Lecture 13: Clustering (continued) May 12, 2010

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.

Project Presentation – Details• recommended format: slides with overhead

projector (e.g. power-point)

• sample presentations – e-mailed 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

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.

cx

xc

||

1(c)μ

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)

Sec. 16.4

More formal description of algorithm

6

K Means Example(K=2)

Pick seeds

Reassign clusters

Compute centroids

xx

Reassign clusters

xx xx Compute centroids

Reassign clusters

Converged!

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?

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

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.

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

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.

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.

In the above, if you startwith B and E as centroidsyou converge to {A,B,C}and {D,E,F}If you start with D and Fyou converge to {A,B,D,E} {C,F}

Example showingsensitivity to seeds

Sec. 16.4

Two different K-means Clusterings

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Sub-optimal Clustering

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Optimal Clustering

Original Points

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 = 0.00036 Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t

Initial Centroids not well chosen

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 6

A seemingly better initial choice

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

x

y

Iteration 4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

xy

Iteration 5

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

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

K

i Cxi

i

xmdistSSE1

2 ),(

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

Sec. 16.4

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.

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

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.

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.

Error as a function of k

25

Pre-processing and Post-processing

Pre-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

Limitations of K-means: Non-globular Shapes

Original Points K-means (2 Clusters)

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.

Overcoming K-means Limitations

Original Points K-means Clusters

Overcoming K-means Limitations

Original Points K-means Clusters

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

Ch. 17

Dendrogram: Hierarchical Clustering

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

32

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.

Sec. 17.1

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

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:

),(max),(,

yxsimccsimji cycx

ji

)),(),,(max()),(( kjkikji ccsimccsimcccsim

Sec. 17.2

Single Link ExampleSec. 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:

),(min),(,

yxsimccsimji cycx

ji

)),(),,(min()),(( kjkikji ccsimccsimcccsim

Ci Cj Ck

Sec. 17.2

Complete Link ExampleSec. 17.2

Simple hierarchical clustering algorithm

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 N2 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

Sec. 17.2.1

Efficient hierarchical clustering algorithm

Efficient single-link clustering algorithm

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

)( :)(

),()1(

1),(

ji jiccx xyccyjiji

ji yxsimcccc

ccsim

Sec. 17.3

Computing Group Average Similarity

Always maintain sum of vectors in each cluster.

Compute similarity of clusters in constant time:

jcx

j xcs

)(

)1||||)(|||(|

|)||(|))()(())()((),(

jiji

jijijiji cccc

cccscscscsccsim

Sec. 17.3

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

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.

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)

Cjnn

Purity ijji

i )(max1

)(

Sec. 16.3

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

Purity exampleSec. 16.3

Rand Index measures between pair decisions

Number of points

Same Cluster in clustering

Different Clusters in clustering

Same class in ground truth

20 24

Different classes in ground truth

20 72

Sec. 16.3

Here RI = 0.68

Rand index and Cluster F-measure

BA

AP

DCBA

DARI

CA

AR

Compare with standard Precision and Recall:

People also define and use a cluster F-measure, which is probably a better measure.

Sec. 16.3

An application of clustering

51

Example: Color quantization of Images ProblemConvert a 24 bit RGB image into a indexed imagewith 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.

Image Examples

Mnp: 30, percent 0.05, cluster number 4

Mnp : 20, percent 0.05, cluster number 7

Original pictures segmented pictures

Project by Qifong Xu, Penn

Image examples 2

Mnp: 10, percent 0.05, cluster number: 9

Mnp: 50, percent 0.05, cluster number: 3

Original pictures Segmented pictures

Mnp: 30, Cluster number: 4

Mnp:10, cluster number: 15

Original picture

Effect of cluster size

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.

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.