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The goal of bootPLS is to provide several non-parametric stable bootstrap-based techniques to determine the numbers of components in Partial Least Squares and sparse Partial Least Squares linear or generalized linear regression.
bootPLS
implements several algorithms that were
published as a book chapter and two articles.
A new bootstrap-based stopping criterion in PLS component construction, J. Magnanensi, M. Maumy-Bertrand, N. Meyer and F. Bertrand (2016), in The Multiple Facets of Partial Least Squares and Related Methods. doi:10.1007/978-3-319-40643-5_18.
A new universal resample-stable bootstrap-based stopping criterion for PLS component construction, J. Magnanensi, F. Bertrand, M. Maumy-Bertrand and N. Meyer, (2017), Statistics and Computing, 27, 757–774. doi:10.1007/s11222-016-9651-4.
New developments in Sparse PLS regression, J. Magnanensi, M. Maumy-Bertrand, N. Meyer and F. Bertrand, (2021), Frontiers in Applied Mathematics and Statistics. doi:10.3389/fams.2021.693126.
Support for parallel computation and GPU is being developed.
This website and these examples were created by F. Bertrand and M. Maumy-Bertrand.
You can install the released version of bootPLS from CRAN with:
install.packages("bootPLS")
You can install the development version of bootPLS from github with:
::install_github("fbertran/bootPLS") devtools
Load and display the pinewood worm dataset.
library(bootPLS)
library(plsRglm)
data(pine, package = "plsRglm")
<-pine[,1:10]
Xpine<-log(pine[,11]) ypine
pairs(pine)
Michel Tenenhaus’ reported in his book, La régression PLS (1998) Technip, Paris, that most of the expert biologists claimed that this dataset features two latent variables, which is tantamount to the PLS model having two components.
Leave one out CV (K=nrow(pine)
) one time
(NK=1
).
<- plsRglm::cv.plsR(log(x11)~.,data=pine,nt=6,K=nrow(pine),NK=1,verbose=FALSE)
bbb ::cvtable(summary(bbb))
plsRglm#> ____************************************************____
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Component____ 4 ____
#> ____Component____ 5 ____
#> ____Component____ 6 ____
#> ____Predicting X without NA neither in X nor in Y____
#> Loading required namespace: plsdof
#> Error in loadNamespace(x): there is no package called 'plsdof'
Set up 6-fold CV (K=6
), 100 times (NK=100
),
and use random=TRUE
to randomly create folds for repeated
CV.
<- plsRglm::cv.plsR(log(x11)~.,data=pine,nt=6,K=6,NK=100,verbose=FALSE) bbb2
Display the results of the cross-validation.
::cvtable(summary(bbb2))
plsRglm#> ____************************************************____
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Component____ 4 ____
#> ____Component____ 5 ____
#> ____Component____ 6 ____
#> ____Predicting X without NA neither in X nor in Y____
#> Loading required namespace: plsdof
#> Error in loadNamespace(x): there is no package called 'plsdof'
The \(Q^2\) criterion is recommended in that PLSR setting without missing data. A model with 1 component is selected by the cross-validation as displayed by the following figure. Hence the \(Q^2\) criterion (1 component) does not agree with the experts (2 components).
plot(plsRglm::cvtable(summary(bbb2)),type="CVQ2")
#> ____************************************************____
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Component____ 4 ____
#> ____Component____ 5 ____
#> ____Component____ 6 ____
#> ____Predicting X without NA neither in X nor in Y____
#> Loading required namespace: plsdof
#> Error in loadNamespace(x): there is no package called 'plsdof'
As for the CV Press criterion it is unable to point out a unique number of components.
plot(plsRglm::cvtable(summary(bbb2)),type="CVPress")
#> ____************************************************____
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Component____ 4 ____
#> ____Component____ 5 ____
#> ____Component____ 6 ____
#> ____Predicting X without NA neither in X nor in Y____
#> Loading required namespace: plsdof
#> Error in loadNamespace(x): there is no package called 'plsdof'
The package features our bootstrap based algorithm to select the
number of components in plsR regression. It is implemented with the
nbcomp.bootplsR
function.
set.seed(4619)
nbcomp.bootplsR(Y=ypine,X=Xpine,R =500)
#> [1] 1
#> ____************************************************____
#> ____Component____ 1 ____
#> ____Predicting X without NA neither in X nor in Y____
#> Loading required namespace: plsdof
#> Error in loadNamespace(x): there is no package called 'plsdof'
The verbose=FALSE
option suppresses messages output
during the algorithm, which is useful to replicate the bootstrap
technique. To set up parallel computing, you can use the
parallel
and the ncpus
options.
set.seed(4619)
<- replicate(20,nbcomp.bootplsR(Y=ypine,X=Xpine,R =500,verbose =FALSE,parallel = "multicore",ncpus = 2))
res_boot_rep #> Loading required namespace: plsdof
#> Error in loadNamespace(x): there is no package called 'plsdof'
It is easy to display the results with the barplot
function.
barplot(table(res_boot_rep))
#> Error in eval(expr, envir, enclos): object 'res_boot_rep' not found
A model with two components should be selected using our bootstrap based algorithm to select the number of components. Hence the number of component selected with our algorithm agrees with what was stated by the experts.
The package also features our bootstrap based algorithm to select,
for a given \(\eta\) value, the number
of components in spls regression. It is implemented with the
nbcomp.bootspls
function.
nbcomp.bootspls(x=Xpine,y=ypine,eta=.5)
#> eta = 0.5
#> [1] 1
#> [1] 2
#> [1] 3
#>
#> Optimal parameters: eta = 0.5, K = 2
#> $mspemat
#>
#> eta= 0.5 , K= 2 1.187203
#>
#> $eta.opt
#> [1] 0.5
#>
#> $K.opt
#> [1] 2
A doParallel
and foreach
based parallel
computing version of the algorithm is implemented as the
nbcomp.bootspls.para
function.
nbcomp.bootspls.para(x=Xpine,y=ypine,eta=.5)
#> [1] "eta = 0.5"
#> [1] 2
#> [1] 3
#>
#> Optimal parameters: eta = 0.5, K = 2
#> $mspemat
#>
#> eta= 0.5 ; K= 2 1.085205
#>
#> $eta.opt
#> [1] 0.5
#>
#> $K.opt
#> [1] 2
nbcomp.bootspls.para(x=Xpine,y=ypine,eta=c(.2,.5))
#> [1] "eta = 0.2"
#> [1] 2
#> [1] 3
#> [1] "eta = 0.5"
#> [1] 2
#> [1] 3
#>
#> Optimal parameters: eta = 0.5, K = 2
#> $mspemat
#>
#> eta= 0.2 ; K= 2 1.261797
#> eta= 0.5 ; K= 2 1.257117
#>
#> $eta.opt
#> [1] 0.5
#>
#> $K.opt
#> result.2
#> 2
Pinewood worm data reloaded.
library(bootPLS)
library(plsRglm)
data(pine, package = "plsRglm")
<-pine[,1:10]
Xpine<-log(pine[,11])
ypine<- cbind(ypine,Xpine) datasetpine
coefs.plsR.adapt.ncomp(datasetpine,sample(1:nrow(datasetpine)))
#> Loading required namespace: plsdof
#> [1] 4.000000000 10.934323181 -0.004164703 -0.061586032
#> [5] 0.038677103 -0.568792403 0.135567126 0.447111779
#> [9] -0.885736030 -0.110684199 -1.141903333 -0.397050615
Replicate the results to get the bootstrap distributions of the selected number of components and the coefficients.
replicate(20,coefs.plsR.adapt.ncomp(datasetpine,sample(1:nrow(datasetpine))))
#> [,1] [,2] [,3] [,4]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
#> [,5] [,6] [,7] [,8]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
#> [,9] [,10] [,11] [,12]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
#> [,13] [,14] [,15] [,16]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
#> [,17] [,18] [,19] [,20]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
Parallel computing support with the ncpus
and
parallel="multicore"
options.
coefs.plsR.adapt.ncomp(datasetpine,sample(1:nrow(datasetpine)),ncpus=2,parallel="multicore")
#> [1] 2.000000000 8.064227044 -0.003019447 -0.054389603
#> [5] -0.005212285 -0.109946405 0.038799785 -0.078324545
#> [9] -1.334080678 -0.045021804 -0.390730689 0.054696227
replicate(20,coefs.plsR.adapt.ncomp(datasetpine,sample(1:nrow(datasetpine)),ncpus=2,parallel="multicore"))
#> [,1] [,2] [,3] [,4]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
#> [,5] [,6] [,7] [,8]
#> [1,] 4.000000000 2.000000000 2.000000000 4.000000000
#> [2,] 10.934323181 8.064227044 8.064227044 10.934323181
#> [3,] -0.004164703 -0.003019447 -0.003019447 -0.004164703
#> [4,] -0.061586032 -0.054389603 -0.054389603 -0.061586032
#> [5,] 0.038677103 -0.005212285 -0.005212285 0.038677103
#> [6,] -0.568792403 -0.109946405 -0.109946405 -0.568792403
#> [7,] 0.135567126 0.038799785 0.038799785 0.135567126
#> [8,] 0.447111779 -0.078324545 -0.078324545 0.447111779
#> [9,] -0.885736030 -1.334080678 -1.334080678 -0.885736030
#> [10,] -0.110684199 -0.045021804 -0.045021804 -0.110684199
#> [11,] -1.141903333 -0.390730689 -0.390730689 -1.141903333
#> [12,] -0.397050615 0.054696227 0.054696227 -0.397050615
#> [,9] [,10] [,11] [,12]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
#> [,13] [,14] [,15] [,16]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
#> [,17] [,18] [,19] [,20]
#> [1,] 2.000000000 2.000000000 2.000000000 2.000000000
#> [2,] 8.064227044 8.064227044 8.064227044 8.064227044
#> [3,] -0.003019447 -0.003019447 -0.003019447 -0.003019447
#> [4,] -0.054389603 -0.054389603 -0.054389603 -0.054389603
#> [5,] -0.005212285 -0.005212285 -0.005212285 -0.005212285
#> [6,] -0.109946405 -0.109946405 -0.109946405 -0.109946405
#> [7,] 0.038799785 0.038799785 0.038799785 0.038799785
#> [8,] -0.078324545 -0.078324545 -0.078324545 -0.078324545
#> [9,] -1.334080678 -1.334080678 -1.334080678 -1.334080678
#> [10,] -0.045021804 -0.045021804 -0.045021804 -0.045021804
#> [11,] -0.390730689 -0.390730689 -0.390730689 -0.390730689
#> [12,] 0.054696227 0.054696227 0.054696227 0.054696227
Loading the data and creating the data frames.
data(aze_compl)
<-aze_compl[,2:34]
Xaze_compl<-aze_compl$y
yaze_compl<- cbind(y=yaze_compl,Xaze_compl) dataset
Fitting a logistic PLS regression model with 10 components. You have to use the family option when fitting the plsRglm.
<- plsRglm(y~.,data=dataset,10,modele="pls-glm-family",family="binomial")
modplsglm #> ____************************************************____
#>
#> Family: binomial
#> Link function: logit
#>
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Component____ 4 ____
#> ____Component____ 5 ____
#> ____Component____ 6 ____
#> ____Component____ 7 ____
#> ____Component____ 8 ____
#> ____Component____ 9 ____
#> ____Component____ 10 ____
#> ____Predicting X without NA neither in X or Y____
#> ****________________________________________________****
Perform the bootstrap based algorithm with the
nbcomp.bootplsRglm
function. By default 250 resamplings are
carried out.
set.seed(4619)
<- suppressWarnings(nbcomp.bootplsRglm(modplsglm)) aze_compl.bootYT
Plotting the bootstrap distributions of the coefficients of the components.
::boxplots.bootpls(aze_compl.bootYT) plsRglm
Computing the bootstrap based confidence intervals of the coefficients of the components.
::confints.bootpls(aze_compl.bootYT)
plsRglm#> Warning in norm.inter(t, adj.alpha): extreme order statistics used
#> as endpoints
#> Warning in norm.inter(t, adj.alpha): extreme order statistics used
#> as endpoints
#> Warning in norm.inter(t, adj.alpha): extreme order statistics used
#> as endpoints
#>
#> [1,] -0.1909709 2.3581748 -0.4489036 1.9697388 1.11236461
#> [2,] -0.1514302 0.8243332 -0.3376837 0.7471037 0.20267973
#> [3,] -0.4319055 1.7434199 -0.6285256 1.5594694 0.22338048
#> [4,] -0.3748813 1.0973698 -0.5490521 0.9657205 0.05733105
#> [5,] -0.3963722 0.8830876 -0.5130832 0.8905919 -0.22108731
#> [6,] -0.6441165 1.2480090 -0.9267788 1.1913771 -0.47946992
#> [7,] -0.6863159 1.0264625 -0.8829492 1.0706084 -0.60361567
#> [8,] -1.1970167 1.4738770 -1.2043483 1.7957092 -1.37023450
#> [9,] -0.8340372 0.9057530 -1.0433636 0.9124285 -0.72858196
#> [10,] -1.0786394 1.2056122 -1.1600493 1.2398892 -1.11385399
#>
#> [1,] 3.531007 0.71686088 2.1861124
#> [2,] 1.287467 0.07412900 0.8286440
#> [3,] 2.411375 -0.03355241 1.6424348
#> [4,] 1.572104 -0.45493576 1.1619803
#> [5,] 1.182588 -0.38904916 0.9004701
#> [6,] 1.638686 -0.53350462 1.4582694
#> [7,] 1.349942 -0.75368847 1.0112706
#> [8,] 1.629823 -1.54876392 1.3140140
#> [9,] 1.227210 -0.74456869 1.1516065
#> [10,] 1.286084 -1.02902293 1.3514701
#> attr(,"typeBCa")
#> [1] TRUE
Computing the bootstrap based confidence intervals of the coefficients of the components.
suppressWarnings(plsRglm::plots.confints.bootpls(plsRglm::confints.bootpls(aze_compl.bootYT)))
The package also features our bootstrap based algorithm to select,
for a given \(\eta\) value, the number
of components in sgpls regression. It is implemented in the
nbcomp.bootsgpls
.
set.seed(4619)
data(prostate, package="spls")
nbcomp.bootsgpls((prostate$x)[,1:30], prostate$y, R=250, eta=0.2, typeBCa = FALSE)
#> [1] "eta = 0.2"
#> [1] "K = 1"
#> [1] "K = 2"
#> [1] "K = 3"
#> [1] "K = 4"
#> [1] "K = 5"
#> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
#>
#> Optimal parameters: eta = 0.2, K = 4
#> $err.mat
#>
#> eta= 0.2 ; K= 4 0.2554545
#>
#> $eta.opt
#> [1] 0.2
#>
#> $K.opt
#> [1] 4
#>
#> $cands
#> [,1] [,2] [,3] [,4]
#> [1,] 0.2554545 30 0.2 4
A doParallel
and foreach
based parallel
computing version of the algorithm is implemented as the
nbcomp.bootspls.para
function.
nbcomp.bootsgpls.para((prostate$x)[,1:30], prostate$y, R=250, eta=c(.2,.5), maxnt=10, typeBCa = FALSE)
#> [1] "eta = 0.2"
#> [1] "K = 1"
#> [1] "K = 2"
#> [1] "K = 3"
#> [1] "K = 4"
#> [1] "K = 5"
#> [1] "eta = 0.5"
#> [1] "K = 1"
#> [1] "K = 2"
#> [1] "K = 3"
#> [1] "K = 4"
#> [1] "K = 5"
#> Warning: glm.fit: algorithm did not converge
#> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
#> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
#> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
#>
#> Optimal parameters: eta = 0.5, K = 4
#> $err.mat
#>
#> eta= 0.2 ; K= 4 0.2727273
#> eta= 0.5 ; K= 4 0.2636364
#>
#> $eta.opt
#> [1] 0.5
#>
#> $K.opt
#> [1] 4
#>
#> $cands
#> [,1] [,2] [,3] [,4]
#> [1,] 0.2636364 27.4 0.5 4
rm(list=c("Xpine","ypine","bbb","bbb2","aze_compl","Xaze_compl","yaze_compl","aze_compl.bootYT","dataset","modplsglm","pine","datasetpine","prostate","res_boot_rep"))
#> Warning in rm(list = c("Xpine", "ypine", "bbb", "bbb2",
#> "aze_compl", "Xaze_compl", : object 'res_boot_rep' not found
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.