From: Benjamin Auder Date: Sun, 2 Apr 2017 13:17:21 +0000 (+0200) Subject: no need to generate random IO params: migrate in test. Add roxygen2 NAMESPACE-generat... X-Git-Url: https://git.auder.net/app_dev.php/current/doc/html/%24%7BgetWhatsApp%28link%29%7D?a=commitdiff_plain;h=086ca318ed5580e961ceda3f1e122a2da58e4427;p=valse.git no need to generate random IO params: migrate in test. Add roxygen2 NAMESPACE-generator. roxygenize pkg a bit more --- diff --git a/pkg/R/A_NAMESPACE.R b/pkg/R/A_NAMESPACE.R new file mode 100644 index 0000000..62989ce --- /dev/null +++ b/pkg/R/A_NAMESPACE.R @@ -0,0 +1,4 @@ +#' @useDynLib valse +#' +#' @importFrom parallel makeCluster parLapply stopCluster clusterExport +NULL diff --git a/pkg/R/gridLambda.R b/pkg/R/computeGridLambda.R similarity index 63% rename from pkg/R/gridLambda.R rename to pkg/R/computeGridLambda.R index 35c412a..f89b2a3 100644 --- a/pkg/R/gridLambda.R +++ b/pkg/R/computeGridLambda.R @@ -1,4 +1,7 @@ +#' computeGridLambda +#' #' Construct the data-driven grid for the regularization parameters used for the Lasso estimator +#' #' @param phiInit value for phi #' @param rhoInit value for rho #' @param piInit value for pi @@ -6,20 +9,20 @@ #' @param X matrix of covariates (of size n*p) #' @param Y matrix of responses (of size n*m) #' @param gamma power of weights in the penalty -#' @param mini minimum number of iterations in EM algorithm -#' @param maxi maximum number of iterations in EM algorithm -#' @param tau threshold to stop EM algorithm +#' @param mini minimum number of iterations in EM algorithm +#' @param maxi maximum number of iterations in EM algorithm +#' @param tau threshold to stop EM algorithm +#' #' @return the grid of regularization parameters +#' #' @export -#----------------------------------------------------------------------- -gridLambda = function(phiInit, rhoInit, piInit, gamInit, X, Y, gamma, mini, maxi, tau) +computeGridLambda = function(phiInit, rhoInit, piInit, gamInit, X, Y, gamma, mini, maxi, tau) { n = nrow(X) p = dim(phiInit)[1] m = dim(phiInit)[2] k = dim(phiInit)[3] - #list_EMG = .Call("EMGLLF_core",phiInit,rhoInit,piInit,gamInit,mini,maxi,1,0,X,Y,tau) list_EMG = EMGLLF(phiInit,rhoInit,piInit,gamInit,mini,maxi,1,0,X,Y,tau) grid = array(0, dim=c(p,m,k)) for (i in 1:p) @@ -29,6 +32,5 @@ gridLambda = function(phiInit, rhoInit, piInit, gamInit, X, Y, gamma, mini, maxi } grid = unique(grid) grid = grid[grid <=1] - - return(grid) + grid } diff --git a/pkg/R/generateXY.R b/pkg/R/generateXY.R new file mode 100644 index 0000000..496d4e5 --- /dev/null +++ b/pkg/R/generateXY.R @@ -0,0 +1,39 @@ +#' generateXY +#' +#' Generate a sample of (X,Y) of size n +#' +#' @param n sample size +#' @param π proportion for each cluster +#' @param meanX matrix of group means for covariates (of size p) +#' @param covX covariance for covariates (of size p*p) +#' @param β regression matrix, of size p*m*k +#' @param covY covariance for the response vector (of size m*m*K) +#' +#' @return list with X and Y +#' +#' @export +generateXY = function(n, π, meanX, β, covX, covY) +{ + p <- dim(covX)[1] + m <- dim(covY)[1] + k <- dim(covY)[3] + + X <- matrix(nrow=0, ncol=p) + Y <- matrix(nrow=0, ncol=m) + + #random generation of the size of each population in X~Y (unordered) + sizePop <- rmultinom(1, n, pi) + class <- c() #map i in 1:n --> index of class in 1:k + + for (i in 1:k) + { + class <- c(class, rep(i, sizePop[i])) + newBlockX <- MASS::mvrnorm(sizePop[i], meanX, covX) + X <- rbind( X, newBlockX ) + Y <- rbind( Y, apply( newBlockX, 1, function(row) + mvrnorm(1, row %*% beta[,,i], covY[,,i]) ) ) + } + + shuffle = sample(n) + list("X"=X[shuffle,], "Y"=Y[shuffle,], "class"=class[shuffle]) +} diff --git a/pkg/R/main.R b/pkg/R/main.R index 1908021..f080954 100644 --- a/pkg/R/main.R +++ b/pkg/R/main.R @@ -1,212 +1,120 @@ -#' @useDynLib valse - -Valse = setRefClass( - Class = "Valse", - - fields = c( - # User defined - - # regression data (size n*p, where n is the number of observations, - # and p is the number of regressors) - X = "matrix", - # response data (size n*m, where n is the number of observations, - # and m is the number of responses) - Y = "matrix", - - # Optionally user defined (some default values) - - # power in the penalty - gamma = "numeric", - # minimum number of iterations for EM algorithm - mini = "integer", - # maximum number of iterations for EM algorithm - maxi = "integer", - # threshold for stopping EM algorithm - eps = "numeric", - # minimum number of components in the mixture - kmin = "integer", - # maximum number of components in the mixture - kmax = "integer", - # ranks for the Lasso-Rank procedure - rank.min = "integer", - rank.max = "integer", - - # Computed through the workflow - - # initialisation for the reparametrized conditional mean parameter - phiInit = "numeric", - # initialisation for the reparametrized variance parameter - rhoInit = "numeric", - # initialisation for the proportions - piInit = "numeric", - # initialisation for the allocations probabilities in each component - tauInit = "numeric", - # values for the regularization parameter grid - gridLambda = "numeric", - # je ne crois pas vraiment qu'il faille les mettre en sortie, d'autant plus qu'on construit - # une matrice A1 et A2 pour chaque k, et elles sont grandes, donc ca coute un peu cher ... - A1 = "integer", - A2 = "integer", - # collection of estimations for the reparametrized conditional mean parameters - Phi = "numeric", - # collection of estimations for the reparametrized variance parameters - Rho = "numeric", - # collection of estimations for the proportions parameters - Pi = "numeric", - - #immutable (TODO:?) - thresh = "numeric" - ), - - methods = list( - ####################### - #initialize main object - ####################### - initialize = function(X,Y,...) - { - "Initialize Valse object" - - callSuper(...) - - X <<- X - Y <<- Y - gamma <<- ifelse (hasArg("gamma"), gamma, 1.) - mini <<- ifelse (hasArg("mini"), mini, as.integer(5)) - maxi <<- ifelse (hasArg("maxi"), maxi, as.integer(10)) - eps <<- ifelse (hasArg("eps"), eps, 1e-6) - kmin <<- ifelse (hasArg("kmin"), kmin, as.integer(2)) - kmax <<- ifelse (hasArg("kmax"), kmax, as.integer(3)) - rank.min <<- ifelse (hasArg("rank.min"), rank.min, as.integer(2)) - rank.max <<- ifelse (hasArg("rank.max"), rank.max, as.integer(3)) - thresh <<- 1e-15 #immutable (TODO:?) - }, - - ################################## - #core workflow: compute all models - ################################## - - initParameters = function(k) +#' valse +#' +#' Main function +#' +#' @param X matrix of covariates (of size n*p) +#' @param Y matrix of responses (of size n*m) +#' @param procedure among 'LassoMLE' or 'LassoRank' +#' @param selecMod method to select a model among 'DDSE', 'DJump', 'BIC' or 'AIC' +#' @param gamma integer for the power in the penaly, by default = 1 +#' @param mini integer, minimum number of iterations in the EM algorithm, by default = 10 +#' @param maxi integer, maximum number of iterations in the EM algorithm, by default = 100 +#' @param eps real, threshold to say the EM algorithm converges, by default = 1e-4 +#' @param kmin integer, minimum number of clusters, by default = 2 +#' @param kmax integer, maximum number of clusters, by default = 10 +#' @param rang.min integer, minimum rank in the low rank procedure, by default = 1 +#' @param rang.max integer, maximum rank in the +#' +#' @return a list with estimators of parameters +#' +#' @examples +#' #TODO: a few examples +#' @export +valse = function(X,Y,procedure = 'LassoMLE',selecMod = 'DDSE',gamma = 1,mini = 10, + maxi = 50,eps = 1e-4,kmin = 2,kmax = 2, + rang.min = 1,rang.max = 10) +{ + #################### + # compute all models + #################### + + p = dim(X)[2] + m = dim(Y)[2] + n = dim(X)[1] + + model = list() + tableauRecap = array(0, dim=c(1000,4)) + cpt = 0 + print("main loop: over all k and all lambda") + + for (k in kmin:kmax) + { + print(k) + print("Parameters initialization") + #smallEM initializes parameters by k-means and regression model in each component, + #doing this 20 times, and keeping the values maximizing the likelihood after 10 + #iterations of the EM algorithm. + init = initSmallEM(k, X, Y) + phiInit <- init$phiInit + rhoInit <- init$rhoInit + piInit <- init$piInit + gamInit <- init$gamInit + grid_lambda <- computeGridLambda(phiInit, rhoInit, piInit, gamInit, X, Y, gamma, mini, maxi, eps) + + if (length(grid_lambda)>100) + grid_lambda = grid_lambda[seq(1, length(grid_lambda), length.out = 100)] + print("Compute relevant parameters") + #select variables according to each regularization parameter + #from the grid: A1 corresponding to selected variables, and + #A2 corresponding to unselected variables. + + params = selectiontotale(phiInit,rhoInit,piInit,gamInit,mini,maxi,gamma,grid_lambda,X,Y,1e-8,eps) + #params2 = selectVariables(phiInit,rhoInit,piInit,gamInit,mini,maxi,gamma,grid_lambda[seq(1,length(grid_lambda), by=3)],X,Y,1e-8,eps) + ## etrange : params et params 2 sont différents ... + selected <- params$selected + Rho <- params$Rho + Pi <- params$Pi + + if (procedure == 'LassoMLE') { - "Parameters initialization" - - #smallEM initializes parameters by k-means and regression model in each component, - #doing this 20 times, and keeping the values maximizing the likelihood after 10 - #iterations of the EM algorithm. - init = initSmallEM(k,X,Y) - phiInit <<- init$phi0 - rhoInit <<- init$rho0 - piInit <<- init$pi0 - tauInit <<- init$tau0 - }, - - computeGridLambda = function() + print('run the procedure Lasso-MLE') + #compute parameter estimations, with the Maximum Likelihood + #Estimator, restricted on selected variables. + model[[k]] = constructionModelesLassoMLE(phiInit, rhoInit,piInit,gamInit,mini,maxi,gamma,X,Y,thresh,eps,selected) + llh = matrix(ncol = 2) + for (l in seq_along(model[[k]])) + llh = rbind(llh, model[[k]][[l]]$llh) + LLH = llh[-1,1] + D = llh[-1,2] + } + else { - "computation of the regularization grid" - #(according to explicit formula given by EM algorithm) - - gridLambda <<- gridLambda(phiInit,rhoInit,piInit,tauInit,X,Y,gamma,mini,maxi,eps) - }, - - computeRelevantParameters = function() - { - "Compute relevant parameters" - - #select variables according to each regularization parameter - #from the grid: A1 corresponding to selected variables, and - #A2 corresponding to unselected variables. - params = selectiontotale( - phiInit,rhoInit,piInit,tauInit,mini,maxi,gamma,gridLambda,X,Y,thresh,eps) - A1 <<- params$A1 - A2 <<- params$A2 - Rho <<- params$Rho - Pi <<- params$Pi - }, - - runProcedure1 = function() - { - "Run procedure 1 [EMGLLF]" - - #compute parameter estimations, with the Maximum Likelihood - #Estimator, restricted on selected variables. - return ( constructionModelesLassoMLE( - phiInit,rhoInit,piInit,tauInit,mini,maxi,gamma,gridLambda,X,Y,thresh,eps,A1,A2) ) - }, - - runProcedure2 = function() - { - "Run procedure 2 [EMGrank]" - - #compute parameter estimations, with the Low Rank - #Estimator, restricted on selected variables. - return ( constructionModelesLassoRank(Pi,Rho,mini,maxi,X,Y,eps, - A1,rank.min,rank.max) ) - }, - - run = function() - { - "main loop: over all k and all lambda" - - # Run the whole procedure, 1 with the - #maximum likelihood refitting, and 2 with the Low Rank refitting. - p = dim(phiInit)[1] - m = dim(phiInit)[2] - for (k in kmin:kmax) - { - print(k) - initParameters(k) - computeGridLambda() - computeRelevantParameters() - if (procedure == 1) - { - r1 = runProcedure1() - Phi2 = Phi - Rho2 = Rho - Pi2 = Pi - p = ncol(X) - m = ncol(Y) - if (is.null(dim(Phi2))) #test was: size(Phi2) == 0 - { - Phi[,,1:k] <<- r1$phi - Rho[,,1:k] <<- r1$rho - Pi[1:k,] <<- r1$pi - } else - { - Phi <<- array(0., dim=c(p,m,kmax,dim(Phi2)[4]+dim(r1$phi)[4])) - Phi[,,1:(dim(Phi2)[3]),1:(dim(Phi2)[4])] <<- Phi2 - Phi[,,1:k,dim(Phi2)[4]+1] <<- r1$phi - Rho <<- array(0., dim=c(m,m,kmax,dim(Rho2)[4]+dim(r1$rho)[4])) - Rho[,,1:(dim(Rho2)[3]),1:(dim(Rho2)[4])] <<- Rho2 - Rho[,,1:k,dim(Rho2)[4]+1] <<- r1$rho - Pi <<- array(0., dim=c(kmax,dim(Pi2)[2]+dim(r1$pi)[2])) - Pi[1:nrow(Pi2),1:ncol(Pi2)] <<- Pi2 - Pi[1:k,ncol(Pi2)+1] <<- r1$pi - } - } else - { - phi = runProcedure2()$phi - Phi2 = Phi - if (dim(Phi2)[1] == 0) - { - Phi[,,1:k,] <<- phi - } else - { - Phi <<- array(0., dim=c(p,m,kmax,dim(Phi2)[4]+dim(phi)[4])) - Phi[,,1:(dim(Phi2)[3]),1:(dim(Phi2)[4])] <<- Phi2 - Phi[,,1:k,-(1:(dim(Phi2)[4]))] <<- phi - } - } - } - } - - ################################################## - #TODO: pruning: select only one (or a few best ?!) model - ################################################## - # - # function[model] selectModel( - # #TODO - # #model = odel(...) - # end - # Give at least the slope heuristic and BIC, and AIC ? - - ) -) + print('run the procedure Lasso-Rank') + #compute parameter estimations, with the Low Rank + #Estimator, restricted on selected variables. + model = constructionModelesLassoRank(Pi, Rho, mini, maxi, X, Y, eps, + A1, rank.min, rank.max) + + ################################################ + ### Regarder la SUITE + phi = runProcedure2()$phi + Phi2 = Phi + if (dim(Phi2)[1] == 0) + Phi[, , 1:k,] <- phi + else + { + Phi <- array(0, dim = c(p, m, kmax, dim(Phi2)[4] + dim(phi)[4])) + Phi[, , 1:(dim(Phi2)[3]), 1:(dim(Phi2)[4])] <<- Phi2 + Phi[, , 1:k,-(1:(dim(Phi2)[4]))] <<- phi + } + } + tableauRecap[(cpt+1):(cpt+length(model[[k]])), ] = matrix(c(LLH, D, rep(k, length(model[[k]])), 1:length(model[[k]])), ncol = 4) + cpt = cpt+length(model[[k]]) + } + print('Model selection') + tableauRecap = tableauRecap[rowSums(tableauRecap[, 2:4])!=0,] + tableauRecap = tableauRecap[(tableauRecap[,1])!=Inf,] + data = cbind(1:dim(tableauRecap)[1], tableauRecap[,2], tableauRecap[,2], tableauRecap[,1]) + require(capushe) + modSel = capushe(data, n) + indModSel <- + if (selecMod == 'DDSE') + as.numeric(modSel@DDSE@model) + else if (selecMod == 'Djump') + as.numeric(modSel@Djump@model) + else if (selecMod == 'BIC') + modSel@BIC_capushe$model + else if (selecMod == 'AIC') + modSel@AIC_capushe$model + model[[tableauRecap[indModSel,3]]][[tableauRecap[indModSel,4]]] +} diff --git a/pkg/R/valse.R b/pkg/R/valse.R deleted file mode 100644 index d5d10ce..0000000 --- a/pkg/R/valse.R +++ /dev/null @@ -1,113 +0,0 @@ -#' Main function -#' -#' @param X matrix of covariates (of size n*p) -#' @param Y matrix of responses (of size n*m) -#' @param procedure among 'LassoMLE' or 'LassoRank' -#' @param selecMod method to select a model among 'DDSE', 'DJump', 'BIC' or 'AIC' -#' @param gamma integer for the power in the penaly, by default = 1 -#' @param mini integer, minimum number of iterations in the EM algorithm, by default = 10 -#' @param maxi integer, maximum number of iterations in the EM algorithm, by default = 100 -#' @param eps real, threshold to say the EM algorithm converges, by default = 1e-4 -#' @param kmin integer, minimum number of clusters, by default = 2 -#' @param kmax integer, maximum number of clusters, by default = 10 -#' @param rang.min integer, minimum rank in the low rank procedure, by default = 1 -#' @param rang.max integer, maximum rank in the -#' @return a list with estimators of parameters -#' @export -#----------------------------------------------------------------------- -valse = function(X,Y,procedure = 'LassoMLE',selecMod = 'DDSE',gamma = 1,mini = 10, - maxi = 50,eps = 1e-4,kmin = 2,kmax = 2, - rang.min = 1,rang.max = 10) { - ################################## - #core workflow: compute all models - ################################## - - p = dim(X)[2] - m = dim(Y)[2] - n = dim(X)[1] - - model = list() - tableauRecap = array(0, dim=c(1000,4)) - cpt = 0 - print("main loop: over all k and all lambda") - - for (k in kmin:kmax){ - print(k) - print("Parameters initialization") - #smallEM initializes parameters by k-means and regression model in each component, - #doing this 20 times, and keeping the values maximizing the likelihood after 10 - #iterations of the EM algorithm. - init = initSmallEM(k, X, Y) - phiInit <<- init$phiInit - rhoInit <<- init$rhoInit - piInit <<- init$piInit - gamInit <<- init$gamInit - grid_lambda <<- gridLambda(phiInit, rhoInit, piInit, gamInit, X, Y, gamma, mini, maxi, eps) - - if (length(grid_lambda)>100){ - grid_lambda = grid_lambda[seq(1, length(grid_lambda), length.out = 100)] - } - print("Compute relevant parameters") - #select variables according to each regularization parameter - #from the grid: A1 corresponding to selected variables, and - #A2 corresponding to unselected variables. - - params = selectiontotale(phiInit,rhoInit,piInit,gamInit,mini,maxi,gamma,grid_lambda,X,Y,1e-8,eps) - #params2 = selectVariables(phiInit,rhoInit,piInit,gamInit,mini,maxi,gamma,grid_lambda[seq(1,length(grid_lambda), by=3)],X,Y,1e-8,eps) - ## etrange : params et params 2 sont différents ... - selected <<- params$selected - Rho <<- params$Rho - Pi <<- params$Pi - - if (procedure == 'LassoMLE') { - print('run the procedure Lasso-MLE') - #compute parameter estimations, with the Maximum Likelihood - #Estimator, restricted on selected variables. - model[[k]] = constructionModelesLassoMLE(phiInit, rhoInit,piInit,gamInit,mini,maxi,gamma,X,Y,thresh,eps,selected) - llh = matrix(ncol = 2) - for (l in seq_along(model[[k]])){ - llh = rbind(llh, model[[k]][[l]]$llh) - } - LLH = llh[-1,1] - D = llh[-1,2] - } else { - print('run the procedure Lasso-Rank') - #compute parameter estimations, with the Low Rank - #Estimator, restricted on selected variables. - model = constructionModelesLassoRank(Pi, Rho, mini, maxi, X, Y, eps, - A1, rank.min, rank.max) - - ################################################ - ### Regarder la SUITE - phi = runProcedure2()$phi - Phi2 = Phi - if (dim(Phi2)[1] == 0) - { - Phi[, , 1:k,] <<- phi - } else - { - Phi <<- array(0, dim = c(p, m, kmax, dim(Phi2)[4] + dim(phi)[4])) - Phi[, , 1:(dim(Phi2)[3]), 1:(dim(Phi2)[4])] <<- Phi2 - Phi[, , 1:k,-(1:(dim(Phi2)[4]))] <<- phi - } - } - tableauRecap[(cpt+1):(cpt+length(model[[k]])), ] = matrix(c(LLH, D, rep(k, length(model[[k]])), 1:length(model[[k]])), ncol = 4) - cpt = cpt+length(model[[k]]) - } - print('Model selection') - tableauRecap = tableauRecap[rowSums(tableauRecap[, 2:4])!=0,] - tableauRecap = tableauRecap[(tableauRecap[,1])!=Inf,] - data = cbind(1:dim(tableauRecap)[1], tableauRecap[,2], tableauRecap[,2], tableauRecap[,1]) - require(capushe) - modSel = capushe(data, n) - if (selecMod == 'DDSE') { - indModSel = as.numeric(modSel@DDSE@model) - } else if (selecMod == 'Djump') { - indModSel = as.numeric(modSel@Djump@model) - } else if (selecMod == 'BIC') { - indModSel = modSel@BIC_capushe$model - } else if (selecMod == 'AIC') { - indModSel = modSel@AIC_capushe$model - } - return(model[[tableauRecap[indModSel,3]]][[tableauRecap[indModSel,4]]]) -} diff --git a/test/generate_test_data/generateRunSaveTest_EMGLLF.R b/test/generate_test_data/generateRunSaveTest_EMGLLF.R index e0bb3b8..bf68ab3 100644 --- a/test/generate_test_data/generateRunSaveTest_EMGLLF.R +++ b/test/generate_test_data/generateRunSaveTest_EMGLLF.R @@ -1,4 +1,5 @@ source("EMGLLF.R") +source("helper.R") generateRunSaveTest_EMGLLF = function(n=200, p=15, m=10, k=3, mini=5, maxi=10, gamma=1., lambda=0.5, tau=1e-6) diff --git a/test/generate_test_data/generateRunSaveTest_EMGrank.R b/test/generate_test_data/generateRunSaveTest_EMGrank.R index f75c8d1..f348d71 100644 --- a/test/generate_test_data/generateRunSaveTest_EMGrank.R +++ b/test/generate_test_data/generateRunSaveTest_EMGrank.R @@ -1,4 +1,5 @@ source("EMGrank.R") +source("helper.R") generateRunSaveTest_EMGrank = function(n=200, p=15, m=10, k=3, mini=5, maxi=10, gamma=1.0, rank = c(1,2,4)) diff --git a/pkg/R/generateSampleInputs.R b/test/generate_test_data/helper.R similarity index 63% rename from pkg/R/generateSampleInputs.R rename to test/generate_test_data/helper.R index c7aa3c6..49cd1b5 100644 --- a/pkg/R/generateSampleInputs.R +++ b/test/generate_test_data/helper.R @@ -1,34 +1,3 @@ -#' Generate a sample of (X,Y) of size n -#' @param meanX matrix of group means for covariates (of size p) -#' @param covX covariance for covariates (of size p*p) -#' @param covY covariance for the response vector (of size m*m*K) -#' @param pi proportion for each cluster -#' @param beta regression matrix, of size p*m*k -#' @param n sample size -#' -#' @return list with X and Y -#' @export -generateXY = function(meanX, covX, covY, pi, beta, n) -{ - p = dim(covX)[1] - m = dim(covY)[1] - k = dim(covY)[3] - - X = matrix(nrow=n,ncol=p) - Y = matrix(nrow=n,ncol=m) - class = matrix(nrow = n) - - require(MASS) #simulate from a multivariate normal distribution - for (i in 1:n) - { - class[i] = sample(1:k, 1, prob=pi) - X[i,] = mvrnorm(1, meanX, covX) - Y[i,] = mvrnorm(1, X[i,] %*% beta[,,class[i]], covY[,,class[i]]) - } - - return (list(X=X,Y=Y, class = class)) -} - #' Generate a sample of (X,Y) of size n with default values #' @param n sample size #' @param p number of covariates @@ -42,9 +11,7 @@ generateXYdefault = function(n, p, m, k) covX = diag(p) covY = array(dim=c(m,m,k)) for(r in 1:k) - { covY[,,r] = diag(m) - } pi = rep(1./k,k) #initialize beta to a random number of non-zero random value beta = array(0, dim=c(p,m,k))