Data Mining in Bioinformatics Day 7: Clustering in Bioinformatics

Data Mining in Bioinformatics

Day 7: Clustering in Bioinformatics

Karsten Borgwardt

February 25 to March 10

Clustering in bioinformatics

Microarrays

Clustering is a widely used tool in microarray analysis Class discovery is an important problem in microarray studies for two reasons:

either the classes are completely unknown before-hand

Clustering in bioinformatics

Examples

Classes unknown:

Does a disease affect gene expression in a particular tissue?

Does gene expression differ between two groups in a particular condition?

Subclasses unknown:

Are there subtypes of a disease?

Clustering in bioinformatics

Popularity

Clustering tools are available in the large microarray database NCBI Gene Expression Omnibus (GEO)

http://www.ncbi.nlm.nih.gov/geo/

Distance metrics

Euclidean distance

Euclidean distance of gene x and y of n samples or sam-ple x and y of n genes:

Distance metrics

Un-centered correlation coefficient

Un-centered correlation coefficient of gene x and y of n samples or sample x and y of n genes:

r_xyu =

Pn

i=1 xiyi

pPn

i=1 x2i

pPn

i=1 yi2

Clustering algorithms

Hierarchical Clustering

Single linkage: The linking distance is the minimum dis-tance between two clusters.

Complete linkage: The linking distance is the maximum distance between two clusters.

Average linkage/UPGMA (The linking distance is the av-erage of all pair-wise distances between members of the two clusters. Since all genes and samples carry equal weight, the linkage is an Unweighted Pair Group Method with Arithmetic Means (UPGMA))

‘Flat’ Clustering

The two-sample problem

Interpretation of clusters

Clustering introduces ‘structure’ into microarray datasets

But is there a statistical or biomedical meaning of these classes?

Biomedical meaning has to be established in experi-ments

‘Statistical meaning’ can be measured using statistical tests, by a so-called two-sample test

The two-sample problem

Data diversity

Molecular biology produces a wealth of information The problem is that these data are generated

on different platforms and by different protocols

under different levels of noise

Hence data from different labs show different scales

different ranges

different distributions Main problem:

The two-sample problem

The two-sample problem

Given two samples X and Y .

Were they generated by the same distribution? Previous approaches

The two-sample problem

t-test

A test of the null hypothesis that the means of two nor-mally distributed populations are equal

unpaired/independent (versus paired)

For equal sample sizes and equal variances, the t statis-tic to test whether the means are different can be calcu-lated as follows:

The two-sample problem

New challenges in bioinformatics high-dimensional

structured (strings and graphs) low sample size

MMD key idea

Key Idea

Avoid density estimator, use means in feature spaces Maximum Mean Discrepancy (Fortet and Mourier, 1953)

D(p, q,F_{) := sup}

f∈F

E_{p[f(x)] −} E_{q [f(y)]}

Theorem

D(p, q,F_{) = 0 iff p = q, when} F _{= C}0₍X_).

Follows directly, e.g. from Dudley, 1984.

MMD statistic

Goal: Estimate D(p, q,F₎

E_{p,pk(x, x}′_{) − 2}E_{p,qk(x, y) +} E_{q,qk(y, y}′₎

U-Statistic: Empirical estimate D(X, Y,F₎

1

Estimate σ2 from data.

Reject null hypothesis that p = q if D(X, Y,F_{) exceeds}

Attractive for bioinformatics

MMD

two-sample test in terms of kernels Computationally attractive

search infinite space of functions by evaluating one ex-pression

Attractive for bioinformatics

Wide applicability

for one- and higher-dimensional vectorial data, but also for structured data!

two-sample problems can now be tackled on strings: protein and DNA sequences

graphs: molecules, protein interaction networks time series: time series of microarray data

Cross-platform comparability

Data

microarray data from two breast cancer studies one on cDNA platform (Gruvberger et al., 2001)

other on oligonucleotide microarray platform (West et al., 2001)

Task

Can MMD help to find out if two sets of observations were generated by

the same study (both from Gruvberger or both from West)?

Cross-platform comparability

Experiment

sample size each: 25

dimension of each datapoint 2,116 significance level: α = 0.05

100 times: 1 sample from Gruvberger, 1 from West 100 times: both from Gruvberger or both from West report percentage of correct decisions

Kernel-based statistical test

novel statistical test for two-sample problem: easy to implement

non-parametric

first for structured data

best on high-dimensional data

quadratic runtime w.r.t. the number of data points impressive accuracy in our experiments

kernel method for two-sample problem:

all kernels recently defined in molecular biology can be re-used for data integration

Biclustering

Clustering in two dimensions

alternative names: co-clustering, two-mode clustering A bicluster is a subset of genes that show similar activ-ity patterns under a subset of conditions.

Clustering in 2 dimensions

Cluster patients and conditions

Earliest work by Hartigan, 1972: Divide a matrix into submatrices with minimum variance.

Most interesting cases are NP-complete.

References and further reading

References

The end