#A simplex solver for linear programming problem in (W)SIMULE
wsimule.linprogSPar <- function(i, Sigma, W, lambda){
# num of p * N
# pTimesN = nrow(Sigma)
# num of p * (N + 1)
# Get parameters
q = ncol(Sigma)
p = ncol(Sigma) - nrow(Sigma)
N = nrow(Sigma) / p
# Generate e_j
e = rep(0, p * N)
for(j in 1:N){
e[i + (j - 1) * p] = 1
}
# linear programming solution
f.obj = rep(W[i, ], 2 * (N+1))
con1 = cbind(-Sigma, +Sigma)
b1 = lambda - e
b2 = lambda + e
f.con = rbind(-diag(2 * q), con1, -con1)
f.rhs = c(rep(0, 2 * q), b1, b2)
f.dir = rep("<=", length(f.rhs))
lp.out = lp("min", f.obj, f.con, f.dir, f.rhs)
beta = lp.out$solution[1:q] - lp.out$solution[(q + 1):(2 * q)]
if (lp.out$status == 2) warning("No feasible solution! Try a larger tuning parameter!")
return(beta)
}
#' A constrained and weighted l1 minimization approach for estimating multiple
#' Sparse Gaussian or Nonparanormal Graphical Models
#'
#' Estimate multiple, related sparse Gaussian or Nonparanormal graphical models
#' from multiple related datasets using the SIMULE algorithm. Please run
#' demo(wsimule) to learn the basic functions provided by this package.
#' For further details, please read the original paper: Beilun Wang, Ritambhara
#' Singh, Yanjun Qi (2017) doi{10.1007/s10994-017-5635-7}.
#'
#' The SIMULE algorithm is a constrained l1 minimization method that can detect
#' both the shared and the task-specific parts of multiple graphs explicitly
#' from data (through jointly estimating multiple sparse Gaussian graphical
#' models or Nonparanormal graphical models). It solves the following equation:
#' \deqn{ \hat{\Omega}^{(1)}_I, \hat{\Omega}^{(2)}_I, \dots,
#' \hat{\Omega}^{(K)}_I, \hat{\Omega}_S =
#' \min\limits_{\Omega^{(i)}_I,\Omega_S}\sum\limits_i ||W \cdot
#' \Omega^{(i)}_I||_1+ \epsilon K||W \cdot \Omega_S||_1 } Subject to : \deqn{
#' ||\Sigma^{(i)}(\Omega^{(i)}_I + \Omega_S) - I||_{\infty} \le \lambda_{n}, i
#' = 1,\dots,K \nonumber } Please also see the equation (7) in our paper. The
#' \eqn{\lambda_n} is the hyperparameter controlling the sparsity level of the
#' matrices and it is the \code{lambda} in our function. The \eqn{\epsilon} is
#' the hyperparameter controlling the differences between the shared pattern
#' among graphs and the individual part of each graph. It is the \code{epsilon}
#' parameter in our function and the default value is 1. For further details,
#' please see our paper:
#' <http://link.springer.com/article/10.1007/s10994-017-5635-7>.
#'
#' @param X A List of input matrices. They can be data matrices or
#' covariance/correlation matrices. If every matrix in the X is a symmetric
#' matrix, the matrices are assumed to be covariance/correlation matrices. More
#' details at <https://github.com/QData/SIMULE>
#' @param lambda A positive number. The hyperparameter controls the sparsity
#' level of the matrices. The \eqn{\lambda_n} in the following section:
#' Details.
#' @param epsilon A positive number. The hyperparameter controls the
#' differences between the shared pattern among graphs and the individual part
#' of each graph. The \eqn{\epsilon} in the following section: Details. If
#' epsilon becomes larger, the generated graphs will be more similar to each
#' other. The default value is 1, which means that we set the same weights to
#' the shared pattern among graphs and the individual part of each graph.
#' @param W A weight matrix. This matrix uses the prior knowledge of the
#' graphs. For example, if we use wsimule to infer multiple human brain
#' connectome graphs, the \eqn{W} can be the anatomical distance matrix of
#' human brain. The default value is a matrix, whose entries all equals to 1.
#' This means that we do not have any prior knowledge.
#' @param covType A parameter to decide which Graphical model we choose to
#' estimate from the input data.
#'
#' If covType = "cov", it means that we estimate multiple sparse Gaussian
#' Graphical models. This option assumes that we calculate (when input X
#' represents data directly) or use (when X elements are symmetric representing
#' covariance matrices) the sample covariance matrices as input to the simule
#' algorithm.
#'
#' If covType = "kendall", it means that we estimate multiple nonparanormal
#' Graphical models. This option assumes that we calculate (when input X
#' represents data directly) or use (when X elements are symmetric representing
#' correlation matrices) the kendall's tau correlation matrices as input to the
#' simule algorithm.
#' @param intertwined indicate whether to use intertwined covariance matrix
#' @param parallel A boolean. This parameter decides if the package will use
#' the multithreading architecture or not.
#' @return \item{$graphs}{A list of the estimated inverse covariance/correlation
#' matrices.} \item{$share}{The share graph among multiple tasks.}
#' @author Beilun Wang
#' @references Beilun Wang, Ritambhara Singh, Yanjun Qi (2017). A constrained
#' L1 minimization approach for estimating multiple Sparse Gaussian or
#' Nonparanormal Graphical Models.
#' http://link.springer.com/article/10.1007/s10994-017-5635-7
#' @export
#' @import lpSolve
#' @import parallel
#' @import pcaPP
#' @details if labels are provided in the datalist as column names, result will contain labels (to be plotted)
#' @examples
#' library(JointNets)
#' data(exampleData)
#' result = wsimule(X = exampleData , lambda = 0.1, epsilon = 0.45,
#' W = matrix(1,20,20), covType = "cov", FALSE)
#' plot(result)
wsimule <- function(X, lambda, epsilon = 1, W, covType = "cov", intertwined = FALSE, parallel = FALSE ){
N = length(X)
for (i in 1:N){
X[[i]] = compute_cov(X[[i]],covType)
}
if (intertwined){
X = intertwined(X,covType = covType)
}
# initialize the parameters
Graphs = list()
p = ncol(X[[1]])
if (missing(W)){
W = matrix(1, p, p)
}
xt = matrix(0, (N + 1) * p, p)
I = diag(1, p, p)
Z = matrix(0, p, p)
# generate the condition matrix A
A = X[[1]]
for(i in 2:N){
A = cbind(A,Z)
}
A = cbind(A,(1/(epsilon * N))*X[[1]])
for(i in 2:N){
temp = Z
for(j in 2:N){
if (j == i){
temp = cbind(temp,X[[i]])
}
else{
temp = cbind(temp,Z)
}
}
temp = cbind(temp, 1/(epsilon * N) * X[[i]])
A = rbind(A, temp)
}
# define the function f for parallelization
f = function(x) wsimule.linprogSPar(x, A, W, lambda)
if(parallel == TRUE){ # parallel version
# number of cores to collect,
# default number is number cores in your machine - 1,
# you can set your own number by changing this line.
no_cores = detectCores() - 1
cl = makeCluster(no_cores)
# declare variable and function names to the cluster
clusterExport(cl, list("f", "A", "W", "lambda", "wsimule.linprogSPar", "lp"), envir = environment())
result = parLapply(cl, 1:p, f)
#print('Done!')
for (i in 1:p){
xt[,i] = result[[i]]
}
stopCluster(cl)
}else{ # single machine code
for (i in 1 : p){
xt[,i] = f(i)
if (i %% 10 == 0){
cat("=")
if(i %% 100 == 0){
cat("+")
}
}
}
#print("Done!")
}
for(i in 1:N){
# combine the results from each column. (\hat{\Omega}_{tot}^1)
Graphs[[i]] = xt[(1 + (i-1) * p):(i * p),] + 1/(epsilon * N) * xt[(1 + N * p):((N + 1) * p),]
# make it be symmetric
for(j in 1:p){
for(k in j:p){
if (abs(Graphs[[i]][j,k]) < abs(Graphs[[i]][k,j])){
Graphs[[i]][j,k] = Graphs[[i]][j,k]
Graphs[[i]][k,j] = Graphs[[i]][j,k]
}
else{
Graphs[[i]][j,k] = Graphs[[i]][k,j]
Graphs[[i]][k,j] = Graphs[[i]][k,j]
}
}
}
}
share = 1/(epsilon * N) * xt[(1 + N * p):((N + 1) * p),]
for(j in 1:p){
for(k in j:p){
if (abs(share[j,k]) < abs(share[k,j])){
share[j,k] = share[j,k]
share[k,j] = share[j,k]
}
else{
share[j,k] = share[k,j]
share[k,j] = share[k,j]
}
}
}
out = list(graphs = Graphs, share = share)
class(out) = "wsimule"
out = add_name_to_out(out,X)
return(out)
}
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