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Non-linear Principal Manifoldsa Useful Tool in Bioinformatics and Medical Applications
Andrei ZinovyevInstitute des Hautes Etudes
Scientifique,France
Plan of the talk
Object of study Definition of principal manifold
(PM) Constructing PMs: elastic maps Examples of biomedical
applications
Principal manifoldsElastic maps framework
SVM
Principal manifolds
Regression,approximation
Supervisedclassification
K-means
SOM
Clustering
Multidim.scaling
VisualizationPCA
Factor analysis
LLE ISOMAP
Non-linearData-miningmethods
Finite set of objects in RN
X i
i=1..m
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
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
Definition of elastic energy
)()()( REY UUUU
2)(
1
)(
)()(
1 ijp
i Kx
Y yXN
Uij
2)()(
1
)( )0()1( iis
ii
E EEU
r
i
iiii
R RRRU1
2)()()()( )0(2)2()1(.
E (0) E (1)
R (1) R (0) R (2)
y
Xj
00 , ii
Regression and principal manifolds
regression principal component
x
F(x)
min2 ii Pxx min)(
2 ii xFx
Data
Gen.curve
Grid
Medical table1700 patients with infarctus myocarde
128 indicators
Age Numberof infarctusin anamnesis
Stenocardia functionalclass
Codon usage in all genes of one genome
Escherichia coli Bacillus subtilis
Majority of genes
Highly expressed genes
“Foreign” genes
“Hydrophobic” genes
Golub’s leukemia dataset3051 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
Golub’s leukemia datasetmap of samples: AML ALL/B-cell ALL/T-cell
density
Cystatin C Retinoblastomabinding protein P48
CA2 Carbonic anhydrase II
X-linked Helicase II