This vignette was built using CoGAPS version:
packageVersion("CoGAPS")
## [1] '3.2.40'
Coordinated Gene Association in Pattern Sets (CoGAPS) is a technique for latent space learning in gene expression data. CoGAPS is a member of the Nonnegative Matrix Factorization (NMF) class of algorithms. NMFs factorize a data matrix into two related matrices containing gene weights, the Amplitude (A) matrix, and sample weights, the Pattern (P) Matrix. Each column of A or row of P defines a feature and together this set of features defines the latent space among genes and samples, respectively. In NMF, the values of the elements in the A and P matrices are constrained to be greater than or equal to zero. This constraint simultaneously reflects the non-negative nature of gene expression data and enforces the additive nature of the resulting feature dimensions, generating solutions that are biologically intuitive to interpret (Seung and Lee (1999)).
CoGAPS has two extensions that allow it to scale up to large data sets, Genome-Wide CoGAPS (GWCoGAPS) and Single-Cell CoGAPS (scCOGAPS). This package presents a unified R interface for all three methods, with a parallel, efficient underlying implementation in C++.
CoGAPS is a bioconductor package and so the release version can be installed as follows:
source("https://bioconductor.org/biocLite.R")
biocLite("CoGAPS")
The most up-to-date version of CoGAPS can be installed directly from the FertigLab Github Repository:
## Method 1 using biocLite
biocLite("FertigLab/CoGAPS", dependencies = TRUE, build_vignettes = TRUE)
## Method 2 using devtools package
devtools::install_github("FertigLab/CoGAPS")
There is also an option to install the development version of CoGAPS, while this version has the latest experimental features, it is not guaranteed to be stable.
## Method 1 using biocLite
biocLite("FertigLab/CoGAPS", ref="develop", dependencies = TRUE, build_vignettes = TRUE)
## Method 2 using devtools package
devtools::install_github("FertigLab/CoGAPS", ref="develop")
We first give a walkthrough of the package features using a simple, simulated data set. In later sections we provide two example workflows on real data sets.
The only required argument to CoGAPS
is the data set. This can be a matrix
,
data.frame
, SummarizedExperiment
, SingleCellExperiment
or the path of a
file (tsv
, csv
, mtx
, gct
) containing the data.
# load data
data(GIST)
# run CoGAPS
CoGAPS(GIST.matrix)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples with parameters:
##
## -- Standard Parameters --
## nPatterns 7
## nIterations 5000
## seed 180
## singleCell FALSE
## sparseOptimization FALSE
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "7 patterns were learned"
While CoGAPS is running it periodically prints status messages. For example,
20000 of 25000, Atoms: 2932(80), ChiSq: 9728, time: 00:00:29 / 00:01:19
. This
message tells us that CoGAPS is at iteration 20000 out of 25000 for this phase,
and that 29 seconds out of an estimated 1 minute 19 seconds have passed. It
also tells us the size of the atomic domain which is a core component of the
algorithm but can be ignored for now. Finally, the ChiSq value tells us how
closely the A and P matrices reconstruct the original data. In general, we want
this value to go down - but it is not a perfect measurment of how well CoGAPS
is finding the biological processes contained in the data. CoGAPS also prints
a message indicating which phase is currently happening. There are two phases
to the algorithm - Calibration and Sampling.
Most of the time we’ll want to set some parameters before running CoGAPS.
Parameters are managed with a CogapsParams
object. This object will
store all parameters needed to run CoGAPS and provides a simple interface for
viewing and setting the parameter values.
# create new parameters object
params <- new("CogapsParams")
# view all parameters
params
## -- Standard Parameters --
## nPatterns 7
## nIterations 5000
## seed 676
## singleCell FALSE
## sparseOptimization FALSE
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
# get the value for a specific parameter
getParam(params, "nPatterns")
## [1] 7
# set the value for a specific parameter
params <- setParam(params, "nPatterns", 3)
getParam(params, "nPatterns")
## [1] 3
Once we’ve created the parameters object we can pass it along with our data to
CoGAPS
.
# run CoGAPS with specified model parameters
CoGAPS(GIST.matrix, params)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples with parameters:
##
## -- Standard Parameters --
## nPatterns 3
## nIterations 5000
## seed 676
## singleCell FALSE
## sparseOptimization FALSE
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "3 patterns were learned"
The CogapsParams
class manages the model parameters - i.e. the parameters
that affect the result. There are also a few parameters that are passed
directly to CoGAPS
that control things like displaying the status of the run.
# run CoGAPS with specified output frequency
CoGAPS(GIST.matrix, params, outputFrequency=250)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples with parameters:
##
## -- Standard Parameters --
## nPatterns 3
## nIterations 5000
## seed 676
## singleCell FALSE
## sparseOptimization FALSE
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "3 patterns were learned"
There are several other arguments that are passed directly to CoGAPS
which
are covered in later sections.
CoGAPS returns a object of the class CogapsResult
which inherits from LinearEmbeddingMatrix
(defined in the SingleCellExperiment
package). CoGAPS stores the lower dimensional representation of the samples
(P matrix) in the sampleFactors
slot and the weight of the features (A matrix)
in the featureLoadings
slot. CogapsResult
also adds two of its own slots -
sampleStdDev
and featureStdDev
which contain the standard deviation across
sample points for each matrix.
There is also some information in the metadata
slot such as the original
parameters and value for the Chi-Sq statistic. In general, the metadata will
vary depending on how CoGAPS
was called in the first place. The package
provides these functions for querying the metadata in a safe manner:
# run CoGAPS
result <- CoGAPS(GIST.matrix, params, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples
# get the mean ChiSq statistic over all samples
getMeanChiSq(result)
## [1] 7650.984
# get the version number used to create this result
getVersion(result)
## [1] '3.2.40'
# get the original parameters used to create this result
getOriginalParameters(result)
## -- Standard Parameters --
## nPatterns 3
## nIterations 5000
## seed 676
## singleCell FALSE
## sparseOptimization FALSE
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
To convert a CogapsResult
object to a LinearEmbeddingMatrix
use
as(result, "LinearEmbeddingMatrix")
## class: LinearEmbeddingMatrix
## dim: 9 3
## metadata(10): meanChiSq chisq ... params version
## rownames: NULL
## colnames: NULL
## factorData names(0):
The CogapsResult
object can be passed on to the analysis
and plotting functions provided in the package. By default, the plot
function
displays how the patterns vary across the samples. (Note that we pass the
nIterations
parameter here directly, this is allowed for any parameters in
the CogapsParams
class and will always take precedent over the values given
in params
).
# store result
result <- CoGAPS(GIST.matrix, params, nIterations=5000, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples
# plot CogapsResult object returned from CoGAPS
plot(result)
In the example workflows we’ll explore some more analysis functions provided in the package.
Non-Negative Matrix Factorization algorithms typically require long computation times and CoGAPS is no exception. In order to scale CoGAPS up to the size of data sets seen in practice we need to take advantage of modern hardware and parallelize the algorithm.
The simplest way to run CoGAPS in parallel is to provide the nThreads
argument to CoGAPS
. This allows the underlying algorithm to run on multiple
threads and has no effect on the mathematics of the algorithm i.e. this is
still standard CoGAPS. The precise number of threads to use depends on many
things like hardware and data size. The best approach is to play around with
different values and see how it effects the estimated time.
CoGAPS(GIST.matrix, nIterations=10000, outputFrequency=5000, nThreads=1, seed=5)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples with parameters:
##
## -- Standard Parameters --
## nPatterns 7
## nIterations 10000
## seed 5
## singleCell FALSE
## sparseOptimization FALSE
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "7 patterns were learned"
CoGAPS(GIST.matrix, nIterations=10000, outputFrequency=5000, nThreads=4, seed=5)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples with parameters:
##
## -- Standard Parameters --
## nPatterns 7
## nIterations 10000
## seed 5
## singleCell FALSE
## sparseOptimization FALSE
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "7 patterns were learned"
Note this method relies on CoGAPS being compiled with OpenMP support, use
buildReport
to check.
cat(CoGAPS::buildReport())
## Compiled with GCC v5.4
## SIMD: AVX instructions enabled
## Compiled with OpenMP
For large datasets (greater than a few thousand genes or samples) the multi-threaded parallelization isn’t enough. It is more efficient to break up the data into subsets and perform CoGAPS on each subset in parallel, stitching the results back together at the end. The CoGAPS extensions, GWCOGAPS and scCoGAPS, each implement a version of this method (Stein-O’Brien et al. (2017)).
In order to use these extensions, some additional parameters are required.
nSets
specifies the number of subsets to break the data set into. cut
,
minNS
, and maxNS
control the process of matching patterns across subsets
and in general should not be changed from defaults. More information about
these parameters can be found in the original papers. These parameters
need to be set with a different function than setParam
since they depend
on each other. Here we only set nSets
(always required), but we have the
option to pass the other parameters as well.
params <- setDistributedParams(params, nSets=3)
## setting distributed parameters - call this again if you change nPatterns
Setting nSets
requires balancing available hardware and run time against the
size of your data. In general, nSets
should be less than or equal to the
number of nodes/cores that are available. If that is true, then the more subsets
you create, the faster CoGAPS will run - however, some robustness can be lost
when the subsets get too small. The general rule of thumb is to set nSets
so that each subset has between 1000 and 5000 genes or cells. We will see an
example of this on real data in the next two sections.
Once the distributed parameters have been set we can call CoGAPS either by
setting the distributed
parameter or by using the provided wrapper functions.
The following calls are equivalent:
# genome-wide CoGAPS
GWCoGAPS(GIST.matrix, params, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running genome-wide CoGAPS on 1363 genes and 9 samples
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "3 patterns were learned"
# genome-wide CoGAPS
CoGAPS(GIST.matrix, params, distributed="genome-wide", messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running genome-wide CoGAPS on 1363 genes and 9 samples
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "3 patterns were learned"
# single-cell CoGAPS
scCoGAPS(GIST.matrix, params, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running single-cell CoGAPS on 1363 genes and 9 samples
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "2 patterns were learned"
# single-cell CoGAPS
CoGAPS(GIST.matrix, params, distributed="single-cell", singleCell=TRUE, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running single-cell CoGAPS on 1363 genes and 9 samples
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "2 patterns were learned"
Notice that we also set the parameter singleCell=TRUE
. This makes some
adjustments to the algorithm to account for the sparsity in single-cell data.
The scCoGAPS
wrapper automatically sets this parameter for us.
The parallel backend for this computation is managed by the package BiocParallel
and there is an option for the user to specifiy which backend they want. See the
Additional Features
section for more information.
In general it is preferred to pass a file name to GWCoGAPS
/scCoGAPS
since
otherwise the entire data set must be copied across multiple processes which
will slow things down and potentially cause an out-of-memory error. We will
see examples of this in the next two sections.
CoGAPS allows the user to save their progress throughout the run, and restart
from the latest saved “checkpoint”. This is intended so that if the server
crashes in the middle of a long run it doesn’t need to be restarted from the
beginning. Set the checkpointInterval
parameter to save checkpoints and
pass a file name as checkpointInFile
to load from a checkpoint.
if (CoGAPS::checkpointsEnabled())
{
# our initial run
res1 <- CoGAPS(GIST.matrix, params, checkpointInterval=100, checkpointOutFile="vignette_example.out", messages=FALSE)
# assume the previous run crashes
res2 <- CoGAPS(GIST.matrix, checkpointInFile="vignette_example.out", messages=FALSE)
# check that they're equal
all(res1@featureLoadings == res2@featureLoadings)
all(res1@sampleFactors == res2@sampleFactors)
}
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples
## [1] TRUE
If your data is stored as samples x genes, CoGAPS
allows you to pass
transposeData=TRUE
and will automatically read the transpose of your data
to get the required genes x samples configuration.
In addition to providing the data, the user can also specify an uncertainty
measurement - the standard deviation of each entry in the data matrix. By
default, CoGAPS
assumes that the standard deviation matrix is 10% of the
data matrix. This is a reasonable heuristic to use, but for specific types
of data you may be able to provide better information.
# run CoGAPS with custom uncertainty
data(GIST)
result <- CoGAPS(GIST.matrix, params, uncertainty=GIST.uncertainty, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running Standard CoGAPS on 1363 genes and 9 samples
The distributed computation for CoGAPS uses BiocParallel
underneath the hood
to manage the parallelization. The user has the option to specify what the
backend should be. By default, it is MulticoreParam
with the same number
of workers as nSets
. Use the BPPARAM
parameter in CoGAPS
to set the
backend. See the vignette for BiocParallel
for more information about the
different choices for the backend.
# run CoGAPS with serial backend
scCoGAPS(GIST.matrix, params, BPPARAM=BiocParallel::SerialParam(), messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running single-cell CoGAPS on 1363 genes and 9 samples
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "2 patterns were learned"
The default method for subsetting the data is to uniformly break up the rows (cols) of the data. There is an alternative option where the user provides an annotation vector for the rownames (colnames) of the data and gives a weight to each category in the annotation vector. Equal sized subsets are then drawn by sampling all rows (cols) according to the weight of each category.
# sampling with weights
anno <- sample(letters[1:5], size=nrow(GIST.matrix), replace=TRUE)
w <- c(1,1,2,2,1)
names(w) <- letters[1:5]
params <- new("CogapsParams")
params <- setAnnotationWeights(params, annotation=anno, weights=w)
result <- GWCoGAPS(GIST.matrix, params, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running genome-wide CoGAPS on 1363 genes and 9 samples
Finally, the user can set explicitSets
which is a list of character or
numeric vectors indicating which names or indices of the data should be put
into each set. Make sure to set nSets
to the correct value before passing explicitSets
.
# running cogaps with given subsets
sets <- list(1:225, 226:450, 451:675, 676:900)
params <- new("CogapsParams")
params <- setDistributedParams(params, nSets=length(sets))
## setting distributed parameters - call this again if you change nPatterns
result <- GWCoGAPS(GIST.matrix, params, explicitSets=sets, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running genome-wide CoGAPS on 1363 genes and 9 samples
When running GWCoGAPS or scCoGAPS, some additional metadata is returned that relates to the pattern matching process. This process is how CoGAPS stitches the results from each subset back together.
# run GWCoGAPS (subset data so the displayed output is small)
data <- GIST.matrix[,1:8]
params <- new("CogapsParams")
params <- setParam(params, "nPatterns", 3)
params <- setDistributedParams(params, nSets=2)
## setting distributed parameters - call this again if you change nPatterns
result <- GWCoGAPS(data, params, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running genome-wide CoGAPS on 1363 genes and 8 samples
# get the unmatched patterns from each subset
getUnmatchedPatterns(result)
## [[1]]
## Pattern_1 Pattern_2 Pattern_3
## IM00 0.4084548 0.9931554 0.9999683
## IM02 0.4712722 0.9909911 0.9510100
## IM04 0.6403601 0.9359978 0.8786938
## IM06 0.7046725 0.9510491 0.8392541
## IM09 0.8200434 0.9433075 0.7657053
## IM12 0.8383245 0.9935842 0.6701200
## IM18 0.9460258 0.9946362 0.6014773
## IM24 0.9998379 0.9817987 0.5526417
##
## [[2]]
## Pattern_1 Pattern_2 Pattern_3
## IM00 0.5968594 0.9999414 0.9930354
## IM02 0.6346853 0.9570559 0.9960371
## IM04 0.6883784 0.8836405 0.9611753
## IM06 0.7356634 0.8610340 0.9698907
## IM09 0.8028724 0.7991097 0.9612444
## IM12 0.8566020 0.7438221 0.9915519
## IM18 0.9084994 0.7011825 0.9881582
## IM24 1.0000000 0.6290938 0.9718338
# get the clustered patterns from the set of all patterns
getClusteredPatterns(result)
## $`1`
## 1.1 2.2
## IM00 0.4084548 0.5968594
## IM02 0.4712722 0.6346853
## IM04 0.6403601 0.6883784
## IM06 0.7046725 0.7356634
## IM09 0.8200434 0.8028724
## IM12 0.8383245 0.8566020
## IM18 0.9460258 0.9084994
## IM24 0.9998379 1.0000000
##
## $`2`
## 2.1 2.3
## IM00 0.9931554 0.9930354
## IM02 0.9909911 0.9960371
## IM04 0.9359978 0.9611753
## IM06 0.9510491 0.9698907
## IM09 0.9433075 0.9612444
## IM12 0.9935842 0.9915519
## IM18 0.9946362 0.9881582
## IM24 0.9817987 0.9718338
##
## $`3`
## 1.2 1.3
## IM00 0.9999683 0.9999414
## IM02 0.9510100 0.9570559
## IM04 0.8786938 0.8836405
## IM06 0.8392541 0.8610340
## IM09 0.7657053 0.7991097
## IM12 0.6701200 0.7438221
## IM18 0.6014773 0.7011825
## IM24 0.5526417 0.6290938
# get the correlation of each pattern to the cluster mean
getCorrelationToMeanPattern(result)
## $`1`
## [1] 0.996 0.991
##
## $`2`
## [1] 0.989 0.968
##
## $`3`
## [1] 0.999 0.999
# get the size of the subsets used
sapply(getSubsets(result), length)
## [1] 681 682
CoGAPS allows for a custom process for matching the patterns together. If you
have a result object from a previous run of GWCoGAPS/scCoGAPS, the unmatched
patterns for each subset are found by calling getUnmatchedPatterns
. Apply
any method you like as long as the result is a matrix with the number of rows
equal to the number of samples (genes) and the number of columns is equal to
the number of patterns. Then pass the matrix to the fixedPatterns
argument
along with the original parameters for the GWCoGAPS/scCoGAPS run.
# initial run
result <- GWCoGAPS(GIST.matrix, messages=FALSE)
##
## This is CoGAPS version 3.2.40
## Running genome-wide CoGAPS on 1363 genes and 9 samples
# custom matching process (just take matrix from first subset as a dummy)
consensusMatrix <- getUnmatchedPatterns(result)[[1]]
# run with our custom matched patterns matrix
params <- CogapsParams()
params <- setFixedPatterns(params, consensusMatrix, 'P')
GWCoGAPS(GIST.matrix, params, explicitSets=getSubsets(result))
##
## This is CoGAPS version 3.2.40
## Running genome-wide CoGAPS on 1363 genes and 9 samples with parameters:
##
## -- Standard Parameters --
## nPatterns 7
## nIterations 5000
## seed 834
## singleCell FALSE
## sparseOptimization FALSE
## distributed genome-wide
##
## -- Sparsity Parameters --
## alpha 0.01
## maxGibbsMass 100
##
## -- Distributed CoGAPS Parameters --
## nSets 4
## cut 7
## minNS 2
## maxNS 6
##
## Creating subsets...using provided named subsets
## set sizes (min, mean, max): (340, 340.75, 343)
## Running Final Stage...
## [1] "CogapsResult object with 1363 features and 9 samples"
## [1] "7 patterns were learned"
sessionInfo()
## R version 3.5.2 (2018-12-20)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 16.04.5 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.8-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.8-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] BiocParallel_1.16.6 CoGAPS_3.2.40 BiocStyle_2.10.0
##
## loaded via a namespace (and not attached):
## [1] Rcpp_1.0.0 compiler_3.5.2
## [3] RColorBrewer_1.1-2 BiocManager_1.30.4
## [5] GenomeInfoDb_1.18.2 XVector_0.22.0
## [7] bitops_1.0-6 tools_3.5.2
## [9] zlibbioc_1.28.0 SingleCellExperiment_1.4.1
## [11] digest_0.6.18 rhdf5_2.26.2
## [13] evaluate_0.13 lattice_0.20-38
## [15] Matrix_1.2-15 DelayedArray_0.8.0
## [17] yaml_2.2.0 parallel_3.5.2
## [19] xfun_0.5 GenomeInfoDbData_1.2.0
## [21] stringr_1.4.0 knitr_1.21
## [23] cluster_2.0.7-1 caTools_1.17.1.1
## [25] gtools_3.8.1 S4Vectors_0.20.1
## [27] IRanges_2.16.0 stats4_3.5.2
## [29] grid_3.5.2 Biobase_2.42.0
## [31] data.table_1.12.0 rmarkdown_1.11
## [33] bookdown_0.9 gdata_2.18.0
## [35] Rhdf5lib_1.4.2 magrittr_1.5
## [37] gplots_3.0.1.1 htmltools_0.3.6
## [39] matrixStats_0.54.0 BiocGenerics_0.28.0
## [41] GenomicRanges_1.34.0 SummarizedExperiment_1.12.0
## [43] KernSmooth_2.23-15 stringi_1.3.1
## [45] RCurl_1.95-4.11
If you use the CoGAPS package for your analysis, please cite Fertig et al. (2010)
If you use the gene set statistic, please cite Ochs et al. (2009)
Fertig, Elana J., Jie Ding, Alexander V. Favorov, Giovanni Parmigiani, and Michael F. Ochs. 2010. “CoGAPS: An R/C++ Package to Identify Patterns and Biological Process Activity in Transcriptomic Data.” Bioinformatics 26 (21):2792–3. https://doi.org/10.1093/bioinformatics/btq503.
Ochs, Michael F., Lori Rink, Chi Tarn, Sarah Mburu, Takahiro Taguchi, Burton Eisenberg, and Andrew K. Godwin. 2009. “Detection of Treatment-Induced Changes in Signaling Pathways in Gastrointestinal Stromal Tumors Using Transcriptomic Data.” Cancer Research 69 (23):9125–32. https://doi.org/10.1158/0008-5472.CAN-09-1709.
Seung, Sebastian, and Daniel D. Lee. 1999. “Learning the Parts of Objects by Non-Negative Matrix Factorization.” Nature 401 (6755):788–91. https://doi.org/10.1038/44565.
Stein-O’Brien, Genevieve L., Jacob L. Carey, Wai S. Lee, Michael Considine, Alexander V. Favorov, Emily Flam, Theresa Guo, et al. 2017. “PatternMarkers & Gwcogaps for Novel Data-Driven Biomarkers via Whole Transcriptome Nmf.” Bioinformatics 33 (12):1892–4. https://doi.org/10.1093/bioinformatics/btx058.