cluster analysis for gene expression data

Cluster Analysis for Gene Expression Data

Ka Yee Yeunghttp://staff.washington.edu/kayee/research.html

Center for Expression ArraysDepartment of Microbiology

[email protected]

10/18/2002 Ka Yee Yeung, CEA 2

• Snapshot of activities in the cell

• Each chip represents an experiment:– time course– tissue samples

(normal/cancer)

……..

A gene expression data set

p experiments

n genes

Xij


What is clustering?• Group similar objects together• Objects in the same cluster (group)

are more similar to each other than objects in different clusters

• Data exploratory tool: to find patterns in large data sets

• Unsupervised approach: do not make use of prior knowledge of data


Applications of clustering gene expression data

• Cluster the genes functionally related genes

• Cluster the experiments discover new subtypes of tissue samples

• Cluster both genes and experiments find sub-patterns


Examples of clustering algorithms

• Hierarchical clustering algorithms eg. [Eisen et al 1998]

• K-means eg. [Tavazoie et al. 1999]• Self-organizing maps (SOM) eg.

[Tamayo et al. 1999]• CAST [Ben-Dor, Yakhini 1999]• Model-based clustering algorithms eg.

[Yeung et al. 2001]


Overview

• Similarity/distance measures• Hierarchical clustering algorithms

– Made popular by Stanford, ie. [Eisen et al. 1998]

• K-means– Made popular by many groups, eg.

[Tavazoie et al. 1999]

• Model-based clustering algorithms [Yeung et al. 2001]


How to define similarity?

• Similarity measures:– A measure of pairwise similarity or dissimilarity– Examples:

• Correlation coefficient• Euclidean distance

Experiments

gene

s gene

s

genes

X

Y

X

Y

Raw matrix

Similarity matrix

1

n

1 p n

n


Similarity measures(for those of you who enjoy equations…)

• Euclidean distance

• Correlation coefficient

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Example

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2

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1 2 3 4

X

Y

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Correlation (X,Y) = 1 Distance (X,Y) = 4

Correlation (X,Z) = -1 Distance (X,Z) = 2.83

Correlation (X,W) = 1 Distance (X,W) = 1.41


Lessons from the example

• Correlation – direction only• Euclidean distance – magnitude &

direction• Array data is noisy need many

experiments to robustly estimate pairwise similarity


Clustering algorithms

• From pairwise similarities to groups• Inputs:

– Raw data matrix or similarity matrix– Number of clusters or some other

parameters


Hierarchical Clustering [Hartigan 1975]• Agglomerative (bottom-up)

• Algorithm:– Initialize: each item a cluster– Iterate:

• select two most similar clusters• merge them

– Halt: when required number of clusters is reached

dendrogram


Hierarchical: Single Link

• cluster similarity = similarity of two most similar members

- Potentially long and skinny clusters

+ Fast


Example: single link

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Hierarchical: Complete Link

• cluster similarity = similarity of two least similar members

+ tight clusters

- slow


Example: complete link

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Hierarchical: Average Link

• cluster similarity = average similarity of all pairs

+ tight clusters

- slow


Software: TreeView [Eisen et al. 1998]

• Fig 1 in Eisen’s PNAS 99 paper

• Time course of serum stinulation of primary human fibrolasts

• cDNA arrays with approx 8600 spots

• Similar to average-link• Free download at:

http://rana.lbl.gov/EisenSoftware.htm


Overview• Similarity/distance measures• Hierarchical clustering algorithms

– Made popular by Stanford, ie. [Eisen et al. 1998]

• K-means– Made popular by many groups, eg.

[Tavazoie et al. 1999]

• Model-based clustering algorithms [Yeung et al. 2001]


Partitional: K-Means[MacQueen 1965]

1 2

3


Details of k-means• Iterate until converge:

– Assign each data point to the closest centroid

– Compute new centroid

2

1

)(

k

i Cxi

i

CCentroidxObjective function:

Minimize


Properties of k-means• Fast• Proved to converge to local optimum• In practice, converge quickly• Tend to produce spherical, equal-sized

clusters• Related to the model-based approach• Gavin Sherlock’s Xcluster:

http://genome-www.stanford.edu/~sherlock/cluster.html


What we have seen so far..

• Definition of clustering• Pairwise similarity:

– Correlation– Euclidean distance

• Clustering algorithms:– Hierarchical agglomerative– K-means

• Different clustering algorithms different clusters

• Clustering algorithms always spit out clusters


Which clustering algorithm should I use?

• Good question• No definite answer: on-going

research• Our preference: the model-based

approach.


Model-based clustering (MBC)

• Gaussian mixture model:– Assume each cluster is generated by the

multivariate normal distribution– Each cluster k has parameters :

• Mean vector: k – Location of cluster k

• Covariance matrix: k

– volume, shape and orientation of cluster k

• Data transformations & normality assumption


More on the covariance matrixk

(volume, orientation, shape)Equal volume, spherical (EI)

unequal volume, spherical (VI)

Equal volume, orientation, shape (EEE)

Diagonal model Unconstrained (VVV)


Key advantage of the model-based approach:

choose the model and the number of clusters

• Bayesian Information Criterion (BIC) [Schwarz 1978]

– Approximate p(data | model)• A large BIC score indicates strong

evidence for the corresponding model.


Gene expression data sets

• Ovary data [Michel Schummer, Institute of Systems Biology]

– Subset of data : 235 clones (portions of genes)

24 experiments (cancer/normal tissue samples)

– 235 clones correspond to 4 genes (external criterion)


BIC analysis: square root ovary data

• EEE and diagonal models -> first local max at 4 clusters

• Global max -> VI at 8 clusters

-3000

-2500

-2000

-1500

-1000

-500

0

0 2 4 6 8 10 12 14 16

number of clusters

BIC

EIVIdiagonalEEE


How do we know MBC is doing well?

Answer: compare to external info

• Adjusted Rand: max at EEE 4 clusters (> CAST)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16

number of clusters

Ad

jus

ted

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nd

EIVIdiagonalCASTEEE


Take home messages• MBC has superior performance on:

– Quality of clusters– Number of clusters and model chosen (BIC)

• Clusters with high BIC scores tend to produce a high agreement with the external information

• MBC tends to produce better clusters than a leading heuristic-based clustering algorithm (CAST)

• Splus or R versions:http://www.stat.washington.edu/fraley/mclust/

cluster analysis for gene expression data

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