Evaluation of Bioinformatics Metrics with evaluomeR
2020-04-13
Abstract
R package evaluomeR how-to guide
Package
evaluomeR 1.18.0
Contents
1 Introduction
The evaluomeR package permits to evaluate the reliability of bioinformatic metrics by analysing the stability and goodness of the classifications of such metrics. The method takes the measurements of the metrics for the dataset and evaluates the reliability of the metrics according to the following analyses: Correlations, Stability and Goodness of classifications.
Correlations: Calculation of Pearson correlation coefficient between every pair of metrics available in order to quantify their interrelationship degree. The score is in the range [-1,1].
- Perfect correlations: -1 (inverse), and 1 (direct).
Stability: This analysis permits to estimate whether the clustering is meaningfully affected by small variations in the sample (Milligan and Cheng 1996). First, a clustering using the k-means algorithm is carried out. The value of K can be provided by the user. Then, the stability index is the mean of the Jaccard coefficient (Jaccard 1901) values of a number of bootstrap replicates. The values are in the range [0,1], having the following meaning:
- Unstable: [0, 0.60[.
- Doubtful: [0.60, 0.75].
- Stable: ]0.75, 0.85].
- Highly Stable: ]0.85, 1].
Goodness of classifications: The goodness of the classifications are assessed by validating the clusters generated. For this purpose, we use the Silhouette width as validity index. This index computes and compares the quality of the clustering outputs found by the different metrics, thus enabling to measure the goodness of the classification for both instances and metrics. More precisely, this goodness measurement provides an assessment of how similar an instance is to other instances from the same cluster and dissimilar to the rest of clusters. The average on all the instances quantifies how the instances appropriately are clustered. Kaufman and Rousseeuw (Kaufman and Rousseeuw 2009) suggested the interpretation of the global Silhouette width score as the effectiveness of the clustering structure. The values are in the range [0,1], having the following meaning:
- There is no substantial clustering structure: [-1, 0.25].
- The clustering structure is weak and could be artificial: ]0.25, 0.50].
- There is a reasonable clustering structure: ]0.50, 0.70].
- A strong clustering structure has been found: ]0.70, 1].
2 Installation
The installation of evaluomeR package is performed via Bioconductor:
if (!requireNamespace("BiocManager", quietly=TRUE))
install.packages("BiocManager")
BiocManager::install("evaluomeR")2.1 Prerequisites
The package evaluomeR depends on the following CRAN packages for the calculus: cluster (Maechler et al. 2018), corrplot (Wei and Simko 2017). Moreover, this package also depends on grDevices, graphics, stats and utils from R Core (R Core Team 2018) for plotting and on the Bioconductor packages SummarizedExperiment (Morgan et al. 2018), MultiAssayExperiment (Ramos et al. 2017) for input/output data.
3 Using evaluomeR
3.1 Creating an input SummarizedExperiment
The input is a SummarizedExperiment object. The assay contained in
SummarizedExperiment must follow a certain structure, see Table
1: A valid header must be specified. The first column of the
header is the ID or name of the instance of the dataset (e.g., ontology,
pathway, etc.) on which the metrics are measured. The other columns of the
header contains the names of the metrics. The rows contains the measurements
of the metrics for each instance in the dataset.
| ID | MetricNameA | MetricNameB | MetricNameC | … |
|---|---|---|---|---|
| instance1 | 1.2 | 6.4 | 0.5 | … |
| instance2 | 2.4 | 5.4 | 0.8 | … |
| instance3 | 1.9 | 8.9 | 1.1 | … |
3.2 Using input sample data from evaluomeR
In our package we provide three different sample input data:
ontMetrics: Structural ontology metrics, 19 metrics measuring structural aspects of bio-ontologies have been analysed on two different corpora of ontologies: OBO Foundry and AgroPortal (Franco et al. 2019).
rnaMetrics: RNA quality metrics for the assessment of gene expression differences, 2 quality metrics from 16 aliquots of a unique batch of RNA Samples. The metrics are Degradation Factor (DegFact) and RNA Integrity Number (RIN) (Imbeaud et al. 2005).
bioMetrics: Metrics for biological pathways, 2 metrics that quantitative characterizations of the importance of regulation in biochemical pathway systems, including systems designed for applications in synthetic biology or metabolic engineering. The metrics are reachability and efficiency (Davis and Voit 2018).
The user shall run the data built-in method to load evaluomeR sample input
data. This requires to provide the descriptor of the desired dataset. The
datasets availables can take the following values: “ontMetrics”, “rnaMetrics” or
“bioMetrics”.
library(evaluomeR)
data("ontMetrics")
data("rnaMetrics")
data("bioMetrics")3.3 Correlations
We provide the metricsCorrelations function to evaluate the correlations among the
metrics defined in the SummarizedExperiment:
library(evaluomeR)
data("rnaMetrics")
correlationSE <- metricsCorrelations(rnaMetrics, margins = c(4,4,12,10))## ## Data loaded.
## Number of rows: 16
## Number of columns: 3
# Access the correlation matrix via its first assay:
# assay(correlationSE,1)3.4 Stability analysis
The calculation of the stability indices is performed by stability and
stabilityRange functions.
3.4.1 Stability
The stability index analysis is performed by the stability function. For
instance, running a stability analysis for the metrics of rnaMetrics with a
number of 100 bootstrap replicates with a k-means cluster whose k is 2
(note that k must be inside [2,15] range):
stabilityData <- stability(rnaMetrics, k=2, bs = 100)
The stability function returns the stabilityData object, a
ExperimentList that contains the several assays such as the stability mean or the mean, betweenss, totss, tot.swithinss and anova values from the kmeans clustering:
stabilityData## ExperimentList class object of length 9:
## [1] stability_mean: SummarizedExperiment with 2 rows and 2 columns
## [2] cluster_partition: SummarizedExperiment with 2 rows and 2 columns
## [3] cluster_mean: SummarizedExperiment with 2 rows and 2 columns
## [4] cluster_centers: SummarizedExperiment with 2 rows and 2 columns
## [5] cluster_size: SummarizedExperiment with 2 rows and 2 columns
## [6] cluster_betweenss: SummarizedExperiment with 2 rows and 2 columns
## [7] cluster_totss: SummarizedExperiment with 2 rows and 2 columns
## [8] cluster_tot.withinss: SummarizedExperiment with 2 rows and 2 columns
## [9] cluster_anova: SummarizedExperiment with 2 rows and 2 columnsThe stability indices plots shown when getImages = TRUE are generated with the values of the stability mean:
assay(stabilityData, "stability_mean")| Metric | Mean_stability_k_2 |
|---|---|
| RIN | 0.82540873015873 |
| DegFact | 0.873916305916306 |
The plot represents the stability mean from each metric for a given k value.
This mean is calculated by performing the average of every stability index
from kranges [1,k] for each metric.
3.4.2 Stability range
The stabilityRange function is an iterative method of stability function.
It performs a stability analysis for a range of k values (k.range).
For instance, analyzing the stability of rnaMetrics in range [2,4], with
bs=100:
stabilityRangeData = stabilityRange(rnaMetrics, k.range=c(2,4), bs = 100)
Two kind of graphs are plotted in stabilityRange function. The first type
(titled as “St. Indices for k=X across metrics”) shows, for every k value,
the stability indices across the metrics. The second kind (titled as
St. Indices for metric ‘X’ in range [x,y]), shows a plot of the behaviour of
each metric across the k range.
3.5 Goodness of classifications
There are two methods to calculate the goodness of classifications: quality
and qualityRange.
3.5.1 Quality
This method plots how the metrics behave for the current k value, according
to the average silhouette width. Also, it will plot how the clusters are
grouped for each metric (one plot per metric).
For instance, running a quality analysis for the two metrics of rnaMetrics
dataset, being k=4:
qualityData = quality(rnaMetrics, k = 4)
The data of the first plot titled as “Qual. Indices for k=4 across metrics”
according to Silhouette avg. width, is stored in Avg_Silhouette_Width
column from the first assay of the SummarizedExperiment, qualityData. The
other three plots titled by their metric name display the input rows grouped
by colours for each cluster, along with their Silhouette width scores.
The variable qualityData contains information about the clusters of each
metric: The average silhouette width per cluster, the overall average
sihouette width (taking into account all the clusters) and the number of
individuals per cluster:
assay(qualityData,1)| Metric | Cluster_1_SilScore | Cluster_2_SilScore | Cluster_3_SilScore | Cluster_4_SilScore | Avg_Silhouette_Width | Cluster_1_Size | Cluster_2_Size | Cluster_3_Size | Cluster_4_Size |
|---|---|---|---|---|---|---|---|---|---|
| RIN | 0.420502645502646 | 0.696597735248404 | 0.473928571428572 | 0.324731573607137 | 0.488264943810529 | 4 | 4 | 5 | 3 |
| DegFact | 0.759196481622952 | 0.59496499852177 | 0.600198799385732 | 0.521618857725795 | 0.634170498361632 | 5 | 3 | 5 | 3 |
3.5.2 Quality range
The qualityRange function is an iterative method that uses the same
functionality of quality for a range of values (k.range), instead for one
unique k value. This methods allows to analyse the goodness of the
classifications of the metric for different values of the range.
In the next example we will keep using the rnaMetrics dataset, and a
k.range set to [4,6].
k.range = c(4,6)
qualityRangeData = qualityRange(rnaMetrics, k.range)
The qualityRange function also returns two kind of plots, as seen in
Stability range section. One for each k in the
k.range, showing the quality indices (goodness of the classification) across
the metrics, and a second type of plot to show each metric with its respective
quality index in each k value.
The qualityRangeData object returned by qualityRange is a ExperimentList from
MultiAssayExperiment, which is a list of SummarizedExperiment
objects whose size is diff(k.range)+1. In the example shown above, the size of
qualityRangeData is 3, since the array length would contain the dataframes from
k=4 to k=6.
diff(k.range)+1## [1] 3length(qualityRangeData)## [1] 3The user can access a specific dataframe for a given k value in three
different ways: by dollar notation, brackets notation or using our wrapper
method getDataQualityRange. For instance, if the user wishes to retrieve the
dataframe which contains information of k=5, being the k.range [4,6]:
k5Data = qualityRangeData$k_5
k5Data = qualityRangeData[["k_5"]]
k5Data = getDataQualityRange(qualityRangeData, 5)
assay(k5Data, 1)| Metric | Cluster_1_SilScore | Cluster_2_SilScore | Cluster_3_SilScore | Cluster_4_SilScore | Cluster_5_SilScore | Avg_Silhouette_Width | Cluster_1_Size | Cluster_2_Size | Cluster_3_Size | Cluster_4_Size | Cluster_5_Size |
|---|---|---|---|---|---|---|---|---|---|---|---|
| RIN | 0.420502645502646 | 0.674226581940152 | 0.251461988304093 | 0.523616734143049 | 0 | 0.451735613203479 | 4 | 4 | 3 | 4 | 1 |
| DegFact | 0.718287037037037 | 0.0375000000000007 | 0.904545454545454 | 0.585791823535685 | 0.521618857725795 | 0.578190921755929 | 4 | 2 | 2 | 5 | 3 |
Once the user believes to have found a proper k value, then the user can run
the quality function to see further silhouette information on the plots.
3.6 General functionality
In this section we describe a series of parameters that are shared among our
analysis functions: metricsCorrelations, stability, stabilityRange, quality
and qualityRange.
3.6.1 Disabling plotting
The generation of the images can be disabled by setting to FALSE the
parameter getImages:
stabilityData <- stability(rnaMetrics, k=5, bs = 50, getImages = FALSE)This prevents from generating any graph, performing only the calculus. By
default getImages is set to TRUE.
4 Selecting the optimal value of k
evaluomeR analyzes the behavior of the metrics in terms of stability and goodness of the clusters for a range of values of \(k\). In case of wishing to select the optimal value for \(k\) for a metric in a given dataset we have implemented the getOptimalKValue function, which returns a table stating which is the optimal value of k for each metric.
The algorithm works as follows: The highest stability and the highest goodness are obtained for the same value of \(k\). In such case, that value would be the optimal one. On the other hand, the highest stability and the highest goodness are obtained for different values of \(k\). In this case, additional criteria are needed. does not currently aim at providing those criteria, but to provide the data that could permit the user to make decisions. In the use cases described in this paper, we will apply the following criteria for the latter case:
If both values of \(k\) provide at least stable classifications (value >0.75), then we select the value of \(k\) that provides the largest Silhouette width. The same would happen if none provides stable classifications.
If \(k_1\) provides stable classifications and \(k_2\) does not, we will select \(k_1\) if the width of the Silhouette is at least reasonable.
If \(k_1\) provides stable classifications, \(k_2\) does not, and the width of the Silhouette of \(k_1\) is less than reasonable, then we will select the value of \(k\) with the largest Silhouette width.
stabilityData <- stabilityRange(data=ontMetrics, k.range=c(2,4),
bs=20, getImages = FALSE, seed=100)
qualityData <- qualityRange(data=ontMetrics, k.range=c(2,4),
getImages = FALSE, seed=100)
kOptTable <- getOptimalKValue(stabilityData, qualityData)| Metric | Stability_max_k | Stability_max_k_stab | Stability_max_k_qual | Quality_max_k | Quality_max_k_stab | Quality_max_k_qual | Global_optimal_k |
|---|---|---|---|---|---|---|---|
| ANOnto | 2 | 1.0000000 | 0.7548590 | 2 | 1.0000000 | 0.7548590 | 2 |
| AROnto | 4 | 0.9276802 | 0.8138336 | 4 | 0.9276802 | 0.8138336 | 4 |
| CBOOnto | 2 | 0.8786781 | 0.7193928 | 3 | 0.7897443 | 0.7231989 | 3 |
| CBOOnto2 | 2 | 0.8786781 | 0.7193928 | 3 | 0.7897443 | 0.7231989 | 3 |
| CROnto | 4 | 0.8450463 | 0.8450265 | 2 | 0.7831180 | 0.9643551 | 2 |
| DITOnto | 4 | 0.8228266 | 0.5933836 | 2 | 0.8129876 | 0.5971269 | 2 |
| INROnto | 2 | 0.9251030 | 0.7251026 | 2 | 0.9251030 | 0.7251026 | 2 |
| LCOMOnto | 3 | 1.0000000 | 0.6529131 | 3 | 1.0000000 | 0.6529131 | 3 |
| NACOnto | 2 | 0.8807614 | 0.7391430 | 2 | 0.8807614 | 0.7391430 | 2 |
| NOCOnto | 2 | 0.9478757 | 0.9431037 | 2 | 0.9478757 | 0.9431037 | 2 |
| NOMOnto | 2 | 0.8912204 | 0.6529583 | 3 | 0.8335419 | 0.6689736 | 3 |
| POnto | 2 | 1.0000000 | 0.7023466 | 2 | 1.0000000 | 0.7023466 | 2 |
| PROnto | 2 | 0.9941906 | 0.7065895 | 2 | 0.9941906 | 0.7065895 | 2 |
| RFCOnto | 2 | 0.8630298 | 0.6442806 | 2 | 0.8630298 | 0.6442806 | 2 |
| RROnto | 2 | 0.9941906 | 0.7065895 | 2 | 0.9941906 | 0.7065895 | 2 |
| TMOnto | 2 | 0.9664951 | 0.7331493 | 2 | 0.9664951 | 0.7331493 | 2 |
| TMOnto2 | 2 | 1.0000000 | 0.9493957 | 2 | 1.0000000 | 0.9493957 | 2 |
| WMCOnto | 2 | 0.9169334 | 0.9464120 | 2 | 0.9169334 | 0.9464120 | 2 |
| WMCOnto2 | 2 | 0.9208949 | 0.9467042 | 2 | 0.9208949 | 0.9467042 | 2 |
Additionally, you can select another subset of k.range to delimit the range of the optimal k.
kOptTable <- getOptimalKValue(stabilityData, qualityData, k.range=c(3,4))## Processing metric: ANOnto## Stability k '4' is stable but quality k '3' is not## Using '4' since it provides higher stability## Processing metric: AROnto## Maximum stability and quality values matches the same K value: '4'## Processing metric: CBOOnto## Both Ks have a stable classification: '4', '3'## Using '3' since it provides higher silhouette width## Processing metric: CBOOnto2## Both Ks have a stable classification: '4', '3'## Using '3' since it provides higher silhouette width## Processing metric: CROnto## Both Ks have a stable classification: '4', '3'## Using '3' since it provides higher silhouette width## Processing metric: DITOnto## Maximum stability and quality values matches the same K value: '4'## Processing metric: INROnto## Stability k '4' is stable but quality k '3' is not## Using '4' since it provides higher stability## Processing metric: LCOMOnto## Maximum stability and quality values matches the same K value: '3'## Processing metric: NACOnto## Both Ks do not have a stable classification: '4', '3'## Using '3' since it provides higher silhouette width## Processing metric: NOCOnto## Maximum stability and quality values matches the same K value: '3'## Processing metric: NOMOnto## Maximum stability and quality values matches the same K value: '3'## Processing metric: POnto## Stability k '4' is stable but quality k '3' is not## Using '4' since it provides higher stability## Processing metric: PROnto## Maximum stability and quality values matches the same K value: '3'## Processing metric: RFCOnto## Maximum stability and quality values matches the same K value: '3'## Processing metric: RROnto## Maximum stability and quality values matches the same K value: '3'## Processing metric: TMOnto## Both Ks have a stable classification: '4', '3'## Using '3' since it provides higher silhouette width## Processing metric: TMOnto2## Both Ks have a stable classification: '3', '4'## Using '4' since it provides higher silhouette width## Processing metric: WMCOnto## Both Ks have a stable classification: '4', '3'## Using '3' since it provides higher silhouette width## Processing metric: WMCOnto2## Both Ks have a stable classification: '4', '3'## Using '3' since it provides higher silhouette width| Metric | Stability_max_k | Stability_max_k_stab | Stability_max_k_qual | Quality_max_k | Quality_max_k_stab | Quality_max_k_qual | Global_optimal_k |
|---|---|---|---|---|---|---|---|
| ANOnto | 4 | 0.7858551 | 0.7033107 | 3 | 0.6537009 | 0.7367429 | 4 |
| AROnto | 4 | 0.9276802 | 0.8138336 | 4 | 0.9276802 | 0.8138336 | 4 |
| CBOOnto | 4 | 0.8009833 | 0.5843870 | 3 | 0.7897443 | 0.7231989 | 3 |
| CBOOnto2 | 4 | 0.8009833 | 0.5843870 | 3 | 0.7897443 | 0.7231989 | 3 |
| CROnto | 4 | 0.8450463 | 0.8450265 | 3 | 0.8202259 | 0.8553226 | 3 |
| DITOnto | 4 | 0.8228266 | 0.5933836 | 4 | 0.8228266 | 0.5933836 | 4 |
| INROnto | 4 | 0.7889145 | 0.6095614 | 3 | 0.7404618 | 0.6909412 | 4 |
| LCOMOnto | 3 | 1.0000000 | 0.6529131 | 3 | 1.0000000 | 0.6529131 | 3 |
| NACOnto | 4 | 0.7076790 | 0.6271890 | 3 | 0.6647891 | 0.6613224 | 3 |
| NOCOnto | 3 | 0.8972253 | 0.8791838 | 3 | 0.8972253 | 0.8791838 | 3 |
| NOMOnto | 3 | 0.8335419 | 0.6689736 | 3 | 0.8335419 | 0.6689736 | 3 |
| POnto | 4 | 0.8464754 | 0.6763749 | 3 | 0.6300979 | 0.6766154 | 4 |
| PROnto | 3 | 0.9704167 | 0.6686449 | 3 | 0.9704167 | 0.6686449 | 3 |
| RFCOnto | 3 | 0.8192584 | 0.6352988 | 3 | 0.8192584 | 0.6352988 | 3 |
| RROnto | 3 | 0.9704167 | 0.6686449 | 3 | 0.9704167 | 0.6686449 | 3 |
| TMOnto | 4 | 0.9002443 | 0.6944084 | 3 | 0.8665560 | 0.7100906 | 3 |
| TMOnto2 | 3 | 0.9375096 | 0.7246579 | 4 | 0.8971283 | 0.7254086 | 4 |
| WMCOnto | 4 | 0.8539321 | 0.7370700 | 3 | 0.8151443 | 0.8285148 | 3 |
| WMCOnto2 | 4 | 0.8699698 | 0.7294024 | 3 | 0.7801632 | 0.8702324 | 3 |
5 Metric analysis
We provide a series of methods for a further analysis on the metrics. These methods are: plotMetricsMinMax, plotMetricsBoxplot, plotMetricsCluster and plotMetricsViolin.
The plotMetricsMinMax function plots the minimum, maximum and standard deviation of min/max points of the values of each metric:
plotMetricsMinMax(ontMetrics)
The plotMetricsBoxplot method boxplots the value of each metric:
plotMetricsBoxplot(rnaMetrics)## Warning: Use of `data.melt$variable` is discouraged.
## ℹ Use `variable` instead.## Warning: Use of `data.melt$value` is discouraged.
## ℹ Use `value` instead.
Next, the plotMetricsCluster function clusters the values of the metrics by
using the euclidean distance and the method ward.D2 from hclust:
plotMetricsCluster(ontMetrics)
And finally the plotMetricsViolin function:
plotMetricsViolin(rnaMetrics)## Warning: Use of `data.melt$variable` is discouraged.
## ℹ Use `variable` instead.## Warning: Use of `data.melt$value` is discouraged.
## ℹ Use `value` instead.## Warning: Use of `data.melt$variable` is discouraged.
## ℹ Use `variable` instead.## Warning: Use of `data.melt$value` is discouraged.
## ℹ Use `value` instead.
6 Information
6.1 Contact
The source code is available at github. For bug/error reports please refer to evaluomeR github issues https://github.com/neobernad/evaluomeR/issues.
6.2 License
The package ‘evaluomeR’ is licensed under GPL-3.
6.3 How to cite
Currently there is no literature for evaluomeR. Please cite the R package, the github or the website. This package will be updated as soon as a citation is available.
6.4 Additional information
The evaluomeR functionality can also be access through a web interface1 Evaluome web an API REST2 API documentation.
6.5 Session information
sessionInfo()## R version 4.3.1 (2023-06-16)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 22.04.3 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.18-bioc/R/lib/libRblas.so
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_GB 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
##
## time zone: America/New_York
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats4 stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] evaluomeR_1.18.0 flexmix_2.3-19
## [3] lattice_0.22-5 randomForest_4.7-1.1
## [5] fpc_2.2-10 cluster_2.1.4
## [7] MultiAssayExperiment_1.28.0 SummarizedExperiment_1.32.0
## [9] Biobase_2.62.0 GenomicRanges_1.54.0
## [11] GenomeInfoDb_1.38.0 IRanges_2.36.0
## [13] S4Vectors_0.40.0 BiocGenerics_0.48.0
## [15] MatrixGenerics_1.14.0 matrixStats_1.0.0
## [17] magrittr_2.0.3 kableExtra_1.3.4
## [19] BiocStyle_2.30.0
##
## loaded via a namespace (and not attached):
## [1] Rdpack_2.5 bitops_1.0-7 rlang_1.1.1
## [4] compiler_4.3.1 systemfonts_1.0.5 vctrs_0.6.4
## [7] reshape2_1.4.4 rvest_1.0.3 stringr_1.5.0
## [10] pkgconfig_2.0.3 crayon_1.5.2 fastmap_1.1.1
## [13] magick_2.8.1 XVector_0.42.0 labeling_0.4.3
## [16] utf8_1.2.4 rmarkdown_2.25 xfun_0.40
## [19] modeltools_0.2-23 zlibbioc_1.48.0 cachem_1.0.8
## [22] jsonlite_1.8.7 highr_0.10 DelayedArray_0.28.0
## [25] parallel_4.3.1 prabclus_2.3-3 R6_2.5.1
## [28] bslib_0.5.1 stringi_1.7.12 jquerylib_0.1.4
## [31] diptest_0.76-0 Rcpp_1.0.11 bookdown_0.36
## [34] knitr_1.44 Matrix_1.6-1.1 nnet_7.3-19
## [37] tidyselect_1.2.0 rstudioapi_0.15.0 abind_1.4-5
## [40] yaml_2.3.7 tibble_3.2.1 plyr_1.8.9
## [43] withr_2.5.1 evaluate_0.22 xml2_1.3.5
## [46] mclust_6.0.0 kernlab_0.9-32 pillar_1.9.0
## [49] BiocManager_1.30.22 corrplot_0.92 generics_0.1.3
## [52] RCurl_1.98-1.12 ggplot2_3.4.4 munsell_0.5.0
## [55] scales_1.2.1 class_7.3-22 glue_1.6.2
## [58] tools_4.3.1 robustbase_0.99-0 webshot_0.5.5
## [61] grid_4.3.1 plotrix_3.8-2 rbibutils_2.2.15
## [64] colorspace_2.1-0 GenomeInfoDbData_1.2.11 cli_3.6.1
## [67] fansi_1.0.5 S4Arrays_1.2.0 viridisLite_0.4.2
## [70] svglite_2.1.2 ggdendro_0.1.23 dplyr_1.1.3
## [73] gtable_0.3.4 DEoptimR_1.1-3 sass_0.4.7
## [76] digest_0.6.33 SparseArray_1.2.0 farver_2.1.1
## [79] htmltools_0.5.6.1 lifecycle_1.0.3 httr_1.4.7
## [82] MASS_7.3-60Bibliography
Davis, Jacob D, and Eberhard O Voit. 2018. “Metrics for Regulated Biochemical Pathway Systems.” Bioinformatics. https://doi.org/10.1093/bioinformatics/bty942.
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