Kernel methods for data integration in systems biology

Kernel methods for data integration in systems biology
Nathalie Vialaneix
nathalie.vialaneix@inrae.fr
http://www.nathalievialaneix.eu
Séminaire CBI
February 17, 2020 – Toulouse
Nathalie Vialaneix, MIAT, INRAE Toulouse | Kernel methods for data integration in systems biology 1/37

A short bio
trained as a mathematician, statistician
application: research applied to human health (obesity) and animal
genomics
data: mostly transcriptome but also Hi-C and metabolome and (to a
lesser extent) scRNAseq, metagenomics, ATACseq, ...
methods: networks (inference, mining), omics data integration,
machine learning (including random forest, SVM and neural networks)

Examples of past works
inferring and understanding the relations between
gene expression, lipids and phenotypes (weight,
waist circumference, ...) in adipose tissu (Diogenes)
⇒ network inference and mining, data integration,
missing data, ... [Montastier et al., 2015, Imbert et al., 2018]
and R package RNAseqNet

Examples of past works
integrating
expression and location (3D DNA FISH)
for network inference in fetal pig tissus
[Marti-Marimon et al., 2018]

Other activities
including training for biologists in RNAseq data
analysis, basic statistics, graphics with R...
organizer of the working group “Biopuces”
http://www.nathalievialaneix.eu/biopuces and active member of
“Chrocogen” https://groupes.renater.fr/sympa/info/chrocogen

In this talk...
How to integrate multiple omics data from various sources and various
types with kernels?

In this talk...
How to integrate multiple omics data from various sources and various
types with kernels?
Disclaimer: equations included (not necessary to understand the talk but
necessary for the speaker to understand her own work during the talk)

A primer on kernel methods for
biology

Before we start: context and motivations
Data characteristics
a few (paired) samples
information at various levels
... but of heterogeneous types
and, when numeric, with a large
dimension

Before we start: context and motivations
Data characteristics
a few (paired) samples
information at various levels
... but of heterogeneous types
and, when numeric, with a large
dimension
What we want to achieve
integrative analysis
to predict a phenotype, to
understand the typology of the
samples, ...

In short: what are kernels?
Data we are used to...
n samples on which p variables are
measured (xi)i=1,...,n with xi ∈ Rp

From that, we can compute:
centers of gravity: x = 1
n
n
i=1 xi
distances and dot products:
d(xi, xi ) = p
j=1
(xij − xi j)2
and xi, xi = p
j=1
xijxi j

n
n
i=1 xi
d(xi, xi ) = p
j=1
(xij − xi j)2
and xi, xi = p
j=1
xijxi j
Kernels...
The characteristics on the n samples
(xi)i are summarized by pairwise
similarities
More formally: n × n-matrix K, st K is
symmetric and positive deﬁnite

n
n
i=1 xi
d(xi, xi ) = p
j=1
(xij − xi j)2
and xi, xi = p
j=1
xijxi j
Kernels...
The characteristics on the n samples
(xi)i are summarized by pairwise
similarities
More formally: n × n-matrix K, st K is
symmetric and positive deﬁnite
Representer Theorem:

Why are kernels interesting?
1 because they can reduce high dimensional data in small similarity
matrices
2 because they are not restricted to data in Rp
(kernels on graphs,
between graphs, on text, ...) some examples to come
3 because they can embed expert knowledge (i.e., phylogeny between
taxons for instance) some examples to come
4 because they offer a rigorous framework to extend many statistical
methods basic principles to come just after
5 because they offer a clean and common framework for data
integration topic of this talk
Nathalie Vialaneix, MIAT, INRAE Toulouse | Kernel methods for data integration in systems biology
10/37

Why are kernels interesting?
1 because they can reduce high dimensional data in small similarity
matrices
2 because they are not restricted to data in Rp
(kernels on graphs,
between graphs, on text, ...) some examples to come
3 because they can embed expert knowledge (i.e., phylogeny between
taxons for instance) some examples to come
4 because they offer a rigorous framework to extend many statistical
methods basic principles to come just after
5 because they offer a clean and common framework for data
integration topic of this talk
but:
1 the choice of the relevant kernel is still up to you...
2 can strongly increase computational time when n is large...
10/37

Kernel examples
1 Rp
observations: Gaussian kernel Kii = e−γ xi−xi
2
11/37

Kernel examples
1 Rp
2
2 nodes of a graph: [Kondor and Lafferty, 2002]
11/37

Kernel examples
1 Rp
2
2 nodes of a graph: [Kondor and Lafferty, 2002]
3 sequence kernels (used to compute similarities between proteins for
instance): spectrum kernel [Jaakkola et al., 2000] (with HMM),
convolution kernel [Saigo et al., 2004]
4 kernel between graphs (or “structured data”; used in metabolomics to
compute similarities between metabolites based on their
fragmentation trees): [Shen et al., 2014, Brouard et al., 2016]
More examples: [Mariette and Vialaneix, 2019]
11/37

Principles for learning from kernels
Start from any statistical method (PCA, regression, k-means clustering)
and rewrite all quantities using:
K to compute distances and dot products
dot product is: Kii and distance is:
√
Kii + Ki i − 2Kii
(implicit) linear or convex combinations of (φ(xi))i to describe all
unobserved elements (centers of gravity and so on...)
12/37

A simple example: k-means
13/37

1: Initialization: random initialization of P centers ¯xCt
j
∈ Rp
2: for t = 1 to T do
3: Affectation step ∀ i = 1, ..., n
ft+1
(xi) = argmin
j=1,...,P
d(xi, ¯xCt
j
)
4: Representation step
∀ j = 1, . . . , P, ¯xCt
j
=
1
|Ct
j
|
xl∈Ct
j
xl
5: end for Convergence
6: return Partition
14/37

1: Initialization: random initialization of a partition of (xi)i and
¯xC1
j
= 1
|C1
j
| xi∈C1
j
φ(xi)
3: Affectation step ∀ i = 1, ..., n
ft+1
(xi) = argmin
j=1,...,P
d(xi, ¯xCt
j
)
∀ j = 1, . . . , P, ¯xCt
j
=
1
|Ct
j
|
xl∈Ct
j
xl
6: return Partition
14/37

¯xC1
j
= 1
|C1
j
| xi∈C1
j
φ(xi)
3: Affectation step
ft+1
(xi) = argmin
j=1,...,P
φ(xi) − ¯xCt
j
2
H ,
∀ j = 1, . . . , P, ¯xCt
j
=
1
|Ct
j
|
xl∈Ct
j
xl
6: return Partition
14/37

¯xC1
j
= 1
|C1
j
| xi∈C1
j
φ(xi)
3: Affectation step
ft+1
(xi) = argmin
j=1,...,P
φ(xi) − ¯xCt
j
2
H ,
∀ j = 1, . . . , P, ¯xCt
j
=
1
|Ct
j
|
xl∈Ct
j
φ(xl)
6: return Partition
14/37

¯xC1
j
= 1
|C1
j
| xi∈C1
j
φ(xi)
3: Affectation step
ft+1
(xi) = argmin
j=1,...,P
= Kii −
2
|Ct
j
|
xl∈Ct
j
Kil +
1
|Ct
j
|2
xl, xl ∈Ct
j
Kll .
∀ j = 1, . . . , P, ¯xCt
j
=
1
|Ct
j
|
xl∈Ct
j
φ(xl)
6: return Partition
14/37

Beyond kernels: relational data
DNA barcoding
Astraptes fulgerator
optimal matching
(edit) distances to
differentiate species
15/37

DNA barcoding
optimal matching
(edit) distances to
Hi-C data
pairwise measure (similarity) related to
the physical 3D distance between loci in
the cell, at genome scale
[Ambroise et al., 2019,
Randriamihamison et al., 2019]
15/37

DNA barcoding
optimal matching
(edit) distances to
Hi-C data
pairwise measure (similarity) related to
the physical 3D distance between loci in
the cell, at genome scale
[Ambroise et al., 2019,
Randriamihamison et al., 2019]
Metagenomics
dissemblance between
samples is better
captured when
phylogeny between
species is taken into
account (unifrac
distances)
15/37

Combining relational data in an
unsupervised setting
16/37

What are metagenomic data?
Source: [Sommer et al., 2010]
17/37

abundance data sparse
n × p-matrices with count data
of samples in rows and
descriptors (species, OTUs,
KEGG groups, k-mer, ...) in
columns. Generally p n.
17/37

abundance data sparse
n × p-matrices with count data
of samples in rows and
descriptors (species, OTUs,
KEGG groups, k-mer, ...) in
columns. Generally p n.
phylogenetic tree (evolution
history between species,
OTUs...). One tree with p leaves
built from the sequences
collected in the n samples.
17/37

What are metagenomic data used for?
produce a proﬁle of the diversity of a given sample ⇒ allows to
compare diversity between various conditions
used in various ﬁelds: environmental science, microbiote, ...
18/37

What are metagenomic data used for?
produce a proﬁle of the diversity of a given sample ⇒ allows to
compare diversity between various conditions
used in various ﬁelds: environmental science, microbiote, ...
Processed by computing a relevant dissimilarity between samples
(standard Euclidean distance is not relevant) and by using this dissimilarity
in subsequent analyses.
18/37

β-diversity data: dissimilarities between count data
Compositional dissimilarities: (nig) count of species g for sample i
Jaccard: the fraction of species speciﬁc of either sample i or j:
djac =
g I{nig>0,njg=0} + I{njg>0,nig=0}
j I{nig+njg>0}
Bray-Curtis: the fraction of the sample which is speciﬁc of either
sample i or j
dBC =
g |nig − njg|
g(nig + njg)
Other dissimilarities available in the R package philoseq, most of them
not Euclidean.
19/37

β-diversity data: phylogenetic dissimilarities
Phylogenetic dissimilarities
20/37

For each branch e, note le its length and pei
the fraction of counts in sample i
corresponding to species below branch e.
20/37

Unifrac: the fraction of the tree speciﬁc to
either sample i or sample j.
dUF =
e le(I{pei>0,pej=0} + I{pej>0,pei=0})
e leI{pei+pej>0}
20/37

Unifrac: the fraction of the tree speciﬁc to
either sample i or sample j.
dUF =
e le(I{pei>0,pej=0} + I{pej>0,pei=0})
e leI{pei+pej>0}
Weighted Unifrac: the fraction of the
diversity speciﬁc to sample i or to sample j.
dwUF =
e le|pei − pej|
e(pei + pej)
20/37

TARA Oceans datasets
The 2009-2013 expedition
Co-directed by Étienne Bourgois
and Éric Karsenti.
7,012 datasets collected from
35,000 samples of plankton and
water (11,535 Gb of data).
Study the plankton: bacteria,
protists, metazoans and viruses
representing more than 90% of the
biomass in the ocean.
21/37

TARA Oceans datasets
Science (May 2015) - Studies on:
eukaryotic plankton diversity
[de Vargas et al., 2015],
ocean viral communities
[Brum et al., 2015],
global plankton interactome
[Lima-Mendez et al., 2015],
global ocean microbiome
[Sunagawa et al., 2015],
. . . .
→ datasets from different types and
different sources analyzed separately.
22/37

TARA Oceans datasets that we used
[Sunagawa et al., 2015]
Datasets used
environmental dataset: 22 numeric features (temperature, salinity, . . . ).
23/37

Datasets used
bacteria phylogenomic tree: computed from ∼ 35,000 OTUs.
23/37

Datasets used
bacteria functional composition: ∼ 63,000 KEGG orthologous groups.
23/37

[de Vargas et al., 2015]
Datasets used
eukaryotic plankton composition splited into 4 groups pico (0.8 − 5µm),
nano (5 − 20µm), micro (20 − 180µm) and meso (180 − 2000µm).
23/37

[Brum et al., 2015]
Datasets used
eukaryotic plankton composition splited into 4 groups pico (0.8 − 5µm),
nano (5 − 20µm), micro (20 − 180µm) and meso (180 − 2000µm).
virus composition: ∼ 867 virus clusters based on shared gene content.
23/37

Common samples
48 samples,
2 depth layers: surface
(SRF) and deep chlorophyll
maximum (DCM),
31 different sampling
stations.
24/37

From multiple kernels to an integrated kernel
How to combine multiple kernels?
naive approach: K∗ = 1
M m Km
25/37

M m Km
supervised framework: K∗ = m βmKm
with βm ≥ 0 and m βm = 1
with βm chosen so as to minimize the prediction error
[Gönen and Alpaydin, 2011]
25/37

M m Km
unsupervised framework but input space is Rp
[Zhuang et al., 2011]
K∗ = m βmKm
with βm ≥ 0 and m βm = 1 with βm chosen so as to
minimize the distortion between all training data ij K∗
(xi, xj) xi − xj
2
;
AND minimize the approximation of the original data by the kernel
embedding i xi − j K∗
(xi, xj)xj
2
.
25/37

M m Km
unsupervised framework but input space is Rp
[Zhuang et al., 2011]
K∗ = m βmKm
with βm ≥ 0 and m βm = 1 with βm chosen so as to
minimize the distortion between all training data ij K∗
(xi, xj) xi − xj
2
;
AND minimize the approximation of the original data by the kernel
embedding i xi − j K∗
(xi, xj)xj
2
.
Our proposal: 2 UMKL frameworks which do not require data to have
values in Rd
.
25/37

Multi-kernel/distances integration
How to “optimally” combine several
relational datasets in an unsupervised
setting?
for kernels K1
, . . . , KM
obtained on the
same n objects, search: Kβ = M
m=1 βmKm
[Mariette and Villa-Vialaneix, 2018]
Package R mixKernel
https://cran.r-project.org/
package=mixKernel
26/37

STATIS like framework
[L’Hermier des Plantes, 1976, Lavit et al., 1994]
Similarities between kernels:
Cmm =
Km
, Km
F
Km
F Km
F
=
Trace(Km
Km
)
Trace((Km)2)Trace((Km )2)
.
(Cmm is an extension of the RV-coefficient [Robert and Escoufier, 1976] to the
kernel framework)
maximizev
M
m=1
K∗
(v),
Km
Km
F F
= v Cv
for K∗
(v) =
M
m=1
vmKm
and v ∈ RM
such that v 2 = 1.
Solution: first eigenvector of C ⇒ Set β = v
M
m=1 vm
(consensual kernel).
27/37

A kernel preserving the original topology of the data I
Similarly to [Lin et al., 2010], preserve the local geometry of the data in the
feature space.
Proxy of the local geometry
Km
−→ Gm
k
k−nearest neighbors graph
−→ Am
k
adjacency matrix
⇒ W = m I{Am
k
>0} or W = m Am
k
Feature space geometry measured by
∆i(β) = φ∗
β(xi),


φ∗
β(x1)
...
φ∗
β(xn)


=


K∗
β(xi, x1)
...
K∗
β(xi, xn)


28/37

A kernel preserving the original topology of the data II
Sparse version (quadprog in R)
minimizeβ
N
i,j=1
Wij ∆i(β) − ∆j(β)
2
for K∗
β =
M
m=1
βmKm
and β ∈ RM
st βm ≥ 0 and
M
m=1
βm = 1.
Non sparse version (ADMM optimization [Boyd et al., 2011]
minimizev
N
i,j=1
Wij ∆i(β) − ∆j(β)
2
for K∗
v =
M
m=1
vmKm
and v ∈ RM
st vm ≥ 0 and v 2 = 1.
29/37

Application to TARA oceans
Similarity between datasets (STATIS)
Low similarities between meso-plankton (euk.meso) and other
datasets: strong geographical structure of mesoplanktonic
communities [de Vargas et al., 2015].
30/37

Similarity between datasets (STATIS)
Low similarities between meso-plankton (euk.meso) and other
datasets: strong geographical structure of mesoplanktonic
communities [de Vargas et al., 2015].
Strongest similarities between environmental variables and small
organisms than largest ones [de Vargas et al., 2015, Sunagawa et al., 2015].
30/37

Integrating all Tara Oceans data sets
no particular pattern in terms of depth layers but in terms of
geography.
31/37

Important variables
Rhizaria abundance strongly structure the differences between samples (analyses
restricted to some organisms found differences mostly based on water depths)
and waters from Arctic Oceans and Paciﬁc Oceans differ in terms of Rhizaria
abundance
32/37

Conclusions
Kernel methods are useful for:
dealing with different types of data
even when they are high-dimensional
combining them
However, they can be:
computationally intensive to train
not easy to interpret (work-in-progress with Jérôme Mariette and
Céline Brouard on variable selection in unsupervised setting)
33/37

SOMbrero
Madalina Olteanu,
Fabrice Rossi, Marie Cottrell,
Laura Bendhaïba and
Julien Boelaert
SOMbrero and mixKernel
Jérôme Mariette
adjclust and Hi-C
Pierre Neuvial, Nathanaël Randriamihamison,
Sylvain Foissac, Guillem Rigail, Christophe Ambroise and
Shubham Chaturvedi
34/37

Credits for pictures
Slide 3: image based on ENCODE project, by Darryl Leja (NHGRI), Ian Dunham
(EBI) and Michael Pazin (NHGRI)
Slide 8: k-means image from Wikimedia Commons by Weston.pace
Slide 10: Astraptes picture is from
https://www.flickr.com/photos/39139121@N00/2045403823/ by Anne Toal
(CC BY-SA 2.0), Hi-C experiment is taken from the article Matharu et al., 2015
DOI:10.1371/journal.pgen.1005640 (CC BY-SA 4.0) and metagenomics illustration is
taken from the article Sommer et al., 2010 DOI:10.1038/msb.2010.16 (CC BY-NC-SA
3.0)
Other pictures are from articles that I co-authored.
35/37

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Unsupervised multiple kernel clustering.
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Optimization issues
Sparse version writes minβ βT
Sβ st β ≥ 0 and β 1 = m βm = 1 ⇒
standard QP problem with linear constrains (ex: package quadprog
in R).
36/37

Optimization issues
Sβ st β ≥ 0 and β 1 = m βm = 1 ⇒
in R).
Non sparse version writes minβ βT
Sβ st β ≥ 0 and β 2 = 1 ⇒ QPQC
problem (hard to solve).
36/37

Optimization issues
Sβ st β ≥ 0 and β 1 = m βm = 1 ⇒
in R).
Solved using Alternating Direction Method of Multipliers (ADMM
[Boyd et al., 2011]) by replacing the previous optimization problem
with
min
x,z
x Sx + 1{x≥0}(x) + 1{ z 2
2
≥1}(z)
with the constraint x − z = 0.
36/37

Optimization issues
Sβ st β ≥ 0 and β 1 = m βm = 1 ⇒
in R).
Solved using Alternating Direction Method of Multipliers (ADMM
[Boyd et al., 2011])
1 minx x Sx + y (x − z) + λ
2
x − z 2
under the constraint x ≥ 0
(standard QP problem)
2 project on the unit ball z = x
min{ x 2,1}
3 update auxiliary variable y = y + λ(x − z)
36/37

A proposal to improve interpretability of K-PCA in our
framework
Issue: How to assess the importance of a given species in the K-PCA?
37/37

framework
our datasets are either numeric (environmental) or are built from a
n × p count matrix
⇒ for a given species, randomly permute counts and re-do the
analysis (kernel computation - with the same optimized weights - and
K-PCA)
37/37

framework
our datasets are either numeric (environmental) or are built from a
n × p count matrix
⇒ for a given species, randomly permute counts and re-do the
analysis (kernel computation - with the same optimized weights - and
K-PCA)
the inﬂuence of a given species in a given dataset on a given PC
subspace is accessed by computing the Crone-Crosby distance
between these two PCA subspaces [Crone and Crosby, 1995] (∼
Frobenius norm between the projectors)
37/37

Kernel methods for data integration in systems biology

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