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

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Presentation on theme: "Non-linear Principal Manifolds a Useful Tool in Bioinformatics and Medical Applications Andrei Zinovyev Institute des Hautes Etudes Scientifique, France."— Presentation transcript:

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

4 Finite set of objects in R N X i i=1..m IRIS database Petal heght Petal width Sepal width Sepal height SPECIES 4.931.40.2Iris-setosa 4.73.21.30.3Iris-setosa 4.63.11.50.2Iris-setosa 73.24.71.4Iris-versicolor 6.43.24.51.5Iris-versicolor 6.93.14.91.5Iris-versicolor 6.33.362.5Iris-virginica 5.82.7X1.9Iris-virginica 7.135.92.1Iris-virginica 6.32.95.61.8Iris-virginica

5 Mean point K-means clustering

6 Principal “Object”,

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

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 Data points Graph nodes U (Y) U (E), U (R)

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

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

13 Elastic manifold

14 Global minimum and softening 0,  0  10 3 0,  0  10 2 0,  0  10 1 0,  0  10 -1

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

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

17 Colorings: visualize any function

18 Density visualization

19 Example: different topologies RNRN R2R2

20 VIDAExpert tool and elmap C++ package

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

22 Image skeletonization or clustering around curves

23 Approximation of molecular surfaces

24 Application: economical data Gross output Density Profit Growth temp

25 Medical table 1700 patients with infarctus myocarde Lethal cases Patients map, density

26 Medical table 1700 patients with infarctus myocarde 128 indicators Age Numberof infarctus in anamnesis Stenocardia functional class

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

28 Golub’s leukemia dataset 3051 genes, 38 samples (ALL/B-cell,ALL/T-cell,AML) ALL sample AML sample Map of genes: vote for ALL vote for AML used by T.Golub used by W.Lie

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

30 Thank you for your attention! Questions?

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