next class (May 19), at 8:45, take-home finals will be given.
Due: May 26 at 6 PM
project presentation – more in the next slide
project report due: May 26 at the time of presentation.

3. Project Presentation – Details
recommended format: slides with overhead projector (e.g. power-point)
sample presentations – ed and can be found under project link
duration: 15 minutes
should include:
Problem statement
Data set – size, how acquired, processing needed
Algorithm – overview, time and space needs
Result – performance, plots
Challenges
Summary and conclusion

4. K-Means Assumes documents are real-valued vectors.
Sec. 16.4
K-Means
Assumes documents are real-valued vectors.
Clusters based on centroids (aka the center of gravity or mean) of points in a cluster, c:
Reassignment of instances to clusters is based on distance to the current cluster centroids.

5. K-Means Algorithm Select K random docs {s1, s2,… sK} as seeds.
Sec. 16.4
K-Means Algorithm
Select K random docs {s1, s2,… sK} as seeds.
Until clustering converges (or other stopping criterion):
for each doc di:
Assign di to the cluster cj such
that dist(xi, sj) is minimal.
(Next, update the seeds to the centroid of each cluster)
for each cluster cj
sj = (cj)

6. More formal description of algorithm

7. K Means Example (K=2) Pick seeds Reassign clusters Compute centroids
Sec. 16.4
K Means Example (K=2)
Pick seeds
Reassign clusters
Compute centroids x
Reassign clusters x
Compute centroids
Reassign clusters
Converged!

8. Termination conditions
Sec. 16.4
Termination conditions
Several possibilities, e.g.,
A fixed number of iterations.
Doc partition unchanged.
Centroid positions don’t change.
Does this mean that the docs in a cluster are unchanged?

9. Convergence Why should the K-means algorithm ever reach a fixed point?
Sec. 16.4
Convergence
Why should the K-means algorithm ever reach a fixed point?
A state in which clusters don’t change.
K-means is a special case of a general procedure known as the Expectation Maximization (EM) algorithm.
EM is known to converge.
Number of iterations could be large.
But in practice usually isn’t

10. Convergence of K-Means
Sec. 16.4
Convergence of K-Means
Define goodness measure of cluster k as sum of squared distances from cluster centroid:
Gk = Σi (di – ck)2 (sum over all di in cluster k)
G = Σk Gk
Reassignment monotonically decreases G since each vector is assigned to the closest centroid.

11. Convergence of K-Means
Sec. 16.4
Convergence of K-Means
Recomputation monotonically decreases each Gk since (mk is number of members in cluster k):
Σ (di – a)2 reaches minimum for:
Σ –2(di – a) = 0
Σ di = Σ a
mK a = Σ di
a = (1/ mk) Σ di = ck
K-means typically converges quickly

12. Sec. 16.4
Time Complexity
Computing distance between two docs is O(M) where M is the dimensionality of the vectors.
Reassigning clusters: O(KN) distance computations, or O(KNM).
Computing centroids: Each doc gets added once to some centroid: O(NM).
Assume these two steps are each done once for I iterations.
Total time = O(IKNM).
However, it is not clear how to bound I unless it is forced externally.

13. Seed Choice Results can vary based on random seed selection.
Sec. 16.4
Seed Choice
Results can vary based on random seed selection.
Some seeds can result in poor convergence rate, or convergence to sub-optimal clusters.
Select good seeds using a heuristic (e.g., doc least similar to any existing mean)
Try out multiple starting points
Initialize with the results of another method.
Example showing
sensitivity to seeds
In the above, if you start
with B and E as centroids
you converge to {A,B,C}
and {D,E,F}
If you start with D and F
you converge to
{A,B,D,E} {C,F}

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

15. Problem with Selecting Initial centroids
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 ‘right’ way, and sometimes they don’t

16. Initial Centroids not well chosen

17. A seemingly better initial choice

18. 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

19. 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

20. K-means: issues, variations, etc.
Sec. 16.4
K-means: issues, variations, etc.
Recomputing the centroid after every assignment (rather than after all points are re-assigned) can improve speed of convergence of K-means.
Assumes clusters are spherical in vector space
Sensitive to coordinate changes, weighting etc.
Disjoint and exhaustive
Doesn’t have a notion of “outliers” by default
But can add outlier filtering

21. How Many Clusters? Number of clusters K is given
Partition n docs into predetermined number of clusters
Finding the “right” number of clusters is part of the problem
Given docs, partition into an “appropriate” number of subsets.
E.g., for query results - ideal value of K not known up front - though UI may impose limits.
Can usually take an algorithm for one flavor and convert to the other.

22. K not specified in advance
Say, the results of a query.
Solve an optimization problem: penalize having lots of clusters
application dependent, e.g., compressed summary of search results list.
Tradeoff between having more clusters (better focus within each cluster) and having too many clusters

23. K not specified in advance
Given a clustering, define the benefit for a doc to be the cosine similarity to its centroid.
Define the total benefit to be the sum of the individual doc benefits.

24. Penalize lots of clusters
For each cluster, we have a Cost C.
Thus for a clustering with K clusters, the Total Cost is KC.
Define the Value of a clustering to be =
Total Benefit - Total Cost.
Find the clustering of highest value, over all choices of K.
Total benefit increases with increasing K. But can stop when it doesn’t increase by “much”. The Cost term enforces this.

25. Error as a function of k

26. Pre-processing and Post-processing
Normalize the data
Eliminate outliers
Post-processing
Eliminate small clusters that may represent outliers
Split ‘loose’ clusters, i.e., clusters with relatively high SSE
Merge clusters that are ‘close’ and that have relatively low SSE
Can use these steps during the clustering process

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

28. 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.

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

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

31. Hierarchical Clustering
Build a tree-based hierarchical taxonomy (dendrogram) from a set of documents.
One approach: recursive application of a partitional clustering algorithm.
animal
vertebrate
fish reptile amphib. mammal worm insect crustacean
invertebrate

32. Dendrogram: Hierarchical Clustering
Clustering obtained by cutting the dendrogram at a desired level: each connected component forms a cluster.

33. Hierarchical Agglomerative Clustering
Sec. 17.1
Hierarchical Agglomerative Clustering
Starts with each doc in a separate cluster
then repeatedly joins the closest pair of clusters, until there is only one cluster.
The history of merging forms a binary tree or hierarchy.

34. Closest pair of clusters
Sec. 17.2
Closest pair of clusters
Many variants to defining closest pair of clusters
Single-link
Similarity of the most cosine-similar (single-link)
Complete-link
Similarity of the “furthest” points, the least cosine-similar
Centroid
Clusters whose centroids (centers of gravity) are the most cosine-similar
Average-link
Average cosine between pairs of elements

35. Single Link Agglomerative Clustering
Sec. 17.2
Single Link Agglomerative Clustering
Use maximum similarity of pairs:
Can result in “straggly” (long and thin) clusters due to chaining effect.
After merging ci and cj, the similarity of the resulting cluster to another cluster, ck, is:

36. Sec. 17.2
Single Link Example

37. Complete Link Use minimum similarity of pairs:
Sec. 17.2
Complete Link
Use minimum similarity of pairs:
Makes “tighter,” spherical clusters that are typically preferable.
After merging ci and cj, the similarity of the resulting cluster to another cluster, ck, is: Ci Cj Ck

38. Sec. 17.2
Complete Link Example

39. Simple hierarchical clustering algorithm

40. Computational Complexity
Sec
Computational Complexity
In the first iteration, all HAC methods need to compute similarity of all pairs of N initial instances, which is O(N2).
In each of the subsequent N2 merging iterations, compute the distance between the most recently created cluster and all other existing clusters.
In order to maintain an overall O(N2) performance, computing similarity to each other cluster must be done in constant time.
Often O(N3) if done naively or O(N2 log N) if done more cleverly

41. Efficient hierarchical clustering algorithm

42. Efficient single-link clustering algorithm

43. Sec. 17.3
Group Average
Similarity of two clusters = average similarity of all pairs within merged cluster.
Compromise between single and complete link.
Two options:
Averaged across all ordered pairs in the merged cluster
Averaged over all pairs between the two original clusters
No clear difference in efficacy

44. Computing Group Average Similarity
Sec. 17.3
Computing Group Average Similarity
Always maintain sum of vectors in each cluster.
Compute similarity of clusters in constant time:

45. What Is A Good Clustering?
Sec. 16.3
What Is A Good Clustering?
Internal criterion: A good clustering will produce high quality clusters in which:
the intra-class (that is, intra-cluster) similarity is high
the inter-class similarity is low
The measured quality of a clustering depends on both the document representation and the similarity measure used

46. External criteria for clustering quality
Sec. 16.3
External criteria for clustering quality
Quality measured by its ability to discover some or all of the hidden patterns or latent classes in gold standard data
Assesses a clustering with respect to ground truth. (requires labeled data)
Assume documents with C gold standard classes, while our clustering algorithms produce K clusters, ω1, ω2, …, ωK with ni members.

47. External Evaluation of Cluster Quality
Sec. 16.3
External Evaluation of Cluster Quality
Simple measure: purity, the ratio between the dominant class in the cluster πi and the size of cluster ωi
Biased because having n clusters maximizes purity
Others are entropy of classes in clusters (or mutual information between classes and clusters)

48. Purity example                  Cluster I Cluster II
Sec. 16.3
Purity example
 
 
 
 
 
 
 
  
Cluster I
Cluster II
Cluster III
Cluster I: Purity = 1/6 (max(5, 1, 0)) = 5/6
Cluster II: Purity = 1/6 (max(1, 4, 1)) = 4/6
Cluster III: Purity = 1/5 (max(2, 0, 3)) = 3/5

49. Rand Index measures between pair decisions
Sec. 16.3
Rand Index measures between pair decisions
Here RI = 0.68
Number of points
Same Cluster in clustering
Different Clusters in clustering
Same class in ground truth 20 24
Different classes in ground truth 72

50. Rand index and Cluster F-measure
Sec. 16.3
Rand index and Cluster F-measure
Compare with standard Precision and Recall:
People also define and use a cluster F-measure, which is probably a better measure.

51. An application of clustering
Example: Color quantization of Images Problem
Convert a 24 bit RGB image into a indexed image
with a palette of K colors.
Solution
The (r, g, b) values of the pixels are the data points xi
The (r, g, b) values of the K palette colors are the centroids wk.
Initialize the wk with the color of random pixels.
Perform one pass of k-means algorithm.
Each cluster is assigned one color.

52. Image Examples Original pictures segmented pictures
Mnp: 30, percent 0.05, cluster number 4
Mnp : 20, percent 0.05, cluster number 7
Project by Qifong Xu, Penn

53. Image examples 2 Original pictures Segmented pictures
Mnp: 10, percent 0.05, cluster number: 9
Mnp: 50, percent 0.05, cluster number: 3

54. Effect of cluster size Original picture Mnp:10, cluster number: 15

55. Image clustering in archeology
Angkor Wat temple
Angkor Wat contains a gallery of 2000 women.
what facial types are represented in these portraits?
A problem being solved by Prof. Anil Jain of MSU using clustering.

56. summary
In clustering, clusters are inferred from the data without human input (unsupervised learning)
There are many ways of influencing the outcome of clustering (with user input): number of clusters, similarity measure, choice of features.
Many applications including text clustering, grouping genes/species, image processing/vision etc.