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input object.

output ofVariableSelection orBiSelection.

parameter ID with respect to the gridLambda. IfNULL, aggregated coefficients across the models run with thecalibrated parameter are returned.

character string indicating if coefficients of predictors(side="X") or outcomes (side="Y") should be returned. Onlyapplicable to PLS models.

component ID. Only applicable to PLS models.

function to use to aggregate coefficients of visited models overresampling iterations. Recommended functions includemedian ormean.

additional arguments to be passed toFUN.

output ofVariableSelection orGraphicalModel.

matrix of stability scores obtained with different combinations ofparameters where rows correspond to different values of the parametercontrolling the level of sparsity in the underlying feature selectionalgorithm and columns correspond to different values of the threshold inselection proportions. IfS=NULL, argumentstability must beprovided.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

vector encoding the grouping structure among predictors. Thisargument indicates the number of variables in each group. Only used formodels with group penalisation (e.g.implementation=GroupPLS orimplementation=SparseGroupPLS).

optional vector encoding the grouping structure amongoutcomes. This argument indicates the number of variables in each group.Only used ifimplementation=GroupPLS orimplementation=SparseGroupPLS.

matrix of parameters controlling the number of selectedvariables (for sparse PCA/PLS) or groups (for group and sparse group PLS)in X.

matrix of parameters controlling the number of selectedvariables (for sparse PLS) or groups (for group or sparse group PLS) in Y.Only used iffamily="gaussian".

matrix of parameters controlling the level of sparsity withingroups in X. Only used ifimplementation=SparseGroupPLS.

matrix of parameters controlling the level of sparsity withingroups in X. Only used ifimplementation=SparseGroupPLS andfamily="gaussian".

number of components.

logical indicating if the data should be scaled (i.e.transformed so that all variables have a standard deviation of one).

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

type of PLS model. This parameter must be set tofamily="gaussian" for continuous outcomes, or tofamily="binomial" for categorical outcomes. Only used ifydata is provided.

function to use for feature selection. Possiblefunctions are:SparsePCA,SparsePLS,GroupPLS,SparseGroupPLS.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

logical indicating if complementary pair stability selectionshould be done. For this, the algorithm is applied on two non-overlappingsubsets of half of the observations. A feature is considered as selected ifit is selected for both subsamples. With this method, the data is splitK/2 times (K models are fitted). Only used ifPFER_method="MB".

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

number of cores to use for parallel computing (see argumentworkers inmultisession). Usingn_cores>1 is only supported withoptimisation="grid_search".

logical indicating if the input datasetsxdata andydata should be included in the output.

logical indicating if a loading bar and messages should beprinted.

sound indicating the end of the run. Possible values are:NULL (no sound) or an integer between 1 and 11 (see argumentsound inbeep).

additional parameters passed to the functions provided inimplementation orresampling.

amatrix of the best stability scores and corresponding parameterscontrolling the level of sparsity in the underlying algorithm for differentnumbers of components. Possible columns include:comp (componentindex),nx (number of predictors to include, parameter of theunderlying algorithm),alphax (sparsity within the predictor groups,parameter of the underlying algorithm),pix (threshold in selectionproportion for predictors),ny (number of outcomes to include,parameter of the underlying algorithm),alphay (sparsity within theoutcome groups, parameter of the underlying algorithm),piy(threshold in selection proportion for outcomes),S (stabilityscore). Columns that are not relevant to the model are not reported (e.g.alpha_x andalpha_y are not returned for sparse PLS models).

a matrix of the best stability scores for differentcombinations of parameters controlling the sparsity and components.

a binary matrix encoding stably selected predictors.

a matrix of calibrated selection proportions forpredictors.

a binary matrix encoding stably selectedoutcomes. Only returned for PLS models.

a matrix ofcalibrated selection proportions for outcomes. Only returned for PLSmodels.

a binary matrix encoding stable relationshipsbetween predictor and outcome variables. Only returned for PLS models.

a binary matrix encoding stably selected predictors.

a matrix of selection proportions for predictors.

a binary matrix encoding stably selected outcomes.Only returned for PLS models.

a matrix of selectionproportions for outcomes. Only returned for PLS models.

anarray of estimated loadings coefficients for the different components(rows), for the predictors (columns), as obtained across theKvisited models (along the third dimension).

an array ofestimated loadings coefficients for the different components (rows), forthe outcomes (columns), as obtained across theK visited models(along the third dimension). Only returned for PLS models.

alist withtype="bi_selection" and values used for argumentsimplementation,family,scale,resampling,cpss andPFER_method.

a list with values usedfor argumentsK,group_x,group_y,LambdaX,LambdaY,AlphaX,AlphaY,pi_list,tau,n_cat,pk,n (number of observations),PFER_thr,FDP_thr andseed. The datasetsxdataandydata are also included ifoutput_data=TRUE.

average number of features selected by the underlying algorithm.

total number of features.

threshold in selection proportions. If n_cat=3, these values mustbe >0.5 and <1. If n_cat=2, these values must be >0 and <1.

number of resampling iterations.

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

vector or matrix of penalty parameters.

optional vector of penalty parameters to use forother blocks in the iterative multi-block procedure.

a matrix of (block-specific) penaltyparameters. In multi-block stability selection, rows correspond to sets ofpenalty parameters and columns correspond to different blocks.

logical matrix encoding the type of procedurefor data with multiple blocks in stability selection graphical modelling.For multi-block estimation, each block is calibrated separately whileothers blocks are weakly penalised (TRUE only for the blockcurrently being calibrated andFALSE for other blocks). Otherapproaches with joint calibration of the blocks are allowed (all entriesare set toTRUE).

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

matrix of parameters controlling the number of splits in thedecision tree.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

additional parameters passed torpart.

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspondto different model parameters. Columns correspond to predictors. Indicesalong the third dimension correspond to outcome variable(s).

output ofVariableSelection,GraphicalModel orBiSelection.

character string indicating if the box around the plot should bedrawn. Possible values include:"o" (default, the box is drawn), or"n" (no box).

label of the x-axis.

label of the y-axis.

font size for axes.

font size for labels.

font size for text legend entries.

type of point, as inpoints.

logical indicating if the points should be linked by lines. Onlyused ifstability is the output ofBiSelection orClustering.

vector of colours.

logical indicating if the legend should be included.

integer indicating the number of columns in the legend.

output ofVariableSelection,GraphicalModel orBiSelection.

ID of the block to visualise. Only used for multi-blockstability selection graphical models. Ifblock_id=NULL, all blocksare represented in separate panels.

vector of colours.

type of point, as inpoints.

size of point.

displayed range along the x-axis. Only used ifstabilityis the output ofBiSelection.

displayed range along the y-axis. Only used ifstabilityis the output ofBiSelection.

character string indicating if the box around the plot should bedrawn. Possible values include:"o" (default, the box is drawn), or"n" (no box).

logical indicating if the points should be linked by lines. Onlyused ifstability is the output ofBiSelection orClustering.

line type, as inpar. Only used ifstability is the output ofBiSelection.

line width, as inpar. Only used ifstability is the output ofBiSelection.

logical indicating if the calibrated parameter(s) shouldbe indicated by lines.

logical indicating if the calibrated threshold in selectionproportion inX should be written for each point. Only used ifstability is the output ofBiSelection.

logical indicating if the calibrated threshold in selectionproportion inY should be written for each point. Only used ifstability is the output ofBiSelection withpenalisation of the outcomes.

distance between the point and the text, as intext. Only used ifshow_pix=TRUE orshow_piy=TRUE.

logical indicating if the legend should be included.

length of the colour bar. Only used ifstabilityis the output ofVariableSelection orGraphicalModel.

range of the colour bar. Only used ifstability isthe output ofVariableSelection orGraphicalModel.

integer indicating the number of columns in the legend.

label of the x-axis.

label of the y-axis.

label of the z-axis. Only used ifstability is the outputofVariableSelection orGraphicalModel.

orientation of labels on the x-axis, aslas inpar.

orientation of labels on the y-axis, aslas inpar.

orientation of labels on the z-axis, aslas inpar.

font size for labels.

font size for axes.

font size for text legend entries.

logical indicating if a vertical grid should be drawn. Only usedifstability is the output ofBiSelection.

logical indicating if a horizontal grid should be drawn. Onlyused ifstability is the output ofBiSelection.

vector of possible parameters ifstability is of classbi_selection. The order of these parameters defines the order inwhich they are represented. Only used ifstability is the output ofBiSelection.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

logical indicating if a loading bar and messages should beprinted.

data matrix with observations as rows and variables as columns.

vector of penalty parameters for weighted distance calculation.Only used for distance-based clustering, including for exampleimplementation=HierarchicalClustering,implementation=PAMClustering, orimplementation=DBSCANClustering.

number of resampling iterations.

subsample size.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for clustering. Possible functionsincludeHierarchicalClustering (hierarchical clustering),PAMClustering (Partitioning Around Medoids),KMeansClustering (k-means) andGMMClustering(Gaussian Mixture Models). Alternatively, a user-defined function takingxdata andLambda as arguments and returning a binary andsymmetric matrix for which diagonal elements are equal to zero can be used.

logical indicating if the data should be scaled to ensure thatall variables contribute equally to the clustering of the observations.

logical indicating if a loading bar and messages should beprinted.

data matrix with observations as rows and variables as columns.For multi-block stability selection, the variables in data have to beordered by group.

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.IfLambda=NULL andimplementation=PenalisedGraphical,LambdaGridGraphical is used to define a relevant grid.Lambda can be provided as a vector or a matrix withlength(pk) columns.

optional vector of parameters controlling thelevel of sparsity in neighbour blocks for the multi-block procedure. To usejointly a specific set of parameters for each block,lambda_other_blocks must be set toNULL (not recommended).Only used for multi-block stability selection, i.e. iflength(pk)>1.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for graphical modelling. Ifimplementation=PenalisedGraphical, the algorithm implemented inglassoFast is used for regularised estimation ofa conditional independence graph. Alternatively, a user-defined functioncan be provided.

character string indicating if the algorithm should beinitialised at the estimated (inverse) covariance with previous penaltyparameters (start="warm") or not (start="cold"). Usingstart="warm" can speed-up the computations, but could lead toconvergence issues (in particular with smallLambda_cardinal). Onlyused forimplementation=PenalisedGraphical (see argument"start" inglassoFast).

logical indicating if the correlation (scale=TRUE) orcovariance (scale=FALSE) matrix should be used as input ofglassoFast ifimplementation=PenalisedGraphical. Otherwise, this argument must beused in the function provided inimplementation.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

number of values in the grid of parameters controllingthe level of sparsity in the underlying algorithm. Only used ifLambda=NULL.

optional maximum value for the grid in penalty parameters.Iflambda_max=NULL, the maximum value is set to the maximumcovariance in absolute value. Only used ifimplementation=PenalisedGraphical andLambda=NULL.

multiplicative factor used to define the minimumvalue in the grid.

threshold on the density. The grid is defined such thatthe density of the estimated graph does not exceed max_density.

logical indicating if a loading bar and messages should beprinted.

character string indicating the name of the package.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.IfLambda=NULL andimplementation=PenalisedRegression,LambdaGridRegression is used to define a relevant grid.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

function to use for variable selection. Possiblefunctions are:PenalisedRegression,SparsePLS,GroupPLS andSparseGroupPLS. Alternatively, a user-definedfunction can be provided.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

number of values in the grid of parameters controllingthe level of sparsity in the underlying algorithm. Only used ifLambda=NULL.

logical indicating if a loading bar and messages should beprinted.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

radius in density-based clustering, seedbscan. Only used ifimplementation=DBSCANClustering.

vector of penalty parameters for weighted distance calculation.Only used for distance-based clustering, including for exampleimplementation=HierarchicalClustering,implementation=PAMClustering, orimplementation=DBSCANClustering.

number of resampling iterations.

subsample size.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for clustering. Possible functionsincludeHierarchicalClustering (hierarchical clustering),PAMClustering (Partitioning Around Medoids),KMeansClustering (k-means) andGMMClustering(Gaussian Mixture Models). Alternatively, a user-defined function takingxdata andLambda as arguments and returning a binary andsymmetric matrix for which diagonal elements are equal to zero can be used.

logical indicating if the data should be scaled to ensure thatall variables contribute equally to the clustering of the observations.

character string indicating the type of linkage used inhierarchical clustering to define the stable clusters. Possible valuesinclude"complete","single" and"average" (seeargument"method" inhclust for a full list).Only used ifimplementation=HierarchicalClustering.

logical indicating if rows (ifrow=TRUE) or columns (ifrow=FALSE) contain the items to cluster.

character string indicating the type of optimisationmethod to calibrate the regularisation parameter (only used ifLambda is notNULL). Withoptimisation="grid_search"(the default), all values inLambda are visited.Alternatively, optimisation algorithms implemented innloptr can be used withoptimisation="nloptr".By default, we use"algorithm"="NLOPT_GN_DIRECT_L","xtol_abs"=0.1,"ftol_abs"=0.1 and"maxeval" defined aslength(Lambda). These values can bechanged by providing the argumentopts (seenloptr).

number of cores to use for parallel computing (see argumentworkers inmultisession). Usingn_cores>1 is only supported withoptimisation="grid_search".

logical indicating if the input datasetsxdata andydata should be included in the output.

logical indicating if a loading bar and messages should beprinted.

sound indicating the end of the run. Possible values are:NULL (no sound) or an integer between 1 and 11 (see argumentsound inbeep).

additional parameters passed to the functions provided inimplementation orresampling.

a matrixof the best stability scores for different (sets of) parameters controllingthe number of clusters and penalisation of attribute weights.

amatrix of numbers of clusters.

a matrix of regularisationparameters for attribute weights.

a matrix of the average numberof selected attributes by the underlying algorithm with differentregularisation parameters.

an array of consensus matrices.Rows and columns correspond to items. Indices along the third dimensioncorrespond to different parameters controlling the number of clusters andpenalisation of attribute weights.

an array of selectionproportions. Columns correspond to attributes. Rows correspond to differentparameters controlling the number of clusters and penalisation of attributeweights.

a list withtype="clustering" and valuesused for argumentsimplementation,linkage, andresampling.

a list with values used for argumentsK,tau,pk,n (number of observations inxdata), andseed.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, nrow(xdata)).

radius in density-based clustering, seedbscan. Only used ifimplementation=DBSCANClustering.

vector of penalty parameters.

logical indicating if the data should be scaled to ensure thatall variables contribute equally to the clustering of the observations.

logical indicating if rows (ifrow=TRUE) or columns (ifrow=FALSE) contain the items to cluster.

function to use for clustering. Possible functionsincludeHierarchicalClustering (hierarchical clustering),PAMClustering (Partitioning Around Medoids),KMeansClustering (k-means) andGMMClustering(Gaussian Mixture Models). Alternatively, a user-defined function takingxdata andLambda as arguments and returning a binary andsymmetric matrix for which diagonal elements are equal to zero can be used.

additional parameters passed to the function provided inimplementation.

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspond todifferent model parameters. Columns correspond to predictors. Indices alongthe third dimension correspond to outcome variable(s).

array of model coefficients. Rows correspond todifferent model parameters. Columns correspond to predictors. Indices alongthe third dimension correspond to outcome variable(s).

output fromClustering. Alternatively, it can bethe estimated co-membership matrix (seeCoMembership).

output fromSimulateClustering.Alternatively,it can be the true co-membership matrix (seeCoMembership).

additional arguments to be passed toClusters.

number of True Positives (TP)

number of FalseNegatives (TN)

number of False Positives (FP)

numberof True Negatives (TN)

sensitivity, i.e. TP/(TP+FN)

specificity, i.e. TN/(TN+FP)

accuracy,i.e. (TP+TN)/(TP+TN+FP+FN)

precision (p), i.e.TP/(TP+FP)

recall (r), i.e. TP/(TP+FN)

F1-score, i.e. 2*p*r/(p+r)

Rand Index, i.e.(TP+TN)/(TP+FP+TN+FN)

Adjusted Rand Index (ARI), i.e.2*(TP*TN-FP*FN)/((TP+FP)*(TN+FP)+(TP+FN)*(TN+FN))

Jaccardindex, i.e. TP/(TP+FP+FN)

vector of group membership.

output ofVariableSelection.

character string indicating if coefficients of the predictor(side="X") or outcome (side="Y") coefficients should bereturned. Optionside="Y" is only applicable to PLS models.

component ID. Only applicable to PLS models.

vector of iteration IDs. Ifiterations=NULL, thecoefficients of all visited models are returned. This number must besmaller than the number of iterationsK used for the stabilityselection run.

output from a first run ofVariableSelection,GraphicalModel, orClustering.

output from a second run ofVariableSelection,GraphicalModel, orClustering.

logical indicating if the beta coefficients of visitedmodels should be concatenated. Only applicable to variable selection orclustering.

a stability object.

another stability object (optional).

consensus matrix.

matrix of co-sampling counts.

consensus co-membership matrix.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

radius in density-based clustering, seedbscan.

vector of penalty parameters (see argumentlambda incosa2). Unweighted distance matrices are used ifLambda=NULL.

character string indicating the type of distance to use. IfLambda=NULL, possible values include"euclidean","maximum","canberra","binary", and"minkowski" (see argumentmethod indist). Otherwise, possible values include"euclidean" (pwr=2) or"absolute" (pwr=1) (seeargumentpwr incosa2).

additional parameters passed todbscan(except forminPts which is fixed to2),dist, orcosa2. Ifweighted=TRUE, parametersniter (default to 1) andnoit (default to 100) correspond to the number of iterations incosa2 to calculate weights and may need to bemodified.

an array of binary and symmetricco-membership matrices.

a matrix of median weights byfeature.

matrix of dummy variables.

logical indicating if messages should be printed.

output ofVariableSelection.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

a vector of refitted intercepts for theKcalibrated models.

a matrix of beta coefficients from theK calibrated models.

a list ofK models thatcan be used for prediction. These models are of class"lm" iffamily="gaussian" or"glm" iffamily="binomial".

type of regression, extracted fromstability. Possible values are"gaussian" or"binomial".

output ofEnsemble.

matrix of predictors with observations as rows and variables ascolumns.

additional parameters passed topredict.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

optional test set (predictor data).

optional test set (outcome data).

output ofVariableSelection. Ifstability=NULL (the default), a model including all variables inxdata as predictors is fitted. Argumentfamily must beprovided in this case.

type of regression model. Possible values include"gaussian" (linear regression),"binomial" (logisticregression), and"cox" (survival analysis). If provided, thisargument must be consistent with inputstability.

optional function to refit the model. Ifimplementation=NULL andstability is the output ofVariableSelection,lm (linearregression),coxph (Cox regression),glm (logistic regression), ormultinom (multinomial regression) is used.

optional function to compute predicted values from themodel refitted withimplementation.

resampling approach to create the training set. The defaultis"subsampling" for sampling without replacement of a proportiontau of the observations. Alternatively, this argument can be afunction to use for resampling. This function must use arguments nameddata andtau and return the IDs of observations to beincluded in the resampled dataset.

number of training-test splits. Only used ifnew_xdata andnew_ydata are not provided.

proportion of observations used in the training set. Only used ifnew_xdata andnew_ydata are not provided.

value of the seed to ensure reproducibility of the results. Onlyused ifnew_xdata andnew_ydata are not provided.

number of thresholds to use to construct the ROC curve. Ifn_thr=NULL, all predicted probability values are iteratively used asthresholds. For faster computations on large data, less thresholds can beused. Only applicable to logistic regression.

numeric indicating the time for which the survival probabilitiesare computed. Only applicable to Cox regression.

logical indicating if a loading bar and messages should beprinted.

additional parameters passed to the function provided inresampling.

True Positive Rate (for logistic regressiononly).

False Positive Rate (for logistic regression only).

Area Under the Curve (for logistic regression only).

Concordance index (for Cox regression only).

matrix of estimated beta coefficients across theKiterations. Coefficients are extracted using thecoeffunction.

matrix or vector of selection proportions.

Per Family Error Rate.

threshold in selection proportions.

vector or matrix of data. In regression, this should be theoutcome data.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

number of folds.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

additional parameters passed toMclust.

an array of binary and symmetricco-membership matrices.

a matrix of median weights byfeature.

adjacency matrix or output ofGraphicalModel.

optional vector of node labels. This vector must contain asmany entries as there are rows/columns in the adjacency matrix and must bein the same order (the order is used to assign labels to nodes).

optional vector of node colours. This vector must containas many entries as there are rows/columns in the adjacency matrix and mustbe in the same order (the order is used to assign colours to nodes).Integers, named colours or RGB values can be used.

optional vector of node shapes. This vector must contain asmany entries as there are rows/columns in the adjacency matrix and must bein the same order (the order is used to assign shapes to nodes). Possiblevalues are"circle","square","triangle" or"star".

optional character string for edge colour. Integers, namedcolours or RGB values can be used.

optional character string for label colour. Integers,named colours or RGB values can be used.

character string indicating how the adjacency matrix should beinterpreted. Possible values include"undirected" or"directed" (seegraph_from_adjacency_matrix).

indicating if entries of the adjacency matrix should defineedge width. Ifweighted=FALSE, an unweighted igraph object iscreated, all edges have the same width. Ifweighted=TRUE, edge widthis defined by the corresponding value in the adjacency matrix. Ifweighted=NULL, nodes are linked by as many edges as indicated in theadjacency matrix (integer values are needed).

logical indicating if unconnected nodes (satellites) shouldbe included in the igraph object.

first graph. Possible inputs are: adjacency matrix, origraph object, or output ofGraphicalModel,VariableSelection,BiSelection, or output ofSimulateGraphical,SimulateRegression.

second graph.

vector of edge colours. The first entry of the vector definesthe colour of edges ingraph1 only, second entry is for edges ingraph2 only and third entry is for common edges.

vector of line types for edges. The order is defined as forargumentcol.

optional vector of node colours. This vector must containas many entries as there are rows/columns in the adjacency matrix and mustbe in the same order (the order is used to assign colours to nodes).Integers, named colours or RGB values can be used.

logical indicating if the node labels should be displayed.

additional arguments to be passed toGraph.

matrix with observations as rows and variables as columns.

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.IfLambda=NULL andimplementation=PenalisedGraphical,LambdaGridGraphical is used to define a relevant grid.Lambda can be provided as a vector or a matrix withlength(pk) columns.

logical matrix encoding the type of procedure touse for data with multiple blocks in stability selection graphicalmodelling. For multi-block estimation, the stability selection model isconstructed as the union of block-specific stable edges estimated while theothers are weakly penalised (TRUE only for the block currently beingcalibrated andFALSE for other blocks). Other approaches with jointcalibration of the blocks are allowed (all entries are set toTRUE).

logical indicating if the correlation (scale=TRUE) orcovariance (scale=FALSE) matrix should be used as input ofglassoFast ifimplementation=PenalisedGraphical. Otherwise, this argument must beused in the function provided inimplementation.

function to use for graphical modelling. Ifimplementation=PenalisedGraphical, the algorithm implemented inglassoFast is used for regularised estimation ofa conditional independence graph. Alternatively, a user-defined functioncan be provided.

character string indicating if the algorithm should beinitialised at the estimated (inverse) covariance with previous penaltyparameters (start="warm") or not (start="cold"). Usingstart="warm" can speed-up the computations, but could lead toconvergence issues (in particular with smallLambda_cardinal). Onlyused forimplementation=PenalisedGraphical (see argument"start" inglassoFast).

additional parameters passed to the function provided inimplementation.

data matrix with observations as rows and variables as columns.For multi-block stability selection, the variables in data have to beordered by group.

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.IfLambda=NULL andimplementation=PenalisedGraphical,LambdaGridGraphical is used to define a relevant grid.Lambda can be provided as a vector or a matrix withlength(pk) columns.

optional vector of parameters controlling thelevel of sparsity in neighbour blocks for the multi-block procedure. To usejointly a specific set of parameters for each block,lambda_other_blocks must be set toNULL (not recommended).Only used for multi-block stability selection, i.e. iflength(pk)>1.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for graphical modelling. Ifimplementation=PenalisedGraphical, the algorithm implemented inglassoFast is used for regularised estimation ofa conditional independence graph. Alternatively, a user-defined functioncan be provided.

character string indicating if the algorithm should beinitialised at the estimated (inverse) covariance with previous penaltyparameters (start="warm") or not (start="cold"). Usingstart="warm" can speed-up the computations, but could lead toconvergence issues (in particular with smallLambda_cardinal). Onlyused forimplementation=PenalisedGraphical (see argument"start" inglassoFast).

logical indicating if the correlation (scale=TRUE) orcovariance (scale=FALSE) matrix should be used as input ofglassoFast ifimplementation=PenalisedGraphical. Otherwise, this argument must beused in the function provided inimplementation.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

logical indicating if complementary pair stability selectionshould be done. For this, the algorithm is applied on two non-overlappingsubsets of half of the observations. A feature is considered as selected ifit is selected for both subsamples. With this method, the data is splitK/2 times (K models are fitted). Only used ifPFER_method="MB".

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

number of values in the grid of parameters controllingthe level of sparsity in the underlying algorithm. Only used ifLambda=NULL.

optional maximum value for the grid in penalty parameters.Iflambda_max=NULL, the maximum value is set to the maximumcovariance in absolute value. Only used ifimplementation=PenalisedGraphical andLambda=NULL.

multiplicative factor used to define the minimumvalue in the grid.

threshold on the density. The grid is defined such thatthe density of the estimated graph does not exceed max_density.

character string indicating the type of optimisationmethod. Withoptimisation="grid_search" (the default), all values inLambda are visited. Alternatively, optimisation algorithmsimplemented innloptr can be used withoptimisation="nloptr". By default, we use"algorithm"="NLOPT_GN_DIRECT_L","xtol_abs"=0.1,"ftol_abs"=0.1 and"maxeval"=Lambda_cardinal. These valuescan be changed by providing the argumentopts (seenloptr). For stability selection using penalisedregression,optimisation="grid_search" may be faster as it allowsfor warm start.

number of cores to use for parallel computing (see argumentworkers inmultisession). Usingn_cores>1 is only supported withoptimisation="grid_search".

logical indicating if the input datasetsxdata andydata should be included in the output.

logical indicating if a loading bar and messages should beprinted.

sound indicating the end of the run. Possible values are:NULL (no sound) or an integer between 1 and 11 (see argumentsound inbeep).

additional parameters passed to the functions provided inimplementation orresampling.

amatrix of the best stability scores for different (sets of) parameterscontrolling the level of sparsity in the underlying algorithm.

a matrix of parameters controlling the level of sparsity inthe underlying algorithm.

a matrix of the average number ofselected features by the underlying algorithm with different parameterscontrolling the level of sparsity.

a matrix of the calibratednumber of stably selected features with different parameters controllingthe level of sparsity.

a matrix of calibrated thresholds inselection proportions for different parameters controlling the level ofsparsity in the underlying algorithm.

a matrix of upper-boundsin PFER of calibrated stability selection models with different parameterscontrolling the level of sparsity.

a matrix of upper-bounds inFDP of calibrated stability selection models with different parameterscontrolling the level of sparsity.

a matrix of stabilityscores obtained with different combinations of parameters. Columnscorrespond to different thresholds in selection proportions.

a matrix of upper-bounds in FDP obtained with differentcombinations of parameters. Columns correspond to different thresholds inselection proportions. Only returned iflength(pk)=1.

a matrix of upper-bounds in PFER obtained with differentcombinations of parameters. Columns correspond to different thresholds inselection proportions. Only returned iflength(pk)=1.

an array of selection proportions. Rows and columnscorrespond to nodes in the graph. Indices along the third dimensioncorrespond to different parameters controlling the level of sparsity in theunderlying algorithm.

a matrix of signs of Pearson'scorrelations estimated fromxdata.

a list withtype="graphical_model" and values used for argumentsimplementation,start,resampling,cpss andPFER_method.

a list with values used for argumentsK,pi_list,tau,n_cat,pk,n(number of observations inxdata),PFER_thr,FDP_thr,seed,lambda_other_blocks, andSequential_template.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

type of PLS model. Iffamily="gaussian", a group PLSmodel as defined ingPLS is run (for continuousoutcomes). Iffamily="binomial", a PLS-DA model as defined ingPLSda is run (for categorical outcomes).

vector encoding the grouping structure among predictors. Thisargument indicates the number of variables in each group.

optional vector encoding the grouping structure amongoutcomes. This argument indicates the number of variables in each group.

matrix of parameters controlling the number of selected groupsat current component, as defined byncomp.

number of selected groups in previous components. Onlyused ifncomp > 1. The argumentkeepX insgPLS is obtained by concatenatingkeepX_previous andLambda.

number of selected groups of outcome variables. This argument isdefined as insgPLS. Only used iffamily="gaussian".

number of components.

logical indicating if the data should be scaled (i.e.transformed so that all variables have a standard deviation of one). Onlyused iffamily="gaussian".

additional arguments to be passed togPLS orgPLSda.

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspondto different model parameters. Columns correspond to predictors (startingwith "X") or outcomes (starting with "Y") variables for differentcomponents (denoted by "PC").

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

vector of penalty parameters (see argumentlambda incosa2). Unweighted distance matrices are used ifLambda=NULL.

character string indicating the type of distance to use. IfLambda=NULL, possible values include"euclidean","maximum","canberra","binary", and"minkowski" (see argumentmethod indist). Otherwise, possible values include"euclidean" (pwr=2) or"absolute" (pwr=1) (seeargumentpwr incosa2).

character string indicating the type of linkage used inhierarchical clustering to define the stable clusters. Possible valuesinclude"complete","single" and"average" (seeargument"method" inhclust for a full list).Only used ifimplementation=HierarchicalClustering.

additional parameters passed tohclust,dist, orcosa2. Parametersniter (default to 1) andnoit (default to 100) correspond tothe number of iterations incosa2 to calculate weightsand may need to be modified. Argumentpwr incosa2 is ignored, please providedistanceinstead.

an array of binary and symmetricco-membership matrices.

a matrix of median weights byfeature.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

optional test set (predictor data).

optional test set (outcome data).

output ofVariableSelection. Ifstability=NULL (the default), a model including all variables inxdata as predictors is fitted. Argumentfamily must beprovided in this case.

type of regression model. Possible values include"gaussian" (linear regression),"binomial" (logisticregression), and"cox" (survival analysis). If provided, thisargument must be consistent with inputstability.

optional function to refit the model. Ifimplementation=NULL andstability is the output ofVariableSelection,lm (linearregression),coxph (Cox regression),glm (logistic regression), ormultinom (multinomial regression) is used.

optional function to compute predicted values from themodel refitted withimplementation.

resampling approach to create the training set. The defaultis"subsampling" for sampling without replacement of a proportiontau of the observations. Alternatively, this argument can be afunction to use for resampling. This function must use arguments nameddata andtau and return the IDs of observations to beincluded in the resampled dataset.

number of predictors to consider.

number of training-test splits. Only used ifnew_xdata andnew_ydata are not provided.

proportion of observations used in the training set. Only used ifnew_xdata andnew_ydata are not provided.

value of the seed to ensure reproducibility of the results. Onlyused ifnew_xdata andnew_ydata are not provided.

number of thresholds to use to construct the ROC curve. Ifn_thr=NULL, all predicted probability values are iteratively used asthresholds. For faster computations on large data, less thresholds can beused. Only applicable to logistic regression.

numeric indicating the time for which the survival probabilitiesare computed. Only applicable to Cox regression.

logical indicating if a loading bar and messages should beprinted.

additional parameters passed to the function provided inresampling.

A list with, for each ofthe models (sequentially added predictors), the False Positive Rates fordifferent thresholds (columns) and different data splits (rows).

A list with, for each of the models (sequentially addedpredictors), the True Positive Rates for different thresholds (columns) anddifferent data splits (rows).

A list with, for each of themodels (sequentially added predictors), a vector of Area Under the Curve(AUC) values obtained with different data splits.

Estimatedregression coefficients from visited models.

Names of thepredictors by order of inclusion.

Binary vector indicatingwhich predictors are stably selected. Only returned ifstability isprovided.

A list with, for eachof the models (sequentially added predictors), a vector of concordanceindices obtained with different data splits.

Estimatedregression coefficients from visited models.

Names of thepredictors by order of inclusion.

Binary vector indicatingwhich predictors are stably selected. Only returned ifstability isprovided.

A list with, for eachof the models (sequentially added predictors), a vector of Q-squaredobtained with different data splits.

Estimated regressioncoefficients from visited models.

Names of the predictors byorder of inclusion.

Binary vector indicating whichpredictors are stably selected. Only returned ifstability isprovided.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

vector of penalty parameters (see argumentwbounds inKMeansSparseCluster).

additional parameters passed tokmeans (ifLambda isNULL) orKMeansSparseCluster.

an array of binary and symmetricco-membership matrices.

a matrix of median weights byfeature.

data matrix with observations as rows and variables as columns.For multi-block stability selection, the variables in data have to beordered by group.

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

optional vector of parameters controlling thelevel of sparsity in neighbour blocks for the multi-block procedure. To usejointly a specific set of parameters for each block,lambda_other_blocks must be set toNULL (not recommended).Only used for multi-block stability selection, i.e. iflength(pk)>1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for graphical modelling. Ifimplementation=PenalisedGraphical, the algorithm implemented inglassoFast is used for regularised estimation ofa conditional independence graph. Alternatively, a user-defined functioncan be provided.

character string indicating if the algorithm should beinitialised at the estimated (inverse) covariance with previous penaltyparameters (start="warm") or not (start="cold"). Usingstart="warm" can speed-up the computations, but could lead toconvergence issues (in particular with smallLambda_cardinal). Onlyused forimplementation=PenalisedGraphical (see argument"start" inglassoFast).

logical indicating if the correlation (scale=TRUE) orcovariance (scale=FALSE) matrix should be used as input ofglassoFast ifimplementation=PenalisedGraphical. Otherwise, this argument must beused in the function provided inimplementation.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

number of values in the grid of parameters controllingthe level of sparsity in the underlying algorithm.

optional maximum value for the grid in penalty parameters.Iflambda_max=NULL, the maximum value is set to the maximumcovariance in absolute value. Only used ifimplementation=PenalisedGraphical.

multiplicative factor used to define the minimumvalue in the grid.

threshold on the density. The grid is defined such thatthe density of the estimated graph does not exceed max_density.

additional parameters passed to the functions provided inimplementation orresampling.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

number of values in the grid of parameters controllingthe level of sparsity in the underlying algorithm.

logical indicating if input values should be checked(recommended).

additional parameters passed to the functions provided inimplementation orresampling.

maximum value in the grid.

minimum value in the grid.

number of values in the grid.

vector of coefficients to assign to entries of the matrix.

binary adjacency matrix of the Directed Acyclic Graph(transpose of the asymmetric matrix A in Reticular Action Model notation).The row and column names of this matrix must be defined.

input vector.

vector of coefficients to assign to entries of the matrix.

binary adjacency matrix of the Directed Acyclic Graph(transpose of the asymmetric matrix A in Reticular Action Model notation).The row and column names of this matrix must be defined.

binary and symmetric matrix encoding the nonzeroentries in the residual covariance matrix (symmetric matrix S in ReticularAction Model notation). By default, this is the identity matrix (noresidual covariance).

binary adjacency matrix of the Directed Acyclic Graph(transpose of the asymmetric matrix A in Reticular Action Model notation).The row and column names of this matrix must be defined.

binary and symmetric matrix encoding the nonzeroentries in the residual covariance matrix (symmetric matrix S in ReticularAction Model notation). By default, this is the identity matrix (noresidual covariance).

optional vector of manifest variable names.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

vector of penalty parameters (see argumentlambda incosa2). Unweighted distance matrices are used ifLambda=NULL.

character string indicating the type of distance to use. IfLambda=NULL, possible values include"euclidean","maximum","canberra","binary", and"minkowski" (see argumentmethod indist). Otherwise, possible values include"euclidean" (pwr=2) or"absolute" (pwr=1) (seeargumentpwr incosa2).

additional parameters passed topam,dist, orcosa2. Ifweighted=TRUE, parametersniter (default to 1) andnoit (default to 100) correspond to the number of iterations incosa2 to calculate weights and may need to bemodified.

an array of binary and symmetricco-membership matrices.

a matrix of median weights byfeature.

average number of features selected by the underlying algorithm.

threshold in selection proportions.

total number of features.

number of resampling iterations.

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

binary matrix of size(ncol(xdata) * ncomp). Thebinary entries indicate which predictors (in rows) contribute to thedefinition of each component (in columns). IfselectedX=NULL, allpredictors are selected for all components.

binary matrix of size(ncol(ydata) * ncomp). Thebinary entries indicate which outcomes (in rows) contribute to thedefinition of each component (in columns). IfselectedY=NULL, alloutcomes are selected for all components.

type of PLS model. Onlyfamily="gaussian" is supported.This corresponds to a PLS model as defined inpls(for continuous outcomes).

number of components.

logical indicating if the data should be scaled (i.e.transformed so that all variables have a standard deviation of one).

matrix of X-weights.

matrix oftransformed X-weights.

matrix of X-loadings.

matrix of Y-weights.

matrix of X-scores.

matrix of Y-scores.

matrix needed forpredictions.

matrix needed for predictions.

vector used for centering of predictors, needed forpredictions.

vector used for scaling of predictors, neededfor predictions.

vector used for centering of outcomes,needed for predictions.

vector used for scaling of outcomes,needed for predictions.

a list withfamily andscale values used for the run.

a list withselectedX andselectedY values used for the run.

matrix with observations as rows and variables as columns.

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

matrix of parameters controlling the level of sparsity.

logical matrix encoding the type of procedure touse for data with multiple blocks in stability selection graphicalmodelling. For multi-block estimation, the stability selection model isconstructed as the union of block-specific stable edges estimated while theothers are weakly penalised (TRUE only for the block currently beingcalibrated andFALSE for other blocks). Other approaches with jointcalibration of the blocks are allowed (all entries are set toTRUE).

logical indicating if the correlation (scale=TRUE) orcovariance (scale=FALSE) matrix should be used as input ofglassoFast ifimplementation=PenalisedGraphical. Otherwise, this argument must beused in the function provided inimplementation.

character string indicating if the algorithm should beinitialised at the estimated (inverse) covariance with previous penaltyparameters (start="warm") or not (start="cold"). Usingstart="warm" can speed-up the computations, but could lead toconvergence issues (in particular with smallLambda_cardinal). Onlyused forimplementation=PenalisedGraphical (see argument"start" inglassoFast).

logical indicating if the estimated precision matricesshould be stored and returned.

additional parameters passed to the function provided inimplementation.

matrix with observations as rows and variables as columns.Column names must be defined and in line with the row and column names ofadjacency.

binary adjacency matrix of the Directed Acyclic Graph(transpose of the asymmetric matrix A in Reticular Action Model notation).The row and column names of this matrix must be defined.

optional binary matrix indicating which coefficients areregularised.

binary and symmetric matrix encoding the nonzeroentries in the residual covariance matrix (symmetric matrix S in ReticularAction Model notation). By default, this is the identity matrix (noresidual covariance).

matrix of parameters controlling the level of sparsity. Onlythe minimum, maximum and length are used inPenalisedOpenMx.

additional parameters passed toOpenMx functions (forPenalisedOpenMx), orglmnet (forPenalisedLinearSystem).

matrix of binary selection status. Rowscorrespond to different regularisation parameters. Columns correspond todifferent parameters to estimated.

matrix of modelcoefficients. Rows correspond to different regularisation parameters.Columns correspond to different parameters to estimated.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

matrix of parameters controlling the level of sparsity.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

type of penalisation to use. Ifpenalisation="classic" (the default), penalised regression is donewith the same regularisation parameter, or usingpenalty.factor, ifspecified. Ifpenalisation="randomised", the regularisation for eachof the variables is uniformly chosen betweenlambda andlambda/gamma. Ifpenalisation="adaptive", the regularisationfor each of the variables is weighted by1/abs(beta)^gamma wherebeta is the regression coefficient obtained from unpenalisedregression.

parameter for randomised or adaptive regularisation. Default isgamma=0.5 for randomised regularisation andgamma=2 foradaptive regularisation. The parametergamma should be between0 and1 for randomised regularisation.

additional parameters passed toglmnet.

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspondto different model parameters. Columns correspond to predictors. Indicesalong the third dimension correspond to outcome variable(s).

matrix of predictors with observations as rows and variables ascolumns.

output ofPLS.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

output ofVariableSelection orBiSelection. Ifstability=NULL (the default), a modelincluding all variables inxdata as predictors is fitted. Argumentfamily must be provided in this case.

type of regression model. Possible values include"gaussian" (linear regression),"binomial" (logisticregression),"multinomial" (multinomial regression), and"cox" (survival analysis). If provided, this argument must beconsistent with inputstability.

optional function to refit the model. Ifstability is the output ofVariableSelection, a regression model is refitted.Ifimplementation=NULL andLambda=0, this is done usinglm (for linearregression),coxph (Cox regression),glm (logistic regression), ormultinom (multinomial regression).IfLambda=NULL, a Ridge regression is fitted and calibrated by cross validation usingcv.glmnet.The functionPLS is used ifstability is the output ofBiSelection.

optional vector of penalty parameters.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

logical indicating if a loading bar and messages should beprinted.

additional arguments to be passed to the function provided inimplementation.

forlinear regression ("gaussian" family).

for Cox regression ("cox"family).

for logistic regression("binomial" family).

formultinomial regression ("multinomial" family).

vector or matrix of data. In regression, this should be theoutcome data.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

additional parameters passed to the function provided inresampling.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.IfLambda=NULL andimplementation=PenalisedRegression,LambdaGridRegression is used to define a relevant grid.

vector encoding the grouping structure among predictors. Thisargument indicates the number of variables in each group. Only used formodels with group penalisation (e.g.implementation=GroupPLS orimplementation=SparseGroupPLS).

logical indicating if the predictor data should be scaled.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

function to use for variable selection. Possiblefunctions are:PenalisedRegression,SparsePLS,GroupPLS andSparseGroupPLS. Alternatively, a user-definedfunction can be provided.

additional parameters passed to the function provided inimplementation.

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspondto different model parameters. Columns correspond to predictors. Indicesalong the third dimension correspond to outcome variable(s).

output fromVariableSelection,BiSelection, orGraphicalModel. Alternatively,it can be a binary matrix of selected variables (in variable selection) ora binary adjacency matrix (in graphical modelling)

output fromSimulateRegression,SimulateComponents, orSimulateGraphical.Alternatively, it can be a binary matrix of true predictors (in variableselection) or the true binary adjacency matrix (in graphical modelling).

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

optional correlation matrix. Only used in graphical modelling.

optional threshold in correlation. Only used in graphicalmodelling and when argument "cor" is not NULL.

number of True Positives (TP)

number of FalseNegatives (TN)

number of False Positives (FP)

numberof True Negatives (TN)

sensitivity, i.e. TP/(TP+FN)

specificity, i.e. TN/(TN+FP)

accuracy,i.e. (TP+TN)/(TP+TN+FP+FN)

precision (p), i.e.TP/(TP+FP)

recall (r), i.e. TP/(TP+FN)

F1-score, i.e. 2*p*r/(p+r)

binary adjacency matrix or output ofGraphicalModel,VariableSelection, orBiSelection.

true binary adjacency matrix or output ofSimulateGraphical orSimulateRegression.

vector of edge colours. The first entry of the vector definesthe colour of False Positive edges, second entry is for True Negatives andthird entry is for True Positives.

vector of line types for edges. The order is defined as forargumentcol.

optional vector of node colours. This vector must containas many entries as there are rows/columns in the adjacency matrix and mustbe in the same order (the order is used to assign colours to nodes).Integers, named colours or RGB values can be used.

logical indicating if the node labels should be displayed.

additional arguments to be passed toGraph.

vector (in variable selection) or matrix (in graphical modelling)containing values of0,1,2 or3.

optional correlation matrix. Only used in graphical modelling.

optional threshold in correlation. Only used in graphicalmodelling and when argument "cor" is not NULL.

number of True Positives (TP)

number of FalseNegatives (TN)

number of False Positives (FP)

numberof True Negatives (TN)

sensitivity, i.e. TP/(TP+FN)

specificity, i.e. TN/(TN+FP)

accuracy,i.e. (TP+TN)/(TP+TN+FP+FN)

precision (p), i.e.TP/(TP+FP)

recall (r), i.e. TP/(TP+FN)

F1-score, i.e. 2*p*r/(p+r)

output ofVariableSelection,GraphicalModel,BiSelection, orClustering.

optional indices of hyper-parameters. Ifargmax_id=NULL, the calibrated hyper-parameters are used.

output ofGraphicalModel.

optional indices of hyper-parameters. Ifargmax_id=NULL, the calibrated hyper-parameters are used.

output ofVariableSelection.

optional indices of hyper-parameters. Ifargmax_id=NULL, the calibrated hyper-parameters are used.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

radius in density-based clustering, seedbscan. Only used ifimplementation=DBSCANClustering.

vector of penalty parameters for weighted distance calculation.Only used for distance-based clustering, including for exampleimplementation=HierarchicalClustering,implementation=PAMClustering, orimplementation=DBSCANClustering.

number of resampling iterations.

subsample size.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for clustering. Possible functionsincludeHierarchicalClustering (hierarchical clustering),PAMClustering (Partitioning Around Medoids),KMeansClustering (k-means) andGMMClustering(Gaussian Mixture Models). Alternatively, a user-defined function takingxdata andLambda as arguments and returning a binary andsymmetric matrix for which diagonal elements are equal to zero can be used.

logical indicating if the data should be scaled to ensure thatall variables contribute equally to the clustering of the observations.

character string indicating the type of linkage used inhierarchical clustering to define the stable clusters. Possible valuesinclude"complete","single" and"average" (seeargument"method" inhclust for a full list).Only used ifimplementation=HierarchicalClustering.

logical indicating if rows (ifrow=TRUE) or columns (ifrow=FALSE) contain the items to cluster.

logical indicating if the input datasetsxdata andydata should be included in the output.

logical indicating if a loading bar and messages should beprinted.

additional parameters passed to the functions provided inimplementation orresampling.

a matrixof the best stability scores for different (sets of) parameters controllingthe number of clusters and penalisation of attribute weights.

amatrix of numbers of clusters.

a matrix of regularisationparameters for attribute weights.

a matrix of the average numberof selected attributes by the underlying algorithm with differentregularisation parameters.

an array of consensus matrices.Rows and columns correspond to items. Indices along the third dimensioncorrespond to different parameters controlling the number of clusters andpenalisation of attribute weights.

an array of selectionproportions. Columns correspond to attributes. Rows correspond to differentparameters controlling the number of clusters and penalisation of attributeweights.

a list withtype="clustering" and valuesused for argumentsimplementation,linkage, andresampling.

a list with values used for argumentsK,tau,pk,n (number of observations inxdata), andseed.

data matrix with observations as rows and variables as columns.For multi-block stability selection, the variables in data have to beordered by group.

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.Ifimplementation="glassoFast",Lambda contains penaltyparameters.

optional vector of parameters controlling thelevel of sparsity in neighbour blocks for the multi-block procedure. To usejointly a specific set of parameters for each block,lambda_other_blocks must be set toNULL (not recommended).Only used for multi-block stability selection, i.e. iflength(pk)>1.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for graphical modelling. Ifimplementation=PenalisedGraphical, the algorithm implemented inglassoFast is used for regularised estimation ofa conditional independence graph. Alternatively, a user-defined functioncan be provided.

character string indicating if the algorithm should beinitialised at the estimated (inverse) covariance with previous penaltyparameters (start="warm") or not (start="cold"). Usingstart="warm" can speed-up the computations, but could lead toconvergence issues (in particular with smallLambda_cardinal). Onlyused forimplementation=PenalisedGraphical (see argument"start" inglassoFast).

logical indicating if the correlation (scale=TRUE) orcovariance (scale=FALSE) matrix should be used as input ofglassoFast ifimplementation=PenalisedGraphical. Otherwise, this argument must beused in the function provided inimplementation.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

logical indicating if complementary pair stability selectionshould be done. For this, the algorithm is applied on two non-overlappingsubsets of half of the observations. A feature is considered as selected ifit is selected for both subsamples. With this method, the data is splitK/2 times (K models are fitted). Only used ifPFER_method="MB".

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

logical indicating if the input datasetsxdata andydata should be included in the output.

logical indicating if a loading bar and messages should beprinted.

additional parameters passed to the functions provided inimplementation orresampling.

a matrix of the best stability scores fordifferent (sets of) parameters controlling the level of sparsity in theunderlying algorithm.

a matrix of parameters controlling thelevel of sparsity in the underlying algorithm.

a matrix of theaverage number of selected features by the underlying algorithm withdifferent parameters controlling the level of sparsity.

amatrix of the calibrated number of stably selected features with differentparameters controlling the level of sparsity.

a matrix ofcalibrated thresholds in selection proportions for different parameterscontrolling the level of sparsity in the underlying algorithm.

a matrix of upper-bounds in PFER of calibrated stabilityselection models with different parameters controlling the level ofsparsity.

a matrix of upper-bounds in FDP of calibratedstability selection models with different parameters controlling the levelof sparsity.

a matrix of stability scores obtained withdifferent combinations of parameters. Columns correspond to differentthresholds in selection proportions.

a matrix ofupper-bounds in FDP obtained with different combinations of parameters.Columns correspond to different thresholds in selection proportions. Onlyreturned iflength(pk)=1.

a matrix of upper-bounds inPFER obtained with different combinations of parameters. Columns correspondto different thresholds in selection proportions. Only returned iflength(pk)=1.

an array of selection proportions.Rows and columns correspond to nodes in the graph. Indices along the thirddimension correspond to different parameters controlling the level ofsparsity in the underlying algorithm.

a matrix of signs ofPearson's correlations estimated fromxdata.

a listwithtype="graphical_model" and values used for argumentsimplementation,start,resampling,cpss andPFER_method.

a list with values used for argumentsK,pi_list,tau,n_cat,pk,n(number of observations inxdata),PFER_thr,FDP_thr,seed,lambda_other_blocks, andSequential_template.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.Withimplementation="glmnet",Lambda contains penaltyparameters.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

function to use for variable selection. Possiblefunctions are:PenalisedRegression,SparsePLS,GroupPLS andSparseGroupPLS. Alternatively, a user-definedfunction can be provided.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

logical indicating if complementary pair stability selectionshould be done. For this, the algorithm is applied on two non-overlappingsubsets of half of the observations. A feature is considered as selected ifit is selected for both subsamples. With this method, the data is splitK/2 times (K models are fitted). Only used ifPFER_method="MB".

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

vector encoding the grouping structure among predictors. Thisargument indicates the number of variables in each group. Only used formodels with group penalisation (e.g.implementation=GroupPLS orimplementation=SparseGroupPLS).

logical indicating if a group penalisation shouldbe considered in the stability score. The use ofgroup_penalisation=TRUE strictly applies to group (not sparse-group)penalisation.

logical indicating if the input datasetsxdata andydata should be included in the output.

logical indicating if a loading bar and messages should beprinted.

additional parameters passed to the functions provided inimplementation orresampling.

a matrix of the best stability scores fordifferent parameters controlling the level of sparsity in the underlyingalgorithm.

a matrix of parameters controlling the level ofsparsity in the underlying algorithm.

a matrix of the averagenumber of selected features by the underlying algorithm with differentparameters controlling the level of sparsity.

a matrix of thecalibrated number of stably selected features with different parameterscontrolling the level of sparsity.

a matrix of calibratedthresholds in selection proportions for different parameters controllingthe level of sparsity in the underlying algorithm.

a matrix ofupper-bounds in PFER of calibrated stability selection models withdifferent parameters controlling the level of sparsity.

amatrix of upper-bounds in FDP of calibrated stability selection models withdifferent parameters controlling the level of sparsity.

amatrix of stability scores obtained with different combinations ofparameters. Columns correspond to different thresholds in selectionproportions.

a matrix of upper-bounds in FDP obtained withdifferent combinations of parameters. Columns correspond to differentthresholds in selection proportions.

a matrix ofupper-bounds in PFER obtained with different combinations of parameters.Columns correspond to different thresholds in selection proportions.

a matrix of selection proportions. Columns correspond topredictors fromxdata.

an array of model coefficients.Columns correspond to predictors fromxdata. Indices along the thirddimension correspond to different resampling iterations. With multivariateoutcomes, indices along the fourth dimension correspond to outcome-specificcoefficients.

a list withtype="variable_selection"and values used for argumentsimplementation,family,resampling,cpss andPFER_method.

alist with values used for argumentsK,pi_list,tau,n_cat,pk,n (number of observations),PFER_thr,FDP_thr andseed. The datasetsxdataandydata are also included ifoutput_data=TRUE.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

type of PLS model. Iffamily="gaussian", a sparse groupPLS model as defined insgPLS is run (for continuousoutcomes). Iffamily="binomial", a PLS-DA model as defined insgPLSda is run (for categorical outcomes).

vector encoding the grouping structure among predictors. Thisargument indicates the number of variables in each group.

optional vector encoding the grouping structure amongoutcomes. This argument indicates the number of variables in each group.

matrix of parameters controlling the number of selected groupsat current component, as defined byncomp.

vector of parameters controlling the level of sparsity withingroups of predictors.

optional vector of parameters controlling the level ofsparsity within groups of outcomes. Only used iffamily="gaussian".

number of selected groups in previous components. Onlyused ifncomp > 1. The argumentkeepX insgPLS is obtained by concatenatingkeepX_previous andLambda.

number of selected groups of outcome variables. This argument isdefined as insgPLS. Only used iffamily="gaussian".

number of components.

logical indicating if the data should be scaled (i.e.transformed so that all variables have a standard deviation of one). Onlyused iffamily="gaussian".

additional arguments to be passed tosgPLS orsgPLSda.

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspondto different model parameters. Columns correspond to predictors (startingwith "X") or outcomes (starting with "Y") variables for differentcomponents (denoted by "PC").

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

vector of penalty parameters (see argumentwbounds inKMeansSparseCluster).

additional parameters passed toKMeansSparseCluster.

an array of binary and symmetricco-membership matrices.

a matrix of median weights byfeature.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of selectedvariables at current component, as defined byncomp.

number of components.

logical indicating if the data should be scaled (i.e.transformed so that all variables have a standard deviation of one).

number of selected predictors in previous components.Only used ifncomp > 1.

character string indicating the name of the algorithm to use forsparse PCA. Possible values are: "sPCA" (for the algorithm proposed by Zou,Hastie and Tibshirani and implemented inspca) or"rSVD" (for the regularised SVD approach proposed by Shen and Huang andimplemented inspca).

additional arguments to be passed tospca (ifalgorithm="sPCA") orspca (ifalgorithm="rSVD").

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspondto different model parameters. Columns correspond to predictors (startingwith "X") or outcomes (starting with "Y") variables for differentcomponents (denoted by "PC").

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

matrix of parameters controlling the number of selectedpredictors at current component, as defined byncomp.

type of PLS model. Iffamily="gaussian", a sparse PLSmodel as defined insPLS is run (for continuousoutcomes). Iffamily="binomial", a PLS-DA model as defined insPLSda is run (for categorical outcomes).

number of components.

logical indicating if the data should be scaled (i.e.transformed so that all variables have a standard deviation of one). Onlyused iffamily="gaussian".

number of selected predictors in previous components.Only used ifncomp > 1. The argumentkeepX insPLS is obtained by concatenatingkeepX_previous andLambda.

number of selected outcome variables. This argument is definedas insPLS. Only used iffamily="gaussian".

additional arguments to be passed tosPLS orsPLSda.

matrix of binary selection status. Rowscorrespond to different model parameters. Columns correspond topredictors.

array of model coefficients. Rows correspondto different model parameters. Columns correspond to predictors (startingwith "X") or outcomes (starting with "Y") variables for differentcomponents (denoted by "PC").

vector or matrix of data. In regression, this should be theoutcome data.

type of regression model. This argument is defined as inglmnet. Possible values include"gaussian"(linear regression),"binomial" (logistic regression),"multinomial" (multinomial regression), and"cox" (survivalanalysis).

vector of the proportion of observations in each of the sets.

matrix encoding the edges between two types of nodes (rows andcolumns).

array of selection proportions.

optional vector encoding the grouping structure. Only used formulti-block stability selection wherepk indicates the number ofvariables in each group. Ifpk=NULL, single-block stabilityselection is performed.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

vector of block-specific thresholds in PFER forconstrained calibration by error control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrained calibration is used.

vector of block-specific thresholds in the expectedproportion of falsely selected features (or False Discovery Proportion,FDP) for constrained calibration by error control. IfPFER_thr=InfandFDP_thr=Inf, unconstrained calibration is used.

logical matrix encoding the type of procedure touse for data with multiple blocks in stability selection graphicalmodelling. For multi-block estimation, the stability selection model isconstructed as the union of block-specific stable edges estimated while theothers are weakly penalised (TRUE only for the block currently beingcalibrated andFALSE for other blocks). Other approaches with jointcalibration of the blocks are allowed (all entries are set toTRUE).

logical indicating if stability selection is performed in aregression (ifFALSE) or graphical (ifTRUE)framework.

vector encoding the grouping structure among predictors. Thisargument indicates the number of variables in each group and only needs tobe provided for group (but not sparse group) penalisation.

a matrix of the best (block-specific) stabilityscores for different (sets of) penalty parameters. In multi-block stabilityselection, rows correspond to different sets of penalty parameters, (valuesare stored in the output "Lambda") and columns correspond to differentblocks.

a matrix of (block-specific) penalty parameters. Inmulti-block stability selection, rows correspond to sets of penaltyparameters and columns correspond to different blocks.

a matrixof average numbers of (block-specific) edges selected by the underlyingalgorithm for different (sets of) penalty parameters. In multi-blockstability selection, rows correspond to different sets of penaltyparameters, (values are stored in the output "Lambda") and columnscorrespond to different blocks.

a matrix of calibrated numbersof (block-specific) stable edges for different (sets of) penaltyparameters. In multi-block stability selection, rows correspond todifferent sets of penalty parameters, (values are stored in the output"Lambda") and columns correspond to different blocks.

a matrix ofcalibrated (block-specific) thresholds in selection proportions fordifferent (sets of) penalty parameters. In multi-block stability selection,rows correspond to different sets of penalty parameters, (values are storedin the output "Lambda") and columns correspond to different blocks.

a matrix of computed (block-specific) upper-bounds in PFER ofcalibrated graphs for different (sets of) penalty parameters. Inmulti-block stability selection, rows correspond to different sets ofpenalty parameters, (values are stored in the output "Lambda") and columnscorrespond to different blocks.

a matrix of computed(block-specific) upper-bounds in FDP of calibrated stability selectionmodels for different (sets of) penalty parameters. In multi-block stabilityselection, rows correspond to different sets of penalty parameters, (valuesare stored in the output "Lambda") and columns correspond to differentblocks.

an array of (block-specific) stability scores obtainedwith different combinations of parameters. Rows correspond to different(sets of) penalty parameters and columns correspond to different thresholdsin selection proportions. In multi-block stability selection, indices alongthe third dimension correspond to different blocks.

anarray of computed upper-bounds of PFER obtained with different combinationsof parameters. Rows correspond to different penalty parameters and columnscorrespond to different thresholds in selection proportions. Not availablein multi-block stability selection graphical modelling.

anarray of computed upper-bounds of FDP obtained with different combinationsof parameters. Rows correspond to different penalty parameters and columnscorrespond to different thresholds in selection proportions. Not availablein multi-block stability selection graphical modelling.

array of selection proportions.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

vector encoding the grouping structure among predictors. Thisargument indicates the number of variables in each group and only needs tobe provided for group (but not sparse group) penalisation.

output ofVariableSelection,BiSelection,GraphicalModel orClustering.

optional indices of hyper-parameters. Ifargmax_id=NULL, the calibrated hyper-parameters are used.

character string indicating the type of linkage used inhierarchical clustering to define the stable clusters. Possible valuesinclude"complete","single" and"average" (seeargument"method" inhclust for a full list).

matrix with observations as rows and variables as columns.Column names must be defined and in line with the row and column names ofadjacency.

binary adjacency matrix of the Directed Acyclic Graph(transpose of the asymmetric matrix A in Reticular Action Model notation).The row and column names of this matrix must be defined.

binary and symmetric matrix encoding the nonzeroentries in the residual covariance matrix (symmetric matrix S in ReticularAction Model notation). By default, this is the identity matrix (noresidual covariance).

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.

number of resampling iterations.

subsample size. Only used ifresampling="subsampling" andcpss=FALSE.

value of the seed to initialise the random number generator andensure reproducibility of the results (seeset.seed).

computation options for the stability score. Default isNULL to use the score based on a z test. Other possible values are 2or 3 to use the score based on the negative log-likelihood.

function to use for variable selection.

resampling approach. Possible values are:"subsampling" for sampling without replacement of a proportiontau of the observations, or"bootstrap" for sampling withreplacement generating a resampled dataset with as many observations as inthe full sample. Alternatively, this argument can be a function to use forresampling. This function must use arguments nameddata andtau and return the IDs of observations to be included in theresampled dataset.

logical indicating if complementary pair stability selectionshould be done. For this, the algorithm is applied on two non-overlappingsubsets of half of the observations. A feature is considered as selected ifit is selected for both subsamples. With this method, the data is splitK/2 times (K models are fitted). Only used ifPFER_method="MB".

method used to compute the upper-bound of the expectednumber of False Positives (or Per Family Error Rate, PFER). IfPFER_method="MB", the method proposed by Meinshausen and Bühlmann(2010) is used. IfPFER_method="SS", the method proposed by Shah andSamworth (2013) under the assumption of unimodality is used.

threshold in PFER for constrained calibration by errorcontrol. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

threshold in the expected proportion of falsely selectedfeatures (or False Discovery Proportion) for constrained calibration byerror control. IfPFER_thr=Inf andFDP_thr=Inf, unconstrainedcalibration is used (the default).

number of values in the grid of parameters controllingthe level of sparsity in the underlying algorithm. Only used ifLambda=NULL.

character string indicating the type of optimisationmethod. Withoptimisation="grid_search" (the default), all values inLambda are visited. Alternatively, optimisation algorithmsimplemented innloptr can be used withoptimisation="nloptr". By default, we use"algorithm"="NLOPT_GN_DIRECT_L","xtol_abs"=0.1,"ftol_abs"=0.1 and"maxeval"=Lambda_cardinal. These valuescan be changed by providing the argumentopts (seenloptr). For stability selection using penalisedregression,optimisation="grid_search" may be faster as it allowsfor warm start.

number of cores to use for parallel computing (see argumentworkers inmultisession). Usingn_cores>1 is only supported withoptimisation="grid_search".

logical indicating if the input datasetsxdata andydata should be included in the output.

logical indicating if a loading bar and messages should beprinted.

additional parameters passed to the functions provided inimplementation orresampling.

amatrix of the best stability scores for different parameters controllingthe level of sparsity in the underlying algorithm.

a matrixof parameters controlling the level of sparsity in the underlyingalgorithm.

a matrix of the average number of selected features bythe underlying algorithm with different parameters controlling the level ofsparsity.

a matrix of the calibrated number of stably selectedfeatures with different parameters controlling the level of sparsity.

a matrix of calibrated thresholds in selection proportions fordifferent parameters controlling the level of sparsity in the underlyingalgorithm.

a matrix of upper-bounds in PFER of calibratedstability selection models with different parameters controlling the levelof sparsity.

a matrix of upper-bounds in FDP of calibratedstability selection models with different parameters controlling the levelof sparsity.

a matrix of stability scores obtained withdifferent combinations of parameters. Columns correspond to differentthresholds in selection proportions.

a matrix ofupper-bounds in FDP obtained with different combinations of parameters.Columns correspond to different thresholds in selection proportions.

a matrix of upper-bounds in PFER obtained with differentcombinations of parameters. Columns correspond to different thresholds inselection proportions.

a matrix of selection proportions.Columns correspond to predictors fromxdata.

an arrayof model coefficients. Columns correspond to predictors fromxdata.Indices along the third dimension correspond to different resamplingiterations. With multivariate outcomes, indices along the fourth dimensioncorrespond to outcome-specific coefficients.

a list withtype="variable_selection" and values used for argumentsimplementation,family,resampling,cpss andPFER_method.

a list with values used for argumentsK,pi_list,tau,n_cat,pk,n(number of observations),PFER_thr,FDP_thr andseed.The datasetsxdata andydata are also included ifoutput_data=TRUE.

data matrix with observations as rows and variables as columns.

matrix of parameters controlling the number of clusters in theunderlying algorithm specified inimplementation. Ifnc isnot provided, it is set toseq(1, tau*nrow(xdata)).

additional parameters passed tokmeans.

an array of binary and symmetricco-membership matrices.

a matrix of median weights byfeature.

matrix of predictors with observations as rows and variables ascolumns.

optional vector or matrix of outcome(s). Iffamily is setto"binomial" or"multinomial",ydata can be a vectorwith character/numeric values or a factor.

matrix of parameters controlling the level of sparsity in theunderlying feature selection algorithm specified inimplementation.IfLambda=NULL andimplementation=PenalisedRegression,LambdaGridRegression is used to define a relevant grid.

vector of thresholds in selection proportions. Ifn_cat=NULL orn_cat=2, these values must be>0 and<1. Ifn_cat=3, these values must be>0.5 and<1.