Estimating graphical models for count data with applications to single-cell gene network
Anonymous Author(s)
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Abstract
Graphical models such as Gaussian graphical models have been widely applied fordirect interaction inference in many different areas.
In many modern applications,such as single-cell RNA sequencing (scRNA-seq) studies, the observed data arecounts and often contain many small counts.
Traditional graphical models forcontinuous data are inappropriate for network inference of count data.
We considerthe Poisson log-normal (PLN) graphical model for count data and the precisionmatrix of the latent normal distribution represents the network.
We propose a twostep method PLNet to estimate the precision matrix.
PLNet ﬁrst estimates the latentcovariance matrix using the maximum marginal likelihood estimator (MMLE) andthen estimates the precision matrix by minimizing the lasso-penalized D-traceloss function.
We establish the convergence rate of the MMLE of the covariancematrix and further establish the convergence rate and the sign consistency of theproposed PLNet estimator of the precision matrix in the high dimensional setting.
Importantly, although the PLN model is not sub-Gaussian, we show that the PLNetestimator is consistent even if the model dimension goes to inﬁnity exponentiallyas the sample size increases.
The performance of PLNet is evaluated and comparedwith available methods using simulation and gene regulatory network analysis ofreal scRNA-seq data.
Introduction
Gaussian graphical model (GGM) [20, 35] is widely used for understanding the complex interactionsof the observed variables in various ﬁelds [33, 19, 30].
GGM assumes that each sample is drawnfrom a multivariate Gaussian distribution, in which the precision matrix (i e., the inverse of thecovariance matrix) represents the network.
In GGM, two nodes are connected if the correspondingelement in the precision matrix is non-zero.
Based on the sparsity assumption of the precisionmatrix, researchers have proposed many methods for estimating the precision matrix of the GGMand established consistency theories, such as methods by maximizing the penalized log-likelihood[35, 8], by solving an equivalent regression problem with the lasso penalty [20, 24] or by minimizinga smooth convex loss function (called D-trace loss) with the lasso penalty [36].
Due to its highinterpretability, well-established theoretical properties, and computational advantages, GGM hasbeen adopted in many applications.
This paper is motivated by the gene regulatory network analysis based on the single-cell RNAsequencing (scRNA-seq) data.
The recent breakthrough of scRNA-seq technologies has providedtremendous opportunities for understanding transcriptional states and activities and for uncoveringgene regulatory networks at single cell level.
However, gene expressions from scRNA-seq are countdata and the Gaussian assumption is inappropriate.
In particular, for the recent unique molecularidentiﬁer (UMI) based scRNA-seq data [7, 38], the expression counts contain many zeros.
Trans- formations such as taking logarithms cannot make the Gaussian model a good approximation to thedistribution of scRNA-seq expression data and may distort the correlation structure when there aremany 0’s in scRNA-seq data.
Poisson distribution is a natural choice for modeling count data.
Researchers have generalized theunivariate Poisson distribution to multivariate distributions for network analysis [12].
Some suchmodels are the Poisson graphical model [2] and its variants generalized Poisson graphical models[32].
But these generalized Poisson graphical models can not handle over-dispersion often observedin real data.
Negative binomial distributions are often used to account for the over-dispersion [25],but it is more difﬁcult to generalize negative bionomial distributions to describe the network structurein the multivarate count data.
Here, we use the multivariate Poisson log-normal (PLN) distributionfor graphical modeling of count data.
The PLN distribution is a hierarchical model of Poissonand multivariate log-normal distributions [1].
A random vector Y =
( Y 1 , . . .
, Y p ) T is from a PLNdistribution, if conditional on a latent vector X =
( X 1 , . . .
, X p ) T with (log( X 1 ) , . .
, log( X p ))
T ∼
N ( µ , Σ) , elements of Y are independent Poisson random variables with mean parameters X 1 , . . .
, X p .
The major advantage of the PLN model is that, similar to the GGM, the precision matrix Θ =Σ − 1 can represent the network.
Network recovery can be achieved by estimating Θ .
In addition,
PLN distributions allow over-dispersion and hence are suitable for modeling count data with overdispersion.
In gene regulatory network analysis of scRNA-seq data, the PLN graphical model hasa clear biological explanation.
Previous experimental researches showed that gene expressions ofsingle cells follow log-normal distributions [4].
The latent variables X 1 , · · · , X p , representing thetrue expressions of p genes in a single cell, can be reasonably assumed to be jointly log-normallydistributed.
The precision matrix of the log-normal distribution represents gene-gene interactionsin single cells.
The observed variables Y are measurements of the underlying true expressions X .
Largely speaking, the log-normal layer of the PLN model captures the biological ﬂuctuation of geneexpressions and the Poisson layer accounts for the technical and measurement noises.
Only thebiological ﬂuctuation reﬂects gene-gene interactions and the regulatory network is the precisionmatrix of the latent log-normal model.
A few algorithms have been developed for estimating the precision matrix of the PLN model in highdimensional settings.
Compared with GGM, the likelihood of the PLN model is more complicatedsince it involves a high-dimensional integration and does not have a close form.
Maximizing thepenalized log-likelihood of the PLN model is more difﬁcult.
Wu et al.
[31] proposed to use Monte Carlo to approximate the log-likelihood and estimate the precision matrix by maximizing the penalizedapproximated log-likelihood of the PLN model.
Chiquet et al.
[5] developed a computational moreappealing method based on the variational approximation.
However, these methods are all based onthe approximation of the log-likelihood and the accuracies of these approximations need to be furtherelaborated.
More importantly, no convergence theory has been developed for these precision matrixestimators.
In this article, we propose a two-step approach named PLNet that ﬁrst estimates the covariance matrixof the latent log-normal variables in the PLN using the maximum marginal likelihood estimator(MMLE) and then estimates the sparse precision matrix by minimizing the lasso penalized D-traceloss [36].
One advantage of this two-step approach is that it avoids computing high-dimensional numerical integration that is involved in the log-likelihood of the multivariate PLN model.
Minimizingthe penalized D-trace loss is computationally cost-effective.
Thus, this estimator is generally computationally more efﬁcient.
Furthermore, we show that, under mild conditions, the estimator given by PLNet is a consistent estimator of the precision matrix in high dimensional settings.
Comprehensivesimulation studies show that PLNet provides more accurate estimates and is computationally moreefﬁcient than available methods.
We also demonstrate the application of PLNet to scRNA-seq data.
Methods2.1 Notation
For a column vector a with the i -th entry a i , (cid:107) a (cid:107)
=
( a T a ) 1 / 2 and (cid:107) a (cid:107)
= (cid:80) i | a i | .
For a matrix Awith the ( i, j ) -th entry A ij , (cid:107) A (cid:107) ∞ =
max i,j |
A ij | be the element-wise l ∞ -norm, (cid:107) A (cid:107)
= (cid:80) i,j |
A ij |be the l 1 -norm, (cid:107) A (cid:107) 1 , ∞ =
max i ( (cid:80) j | A ij | ) be the l 1 , ∞ -norm, (cid:107) A (cid:107) F = (cid:16)(cid:80) i,j | A ij | 2 (cid:17) 1 / 2 be the
Frobenius norm, (cid:107) A (cid:107) 2 = max (cid:107) v (cid:107) 2 =1 (cid:107) Av (cid:107) 2 be the operator norm, (cid:107) A (cid:107) 1 , off = (cid:80) i (cid:54) =
j | A ij | , (cid:107) A (cid:107) 0 = 89 (cid:80) i,j 1 { A ij (cid:54) =0 } be the number of nonzero entries where 1 {·} is the indicator function, tr ( A )
= (cid:80) i
A iibe the trace, and A (cid:23) 0 means A is semi-positive deﬁnite.
2.2 The PLN graphical model
Let Y =
[ Y ij ] 1 ≤ i ≤ n, 1 ≤ j ≤ p be the observed count matrix, Y i
=
( Y i 1 , . . .
, Y ip ) T be the observedcount data of the i th sample, and X i =
( X i 1 , . . .
, X ip ) T be the latent random vector.
In scRNA-seqdata, Y ij and X ij are the observed expression and the underlying "true" expression of the j th genein the i th cell, respectively.
We assume that conditional on X i , elements of Y i are independent
Poisson random variables with mean S
i X ij ( 1 ≤ j ≤ p ), where S i is a scaling factor sampled fromone distribution g ( S ) and we deﬁne S =
( S 1 , ..., S n ) T as the scaling vector.
In scRNA-seq data,
S i corresponds to the library size of the i th cell.
The library size is related to the total sequencingreads and can be estimated by the sum of counts within each cell or other methods [17, 18, 28].
Thelog( X i ) =
(log( X i 1 ) , . . .
, log( X ip ))
T follows a multivariate normal random vector with mean µ ∗and covariance Σ ∗ .
The precision matrix Θ ∗ =
(Σ ∗ ) − 1 represents the network.
In summary, we havethe following graphical model for count data, for 1 ≤ i ≤ n ,
[Equation]
We denote this PLN ditribution by Y i ∼ PLN( S i ; µ ∗ , Σ ∗ ) .
The above PLN model is a little differentfrom the classical form [1], in which S i = 1 for 1 ≤ i ≤ n .
We assume that the network Θ ∗ is sparseand therefore could use the penalized log-likelihood to estimate Θ ∗ .
However, the likelihood functionin the PLN model involves a p -dimensional integration and is difﬁcult to compute, especially when pis large.
Chiquet et al.
developed a variational algorithm called VPLN to maximize the penalizedlog-likelihood [5].
Although the variational method is computationally more feasible than directlymaximizing the penalized log-likelihood, the estimator’s theoretical properties are difﬁcult to beobtained.
We develop a two-step estimator called PLNet that is computationally efﬁcient and has good theoretical properties.
We ﬁrst use the MMLE method to derive an estimator (cid:98) Σ for Σ ∗ .
Then, we apply the
D-trace method [36] with the covariance estimator (cid:98) Σ to estimate the sparse precision matrix Θ ∗ .
Weshow that the derived estimator (cid:98) Θ is a consistent estimator of Θ ∗ even when the model dimensiongoes to inﬁnity exponentially as the sample size increases.
2.3 The maximum marginal likelihood estimator
We estimate Σ ∗ =
[ σ ∗ jk ] 1 ≤ j,k ≤ p using the marginal log-likelihood to avoid high dimensional integrations.
Let Y · j be the j th column of Y .
For 1 ≤ j ≤ p , we estimate σ ∗ jj by maximizing themarginal log-likelihood of Y · j and denote the estimator as ˜ σ jj . For 1 ≤ j (cid:54) = k ≤ p , we estimate σ ∗ jkby maximizing the marginal log-likelihood of ( Y · j , Y · k ) and denote the estimator as ˜ σ jk .
The speciﬁcforms of these marginal log-likelihoods and optimization problems are shown in the Supplemental Material (S2.1).
Further, we deﬁne the MMLE for Σ ∗ as (cid:101) Σ =
[˜ σ jk ] 1 ≤ j,k ≤ p .
To solve the optimizationproblem for maximizing the marginal log-likelihood, we apply the Newton-Raphson algorithm withthe moment estimator (cid:101) Σ m in the Supplemental Material (S1.1) as the initial value.
The explicitforms of the ﬁrst and second order partial derivatives of the marginal log-likelihood function areshown in the Supplemental Material (S2.1).
The integrations involved in optimizations can be wellapproximated by the method described in [1, 22].
The above MMLE (cid:101) Σ may be not semi-positive deﬁnite.
However, the D-trace method requires theinput covariance matrix estimator to be semi-positive deﬁnite to guarantee the convexity of the lossfunction.
We project (cid:101) Σ to the space of semi-positive deﬁnite matrices and identify ˇΣ that is closest to(cid:101) Σ in the space, i.e.,
[Equation]
Then we add a small positive deﬁnite matrix to ˇΣ and get a positive-deﬁnite estimator (cid:98) Σ of Σ ∗ ,
[Equation]
where I is the identity matrix.
The optimization problem (2) for ˇΣ can be solved by a splitting conicsolver [9].
With the covariance matrix estimator (cid:98) Σ , we apply the D-trace method to estimate the precision matrix,
[Equation]
The above optimization problem (4) can be solved by an alternating direction method of multipliers[36].
In this paper, we use a more efﬁcient algorithm developed in [29] to calculate (cid:98) Θ .
The tuningparameter λ n is selected by minimizing the following approximate Bayesian information criterion(BIC)
[Equation]
Theoretical properties
In this section, we establish the theoretical properties in the high dimensional setting.
We ﬁrstprove that (cid:98) Σ is a consistent estimator of Σ ∗ .
The convergence rate for (cid:98) Σ is similar to that of thesample covariance matrix for random variables with sub-Gaussian distribution.
Then, under the sameirrepresentability condition of the D-trace method in [36], we derive the edge recovery property andconsistency for the PLNet estimator (cid:98) Θ .
3.1 Notation
Let G =
{ ( i, j ) | Θ ∗ ij (cid:54) = 0 } be positions of non-zero elements in Θ ∗ , G c be the complement set of
G , d be the maximum node degree in Θ ∗ , s = (cid:107) Θ ∗ (cid:107) 0 , and θ
min = min
( i,j ) ∈ G (cid:12)(cid:12)
Θ ∗ ij (cid:12)(cid:12) be the minimal absolute value of nonzero elements of Θ ∗ .
Let λ max ( A ) and λ min ( A ) be the largest and smallest eigenvalues of a symmetric matrix A .
We deﬁne Γ ∗ = Γ(Σ ∗ ) =
(Σ ∗ ⊗ I + I ⊗ Σ ∗ ) / 2 ,
(cid:98) Γ =
Γ (cid:16)(cid:98) Σ (cid:17) , where A ⊗ B is the Kronecker product.
For a p × p matrix A , the row ( i − 1) p + j and column ( k − 1) p + l of Γ ( A ) is Γ ( A ) ( i,j ) , ( k,l ) =
(cid:0)
A ik 1 { j = l } + A jl 1 { i = k } (cid:1) / 2 .
For two subsets T 1 and Tof { ( i, j ) | 1 ≤ i, j ≤ p } , we deﬁne Γ ( A ) T 1 ,T 2 be the submatrix of Γ ( A ) whose rows and columns indexed by T 1 and T 2 , respectively.
Other notations are as follows,
[Equation]
3.2 The irrepresentability condition and rate of convergence
We ﬁrst present the necessary irrepresentability condition for establishing the rate of convergence forthe estimator in PLNet.
This irrepresentability condition is from the D-trace method in [36].
Condition 1 (Irrepresentability condition)
There exists 0 < α < 1 , such that
[Equation]
The irrepresentability condition 1 is equivalent to γ ≥ α > 0 .
We also need a boundedness conditionin the PLN model (1).
Condition 2 (Boundedness condition)
There exist positive constants M 1 , M 2 , M 3 , M 4 , and M 5 ,such that the distribution of S i has bounded support [ M 1 , M 2 ] , max 1 ≤ j,k ≤ p (cid:110)(cid:12)(cid:12) µ ∗ j (cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) σ ∗ jk (cid:12)(cid:12)(cid:12)(cid:111) ≤ M 3and M 4 ≤ λ min (Σ ∗ ) ≤ λ max (Σ ∗ ) ≤ M 5 .
Based on the boundedness condition 2, we can establish the convergence rate for the covariancematrix estimator (cid:98) Σ .
Theorem 1 (Rate of convergence for (cid:98) Σ ) Under Condition 2, there exist positive constants A, B , such that for any (cid:15) > 0 ,
[Equation]
After plugging (cid:98) Σ into the lasso penalized D-trace loss, we get a consistent estimator (cid:98) Θ that convergesto Θ ∗ in several matrix norms.
Theorem 2 (Rate of convergence for (cid:98) Θ ) Under the irrepresentability condition 1 and the boundedness condition 2, there exist constants A, B , such that for some η > 2 , ifn >
[Equation]
and λ n = 12 γ − [Equation] then with probability 1 − p 2 − η ,
[Equation]
[Table]
Meanwhile, with a high probability, the sign of the sparse precision matrix Θ ∗ can be recovered by(cid:98) Θ .
We have the following theorem about the sign consistency of (cid:98) Θ .
Theorem 3 (Sign consistency for (cid:98) Θ ) Under the conditions in Theorem 2, for some η > 2 , choosingthe same n and λ n in Theorem 2, then with probability 1 − p 2 − η , (cid:98) Θ recovers all zeros and nonzerosin Θ ∗ .
The rate-of-convergence and sign consistency results in Theorem 2 and 3 are closely related to thetail probability of the MMLE estimator (cid:98) Σ in Theorem 1.
The boundedness condition 2 is assumed toguarantee the convergency of the MMLE.
Ignoring the complicated constants in theorems, for anyη > 2 , if we have n > Cη log p , or in other words, if p tends to inﬁnity not faster than exponentialof the sample size n , (cid:98) Θ is a consistent estimator of Θ ∗ .
Especially, the rate of convergence for (cid:98) Θ is
O (cid:16) [ η (log p ) /n ] 1 / 2 (cid:17) under l ∞ -norm.
Simulation studies
4.1 Simulation settings
We conduct simulations to evaluate the performance of PLNet and compare with the available networkinference methods including VPLN [5], glasso [8], and an estimator called PLNet-MOM by pluggingthe moment estimator (cid:98) Σ m into D-trace loss (See Supplemental Material (S1.1)).
Both PLNet and
VPLN are designed to estimate the precision matrix for count data in the PLN model.
The glassoalgorithm is a classical approach for continuous data in GGM and we apply glasso to the logarithmictransformation of the normalized data, which is deﬁned as ˜ Y ij =
( Y ij + 1) /
ˆ S i , 1 ≤ i ≤ n , where ˆ S iis the estimated library size of i th cell.
In all simulations, we estimate the library size by the total sum scaling (i.e. ˆ S i =
(cid:80) pj =1 Y ij ), which is a classical normalization method for scRNA-seq and isdeﬁned as the sum of counts within each cell.
We consider 48 different simulation scenarios, which are 2 sample size setups ( n = 500 , 2000 ) × 3dimension setups ( p = 100 , 300 , 500 ) × 2 dropout levels (low and high) × 4 graph structures andsimulate count data from the PLN model, where the dropout level represents the proportion of zerosin count matrix.
Details of the simulation settings are shown in the Supplementary Material (S3.1).
In each scenario, we independently repeat simulations 100 times.
The four graph structures are asfollows.
Banded Graph: Pairs ( i, j ) of nodes are connected if | i − j | ≤ 2 , i (cid:54) =
j .
All nonzero edges are set as 0 .
Random Graph: Pairs of nodes are connected with probability 0 .
The nonzero edges areset as 0 . 3 with probability 0 .
8 and as − 0 .
3 with probability 0 .
Scale-free Graph: The Barabasi-Albert model [3] is used to generate a scale-free graph withpower 1 .
The nonzero edges are set as 0 . 3 .
Blocked Graph: The nodes are divided into 5 blocks of equal sizes.
Pairs of nodes in the
same block are connected with probability 0 .
and the nonzero edges are set as 0 . 3 .
Differentblocks have no edge connection.
The diagonal elements of precision matrices are set as 1 plus a positive number to guarantee positivedeﬁniteness.
4.2 Performance comparison
Table 1 and Table 2 show the area under precision and recall curve (AUPR) of each estimator.
AUPRsare calculated by varying the tuning parameters (i.e. the penalty parameters of the four estimators).
As expected, the AUPR decreases as the number of genes increases.
AUPRs in the high-dropout casesare generally smaller than that in the low-dropout cases.
PLNet is the most robust estimation amongthese four estimators and outperforms PLNet-MOM, VPLN, and glasso in most simulation settingsin AUPR, especially for the settings with high dropouts.
For example, for n = 2000 , p = 100 , PLNetachieves an AUPR of 0.88 for the random graph under the scenario of the high dropout rate, while
PLNet-MOM, VPLN and glasso only have AUPRs of 0.82, 0.69, and 0.18, respectively.
Furthermore,
PLNet-MOM outperforms glasso in most settings and outperforms VPLN under the scenario of largesample size.
The results of the area under the receiver operating characteristic curve (AUC) aresimilar and shown in Supplementary Table S1-S2.
To further demonstrate the performance of PLNet, we visualize the mean networks predicted by thefour methods tunned by BIC criterion for the banded graph with n = 2000 , p = 100 over the 100simulations (Figure 1).
More speciﬁcally, we calculate the relative frequency F ij that an algorithmreports edges over the 100 simulations.
For positions ( i, j ) with Θ ij (cid:54) = 0 , F ij is the proportion that analgorithm correctly recovers the edge in the 100 simulations, while for positions ( i, j ) with Θ ij = 0 ,
F ij is the the proportion that an algorithm falsely predicts edges between nodes i and j in the 100simulations.
We plot the relative frequency matrices of the four methods in Figure 1.
We clearly seethat PLNet is able to detect more true positives while having fewer false positives than other methods,especially for the high dropout case.
Supplementary Table S3-S4 show the computational time of the four methods.
PLNet is computationally more efﬁcient than VPLN in most simulation settings.
Also, the computational efﬁciencyof PLNet is roughly the same for different sample size, while computational complexity of VPLNmaintains a linear relationship with sample size.
The detailed comparsions between computationalefﬁciency of the four methods are shown in the Supplementary Material (S3.2.2).
Application to a scRNA-seq dataset
We apply PLNet and VPLN to infer the gene regulatory network of CD14+ Monocytes proﬁled in
Kang et al.
The single cells are proﬁled in two different conditions, IFN- β -treated and control.
IFN- β is a cytokine in the interferon family that inﬂuences the transcriptional proﬁles for many genes,especially that in the JAK/STAT pathway [21].
We focus on the IFN- β -treated cells (2147 cells) anduse the top 200 highly variable genes that are used in [27] for network analysis. [Table] [Table]
[Figure]
We compare networks of PLNet and VPLN with parameters tuned such that the network densities arearound 5%.
Gene Ontology (GO) analysis [15] shows that the 200 genes mainly involve in 4 biologicalprocesses, including "Cytokine-mediated signaling pathway" (Module M 1 ), "Neutrophil mediatedimmunity" (Module M 2 ), "Cellular protein metabolic process" (Module M 3 ), and "Proteolysis"(Module M 4 ).
Figure 2 shows the predicted networks of the genes in the 4 modules by PLNet and
VPLN, where the colors represent the partial correlations between genes.
The partial correlationgiven by PLNet between genes i and j is deﬁned as − (cid:98) Θ ij / ( (cid:98) Θ ii (cid:98) Θ ij ) 1 / 2 .
The partial correlationgiven by VPLN is deﬁned similarly.
We clearly see that the network given by PLNet tends tohave more connections within the modules than VPLN.
To see this more clearly, for each module
M k , we calculate the ratio between within-module and between-module connections R ( M k ) =
Σ i,j ∈ M k W ij / Σ i ∈ M k , j/ ∈ M k W ij , where the weights W ij are set as the absolute partial correlation 244 between genes i and j (weighted within-between connection ratio) or are set as 1 and 0 dependingon whether genes i and j are connected (unweighted within-between connection ratio).
The withinbetween connection ratios of PLNet are much larger than that of VPLN in most cases (Table 3).
Similar results also for networks of other different densities that are shown in the Supplementary
Material (S4.1).
[Table]
Discussion
In this paper, we consider the PLN graphical model for count data.
This model has an intuitiveexplanation for single-cell gene regulatory network analysis.
The Poisson layer is to capture thetechnical noises and the log-normal layer is to model the biological ﬂuctuations in single cells.
Generegulatory network is represented by the precision matrix of the latent log-normal model.
To estimatethe precision matrix, we propose a two-step estimator PLNet, using the MMLE to estimate thecovariance matrix and then minimizing the penalized D-trace loss to estimate the precision matrix.
The simplicity of this estimation procedure allows us to establish the consistency theory for theproposed PLNet estimator for the high dimensional setting.
The numerical analysis also shows thatthe PLNet method outperforms available methods.
The proposed method can be generalized in several ways.
Firstly, a straightforward generalizationis the differential network analysis.
We could use the same method to estimate covariance matricesin two different conditions and use the D-trace loss developed in earlier work [34] for differentialnetwork analysis in single cells.
Secondly, the mean parameters of the latent log-normal randomvariable can depend on covariates.
In such model, we could ﬁrst use regression to estimate themean parameters and derive the corresponding moment estimator of the covariance matrix.
Anothergeneralization is gene regulatory network analysis of mixtures of cell populations.
Different cellpopulations may have different gene regulatory networks and we could jointly model the mixture andinfer the gene regulatory networks for all cell populations. Checklist For all authors...
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