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Non-linear Principal Manifolds a Useful Tool in Bioinformatics and Medical Applications

Non-linear Principal Manifolds a Useful Tool in Bioinformatics and Medical Applications. Andrei Zinovyev Institute des Hautes Etudes Scientifique, France. Plan of the talk. Object of study Definition of principal manifold (PM) Constructing PMs: elastic maps

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Non-linear Principal Manifolds a Useful Tool in Bioinformatics and Medical Applications

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  1. Non-linear Principal Manifoldsa Useful Tool in Bioinformatics and Medical Applications Andrei Zinovyev Institute des Hautes Etudes Scientifique, France

  2. Plan of the talk • Object of study • Definition of principal manifold (PM) • Constructing PMs: elastic maps • Examples of biomedical applications

  3. Principal manifoldsElastic maps framework LLE ISOMAP Clustering Multidim. scaling Principal manifolds PCA K- means Visualization SOM Non-linear Data-mining methods Factor analysis Supervised classification SVM Regression, approximation

  4. Finite set of objects in RN X i i=1..m

  5. K-means clustering Mean point

  6. Principal “Object” ,

  7. 1st Principal axis Maximal dispersion 2nd principal axis Principal Component Analysis ,

  8. Principal manifold

  9. What do we want? • Non-linear surface (1D, 2D, 3D …) • Smooth and not twisted • The data model is unknown • Speed (time linear with Nm) • Uniqueness • Fast way to project datapoints

  10. Metaphor of elasticity U(Y) U(E), U(R) Data points Graph nodes

  11. y E(0) R(1) R(2) E(1) R(0) Constructing elastic nets

  12. Xj y E(0) R(1) R(2) E(1) R(0) Definition of elastic energy .

  13. Elastic manifold

  14. Global minimum and softening 0, 0 103 0, 0 102 0, 0 101 0, 0 10-1

  15. Adaptive algorithms Refining net: Growing net Idea of scaling: Adaptive net

  16. Projection onto the manifold Closest node of the net Closest point of the manifold

  17. Colorings: visualize any function Value of the coordinate

  18. Density visualization

  19. Example: different topologies RN R2

  20. VIDAExpert tool and elmap C++ package

  21. principal component regression F(x) x Regression and principal manifolds

  22. Image skeletonization or clustering around curves

  23. Approximation of molecular surfaces

  24. Application: economical data Density Gross output Profit Growth temp

  25. Medical table1700patients with infarctus myocarde Patients map, density Lethal cases

  26. Medical table1700patients with infarctus myocarde 128 indicators Stenocardia functional class Numberof infarctus in anamnesis Age

  27. Codon usage in all genes of one genome Escherichia coli Bacillus subtilis Majority of genes “Foreign” genes “Hydrophobic” genes Highly expressed genes

  28. Golub’s leukemia dataset3051 genes, 38 samples (ALL/B-cell,ALL/T-cell,AML) Map of genes: vote for ALL vote for AML used by T.Golub used by W.Lie ALL sample AML sample

  29. Golub’s leukemia datasetmap of samples: AML ALL/B-cell ALL/T-cell Retinoblastoma binding protein P48 Cystatin C density CA2 Carbonic anhydrase II X-linked Helicase II

  30. Thank you for your attention! • Questions?

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