Recursive regularization for inferring gene networks from time-course gene expression profiles
- Teppei Shimamura^{1}Email author,
- Seiya Imoto^{1},
- Rui Yamaguchi^{1},
- André Fujita^{1},
- Masao Nagasaki^{1} and
- Satoru Miyano^{1}
DOI: 10.1186/1752-0509-3-41
© Shimamura et al; licensee BioMed Central Ltd. 2009
Received: 08 December 2008
Accepted: 22 April 2009
Published: 22 April 2009
Abstract
Background
Inferring gene networks from time-course microarray experiments with vector autoregressive (VAR) model is the process of identifying functional associations between genes through multivariate time series. This problem can be cast as a variable selection problem in Statistics. One of the promising methods for variable selection is the elastic net proposed by Zou and Hastie (2005). However, VAR modeling with the elastic net succeeds in increasing the number of true positives while it also results in increasing the number of false positives.
Results
By incorporating relative importance of the VAR coefficients into the elastic net, we propose a new class of regularization, called recursive elastic net, to increase the capability of the elastic net and estimate gene networks based on the VAR model. The recursive elastic net can reduce the number of false positives gradually by updating the importance. Numerical simulations and comparisons demonstrate that the proposed method succeeds in reducing the number of false positives drastically while keeping the high number of true positives in the network inference and achieves two or more times higher true discovery rate (the proportion of true positives among the selected edges) than the competing methods even when the number of time points is small. We also compared our method with various reverse-engineering algorithms on experimental data of MCF-7 breast cancer cells stimulated with two ErbB ligands, EGF and HRG.
Conclusion
The recursive elastic net is a powerful tool for inferring gene networks from time-course gene expression profiles.
Background
The inference of gene networks from time-course microarray data can be defined as the process of identifying functional interactions between genes over time. Typically, a gene network is represented by a directed or undirected graph where nodes indicate genes encoded in a given organism of interest and edges represent various functional properties. The elucidation of gene networks has been expected to having an essential role for better understanding of molecular mechanisms and can be useful in the identification of new drug targets [1–5].
- 1.
High computational cost for model selection: When the number of variables is m, there are m × 2^{ m }candidate models in model selection. The best-subset selection is computationally prohibitive when the number of variables is large.
- 2.
High correlation between variables: When the number of variables is much larger than the number of samples, two or more variables tend to be highly correlated [11]. In this situation, the coefficient estimates of the subset selection or the lasso may change erratically in response to small changes in the observed data, and thus the resulting models have poor performances [12, 13]. What is worse that these methods tend to select only one variable from the highly correlated variables [13] which can lead to reducing the number of true positives in gene network inference.
One solution for the above problems is to use a regularization method called elastic net [13] which minimizes a penalized loss function with l_{1}- and l_{2}-penalties of the coefficients. Applying an l_{1}-penalty regularizes the least squares fit and shrinks some coefficients exactly to zero, i.e., achieves automatic variable selection, as the lasso does. Adding of an l_{2}-penalty to an l_{1}-penalty encourages a grouping effect so that highly correlated variables will be in or out of the model together. The elastic net is also capable of selecting a set of relevant variables with low computational effort even when the number of variables is much larger than the number of observations with LARS-EN algorithm [13]. However, although VAR modeling with the elastic net succeeds in increasing the number of true positives, it also results in increasing the number of false positives. This is because the elastic net shrinks the same amount of the l_{1}-penalty to each coefficient without assessing their relative importance.
To increase the capability of the elastic net, i.e., to decrease the number of false positives while keeping the high number of true positives in inferring gene networks based on VAR model, we propose a new class of regularization, called recursive elastic net, by incorporating relative importance of the coefficients. The recursive elastic net is a kind of iterative procedures that can reduce the number of false positives gradually by updating the relative importance of each VAR coefficient. The performance of variable selection strongly depends on the regularization parameters of the l_{1}- and l_{2}-penalty terms in the recursive elastic net. For selecting these regularization parameters automatically, we derive a modified Bayesian information criterion and a modified bias-corrected Akaike information criterion.
Methods
Vector Autoregressive Model
where c is a constant vector, B= [B_{ ij }]_{1≤i, j≤m}is an m × m autoregressive coefficient matrix, ϵ_{ t }is assumed to follow an m-dimensional multivariate normal distribution N(0, Σ), and the notation denotes the transpose of the vector y_{ t }. Throughout the paper, we assume that Σ is a diagonal matrix, that is, . Note that the coefficient B_{ ij }measures the influence that node i exerts on node j and the nonzero B_{ ij }provides a functional connectivity which is related to Granger causality [14]. The Granger causality is a concept widely used in econometrics to analyze the relationships between variables and is based on the intuitive idea that a cause always occurs before its effect. Thus, we can describe a gene network by a directed graph based on the coefficient matrix B where nodes represent genes and edges show functional connectivities. In the estimated graph, node i is linked to node j by the edge i → j if and only if B_{ ij }≠ 0.
Here, we center each input variable so that there is no intercept in (2) and scale each input variable so that the observed standard deviation is equal to one.
The major objective of our analysis is to estimate the coefficient matrix B from observations. Especially, our interest is its structures, that is, we want to estimate which elements of each column of B are zero or nonzero. Let denote the set of indices corresponding the nonzero components of by . The graph inference problem can be formally stated as the problem of predicting from the observations. This formulation is equivalent to m variable selection problems in linear regression where each component in y_{ t }is the response and y_{t-1}is the covariates.
Recursive Elastic Net
From this section, we focus on a variable selection problem in linear regression model with response y_{t, j}and covariates y_{t-1,1}, ..., y_{t-1, m}. For notational simplicity, we omit the suffix j of z_{ j }, β_{ j }, λ_{1j}and λ_{2j}, and use the notation z, β, λ_{1} and λ_{2}.
However, the elastic net suffers from too many false positives in the variable selection problem for extremely high-dimensional data, which will be illustrated in Simulation Section. This is due to the same amount of shrinkage for all the coefficients in the l_{1}-penalty. That is, the same regularization parameter λ_{1} is used for each coefficient without assessing their relative importance. In a typical setting of linear regression, it has been shown that such an excessive l_{1}-penalty can affect the consistency of model selection [15–18].
The modification of the l_{1}-penalty was first proposed by Zou [18]. This type of l_{1}-regularization was applied to parameter estimation for several statistical models, such as Cox's proportional hazard model [19], least absolute deviation regression model [20], and graphical Gaussian model [21].
where L is some specified large value. If w_{ k }is small, tends to be nonzero. While tends to be zero if w_{ k }is large. By using the weight (9) and appropriate values of the regularization parameters, we can choose the true set of variables, i.e., . However, in many situations, one does not know true coefficients, and thus we need some plug-in estimator instead of the true coefficients. In this article, we propose a new multi-step regularization, called recursive elastic net. The proposed method tries to obtain a better estimator than the elastic net estimator by solving the weighted elastic net problem and updating the weights alternatively. The recursive elastic net can be described as:
Recursive Elastic Net
We call the iterative procedure using the elastic net estimator (5) as the initial variable weights and updating the coefficients by the bias-corrected weighted elastic net (11) corrected recursive elastic net. Note that our solution with the iterative procedure is a local optimum when initializing some variable weights and we cannot guarantee the convergence of a global minimum. Thus, it is important to choose a suitable starting point for the variable weights. In the recursive elastic net and the corrected recursive elastic net, we propose to initialize with the naive elastic net and the elastic net estimators, (4) and (5), that is, the unweighted solutions of (10) and (11), respectively. In the recursive elastic net and the corrected recursive elastic net, the regularization parameters determine whether or not the l-th estimator is better than the (l - 1)-th estimator. The selection problem of the regularization parameters will be discussed in the latter section.
Grouping Effect
In this section, we show that the recursive elastic net estimator can lead to a desirable grouping effect for correlated variables. In the framework of linear regression, the grouping effect can be considered as an effect that the coefficients of highly correlated variables should be similar each other. As discussed in Zou and Hastie [13], the lasso does not have the grouping effect and tends to select one among a correlated set. The following Lemma which is quoted from Zou and Hastie [13] guarantees the grouping effect for the recursive elastic net in the situation with identical variables, since the loss function of the recursive elastic net is strictly convex when λ_{2} > 0. The Lemma is given by:
Lemma 1 Assume that x_{ i }= x_{ j }for i, j ∈ {1, ..., m} and is estimated by (10), then for any λ_{2} > 0.
Furthermore, the following theorem provides an upper bound on the difference of the i-th and j-th coefficients for the l-th iteration with the proposed method:
where and is the sample correlation.
The proof of Theorem 1 is given by Appendix. We notice that the grouping effect of the recursive elastic net contributes not only the sample correlation ρ but also the difference between the i-th and j-th coefficients of the (l - 1)-th iterations . If x_{ i }and x_{ j }are highly correlated, i.e., ρ ≐ 1, and the difference is almost zero, then Theorem 1 says that the difference is also almost zero.
Computational Algorithm
In this section, we provide an algorithm for solving the weighted elastic net problem in (6) by modifying the LARS-EN algorithm [13]. The modified algorithm called LARS-WEN can be described as:
Algorithm: LARS-WEN
- 1.
- 2.
- 3.
For a fixed λ_{2}, the LARS-WEN algorithm sequentially updates the locations of the nonzero coefficients and its values, and produces the entire solution path of the weighted elastic net estimator . The R source code is available from the authors upon request. The computational cost of the LARS-WEN algorithm is the same as that of the LARS-EN algorithm except for the treatment of Step 1 and Step 3. If the algorithm for calculating is stopped after u steps, it requires O(u^{3} + mu^{2}) operations.
Selection of Regularization Parameters
Note that does not include λ_{1} and depends on and λ_{2}. Let and be the coefficient vector and the active set of the LARS-WEN step α, respectively, and λ_{α,1}be the corresponding l_{1}-regularization parameter of the step α. Hence, we can use in place of λ_{α,1}as a tuning parameter.
For a given λ_{2}, we choose the optimal step that gives the smallest BIC or the smallest AICc from the candidate models.
VAR Modeling with Recursive Elastic Net
As shown in the previous section, the inference of nonzero components in the VAR model can be formalized as m variable selection problems, i.e., our objective is to estimate . Therefore, we can apply the recursive elastic net directly to estimate the coefficient matrix B in the VAR model. In this section, we provide an algorithm for estimating B in the VAR model with the recursive elastic net and the derived model selection criteria for a fixed λ_{2}. The algorithm is as follows:
Algorithm:RENET-VAR
- 1.
Set a maximum number of iterations to be M and a maximum step size to be u.
- 2.
- 3.For j = 1, ..., m
- (a)
where- (b)
Calculate the values of model selection criterion for u candidate models, say
- (a)
- 4.
Update l = l + 1 and repeat Step 3 until a stopping criterion is satisfied or the number of iterations attains M.
The algorithm stops if SMC^{(l)}(λ_{2}) - SMC^{(l - 1)}(λ_{2}) > 0 (l = 2, 3, ..., M) holds.
We note that our iterative algorithm guarantees the decrease of the sum of the values of model selection criterion and searches the model minimizing (19), although it cannot guarantee the decreasing of the loss function.
Results and Discussion
We tested the performance of the proposed method on both simulated and real gene expression time-course data. Using simulated data, we showed the differences between the proposed method and other l_{1}-regularization approaches in the variable selection problem for linear regression model and evaluated how well the proposed method estimates the coefficient matrix of the VAR model compared with other estimation methods. Using real data, we also compared the proposed method with various reverse-engineering algorithms for inferring gene networks.
Simulation Results
Simulation 1
Results of Simulation 1
Method | TP | FP | TN | FN | TDR | SE |
---|---|---|---|---|---|---|
LA | 4.60 | 25.38 | 969.62 | 0.40 | 0.22 | 0.92 |
NEN | 4.38 | 11.90 | 983.10 | 0.62 | 0.30 | 0.88 |
EN | 4.22 | 7.89 | 987.11 | 0.78 | 0.40 | 0.84 |
REN | 4.07 | 1.01 | 993.99 | 0.93 | 0.84 | 0.81 |
CREN | 3.78 | 1.05 | 993.95 | 1.22 | 0.84 | 0.76 |
The coefficients of the 0-th iteration was equivalent to the naive elastic net estimators where 19 coefficients were identified as nonzero including the 5 true components (true positives) and 14 noisy components (false positives). In the first iteration, among the 19 components selected in the naive elastic net, 7 coefficients including the 5 true components were estimated as nonzero by the recursive elastic net. That is, we succeeded in reducing the number of false positives from 14 to 4 while keeping all the 5 true positives. After 5 iterations, the stopping criterion was fulfilled and the recursive elastic net yielded 5 true positives and only 1 false positive. This reduction of the number of false positives in the linear regression has much effect on the performance of the VAR model which is equivalent to m linear regressions. The performances of the proposed methods in the VAR model will be illustrated in the next section.
Simulation 2
For the second simulation, we used the VAR model with the structure of scale-free networks. We considered five models with (n, m) = (20, 100), (20, 200), (20, 500), (20, 1000) and (20, 2000) where n was the number of time points and m was the number of variables. The topologies of the scale-free networks were generated according to the Barabasi-Albert model [26] by using the R function barabasi.game in the R package igraph [27]. The parameter power was set to 1.2 and the other parameters were set to their default values, and then the numbers of edges were chosen to be 108, 213, 525, 1045 and 2079 for m = 100, 200, 500, 1000 and 2000, respectively. The structures of the simulated networks are illustrated in Additional File 1. The dataset from the VAR model with a given structure was drawn as follows:
- 1.
The nonzero entities for the true VAR coefficients B* were generated from a uniform distribution on the interval [-0.8, -0.4] and [0.4, 0.8].
- 2.
The initial value y_{0} was drawn from a uniform distribution on the interval [10, 100].
- 3.
The data at time points t ∈ {1, 2, ..., 20} was generated recursively according to the VAR model with identity random noise matrix Σ = I.
We simulated 100 different datasets from the above procedure for each combination of (n, m). We compared the recursive elastic net (REN) and the corrected recursive elastic net (CREN) with the lasso (LA), the naive elastic net (NEN), the elastic net (EN), and the Jame-Stein shrinkage (JS) [7] on simulated datasets. Opgen-Rhein and Strimmer [7] proposed an improved estimator of the VAR coefficients by using a James-Stein-type shrinkage approach. All the algorithms were run out on each drawn dataset {y_{1}, ..., y_{20}}. The setting of the l_{2}-regularization parameters was same as given in Simulation 1. The regularization parameters of LA, NEN, EN, REN and CREN were selected by a model selection criterion, BIC or AICc. JS is a test-based approach and a cut-off value for the local false discovery was set to 0.2 (JS-A), which was used in Opgen-Rhein and Strimmer [7]. We also used a cut-off value such that the number of significant edges detected by JS was as same as that of REN with AICc (JS-B).
Results of Simulation 2
number of time points × number of variables | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
20 × 100 | 20 × 200 | 20 × 500 | 20 × 1000 | 20 × 2000 | |||||||
Method | MC | TDR | SE | TDR | SE | TDR | SE | TDR | SE | TDR | SE |
LA | BIC | 0.06 | 0.77 | 0.05 | 0.72 | 0.05 | 0.69 | 0.05 | 0.67 | 0.05 | 0.65 |
LA | AICc | 0.08 | 0.78 | 0.07 | 0.76 | 0.07 | 0.73 | 0.06 | 0.71 | 0.06 | 0.69 |
NEN | BIC | 0.08 | 0.93 | 0.07 | 0.94 | 0.06 | 0.91 | 0.06 | 0.88 | 0.06 | 0.84 |
NEN | AICc | 0.12 | 0.93 | 0.10 | 0.93 | 0.08 | 0.91 | 0.08 | 0.87 | 0.07 | 0.83 |
EN | BIC | 0.25 | 0.91 | 0.19 | 0.91 | 0.15 | 0.89 | 0.12 | 0.85 | 0.11 | 0.82 |
EN | AICc | 0.25 | 0.91 | 0.20 | 0.91 | 0.15 | 0.89 | 0.13 | 0.85 | 0.11 | 0.82 |
REN | BIC | 0.74 | 0.73 | 0.71 | 0.70 | 0.68 | 0.68 | 0.65 | 0.65 | 0.61 | 0.61 |
REN | AICc | 0.77 | 0.75 | 0.73 | 0.72 | 0.70 | 0.69 | 0.67 | 0.66 | 0.63 | 0.62 |
CREN | BIC | 0.54 | 0.82 | 0.44 | 0.81 | 0.33 | 0.81 | 0.27 | 0.78 | 0.22 | 0.76 |
CREN | AICc | 0.56 | 0.82 | 0.45 | 0.81 | 0.34 | 0.81 | 0.28 | 0.78 | 0.23 | 0.76 |
JS-A | - | 0.09 | 0.79 | 0.05 | 0.74 | 0.02 | 0.75 | 0.01 | 0.74 | 4.7 × 10^{-2} | 0.74 |
JS-B | - | 0.29 | 0.29 | 0.19 | 0.19 | 0.11 | 0.10 | 0.05 | 0.05 | 0.03 | 0.03 |
- 1.
The sensitivity and true discovery rate of each algorithm were gradually decreasing as increasing the number of variables.
- 2.
NEN and EN had higher sensitivities than the other methods, but their true discovery rates were much lower than those of REN and CREN.
- 3.
The sensitivity of LA was slightly higher than that of REN, while the true discovery rate of REN was over ten times higher than that of LA.
- 4.
Compared with the same significant edges of REN and JS-B, REN outperformed JS-B in both of sensitivity and true discovery rate.
- 5.
CREN succeeded in higher sensitivity than REN, but it suffered from its low true discovery rate compared with REN when the number of variables was large.
- 6.
REN had the highest true discovery rate between all the algorithms in all cases while its sensitivity was almost same as that of LA, JS-A and CREN.
- 7.
REN with AICc was slightly better than REN with BIC.
We also compared our proposed method with other competing methods on simulated dataset by using the VAR model with hierarchical structures. These results are given in Additional File 2.
Experimental Results
In this section we used experimental data of MCF-7 breast cancer cells stimulated with two ErbB ligands, epidermal growth factor (EGF) and heregulin (HRG), to compare the proposed method with various reverse-engineering algorithms. This dataset is available from Gene Expression Omnibus (GEO, Accession Number GSE6462) [28]. In GSE6462, cells were stimulated with 0.1, 0.5, 1, or 10 nM of either EGF or HRG and the expression values were measured at eight time points (0, 5, 10, 15, 30, 45, 60, and 90 minutes) by Affymetrix GeneChip U133A-2. The expressions with 10 nM of EGF at 60 minutes were not observed, and thus its values were estimated with linear interpolation. These microarray datasets were also normalized by faster cyclic loess [29].
In addition to the lasso, the naive elastic net, the elastic net, and the James-Stein shrinkage which were used in Simulation Section, we considered MRNET [30], CLR [2], and ARACNE [31] as competing methods. These methods are available from the R package minet [32]. Note that they do not intend to analyze time-course data. To increase their capabilities, we considered a lagged version of a time series matrix, denoted by D, whose (i, j)-element indicates D_{i, j}= y_{i+1, j}- y_{i, j}and used it as an input of MRNET, CLR and ARACNE. This modification enables us to extract gene-gene interactions between time steps i + 1 and i. To distinguish between the original version and the modified version, we refer the original MRNET, CLR and ARACNE as MRN, CLR and ARA, and the modified MRNET, CLR and ARACNE as MRN-L, CLR-L and ARA-L, respectively. In our analysis, we assumed that time points were equidistant and that the structure of gene networks did not change with different dosages of EGF or HRG. Thus, the datasets from different dose conditions of EGF or HRG were considered as replicated observations. We validated the performances of these reverse-engineering algorithms in terms of how to identify pairwise gene-gene interaction edges by using TRANSPATH database [33]. Although we do not have perfect knowledge of true network, we used biological interactions obtained from TRANSPATH database as true edges. We started by focusing on 254 probes by Nagashima et al. [34]. Among these probes, 233 probes were mapped to unique Entrez Gene IDs, 7 probes were mapped to more than one Entrez Gene IDs, and 14 probes were not mapped, respectively. We only used 233 probes with 228 unique Entrez Gene IDs. To correspond each Entrez Gene ID to unique probe, we selected a probe for each Entrez Gene ID with the largest mean value of expressions through all the experiments. Among these 228 Entrez Gene IDs, TRANSPATH database identified a biological pathway which includes 110 Entrez Gene IDs. While the other 128 Entrez Gene IDs do not connect each other in TRANSPATH database. Thus, all the algorithms were run on time-course expression profiles of the 110 Entrez Gene IDs for EGF-stimulation and HRG-stimulation, respectively.
Results of the applications of network inference algorithms on the experimental datasets
Method | EGF | HRG | ||||
---|---|---|---|---|---|---|
TP | TDR | p-value | TP | TDR | p-value | |
RAND | 8.73 | 0.05 | - | 8.29 | 0.05 | |
MRN | 8 | 0.05 | 0.65 | 8 | 0.05 | 0.59 |
CLR | 4 | 0.03 | 0.98 | 7 | 0.05 | 0.73 |
ARA | 9 | 0.06 | 0.51 | 7 | 0.05 | 0.73 |
MRN-L | 12 | 0.08 | 0.16 | 10 | 0.07 | 0.32 |
CLR-L | 6 | 0.04 | 0.87 | 9 | 0.06 | 0.45 |
ARA-L | 6 | 0.04 | 0.87 | 9 | 0.06 | 0.45 |
JS | 6 | 0.04 | 0.87 | 8 | 0.05 | 0.59 |
LA | 9 | 0.06 | 0.51 | 7 | 0.05 | 0.73 |
NEN | 10 | 0.06 | 0.38 | 9 | 0.06 | 0.45 |
EN | 8 | 0.05 | 0.65 | 17 | 0.11 | 4.0 × 10^{ -3 } |
REN | 16 | 0.10 | 0.01 | 19 | 0.13 | 6.7 × 10^{ -4 } |
CREN | 11 | 0.07 | 0.25 | 16 | 0.11 | 8.8 × 10^{ -3 } |
Next we applied our method to time-course expression profiles of the 228 Entrez Gene IDs stimulated with EGF and HRG, respectively, in order to investigate the difference between the EGF- and HRG-induced gene networks. In the ErbB signaling network, two different ErbB receptor ligands, EGF and HRG, play a key role in controlling diverse cell fates. In MCF-7 cells, stimulation with EGF causes sustained network activation and leads to cell differentiation, while stimulation with HRG causes transient network activation and leads to cell proliferation.
These figures were illustrated by Cell Illustrator [35, 36]. We observed that the two directed graphs had different network topologies each other. We compared with the two graphs from characteristics of node degree, that is, the number of edges that they had. In directed networks, we distinguish between the in-degree, the number of directed edges that point toward the node, and the out-degree, the number of directed edges that start at the node. The in-degree distributions and the out-degree distributions of the EGF- and HRG-induced networks are illustrated in Additional File 3. We found that each of the in-degree distributions concentrated around its average value, whereas the out-degree distributions had long tails. In particular, the out-degree distribution of the HRG-induced network was more heavy tailed than that of the EGF-induced network. Furthermore, in the HRG-induced network, we found that three hub genes, FOS, MAP3K8 and PDLIM5, had larger out-degrees (30, 68 and 75, respectively) than the other genes. While, in the EGF-induced network, each of FOS, MAP3K8 and PDLIM5 had only one outgoing edge (one out-degree). Thus, these hub genes are thought to be essential in only the HRG-induced network since their disruptions do not lead to a major loss of connectivity in the EGF-induced network, but the loss of them causes the breakdown of the HRG-induced network into isolated clusters. Our analysis based on the VAR model with the proposed method produced the hypothesis that these hub genes might be implicated why only HRG allows MCF-7 cells to be differentiated.
Conclusion
In this article, we have proposed a new class of l_{1}-regularization, called recursive elastic net for inferring gene networks from time-course microarray data based on the VAR model. The recursive elastic net addresses the drawback of the elastic net and has higher true discovery rate than other competing methods in gene network inference. Numerical simulations demonstrated that the proposed method succeeded in reducing the number of false positives drastically while keeping the large number of true positives and achieved two or more times higher true discovery rate than the lasso, the naive elastic net, the elastic net and the James-Stein shrinkage even when the number of time points was small. We also compared our method with various reverse-engineering algorithms including MRNET, CLR and ARACNE by using experimental data of MCF-7 breast cancer cells and TRANSPATH database. As a result, we found that the proposed method had the best performance between them and provided some differences between the EGF- and HRG-induced networks.
An interesting direction for future work would be to incorporate prior information elicited from the known biological networks. In the recursive elastic net, we can incorporate the topology of the known network directly as the initial variable weights (10). For example, if an edge from gene i to gene j exists the known biological pathway, we impose a small weight less than one on the corresponding coefficient. This incorporation would further provide higher sensitivity and higher true discovery rate of the proposed method in gene network inference. Another promising direction for future work would be the extension of the recursive elastic net to other type of mathematical models. A limitation of the VAR model comes from their equidistant assumption which is not true in many real situations. We might solve the problem by applying the proposed method to differential equation models [37, 38].
Appendix: Proof of Theorem 1
This completes the proof.
Declarations
Acknowledgements
We are grateful to Dr. Rainer Opgen-Rhein for providing the R program code of the James-Stein procedure and to anonymous reviewers for many constructive comments and suggestions. This work was supported by the Next-Generation Supercomputer Project under RIKEN and MEXT, Japan. The computational resource was also provided by the Super Computer System, Human Genome Center, Institute of Medical Science, University of Tokyo.
Authors’ Affiliations
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