Anindya Bhattacharya and Rajat K. De Bioinformatics, 2008.

Anindya Bhattacharya and Rajat K. DeBioinformatics, 2008

IntroductionDivisive Correlation Clustering

AlgorithmResultsConclusions

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Correlation Clustering

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Correlation clustering is proposed by Bansal et al. in Machine Learning, 2004.

It is basically based on the notion of graph partitioning.

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How to construct the graph? Nodes: genes. Edges: correlation between the genes.

Two types of edges: Positive edge. Negative edge.

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For example:

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XX YY Positive correlation coefficient: Positive edge( )

XX YY Negative correlation coefficient: Negative edge( )

CC

GG

BB

DD

AA

HH

GG

FF

EE

Cluster 1

Cluster 2

Graph Construction

Graph Partitioning CC

GG

BB

DD

AA

HH

GG

FF

EE

How to measure the quality of clusters? The number of agreements. The number of disagreements.

The number of agreements: the number of genes that are in correct clusters.

The number of disagreements: the number of genes wrongly clustered.

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For example:

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AA

CC

DD EE

BB

Cluster 1

Cluster 2

The measure of agreements is the sum of:(1) # of positive edges in the same clusters(2) # of negative edges in different clustersThe measure of disagreements is the sum of:(1) # of negative edges in the same clusters(2) # of positive edges in different clusters

4 + 4 = 8

0 + 2 = 2

Minimization of disagreements or equivalently Maximization of agreements!

However, it’s NP-Complete proved by Bansal et al., 2004.

Another problem is without the magnitude of correlation coefficients.

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Pearson correlation coefficientTerms and measurements used in

DCCADivisive Correlation Clustering

Algorithm

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Consider a set of genes, , for each of which expression values are given.

The Pearson correlation coefficient between two genes and is defined as:

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lth sample value of gene

mean value of gene from samples

: and are positively correlated with the degree of correlation as its magnitude.

: and are negatively correlated with value .

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We define some terms and measurements used in DCCA: Attraction Repulsion Attraction/Repulsion value Average correlation value

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Attraction: There’s an attraction between and if .

Repulsion: There’s a repulsion between and if .

Attraction/Repulsion value: Magnitude of

is the strength of attraction or repulsion.

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The genes will be grouped into disjoint clusters .

Average correlation value: Average correlation value for a gene with respect to cluster is defined as:

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the number of data points in

indicates that the average correlation for a gene with other genes inside the cluster .

Average correlation value reflects the degree of inclusion of to cluster .

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Divisive Correlation Clustering Algorithm

11 mm

m samples

11 mm

n genes

DCCA

C1C1 C2C2 CkCk

K disjoint clustersX1

Xn

Step 1:

Step 2: for each iteration, do: Step 2-i:

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Step 2: Step 2-ii:

Step 2-iii:

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C1C1 C2C2 CpCp

Which cluster exists the most repulsion value?

Cluster C!

Step 2-iv:

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xixi

xjxj

xk

xk

xk

xk

xk

xk

xk

xkx

k

xk

xk

xk

xk

xk

Cluster C

xjxj

xixi

Cp

Cq

Step 2-v:

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xk

xk

C1C1 C2C2 CKCK

The highest average correlation value!

C1C1 C2C2 CKCKxk

xk

Place a copy of xk

CNEW: new clusters

Step 2-vi:

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C1C1 C2C2 CKCK

C1C1 C2C2 CKCK

CNEW: new clusters

Any change?



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Performance comparison A synthetic dataset ADS Nine gene expression datasets

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A synthetic dataset ADS:

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Three groups.

Experimental results:

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Clustering correctly.


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Undesired Clusters.

Undesired Clusters.

Five yeast datasets: Yeast ATP, Yeast PHO, Yeast AFR, Yeast

AFRt, Yeast Cho et al.Four mammalian datasets:

GDS958 Wild type, GDS958 Knocked out, GDS1423, GDS2745.

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Performance comparison: z-score is calculated by observing the relation between a clustering result and the functional annotation of the genes in the cluster.

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Attributes

Mutual information

The entropies for each cluster-attribute pair.

The entropies for clustering result independent of attributes.

The entropies for each of the NA attributes independent of clusters.

z-score is defined as:

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The computed MI for the clustered data, using the

attribute database.

MIrandom is computed by computing MI for a clustering obtained by randomly assigning genes to clusters of uniform size and repeating until a distribution of values is obtained.

Mean of these MI-values.

The standard deviation of these MI-values.

A higher value of z indicates that genes would be better clustered by function, indicating a more biologically relevant clustering result.

Gibbons ClusterJudge tool is used to calculating z-score for five yeast datasets.

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Pros: DCCA is able to obtain clustering

solution from gene-expression dataset with high biological significance.

DCCA detects clusters with genes in similar variation pattern of expression profiles, without taking the expected number of clusters as an input.

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Cons: The computation cost for repairing any

misplacement occurring in clustering step is high.

DCCA will not work if dataset contains less than 3 samples. The correlation value will be either +1 or -1.

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