Reconstruction of ecological interaction networks by Raphaëlle Momal-Leisenring, PhD Candidate in Applied Mathematics @AgroParisTech

Inference of species interaction networks from
abundances
Rapha¨elle Momal
Supervision: S. Robin1
and C. Ambroise2
1UMR AgroParisTech / INRA MIA-Paris
2LaMME, Evry
September 26th, 2019
Rapha¨elle Momal WiMLDS Meetup 2019 September 26th
, 2019 1 / 23

The thesis project
In a few words
A project using mathematics, statistical modelling and machine learning
techniques for applications in microbiology, metagenomics, or ecology.
Direction:
St´ephane Robin Chistophe Ambroise
Supports:
, 2019 2 / 23

The thesis project
Network example in ecology
Pocock et. al 2012
Tool to better
understand species
interactions,
eco-systems
organizations
Allows for resilience
analyses, pathogens
control, ecosystem
comparison, response
prediction...
, 2019 3 / 23

The thesis project
Aim of network inference from abundance data
date site
apr93 km03
apr93 km03
apr93 km03
apr93 km03
apr93 km17
apr93 km17
...
...
EFI ELA GDE GME HFA
71 1 5 6 0
118 2 3 0 0
69 0 6 2 0
56 0 0 0 0
0 1 1 0 0
0 0 2 0 0
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.
.
.
.
.
.
.
.
.
.
.
.
.
.
(a) covariates X (b) species abundances Y (c) inferred network
Data sample from the Fatala river dataset (ade4 R package).
Unknown underlying structure
Unobserved interaction data
, 2019 4 / 23

Statistical modelling The dependency structure
Graphical models: a statistical framework for conditional
dependence
Example:
A1
A2
A3
A4
Connected: all variables are
dependant
Direct dependence or
conditional independence
A4 is independent from (A1, A3)
conditionally on A2
, 2019 5 / 23

Statistical modelling The dependency structure
Graphical models: a statistical framework for conditional
dependence
Example:
A1
A2
A3
A4
Connected: all variables are
dependant
Direct dependence or
conditional independence
A4 is independent from (A1, A3)
conditionally on A2
P(A1, . . . , Ap) ∝
C∈CG
ψC (AC )
where CG = set of maximal cliques of G.
, 2019 5 / 23

Statistical modelling The observed counts
P N model
Yij ∼ P (exp(oij + xi θj + Zij)) .
A latent variable model
easy handling of multi-variate data, oﬀsets and covariates
Random eﬀects Z add dependence among species. Classically (Aitchison
and Ho, 1989):
Z ∼ N(0, Σ)
We foster sparsity with a mixture of tree structures:
Z ∼ p(T)N(0, ΣT ), T ∼
jk
βjk/B
, 2019 6 / 23

Inference EM algorithm
Maximum likelihood with hidden data
observations Y
hidden parameters H
⇒ log p(Y ) intractable.
EM algorithm maximizes a surrogate for the log-likelihood :
Q = E[log p(Y , H)|Y ] = log p(Y , h)p(h|Y )dh
In most cases the conditional density p(h|Y ) is intractable.
Variational EM (VEM) resorts to a proxy q(h) = ˜p(h|Y ).
, 2019 7 / 23

Inference EM algorithm
Two hidden quantities
Our model includes two hidden layers of parameters. We need to compute
conditional probabilities:
p(T|Y ): computationally complex but tractable thanks to an
algebraic mathematical tool (E: Kirshner (2008), M: Meil˘a and
Jaakkola (2006)).
p(Z|Y ): no close form, a VEM gives ˆΣ and ˆθ (VEM: Chiquet et al.
(2017)).
, 2019 8 / 23

Inference Using trees
Mixture of trees: sparse and eﬃcient
Sparse structures:
#Gp = 2
p(p−1)
2 reduced to #Tp = p(p−2)
, 2019 9 / 23

Mixture of trees: sparse and eﬃcient
Sparse structures:
#Gp = 2
p(p−1)
2 reduced to #Tp = p(p−2)
Suitable algebraic tool:
Matrix tree theorem (Chaiken and Kleitman, 1978)
T∈T (k,l)∈T
ψk,l (Y ) = det(Lψ(Y )) → Θ(p3
)
Approach: infer the network by averaging spanning trees
, 2019 9 / 23

Concept of tree averaging
Z1 Z2
Z3Z4
P{T = t1|Y }
Z1 Z2
Z3Z4
P{T = t2|Y }
Z1 Z2
Z3Z4
P{T = t3|Y }
Z1 Z2
Z3Z4
P{T = t4|Y }
...
Compute edge
probabilities: Z1 Z2
Z3Z4
P{(j, k) ∈ T|Y }
Thresholding
probabilities:
Z1 Z2
Z3Z4
, 2019 10 / 23

EMtree algorithm
Input: Abundance data, covariates, oﬀsets
1rst step: VEM algorithm to ﬁt PLN model ⇒ ˆθ, ˆΣZ .
2nd step: EM algorithm to update the βjk ⇒ conditional probabilities
for all edges.
Yij ∼ P exp(oij + xi θj + Zij ) .
Z ∼ p(T)N(0, ΣT ), T ∼
jk
βjk/B
, 2019 11 / 23

EMtree algorithm
Input: Abundance data, covariates, oﬀsets
1rst step: VEM algorithm to ﬁt PLN model ⇒ ˆθ, ˆΣZ .
2nd step: EM algorithm to update the βjk ⇒ conditional probabilities
for all edges.
Thresholding: Select edges with probability above the probability of
edges in a tree drawn uniformly (2/p)
Resampling: Strengthen the results: only edges selected in more than
80% of S sub-samples are kept.
Available for download at https://github.com/Rmomal/EMtree
, 2019 11 / 23

Application Fatala River ﬁshes
Inferred networks
, 2019 12 / 23

Evaluating performances
Evaluation strategy
Alternatives:
Two methods on transformed counts, no covariates:
SpiecEasi algorithm Kurtz et al. (2015)
gCoda Fang et al. (2017)
One taking raw counts and covariates:
MInt Biswas et al. (2016) (uses PLN model)
, 2019 13 / 23

Evaluating performances
Evaluation strategy
Alternatives:
Two methods on transformed counts, no covariates:
SpiecEasi algorithm Kurtz et al. (2015)
gCoda Fang et al. (2017)
One taking raw counts and covariates:
MInt Biswas et al. (2016) (uses PLN model)
Simulation design:
1 Choose G and deﬁne ΣG accordingly
2 Sample count data Y from P N(X, ΣG )
3 Infer the network with EMtree, SpiecEasi, gCoda, and MInt
4 Compare results with presence/absence of edges (FDR, AUC)
, 2019 13 / 23

Evaluating performances Results
Diﬃculty level
False Discovery Rate (FDR): how many false edges there is among what is detected ?
ratio: number of detections over the number of true edges
EMtree is a sparser approach than MInt
, 2019 14 / 23

Network density
Area under the (ROC) curve (AUC): ”how good is a classifier to rank true positives
higher”
100 observations, 20 species:
Effect of graph density on the evolution of AUC median and inter-quartile intervals in Erdös and
Cluster structures.
, 2019 15 / 23

To be published soon
, 2019 16 / 23

Conclusion
Contributions:
Formal probabilistic model for network inference from
count data
R package: https://github.com/Rmomal/EMtree
Preprint: Tree-based Reconstruction of Ecological Network from
Abundance Data. https://arxiv.org/pdf/1905.02452.pdf
Perspectives:
Sign and strength of interactions according to graphical
models theory
Missing major actor (species/covariates)
More collaborations with experts in macro-ecology ﬁeld
, 2019 17 / 23

Thank you
Contact :
email raphaelle.momal@agroparistech.fr
Web Rmomal.github.io
Twitter @MomalRaphaelle
, 2019 18 / 23

References I
Aitchison, J. and Ho, C. (1989). The multivariate Poisson-log normal distribution. Biometrika, 76(4):643–653.
Biswas, S., McDonald, M., Lundberg, D. S., Dangl, J. L., and Jojic, V. (2016). Learning microbial interaction networks from
metagenomic count data. Journal of Computational Biology, 23(6):526–535.
Chaiken, S. and Kleitman, D. J. (1978). Matrix tree theorems. Journal of combinatorial theory, Series A, 24(3):377–381.
Chiquet, J., Mariadassou, M., and Robin, S. (2017). Variational inference for probabilistic Poisson PCA. Technical report,
arXiv:1703.06633. to appear in Annals of Applied Statistics.
Chow, C. and Liu, C. (1968). Approximating discrete probability distributions with dependence trees. IEEE Transactions on
Information Theory, 14(3):462–467.
Fang, H., Huang, C., Zhao, H., and Deng, M. (2017). gcoda: conditional dependence network inference for compositional data.
Journal of Computational Biology, 24(7):699–708.
Kirshner, S. (2008). Learning with tree-averaged densities and distributions. In Advances in Neural Information Processing
Systems, pages 761–768.
Kurtz, Z. D., M¨uller, C. L., Miraldi, E. R., Littman, D. R., Blaser, M. J., and Bonneau, R. A. (2015). Sparse and
compositionally robust inference of microbial ecological networks. PLoS computational biology, 11(5):e1004226.
Meil˘a, M. and Jaakkola, T. (2006). Tractable bayesian learning of tree belief networks. Statistics and Computing, 16(1):77–92.
Meil˘a, M. and Jordan, M. I. (2000). Learning with mixtures of trees. Journal of Machine Learning Research, 1:1–48.
, 2019 19 / 23

Conditional probability computation
Kirchhoﬀ’s theorem (matrix tree, Aitchison and Ho (1989))
For all W = (akl )k,l a symmetric matrix, the corresponding Laplacian Q(W ) is deﬁned
as follows:
Quv (W ) =
−auv 1 ≤ u < v ≤ n
n
i=1 avi 1 ≤ u = v ≤ n.
Then for all u et v:
|Q∗
uv (W )| =
T∈T {k,l}∈ET
akl
P((k, l) ∈ T|Z) =
T∈T :(k,l)∈T
P(T|Z) =
(k,l)∈T P(T)P(Z|T)
T P(T)P(Z|T)
= 1 −
|Q∗
uv (βΨ−kl
)|
|Q∗
uv (βΨ)|
= τkl
, 2019 19 / 23

Tree structured data
Data dependency structure relies on a tree
Likelihood factorizes on nodes and edges
(Chow and Liu, 1968):
P(Z|T) =
d
j=1
P(Zj )
k,l∈T
ψkl (Z) ,
Where
ψkl (Z) =
P(Zk, Zl )
P(Zk) × P(Zl )
.
Rmq : with standardised gaussian data, ˆΨ = [ ˆψkl ] ∝ (1 − ˆρZ
2
)−1/2
, 2019 20 / 23

Direct EM algorithm ?
Complete likelihood :
P(Y , Z, T) = P(T) × P(Z|T) × P(Y |Z)
log(P(Y , Z, T)) =
k,l
1{(k,l)∈T}(log(βkl ) + log(ψkl (Z))) − log(B)
+
k
(log(P(Zk )) + log(P(Yk |Zk )))
, 2019 21 / 23

Direct EM algorithm ?
Complete likelihood :
P(Y , Z, T) = P(T) × P(Z|T) × P(Y |Z)
log(P(Y , Z, T)) =
k,l
1{(k,l)∈T}(log(βkl ) + log(ψkl (Z))) − log(B)
+
k
(log(P(Zk )) + log(P(Yk |Zk )))
Conditional expectation :
Eθ[log(P(Y , Z, T))|Y ] =
k,l∈V
P((k, l) ∈ T|Y ) log(βkl ) + E[1{(k,l)∈T}log(ψkl (Z)|Y )]
+
k
E[log(P(Zk ))|Y ] + E[log(P(Yk |Zk ))|Y ] − log(B)
, 2019 21 / 23

M step
M step
Goal : optimization of weights βkl .
argmax
βkl



k,l∈V
τkl (log(βkl ) + log(ψkl )) − log(B) +
k
log(P(Zk))



With high combinatorial complexity of B =
T∈T k,l∈T
βkl
How to compute
∂B
∂βkl
?
, 2019 22 / 23

M step
βkl update
A result from Meil˘a Meil˘a and Jordan (2000)
Inverting a minor of the laplacien Q, we deﬁne M :



Muv = [Q∗−1
]uu + [Q∗−1
]vv − 2[Q∗−1
]uv u, v < n
Mnv = Mvn = [Q∗−1
]vv v < n
Mvv = 0.
On peut montrer que :
∂|Q∗
uv (W )|
∂βkl
= Mkl × |Q∗
uv (W )|
∂Eθ[log(P(Z, T))|Z]
∂βkl
=
τkl
βkl
−
1
B
∂B
∂βkl
ˆβh+1
kl =
τh
kl
Mh
kl
, 2019 23 / 23

Reconstruction of ecological interaction networks by Raphaëlle Momal-Leisenring, PhD Candidate in Applied Mathematics @AgroParisTech

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