Robust Clustering Techniques in Bioinformatics. Rob Beverly Fall 2004

Transcription

1 Robus Clusering Techniques in Bioinformaics Rob Beverly 8.47 Fall 004

2 Why Clusering? Class iscovery Given jus he daa, can one find inheren classes/clusers Class redicion Given an exising clusering, predic class of new elemens

3 k-means Clusering Bad k-means clusering ofen used Simple Fas Cenroids force spherical inerpreaion of he daa Easy o consruc degenerae examples:

4 k- Clusering

5 k- Clusering

6 Specral Clusering High-level: Consruc a neighbor graph k-neares neighbor hreshold Assign weighs o edges efine ransiion probabiliy over edges Cluser based on eigenvecors of probabiliy marix

7 Specral Clusering Assign weighs based on Euclidian disance in d-dimensional space wih exponenial fall-off: If an edge exiss beween verices i and j in he graph, hen assign weigh: W ij exp{-β x i x j }

8 Specral Clusering efine a Markov random walk over he graph by normalizing edge weighs o form ransiion probabiliies e be a diagonal marix wih elemens ii equal o he sum of weighs for node I Then: - W And: Wij ij Wij j

9 Specral Clusering isribuion of poins afer random seps converges as increases If graph is conneced and ergodic, he disribuion becomes independen of saring poin Recover his effec from he eigenvecors

10 Specral Clusering Compuing random walk: Find eigenvecors corresponding o second larges eigenvalue (larges correcion o asympoic limi) of eiher: Sochasic marix: - W aplacian: W W W W W W W W W W W W W W W W W W W

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13 Research Quesions : oes specral clusering ouperform radiional mehods on real daa ses

14 Canonical A/AM aase [Golub e al, 999] Gene expression paerns (~7000) from Microarrays of 38 paiens wih leukemia Aracive because here are wo inheren ypes of leukemia: A and AM aper uses k-means based Self-Organizing Maps (SOMs) o cluser

15 A/AM Golub Resuls: Cluser : 4/5 A Cluser : 0/3 AM False osiive, 3 False Negaives Toal 4 misclassificaions: ~0% oes specral clusering perform beer? Yes misclassificaions

16 A/AM

17 A/AM

18 Specral Clusering Finding more han wo clusers? Recursive subdivide unil correc number of clusers Mulicu: Find k eigenvecors corresponding o he k larges eigenvalues Run k-means clusering on resuling marix

19 Number of Clusers How can we know a priori he number of clusers in he daa? Explored a divisive clusering algorihm [Newman 003]

20 ivisive Clusering Sar wih k-neares neighbors graph Compue all-pairs shores pahs Ierae unil graph is empy: Find edge e wih larges number of Ss raversing i Remove e Compue modulariy score Q Graph wih highes modulariy score is seleced as represening he inheren clusers

21 Modulariy Score a i j e ij Q ) (e a i ii i e ij is he fracion of edges from cluser i o cluser j Inuiion: edges wihin a cluser minus expeced value if edges fall a random Q0 implies random number of wihin cluser edges

22 Research Quesions : oes specral clusering ouperform radiional mehods on real daa ses : Can we infer he correc number of clusers

23 A/AM ivisive Clusering k Neares Neighbors Resuling Clusers

24 A/AM ivisive Clusering k6 Neares Neighbors Resuling Clusers

25 o our Resuls Generalize A/AM an older, well-sudied daa-se Relaively easy o do well on More recen: Gene expression-based classificaion of malignan gliomas [Nu e al, 003]

26 Malignan Gliomas Sudy wo differen brain cancers wih differen courses of reamen: Glioblasomas Anaplasic Oligodendrogliomas isinguishing beween hem is diagnosically challenging Gene expression paerns (~,000) from 50 gliomas

27 Clusering Malignan Gliomas Firs aemp: poor error raes Read paper more carefully: Variaion filering sep o reduce noise Genes wih less han 00 unis of variaion removed Reduced daa se from ~,000 genes o ~5,000

28 Clusering Malignan Gliomas

29 Clusering Malignan Gliomas

30 Clusering Malignan Gliomas k3 Neares Neighbors Resuling Clusers

31 Clusering Malignan Gliomas k4 Neares Neighbors Resuling Clusers

32 Conclusions All mehods require some knowledge of underlying daa o une parameers Specral clusering offers (demonsrably) beer resuls on gene expression daases No clear number of clusers in Gliomas sudy Thanks!