1 Introduction

Among the main challenges in mass spectrometric metabolomic analysis is the high-throughput analysis of metabolic features, their fast detection and annotation. By contrast to the screening of known, previously characterized, metabolic features in these data, the putative annotation of unknown features is often cumbersome and requires a lot of manual work, hindering the biological information retrieval of these data. High-resolution mass spectrometric data is often very rich in information content and metabolic conversions, and reactions can be derived from structural properties of features (Breitling et al. 2006). In addition to that, statistical associations between features (based on their intensity values) can be a valuable ressource to find co-synthesised or co-regulated metabolites, which are synthesised in the same biosynthetic pathways. Given that an analysis tool within the R framework is still lacking that is integrating the two features of mass spectrometric information commonly acquired with mass spectrometers (m/z and intensity values), I developed MetNet to close this gap. The MetNet package comprises functionalities to infer network topologies from high-resolution mass spectrometry data. MetNet combines information from both structural data (differences in m/z values of features) and statistical associations (intensity values of features per sample) to propose putative metabolic networks that can be used for further exploration.

The idea of using high-resolution mass spectrometry data for network construction was first proposed in Breitling et al. (2006) and followed soon afterwards by a Cytoscape plugin, MetaNetter (Jourdan et al. 2007), that is based on the inference of metabolic networks on molecular weight differences and correlation (Pearson correlation and partial correlation).

Inspired by the paper of Marbach et al. (2012) different algorithms for network were implemented in MetNet to account for biases that are inherent in these statistical methods, followed by the calculation of a consensus adjacency matrix using the differently computed individual adjacency matrices.

The two main functionalities of the package include the creation of an adjacency matrix from structual properties, based on losses/addition of functional groups defined by the user, and statistical associations. Currently, the following statistical models are implemented to infer a statistical adjacency matrix: Least absolute shrinkage and selection operator (LASSO, L1-norm regression, (Tibshirani 1994)), Random Forest (Breiman 2001), Pearson and Spearman correlation (including partial and semipartial correlation, see Steuer (2006) for a discussion on correlation-based metabolic networks), context likelihood of relatedness (CLR, (Faith et al. 2007)), the algorithm for the reconstruction of accurate cellular networks (ARACNE, (Margolin et al. 2006)) and constraint-based structure learning (Bayes, (Scutari 2010)). Since all of these methods have advantages and disadvantages, the user has the possibility to select several of these methods, compute adjacency matrices from these models and create a consensus matrix from the different statistical frameworks.

After creating the statistical and structural adjaceny matrices these two matrices can be combined to form a consensus matrix that has both information from structural and statistical properties of the data. This can be followed by further network analyses (e.g. calculation of topological parameters), integration with other data sources (e.g. genomic information or transcriptomic data) and/or visualization.

Questions and bugs

MetNet is currently under active development. If you discover any bugs, typos or develop ideas of improving MetNet feel free to raise an issue via Github or send a mail to the developer.

2 Prepare the environment and load the data

To install MetNet enter the following to the R console

install.packages("BiocManager")
BiocManager::install("MetNet")

Before starting with the analysis, load the MetNet package. This will also load the required packages glmnet, stabs, randomForest, rfPermute, mpmi, parmigene, WGCNA and bnlearn that are needed for functions in the statistical adjacency matrix inference.

library(MetNet)

The data format that is compatible with the MetNet framework is in the xcms/CAMERA output-like \(m~\times~n\) matrix, where columns denote the different samples \(n\) and where \(m\) features are present. In such a matrix, information about the masses of the features and quantitative information of the features (intensity or concentration values) are needed. The information about the m/z values has to be stored in a vector of length \(\vert m \vert\) in the column "mz".

MetNet does not impose any requirements for data normalization, filtering, etc. However, the user has to make sure that the data is properly preprocessed. These include division by internal standard, log2 transformation, noise filtering, removal of features that do not represent mass features/metabolites, removal of isotopes, etc.

We will load here the object x_test that contains m/z values (in the column "mz"), together with the corresponding retention time (in the column "rt") and intensity values. We will use here the object x_test for guidance through the workflow of MetNet.

data("x_test", package = "MetNet")
x_test <- as.matrix(x_test)

3 Creating the structural matrix

The function structural will create the adjacency matrix based on structual properties (m/z values) of the features. The function expects a matrix with a column "mz" that contains the mass information of a feature (typically the m/z value). Furthermore, structural takes a data.frame object as argument transformations with the colnames "mass", "name" and additional columns (e.g. "formula"). structural looks for transformation (in the sense of additions/losses of functional groups mediated by biochemical, enzymatic reactions) in the data using the mass information.

Following the work of (Breitling et al. 2006) and (Jourdan et al. 2007), molecular weight difference wX is defined by \(w_X = \vert w_A - w_B \vert\)

where wA is the molecular weight of substrate A, and wB is the molecular weight of product B (typically, m/z values will be used as a proxy for the molecular weight since the molecular weight is not directly derivable from mass spectrometric data). As examplified in (Jourdan et al. 2007) specific enzymatic reactions refer to specific changes in the molecular weight, e.g. carboxylation reactions will result in a mass difference of 43.98983 (molecular weight of CO2) between metabolic features.

The search space for these transformation is adjustable by the transformation argument in structural allowing to look for specific enzymatic transformations in mind. Hereby, structural will take into account the ppm value, to adjust for inaccuracies in m/z values due to technical reasons according to the formula

\[ppm = \frac{m_{exp} - m_{calc}}{m_{exp}} \cdot 10^{-6}\]

with mexp the experimentally determined m/z value and mcalc the calculated accurate mass of a molecule. Within the function, a lower and upper range is calculated depending on the supplied ppm value, differences between the m/z feature values are calculated and matched against the "mass"es of the transformations argument. If any of the additions/losses defined in transformations is found in the data, it will be reported as an (unweighted) connection in the returned adjacency matrix. Together with the adjacency matrix the type of connection (derived from the column "name" in the transformations) will be written to a character matrix. These two matrices will be returned as a list (first entry: numerical adjacency matrix, second entry: character matrix) by the function structural.

Before calculating the structural matrix, one must define the search space, i.e. these transformation that will be looked for in the mass spectrometric data by creating the transformations object.

## define the search space for biochemical transformation 
transformations <- rbind(
    c("Hydroxylation (-H)", "O", 15.9949146221, "-"),
    c("Malonyl group (-H2O)", "C3H2O3", 86.0003939305, "+"),
    c("D-ribose (-H2O) (ribosylation)", "C5H8O4", 132.0422587452, "-"),
    c("C6H10O6", "C6H10O6", 178.0477380536, "-"),
    c("Rhamnose (-H20)", "C6H10O4", 146.057910, "-"),
    c("Monosaccharide (-H2O)", "C6H10O5", 162.0528234315, "-"),
    c("Disaccharide (-H2O) #1", "C12H20O10", 324.105649, "-"),
    c("Disaccharide (-H2O) #2", "C12H20O11", 340.1005614851, "-"),
    c("Trisaccharide (-H2O)", "C18H30O15", 486.1584702945, "-"),
    c("Glucuronic acid (-H2O)", "C6H8O6", 176.0320879894, "?"),
    c("coumaroyl (-H2O)", "C9H6O2", 146.0367794368, "?"),
    c("feruloyl (-H2O)", "C9H6O2OCH2", 176.0473441231, "?"),
    c("sinapoyl (-H2O)", "C9H6O2OCH2OCH2", 206.0579088094, "?"),
    c("putrescine to spermidine (+C3H7N)", "C3H7N", 57.0578492299, "?"))

## convert to data frame
transformations <- data.frame(
    group = transformations[, 1],
    formula = transformations[, 2],
    mass = as.numeric(transformations[, 3]),
    rt = transformations[, 4])

The function structural will then check for those m/z differences that are stored in the column "mass" in the object transformations. To create the adjacency matrix derived from these structural information we enter

struct_adj <- structural(x = x_test, transformation = transformations, ppm = 10)

in the R console.

Refining the structural adjacency matrix (optional)

Depending on the chemical group added the retention time will differ depending on the chemical group added, e.g. an addition of a glycosyl group will usually result in a lower retentiom time in reverse-phase chromatography- This information can be used in refining the adjacency matrix derived from the structural matrix. The rtCorrection does this checking, if predicted transformation correspond to the expected retention time shift, in an automated fashion. It requires information about the expected retention time shift in the data.frame passed to the transformation argument (in the "rt" column). Within this columns, information about retention time shifts is encoded by "-", "+" and "?", which means the feature with higher m/z value has lower, higher or unknown retention time than the feature with the lower m/z value. The values for m/z and retention time will be taken from the object passed to the x argument. In case there is a discrepancy between the transformation and the retention time shift the adjacency matrix at the specific position will be set to 0. rtCorrection will return the updated adjacency matrix and the updated character matrix with the descriptions of the transformation.

To account for retention time shifts we enter

struct_adj <- rtCorrection(structural = struct_adj, 
    x = x_test, transformation = transformations)

in the R console.

4 Creating the statistical matrix

4.1 Creating weighted adjacency matrices using statistical

The function statistical will create the adjacency matrix based on statistical associations. The function will create a list of weighted adjacency matrices using the statistical models defined by the model argument. Currently, the models LASSO (using stabs, (Hofner, Boccuto, and Göker 2015; Thomas et al. 2017)), Random Forest (using GENIE3, CLR, ARACNE (the two latter using the package mpmi to calculate Mutual Information using a nonparametric bias correction by Bias Corrected Mutual Information, and the functions clr and aracne.a from the parmigene package), Pearson and Spearman correlation (based on the stats package), partial and semipartial Pearson and Spearman correlation (using the ppcor package) and score-based structure learning returning the strength of the probabilistic relationships of the arcs of a Bayesian network, as learned from bootstrapped data (using the boot.strength with the Tabu greedy search as default from the bnlearn package (Scutari 2010)).

For further information on the different models take a look on the respective help pages of lasso, randomForest, clr, aracne, correlation and/or bayes. Arguments that are accepted by the respective underlying functions can be passed directly to the statistical function. In addition, arguments that are defined in the functions lasso, randomForest, clr, aracne, correlation and/or bayes can be passed to the functions.

4.2 Creating an unweighted adjacency matrix using threshold

From the list of adjacency matrices the function threshold will create a unweighted adjacency matrix from the weighted adjacency matrices unifying the information present from all statistical models. The reasoning behind this step is to circumvent disadvantages arising from each model and creating a statistically reliable topology that reflects the actual metabolic relations. threshold return an unweighted adjancency matrix with connections inferred from the respective models.

There are four different types implemented how the unweighted adjacency matrix can be created: threshold, top1, top2, mean.

For type = "threshold", threshold values have to be defined for the args argument for each respective statistical model, above or below which the each in each weighted adjacency matrix will be reported as a unweighted each. The unweighted adjacency matrices will be passed to the
consensus function from the sna (Butts 2016) to calculate the unweighted consensus adjacency matrix. The arguments that are accepted by this function can be passed to the threshold function. Furthermore, in args an entry threshold needs to be defined to threshold if the value ai,j of the consensus adjacency matrix will be reported as a connection in the returned matrix (if ai,j \(\geq\) threshold) or not. In the case of the method "central.graph" (default), the argument threshold should be set to 1.

For the other three types (top1, top2, mean) the ranks per statistical model will be calculated and from each respective link the top1, top 2 or mean rank across statistical models will be calculated (cf. (Hase et al. 2013)). The top n unqique ranks (defined by the entry n in args) will be returned as links in the unweighted consensus adjacency matrix.

In the following example, we will create a list of unweighted adjacency matrices using Pearson and Spearman correlation using the intensity values as input data.

x_int <- x_test[, 3:dim(x_test)[2]]
x_int <- as.matrix(x_int)
stat_adj_l <- statistical(x_int, model = c("pearson", "spearman"))
## [1] "pearson finished."
## [1] "spearman finished."

threshold implements four types to obtain an unweighted adjacency matrix. We will create for all types the unweighted consensus adjacency matrices.

## type = "threshold" 
args_thr <- list("pearson" = 0.95, "spearman" = 0.95, threshold = 1)
stat_adj_thr <- threshold(statistical = stat_adj_l, type = "threshold", 
    args = args_thr)
 
## type = "top1" 
args_top <- list(n = 40)
stat_adj_top1 <- threshold(statistical = stat_adj_l, type = "top2", 
    args = args_top)

## type = "top2"
stat_adj_top2 <- threshold(statistical = stat_adj_l, type = "top2", 
    args = args_top)
 
## type = "mean"
stat_adj_mean <- threshold(statistical = stat_adj_l, type = "mean", 
    args = args_top)

5 Combining the structural and statistical matrix

After creating the unweighted structural and unweighted statistical adjacency matrices, it is time to combine these two matrices. The function combine will combine the matrices to the consensus matrix. The function accepts the arguments structure and statistical for the list returned by structural and the matrix returned by threshold, respectively, and the argument threshold, that is a numerical value (default = 1). After adding the matrices, the entries will be checked if they are greater or equal than threshold and 1 or 0 will be returned, respectively. The argument threshold needs to be adjusted by the user if another method than "central.graph" in threshold (type = “threshold”) is used.

We will use here the unweighted statistical adjacency matrix from type = "mean":

cons_adj <- combine(structural = struct_adj, statistical = stat_adj_mean)

6 Visualization and further analyses

To display the created consensus adjacency matrix, existing visualization tools available in the R framework can be employed or any other visualization tool after exporting the consensus matrix as a text file. In this example We will use the igraph (Csardi and Nepusz 2006) package to visualize the adjacency matrix.

combine returns a list of two adjacency matrices, where the first entry contains the unweighted adjacency matrix and the second entry contains a matrix given information on the type of link between features based on structural information. Only the first entry of the list will be passed to the graph_from_adjacency_matrix function:

g <- igraph::graph_from_adjacency_matrix(cons_adj[[1]], mode = "undirected")
plot(g, edge.width = 5, vertex.label.cex = 0.5, edge.color = "grey")
** _Ab initio_ network inferred from structural and quantitative mass spectrometry data.** Vertices are connected that are separated by given metabolic transformation and statistical association.

Figure 1: ** Ab initio network inferred from structural and quantitative mass spectrometry data.** Vertices are connected that are separated by given metabolic transformation and statistical association

Furthermore, the network can be analysed by network analysis techniques (topological parameters such as centrality, degree, clustering indices) that are implemented in different packages in R (e.g. igraph or sna) or other software tools outside of the R environment.

Appendix

Session information

All software and respective versions to build this vignette are listed here:

## R version 4.0.0 (2020-04-24)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.4 LTS
## 
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.11-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.11-bioc/R/lib/libRlapack.so
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=C              
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] MetNet_1.6.0     knitr_1.28       BiocStyle_2.16.0
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.4.6         sna_2.5              highr_0.8           
##  [4] compiler_4.0.0       pillar_1.4.3         BiocManager_1.30.10 
##  [7] ppcor_1.1            GENIE3_1.10.0        tools_4.0.0         
## [10] digest_0.6.25        evaluate_0.14        tibble_3.0.1        
## [13] lifecycle_0.2.0      parmigene_1.0.2      lattice_0.20-41     
## [16] pkgconfig_2.0.3      rlang_0.4.5          igraph_1.2.5        
## [19] mpmi_0.43.1          magick_2.3           yaml_2.2.1          
## [22] parallel_4.0.0       xfun_0.13            coda_0.19-3         
## [25] stringr_1.4.0        stabs_0.6-3          vctrs_0.2.4         
## [28] grid_4.0.0           BiocParallel_1.22.0  bnlearn_4.5         
## [31] rmarkdown_2.1        bookdown_0.18        magrittr_1.5        
## [34] htmltools_0.4.0      ellipsis_0.3.0       MASS_7.3-51.6       
## [37] KernSmooth_2.23-17   stringi_1.4.6        network_1.16.0      
## [40] statnet.common_4.3.0 crayon_1.3.4

Transformations

The list of transformations is taken from Breitling et al. (2006). The numerical m/z values were calculated by using the structural formula and the Biological Magnetic Resonance Data Bank web tool.

transformations <- rbind(
    c("Alanine", "C3H5NO", "71.0371137878"),
    c("Arginine", "C6H12N4O", "156.1011110281"),
    c("Asparagine", "C4H6N2O2", "114.0429274472"),
    c("Guanosine 5-diphosphate (-H2O)", "C10H13N5O10P2", "425.0137646843"),
    c("Guanosine 5-monophosphate (-H2O)", "C10H12N5O7P", "345.0474342759"),
    c("Guanine (-H)", "C5H4N5O", "150.0415847765"),
    c("Aspartic acid", "C4H5NO3", "115.0269430320"),
    c("Guanosine (-H2O)", "C10H11N5O4", "265.0811038675"),
    c("Cysteine", "C3H5NOS", "103.0091844778"),
    c("Deoxythymidine 5'-diphosphate (-H2O)", "C10H14N2O10P2", "384.01236770"),
    c("Cystine", "C6H10N2O3S2", "222.0132835777"),
    c("Thymidine (-H2O)", "C10H12N2O4", "224.0797068840"),
    c("Glutamic acid", "C5H7NO3", "129.0425930962"),
    c("Thymine (-H)", "C5H5N2O2", "125.0351024151"),
    c("Glutamine", "C5H8N2O2", "128.0585775114"),
    c("Thymidine 5'-monophosphate (-H2O)", "C10H13N2O7P", "304.0460372924"),
    c("Glycine", "C2H3NO", "57.0214637236"),
    c("Uridine 5'-diphosphate (-H2O)", "C9H12N2O11P2", "385.9916322587"),
    c("Histidine", "C6H7N3O", "137.0589118624"),
    c("Uridine 5'-monophosphate (-H2O)", "C9H11N2O8P", "306.0253018503"),
    c("Isoleucine", "C6H11NO", "113.0840639804"),
    c("Uracil (-H)", "C4H3N2O2", "111.0194523509"),
    c("Leucine", "C6H11NO", "113.0840639804"),
    c("Uridine (-H2O)", "C9H10N2O5", "226.0589714419"),
    c("Lysine", "C6H12N2O", "128.0949630177"),
    c("Acetylation (-H)", "C2H3O2", "59.0133043405"),
    c("Methionine", "C5H9NOS", "131.0404846062"),
    c("Acetylation (-H2O)", "C2H2O",  "42.0105646863"),
    c("Phenylalanine", "C9H9NO",  "147.0684139162"),
    c("C2H2", "C2H2", "26.0156500642"),
    c("Proline", "C5H7NO", "97.0527638520"),
    c("Carboxylation", "CO2", "43.9898292442"),
    c("Serine", "C3H5NO2", "87.0320284099"),
    c("CHO2", "CHO2", "44.9976542763"),
    c("Threonine",  "C4H7NO2",  "101.0476784741"),
    c("Condensation/dehydration", "H2O", "18.0105646863"),
    c("Tryptophan", "C11H10N2O",  "186.0793129535"),
    c("Diphosphate", "H3O6P2", "160.9404858489"),
    c("Tyrosine", "C9H9NO2", "163.0633285383"),
    c("Ethyl addition (-H2O)", "C2H4", "28.0313001284"),
    c("Valine", "C5H9NO",  "99.0684139162"),
    c("Formic Acid (-H2O)", "CO", "27.9949146221"),
    c("Acetotacetate (-H2O)",  "C4H4O2", "84.0211293726"),
    c("Glyoxylate (-H2O)", "C2O2",  "55.9898292442"),
    c("Acetone (-H)", "C3H5O", "57.0340397826"),
    c("Hydrogenation/dehydrogenation", "H2", "2.0156500642"),
    c("Adenylate (-H2O)", "C10H12N5O6P", "329.0525196538"),
    c("Hydroxylation (-H)", "O", "15.9949146221"),
    c("Biotinyl (-H)", "C10H15N2O3S", "243.0803380482"),
    c("Inorganic phosphate", "P", "30.9737615100"),
    c("Biotinyl (-H2O)", "C10H14N2O2S", "226.0775983940"),
    c("Ketol group (-H2O)", "C2H2O", "42.0105646863"),
    c("Carbamoyl P transfer (-H2PO4)", "CH2ON", "44.0136386915"),
    c("Methanol (-H2O)", "CH2", "14.0156500642"),
    c("Co-enzyme A (-H)", "C21H34N7O16P3S", "765.0995583014"),
    c("Phosphate", "HPO3", "79.9663304084"),
    c("Co-enzyme A (-H2O)", "C21H33N7O15P3S", "748.0968186472"),
    c("Primary amine", "NH2", "16.0187240694"),
    c("Glutathione (-H2O)", "C10H15N3O5S", "289.0732412976"),
    c("Pyrophosphate", "PP", "61.9475230200"),
    c("Isoprene addition (-H)", "C5H7", "67.0547752247"),
    c("Secondary amine", "NH", "15.0108990373"),
    c("Malonyl group (-H2O)", "C3H2O3", "86.0003939305"),
    c("Sulfate (-H2O)", "SO3", "79.9568145563"),
    c("Palmitoylation (-H2O)", "C16H30O", "238.2296655851"),
    c("Tertiary amine", "N", "14.0030740052"),
    c("Pyridoxal phosphate (-H2O)", "C8H8NO5P", "229.0140088825"),
    c("C6H10O5", "C6H10O5", "162.0528234315"),
    c("Urea addition (-H)", "CH3N2O", "59.0245377288"),
    c("C6H10O6", "C6H10O6", "178.0477380536"),
    c("Adenine (-H)", "C5H4N5", "134.0466701544"),
    c("D-ribose (-H2O) (ribosylation)", "C5H8O4", "132.0422587452"),
    c("Adenosine (-H2O)", "C10H11N5O3", "249.0861892454"),
    c("Disaccharide (-H2O) #1", "C12H20O10", "324.105649"),
    c("Disaccharide (-H2O) #2", "C12H20O11", "340.1005614851"),
    c("Adenosine 5'-diphosphate (-H2O)", "C10H13N5O9P2", "409.0188500622"),
    c("Glucose-N-phosphate (-H2O)", "C6H11O8P", "242.0191538399"),
    c("Adenosine 5'-monophosphate (-H2O)", "C10H12N5O6P", "329.0525196538"),
    c("Glucuronic acid (-H2O)", "C6H8O6", "176.0320879894"),
    c("Cytidine 5'-diphosphate (-H2O)", "C9H13N3O10P2", "385.0076166739"),
    c("Monosaccharide (-H2O)", "C6H10O5", "162.0528234315"),
    c("Cytidine 5'-monophsophate (-H2O)", "C9H12N3O7P", "305.0412862655"),
    c("Trisaccharide (-H2O)", "C18H30O15", "486.1584702945"),
    c("Cytosine (-H)", "C4H4N3O",  "110.0354367661"))

transformations <- data.frame(name = transformations[, 1], 
            formula = transformations[, 2],
            mass = as.numeric(transformations[, 3]))

References

Breiman, L. 2001. “Random Forests.” Machine Learning 45:5–32. https://doi.org/10.1023/A:1010933404324.

Breitling, R., S. Ritchie, D. Goodenowe, M.L. Stewart, and M.P. Barrett. 2006. “Ab Initio Prediction of Metabolic Networks Using Fourier Transform Mass Spectrometry Data.” Metabolomics 2:155–64. https://doi.org/10.1007/s11306-006-0029-z.

Butts, Carter T. 2016. Sna: Tools for Social Network Analysis. https://CRAN.R-project.org/package=sna.

Csardi, G., and T. Nepusz. 2006. “The Igraph Software Package for Complex Network Research.” InterJournal Complex Systems:1695. http://igraph.org.

Faith, J.J., B. Hayete, J.T. Thaden, I. Mogno, J. Wierzbowski, G. Cottarel, S. Kasif, J.J. Collins, and T.S. Gardner. 2007. “Large-Scale Mapping and Validation of Escherichia Coli Transcriptional Regulation from a Compendium of Expression Profiles.” Plos Biology 5:e8. https://doi.org/10.1371/journal.pbio.0050008.

Hase, T., S. Ghosh, R. Yamanaka, and H. Kitano. 2013. “Harnessing Diversity Towards the Reconstructing of Large Scale Gene Regulatory Networks.” PLoS Computational Biology 9:e1003361. https://doi.org/10.1371/journal.pcbi.1003361.

Hofner, B., L. Boccuto, and M. Göker. 2015. “Controlling False Discoveries in High-Dimensional Situations: Boosting with Stability Selection.” BMC Bioinformatics 16:144. https://doi.org/10.1186/s12859-015-0575-3.

Jourdan, F., R. Breitling, M.P. Barrett, and D. Gilbert. 2007. “MetaNetter: Inference and Visualization of High-Resolution Metabolomic Networks.” Bioinformatics 24:143–45. https://doi.org/10.1093/bioinformatics/btm536.

Marbach, D., J.C. Costello, R. Küffner, N.M. Vega, R.J. Prill, D.M. Camacho, K.R. Allison, et al. 2012. “Wisdom of Crowds for Robust Gene Network Inference.” Nature Methods 9:796–804. https://doi.org/10.1038/nmeth.2016.

Margolin, A.A., I. Nemenman, K. Basso, C. Wiggins, G. Stolovitzky, R. Dalla Favera, and A. Califano. 2006. “ARACNE: An Algorithm for the Reconstruction of Gene Regulatory Networks in a Mammalian Cellular Context.” BMC Bioinformatics 7:S7. https://doi.org/10.1186/1471-2105-7-S1-S7.

Scutari, M. 2010. “Learning Bayesian Networks with the bnlearn R Package.” Journal of Statistical Software 35:1–22. https://doi.org/10.18637/jss.v035.i03.

Steuer, R. 2006. “On the Analysis and Interpretation of Correlations in Metabolomic Data.” Briefings in Bioinformatics 7:151–58. https://doi.org/10.1093/bib/bbl009.

Thomas, J., A. Mayr, B. Bischl, M. Schmid, A. Smith, and B. Hofner. 2017. “Gradient Boosting for Distributional Regression - Faster Tuning and Improved Variable Selection via Noncyclical Updates.” Statistics and Computing 28:673–87. https://doi.org/10.1007/s11222-017-9754-6.

Tibshirani, R. 1994. “Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society. Series B (Methodological) 58:267–88.