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Cluster Analysis for Gene Expression Data. Ka Yee Yeung http://staff.washington.edu/kayee/research.html Center for Expression Arrays Department of Microbiology kayee@u.washington.edu. A gene expression data set. ……. Snapshot of activities in the cell Each chip represents an experiment:

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Cluster analysis for gene expression data

Cluster Analysis for Gene Expression Data

Ka Yee Yeung

http://staff.washington.edu/kayee/research.html

Center for Expression Arrays

Department of Microbiology

kayee@u.washington.edu


Cluster analysis for gene expression data

A gene expression data set

……..

  • Snapshot of activities in the cell

  • Each chip represents an experiment:

    • time course

    • tissue samples (normal/cancer)

p experiments

n genes

Xij

Ka Yee Yeung, CEA


What is clustering
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

Ka Yee Yeung, CEA


Applications of clustering gene expression 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

Ka Yee Yeung, CEA


Examples of clustering algorithms
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]

Ka Yee Yeung, CEA


Overview
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]

Ka Yee Yeung, CEA


How to define similarity
How to define similarity?

Experiments

X

genes

n

1

p

1

X

  • Similarity measures:

    • A measure of pairwise similarity or dissimilarity

    • Examples:

      • Correlation coefficient

      • Euclidean distance

genes

genes

Y

Y

n

n

Raw matrix

Similarity matrix

Ka Yee Yeung, CEA


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

  • Euclidean distance

  • Correlation coefficient

Ka Yee Yeung, CEA


Example
Example

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

Ka Yee Yeung, CEA


Lessons from the example
Lessons from the example

  • Correlation – direction only

  • Euclidean distance – magnitude & direction

  • Array data is noisy  need many experiments to robustly estimate pairwise similarity

Ka Yee Yeung, CEA


Clustering algorithms
Clustering algorithms

  • From pairwise similarities to groups

  • Inputs:

    • Raw data matrix or similarity matrix

    • Number of clusters or some other parameters

Ka Yee Yeung, CEA


Hierarchical clustering hartigan 1975
Hierarchical Clustering [Hartigan 1975]

  • Agglomerative(bottom-up)

  • Algorithm:

    • Initialize: each item a cluster

    • Iterate:

      • select two most similarclusters

      • merge them

    • Halt: when required number of clusters is reached

dendrogram

Ka Yee Yeung, CEA


Hierarchical single link
Hierarchical: Single Link

  • cluster similarity = similarity of two most similar members

- Potentially long and skinny clusters

+ Fast

Ka Yee Yeung, CEA


Example single link
Example: single link

5

4

3

2

1

Ka Yee Yeung, CEA


Example single link1
Example: single link

5

4

3

2

1

Ka Yee Yeung, CEA


Example single link2
Example: single link

5

4

3

2

1

Ka Yee Yeung, CEA


Hierarchical complete link
Hierarchical: Complete Link

  • cluster similarity = similarity of two least similar members

+ tight clusters

- slow

Ka Yee Yeung, CEA


Example complete link
Example: complete link

5

4

3

2

1

Ka Yee Yeung, CEA


Example complete link1
Example: complete link

5

4

3

2

1

Ka Yee Yeung, CEA


Example complete link2
Example: complete link

5

4

3

2

1

Ka Yee Yeung, CEA


Hierarchical average link
Hierarchical: Average Link

  • cluster similarity = average similarity of all pairs

+ tight clusters

- slow

Ka Yee Yeung, CEA


Software treeview eisen et al 1998
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

Ka Yee Yeung, CEA


Overview1
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]

Ka Yee Yeung, CEA


Partitional k means macqueen 1965
Partitional: K-Means[MacQueen 1965]

2

1

3

Ka Yee Yeung, CEA


Details of k means
Details of k-means

  • Iterate until converge:

    • Assign each data point to the closest centroid

    • Compute new centroid

Objective function:

Minimize

Ka Yee Yeung, CEA


Properties of k means
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

Ka Yee Yeung, CEA


What we have seen so far
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

Ka Yee Yeung, CEA


Which clustering algorithm should i use
Which clustering algorithm should I use?

  • Good question

  • No definite answer: on-going research

  • Our preference: the model-based approach.

Ka Yee Yeung, CEA


Model based clustering mbc
Model-based clustering (MBC)

  • Gaussian mixture model:

    • Assume each cluster is generated by the multivariate normal distribution

    • Each cluster k has parameters :

      • Mean vector: mk

        • Location of cluster k

      • Covariance matrix: Sk

        • volume, shape and orientation of cluster k

  • Data transformations & normality assumption

Ka Yee Yeung, CEA


More on the covariance matrix s k volume orientation shape
More on the covariance matrix Sk(volume, orientation, shape)

Equal volume, spherical (EI)

unequal volume, spherical (VI)

Equal volume, orientation, shape (EEE)

Diagonal model

Unconstrained (VVV)

Ka Yee Yeung, CEA


Key advantage of the model based approach choose the model and the number of clusters
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.

Ka Yee Yeung, CEA


Gene expression data sets
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)

Ka Yee Yeung, CEA


Bic analysis square root ovary data
BIC analysis: square root ovary data

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

  • Global max -> VI at 8 clusters

Ka Yee Yeung, CEA


How do we know mbc is doing well answer compare to external info
How do we know MBC is doing well?Answer: compare to external info

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

Ka Yee Yeung, CEA


Take home messages
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/

Ka Yee Yeung, CEA