Non-linear%20Principal%20Manifolds%20a%20Useful%20Tool%20in%20Bioinformatics%20and%20Medical%20Applications

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Non-linear Principal Manifoldsa Useful Tool in
Bioinformatics and Medical Applications

• Andrei Zinovyev
• Institute des Hautes Etudes Scientifique,
• France

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Plan of the talk

• Object of study
• Definition of principal manifold (PM)
• Constructing PMs elastic maps
• Examples of biomedical applications

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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
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Finite set of objects in RN
IRIS database IRIS database IRIS database IRIS database

Petal heght Petal width Sepal width Sepal height SPECIES
4.9 3 1.4 0.2 Iris-setosa
4.7 3.2 1.3 0.3 Iris-setosa
4.6 3.1 1.5 0.2 Iris-setosa
7 3.2 4.7 1.4 Iris-versicolor
6.4 3.2 4.5 1.5 Iris-versicolor
6.9 3.1 4.9 1.5 Iris-versicolor
6.3 3.3 6 2.5 Iris-virginica
5.8 2.7 X 1.9 Iris-virginica
7.1 3 5.9 2.1 Iris-virginica
6.3 2.9 5.6 1.8 Iris-virginica
X i
i1..m
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Mean point
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Principal Object
,
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Principal Component Analysis
,
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Principal manifold
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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

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Metaphor of elasticity
U(Y)
U(E), U(R)
Data points
Graph nodes
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Constructing elastic nets
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Definition of elastic energy
.
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Elastic manifold

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Global minimum and softening
?0, ?0 ? 103
?0, ?0 ? 102
?0, ?0 ? 101
?0, ?0 ? 10-1
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Adaptive algorithms
Refining net
Growing net
Idea of scaling
Adaptive net
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Projection onto the manifold


Closest node of the net
Closest point of the manifold
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Colorings visualize any function
Value of the coordinate

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Density visualization
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Example different topologies
RN
R2
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VIDAExpert tool and elmap C package
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Regression and principal manifolds
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Image skeletonization or clustering around curves
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Approximation of molecular surfaces
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Application economical data
Density
Gross output
Profit
Growth temp
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Medical table1700 patients with infarctus
myocarde
Patients map, density
Lethal cases
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Medical table1700 patients with infarctus
myocarde
128 indicators
Stenocardia functional class
Numberof infarctus in anamnesis
Age
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Codon usage in all genes of one genome
Escherichia coli
Bacillus subtilis
Majority of genes
Foreign genes
Hydrophobic genes
Highly expressed genes
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Golubs 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
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Golubs 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
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Thank you for your attention!

• Questions?