YAPSA: Yet Another Package for Signature Analysis

Documented in LCD LCD_complex_cutoff LCD_complex_cutoff_perPID

# Copyright © 2014-2016  The YAPSA package contributors
# This file is part of the YAPSA package. The YAPSA package is licenced under
# GPL-3

#' Linear Combination Decomposition
#'
#'\code{LCD} performs a mutational signatures decomposition of a given
#'mutational catalogue \code{V} with known signatures \code{W} by 
#'solving the minimization problem \eqn{min(||W*H - V||)} 
#'with additional constraints of non-negativity on H where W and V
#'are known
#'
#' @param in_mutation_catalogue_df A numeric data frame \code{V} with \code{n}
#'  rows and \code{m} columns, \code{n} being the number of features and 
#'  \code{m} being the number of samples
#' @param in_signatures_df A numeric data frame \code{W} with \code{n} rows and
#'  \code{l} columns, \code{n} being the number of features and \code{l} being
#'  the number of signatures
#' @param in_per_sample_cutoff
#'  A numeric value less than 1. Signatures from within \code{W}
#'  with an exposure per sample less than \code{in_cutoff} will be
#'  discarded.
#'                          
#' @return The exposures \code{H}, a numeric data frame with \code{l} rows and
#'          \code{m} columns, \code{l} being the number of signatures and
#'          \code{m} being the number of samples
#' 
#' @seealso \code{\link[lsei]{lsei}}
#' 
#' @examples
#' ## define raw data
#' W_prim <- matrix(c(1,2,3,4,5,6),ncol=2) 
#' W_prim_df <- as.data.frame(W_prim)
#' W_df <- YAPSA:::normalize_df_per_dim(W_prim_df,2) # corresponds to the sigs
#' W <- as.matrix(W_df) 
#' ## 1. Simple case: non-negativity already in raw data
#' H <- matrix(c(2,5,3,6,1,9,1,2),ncol=4)
#' H_df <- as.data.frame(H) # corresponds to the exposures
#' V <- W %*% H # matrix multiplication
#' V_df <- as.data.frame(V) # corresponds to the mutational catalogue
#' exposures_df <- YAPSA:::LCD(V_df,W_df)
#' ## 2. more complicated: raw data already contains negative elements
#' ## define indices where sign is going to be swapped
#' sign_ind <- c(5,7)
#' ## now compute the indices of the other fields in the columns affected
#' ## by the sign change
#' row_ind <- sign_ind %% dim(H)[1]
#' temp_ind <- 2*row_ind -1
#' other_ind <- sign_ind + temp_ind
#' ## alter the matrix H to yield a new mutational catalogue
#' H_compl <- H
#' H_compl[sign_ind] <- (-1)*H[sign_ind]
#' H_compl_df <- as.data.frame(H_compl) # corresponds to the exposures
#' V_compl <- W %*% H_compl # matrix multiplication
#' V_compl_df <- as.data.frame(V_compl) # corresponds to the mutational catalog
#' exposures_df <- YAPSA:::LCD(V_compl_df,W_df)
#' exposures <- as.matrix(exposures_df)
#' 
#' @importFrom lsei lsei
#' @export
#' 
LCD <- function(in_mutation_catalogue_df,
                in_signatures_df,
                in_per_sample_cutoff=0){
  signatures_matrix <- as.matrix(in_signatures_df)
  out_exposures_df <- data.frame()
  G <- diag(dim(signatures_matrix)[2])
  H <- rep(0,dim(signatures_matrix)[2])
  for (i in seq_len(ncol(in_mutation_catalogue_df))) {
    # temp_fractions <- limSolve::lsei(A = signatures_matrix,
    #                                  B = in_mutation_catalogue_df[,i],
    #                                  G=G, H=H, verbose=FALSE)
    # temp_exposures_vector <- as.vector(temp_fractions$X)
    temp_fractions <- lsei::lsei(a = signatures_matrix, 
                                 b = in_mutation_catalogue_df[,i],
                                 e=G, f=H)
    temp_exposures_vector <- round(temp_fractions,digits = 6)
    names(temp_exposures_vector) <- names(in_signatures_df)
    rel_exposures_vector <- temp_exposures_vector/sum(temp_exposures_vector)
    deselect_ind <- which(rel_exposures_vector<in_per_sample_cutoff)
    temp_exposures_vector[deselect_ind] <- 0
    out_exposures_df[seq(1,dim(signatures_matrix)[2],1),i] <- 
      temp_exposures_vector
    rm(temp_fractions)
  }
  colnames(out_exposures_df) <- colnames(in_mutation_catalogue_df)
  rownames(out_exposures_df) <- colnames(in_signatures_df)
  return(out_exposures_df) 
}


#' @importFrom lsei lsei
#' 
LCD_cutoff <- function(in_mutation_catalogue_df,in_signatures_df,
                       in_cutoff=0.01,in_filename=NULL,
                       in_method="abs",in_convention="weak",
                       in_per_sample_cutoff=0) {
  # first run analysis without cutoff
  if(in_convention=="strict"){
    all_exposures_df <- LCD(in_mutation_catalogue_df,in_signatures_df)
  } else{
    all_exposures_df <- LCD(in_mutation_catalogue_df,
                            in_signatures_df,
                            in_per_sample_cutoff=in_per_sample_cutoff)
  }
  # now apply cutoff criteria to choose the main signatures
  if(in_method=="rel"){
    rel_all_exposures_df <- normalize_df_per_dim(all_exposures_df,2)
    average_rel_exposure_vector <- average_over_present(rel_all_exposures_df,1)
    sig_choice_ind <- which(average_rel_exposure_vector >= in_cutoff)
  } else {
    all_exposures_sum_df <- data.frame(sum=apply(all_exposures_df,1,sum))
    all_exposures_sum_df$sum_norm <- 
      all_exposures_sum_df$sum/sum(all_exposures_sum_df$sum)
    sig_choice_ind <- which(all_exposures_sum_df$sum_norm >= in_cutoff)
    if (!is.null(in_filename)) {
      break_vector <- seq(0,0.5,0.01)
      png(in_filename,width=400,height=400)
      hist(all_exposures_sum_df$sum_norm,breaks=break_vector,
           xlab="exposures per sig",main="Sum over whole cohort")    
      dev.off()
    }
  }
  choice_signatures_df <- in_signatures_df[,sig_choice_ind,drop=FALSE]
  # now rerun the decomposition with only the chosen signatures
  if(in_convention=="strict"){
    #out_exposures_df <- LCD_strict(in_mutation_catalogue_df,
    #                               choice_signatures_df)    
    out_exposures_df <- LCD(in_mutation_catalogue_df,choice_signatures_df)    
  } else{
    out_exposures_df <- LCD(in_mutation_catalogue_df,
                            choice_signatures_df,
                            in_per_sample_cutoff=in_per_sample_cutoff)
  }
  out_exposures_sum_df <- data.frame(sum=apply(out_exposures_df,1,sum))
  out_exposures_sum_df$sum_norm <- 
    out_exposures_sum_df$sum/sum(out_exposures_sum_df$sum)
  exposure_order <- order(out_exposures_sum_df$sum,decreasing=TRUE)
  # compute QC and error measures
  fit_catalogue_df <- as.data.frame(
    as.matrix(choice_signatures_df) %*% as.matrix(out_exposures_df))
  residual_catalogue_df <- in_mutation_catalogue_df - fit_catalogue_df
  rss <- sum(residual_catalogue_df^2)
  cosDist_fit_orig_per_matrix <- cosineDist(in_mutation_catalogue_df,
                                            fit_catalogue_df)
  cosDist_fit_orig_per_col <- rep(0,dim(in_mutation_catalogue_df)[2])
  for(i in dim(in_mutation_catalogue_df)[2]){
    cosDist_fit_orig_per_col[i] <- cosineDist(in_mutation_catalogue_df[,i],
                                              fit_catalogue_df[,i])
  }
  total_counts <- colSums(in_mutation_catalogue_df)
  sum_ind <- rev(order(total_counts))
  return(list(exposures=out_exposures_df,
              signatures=choice_signatures_df,
              choice=sig_choice_ind,
              order=exposure_order,
              residual_catalogue=residual_catalogue_df,
              rss=rss,
              cosDist_fit_orig_per_matrix=cosDist_fit_orig_per_matrix,
              cosDist_fit_orig_per_col=cosDist_fit_orig_per_col,
              sum_ind=sum_ind))
}


#' LCD with a signature-specific cutoff on exposures
#'
#'\code{LCD_cutoff} performs a mutational signatures decomposition by 
#'Linear Combination Decomposition (LCD) of a given
#'mutational catalogue \code{V} with known signatures \code{W} by 
#'solving the minimization problem \eqn{min(||W*H - V||)} 
#'with additional constraints of non-negativity on H where W and V
#'are known, but excludes signatures with an overall contribution less than
#'a given signature-specific cutoff (and thereby accounting for a background
#'model) over the whole cohort.
#'
#' @param in_mutation_catalogue_df
#'  A numeric data frame \code{V} with \code{n} rows and \code{m} columns, 
#'  \code{n} being the number of features and \code{m} being the number of
#'  samples
#' @param in_signatures_df
#'  A numeric data frame \code{W} with \code{n} rows and \code{l} columns,
#'  \code{n} being the number of features and \code{l} being the number of
#'  signatures
#' @param in_cutoff_vector
#'  A numeric vector of values less than 1. Signatures from within \code{W}
#'  with an overall exposure less than the respective value in 
#'  \code{in_cutoff_vector} will be discarded.
#' @param in_filename
#'  A path to generate a histogram of the signature exposures if non-NULL
#' @param in_method
#'  Indicate to which data the cutoff shall be applied: absolute exposures,
#'  relative exposures
#' @param in_per_sample_cutoff
#'  A numeric value less than 1. Signatures from within \code{W}
#'  with an exposure per sample less than \code{in_cutoff} will be
#'  discarded.
#' @param in_rescale
#'  Boolean, if TRUE (default) the exposures are rescaled such that colSums 
#'  over exposures match colSums over mutational catalogue
#' @param in_sig_ind_df
#'  Data frame of type signature_indices_df, i.e. indicating name,
#'  function and meta-information of the signatures. Default is NULL.
#' @param in_cat_list
#'  List of categories for aggregation. Have to be among the column names of 
#'  \code{in_sig_ind_df}. Default is NULL.
#'  
#' @return A list with entries:
#' \itemize{
#'  \item \code{exposures}:
#'    The exposures \code{H}, a numeric data frame with 
#'    \code{l} rows and \code{m} columns, \code{l} being
#'    the number of signatures and \code{m} being the number
#'    of samples
#'  \item \code{norm_exposures}:
#'    The normalized exposures \code{H}, a numeric data frame with 
#'    \code{l} rows and \code{m} columns, \code{l} being
#'    the number of signatures and \code{m} being the number
#'    of samples
#'  \item \code{signatures}:
#'    The reduced signatures that have exposures bigger
#'    than \code{in_cutoff}
#'  \item \code{choice}:
#'    Index vector of the reduced signatures in the input
#'    signatures
#'  \item \code{order}: Order vector of the signatures by exposure
#'  \item \code{residual_catalogue}:
#'    Numerical data frame (matrix) of the difference between fit (product of
#'    signatures and exposures) and input mutational catalogue
#'  \item \code{rss}:
#'    Residual sum of squares (i.e. sum of squares of the residual catalogue)
#'  \item \code{cosDist_fit_orig_per_matrix}:
#'    Cosine distance between the fit (product of signatures and exposures) and
#'    input mutational catalogue computed after putting the matrix into vector
#'    format (i.e. one scaler product for the whole matrix)
#'  \item \code{cosDist_fit_orig_per_col}:
#'    Cosine distance between the fit (product of signatures and exposures) and
#'    input mutational catalogue computed per column (i.e. per sample, i.e. as
#'    many scaler products as there are samples in the cohort)
#'  \item \code{sum_ind}:
#'    Decreasing order of mutational loads based on the input mutational
#'    catalogue
#'  \item \code{out_sig_ind}:
#'    Data frame of the type \code{signature_indices_df}, i.e. indicating name,
#'    function and meta-information of the signatures. Default is NULL, 
#'    non-NULL only if \code{in_sig_ind_df} is non-NULL.
#'  \item \code{aggregate_exposures_list}:
#'    List of exposure data frames aggregated over different categories. 
#'    Default is NULL, non-NULL only if \code{in_sig_ind_df} and 
#'    \code{in_cat_list} are non-NULL and if the categories specified in 
#'    \code{in_cat_list} are among the column names of \code{in_sig_ind_df}.
#' }
#' 
#' @seealso \code{\link{LCD}}
#' @seealso \code{\link{aggregate_exposures_by_category}}
#' @seealso \code{\link[lsei]{lsei}}
#' 
#' @examples
#'  NULL
#' 
#' @importFrom lsei lsei
#' @export
#' 
LCD_complex_cutoff <- function(in_mutation_catalogue_df,
                               in_signatures_df,
                               in_cutoff_vector=NULL,
                               in_filename=NULL,
                               in_method="abs",
                               in_per_sample_cutoff=0,
                               in_rescale=TRUE,
                               in_sig_ind_df=NULL,
                               in_cat_list=NULL) {
  # first run analysis without cutoff
  all_exposures_df <- LCD(in_mutation_catalogue_df,
                          in_signatures_df,
                          in_per_sample_cutoff=in_per_sample_cutoff)
  # now apply cutoff criteria to choose the main signatures
  if(in_method=="rel"){
    rel_all_exposures_df <- normalize_df_per_dim(all_exposures_df,2)
    average_rel_exposure_vector <- average_over_present(rel_all_exposures_df,1)
    sig_choice_ind <- which(average_rel_exposure_vector >= in_cutoff_vector &
                              average_rel_exposure_vector > 0)
  } else {
    all_exposures_sum_df <- data.frame(sum=apply(all_exposures_df,1,sum))
    all_exposures_sum_df$sum_norm <- 
      all_exposures_sum_df$sum/sum(all_exposures_sum_df$sum)
    sig_choice_ind <- which(all_exposures_sum_df$sum_norm >= in_cutoff_vector &
                              all_exposures_sum_df$sum_norm > 0)
    if (!is.null(in_filename)) {
      break_vector <- seq(0,0.5,0.01)
      png(in_filename,width=400,height=400)
      hist(all_exposures_sum_df$sum_norm,breaks=break_vector,
           xlab="exposures per sig",main="Sum over whole cohort")    
      dev.off()
    }
  }
  choice_signatures_df <- in_signatures_df[,sig_choice_ind,drop=FALSE]
  # now rerun the decomposition with only the chosen signatures
  out_exposures_df <- LCD(in_mutation_catalogue_df,
                          choice_signatures_df,
                          in_per_sample_cutoff=in_per_sample_cutoff)
  out_norm_exposures_df <- normalize_df_per_dim(out_exposures_df,2)
  total_counts <- colSums(in_mutation_catalogue_df)
  sum_ind <- rev(order(total_counts))
  if(in_rescale){
    out_exposures_df <- as.data.frame(t(t(out_norm_exposures_df)*total_counts))
  }
  out_exposures_sum_df <- data.frame(sum=apply(out_exposures_df,1,sum))
  out_exposures_sum_df$sum_norm <-
    out_exposures_sum_df$sum/sum(out_exposures_sum_df$sum)
  exposure_order <- order(out_exposures_sum_df$sum,decreasing=TRUE)
  # compute QC and error measures
  fit_catalogue_df <- as.data.frame(
    as.matrix(choice_signatures_df) %*% as.matrix(out_exposures_df))
  residual_catalogue_df <- in_mutation_catalogue_df - fit_catalogue_df
  rss <- sum(residual_catalogue_df^2)
  cosDist_fit_orig_per_matrix <- cosineDist(in_mutation_catalogue_df,
                                            fit_catalogue_df)
  cosDist_fit_orig_per_col <- rep(0,dim(in_mutation_catalogue_df)[2])
  for(i in dim(in_mutation_catalogue_df)[2]){
    cosDist_fit_orig_per_col[i] <- cosineDist(in_mutation_catalogue_df[,i],
                                              fit_catalogue_df[,i])
  }
  out_sig_ind_df <- NULL
  aggregate_exposures_list <- NULL
  if(!is.null(in_sig_ind_df)){
    out_sig_ind_df <- in_sig_ind_df[sig_choice_ind,]
    if(!is.null(in_cat_list)){
      aggregate_exposures_list <- lapply(
        in_cat_list,FUN=function(current_category){
          aggregate_exposures_by_category(
            out_exposures_df,out_sig_ind_df,current_category)
        })   
      names(aggregate_exposures_list) <- in_cat_list
    }
  }
  return(list(exposures=out_exposures_df,
              norm_exposures=out_norm_exposures_df,
              signatures=choice_signatures_df,
              choice=sig_choice_ind,
              order=exposure_order,
              residual_catalogue=residual_catalogue_df,
              rss=rss,
              cosDist_fit_orig_per_matrix=cosDist_fit_orig_per_matrix,
              cosDist_fit_orig_per_col=cosDist_fit_orig_per_col,
              sum_ind=sum_ind,
              out_sig_ind_df=out_sig_ind_df,
              aggregate_exposures_list=aggregate_exposures_list))
}


#' LCD, signature-specific cutoff on exposures, per PID
#'
#'\code{LCD_complex_cutoff_perPID} is a wrapper for
#'\code{\link{LCD_complex_cutoff}} and runs individually for every PID.
#'
#' @export
#' @rdname LCD_complex_cutoff
#' 
LCD_complex_cutoff_perPID <- function(in_mutation_catalogue_df,
                                      in_signatures_df,
                                      in_cutoff_vector=NULL,
                                      in_filename=NULL,
                                      in_method="abs",
                                      in_rescale=TRUE,
                                      in_sig_ind_df=NULL,
                                      in_cat_list=NULL){
  complex_COSMIC_list_list <- lapply(
    seq_along(in_mutation_catalogue_df), FUN=function(current_col){
      current_mut_cat <- in_mutation_catalogue_df[,current_col,drop=FALSE]
      complex_COSMIC_list <- LCD_complex_cutoff(
        current_mut_cat,
        in_signatures_df,
        in_cutoff_vector=in_cutoff_vector,
        in_filename=NULL,
        in_method=in_method,
        in_rescale=in_rescale)
      return(complex_COSMIC_list)
  })
  exposures_list <- lapply(complex_COSMIC_list_list,FUN=function(x) {
    return(x$exposures)})
  exposures_df <- merge_exposures(exposures_list,
                                  in_signatures_df)
  norm_exposures_df <- normalize_df_per_dim(exposures_df,2)
  sig_choice_ind <- match(rownames(exposures_df),names(in_signatures_df))
  choice_signatures_df <- in_signatures_df[,sig_choice_ind,drop=FALSE]
  exposure_order <- order(rowSums(exposures_df),decreasing=TRUE)
  fit_catalogue_df <- as.data.frame(
    as.matrix(choice_signatures_df) %*% as.matrix(exposures_df))
  residual_catalogue_df <- in_mutation_catalogue_df - fit_catalogue_df
  rss <- sum(residual_catalogue_df^2)
  cosDist_fit_orig_per_matrix <- cosineDist(in_mutation_catalogue_df,
                                            fit_catalogue_df)
  cosDist_fit_orig_per_col <- rep(0,dim(in_mutation_catalogue_df)[2])
  for(i in dim(in_mutation_catalogue_df)[2]){
    cosDist_fit_orig_per_col[i] <- cosineDist(in_mutation_catalogue_df[,i],
                                              fit_catalogue_df[,i])
  }
  total_counts <- colSums(in_mutation_catalogue_df)
  sum_ind <- rev(order(total_counts))
  out_sig_ind_df <- NULL
  aggregate_exposures_list <- NULL
  if(!is.null(in_sig_ind_df)){
    out_sig_ind_df <- in_sig_ind_df[sig_choice_ind,]
    if(!is.null(in_cat_list)){
      aggregate_exposures_list <- lapply(
        in_cat_list,FUN=function(current_category){
          aggregate_exposures_by_category(
            exposures_df,out_sig_ind_df,current_category)
        })   
      names(aggregate_exposures_list) <- in_cat_list
    }
  }
  return(list(exposures=exposures_df,
              norm_exposures=norm_exposures_df,
              signatures=choice_signatures_df,
              choice=sig_choice_ind,
              order=exposure_order,
              residual_catalogue=residual_catalogue_df,
              rss=rss,
              cosDist_fit_orig_per_matrix=cosDist_fit_orig_per_matrix,
              cosDist_fit_orig_per_col=cosDist_fit_orig_per_col,
              sum_ind=sum_ind,
              out_sig_ind_df=out_sig_ind_df,
              aggregate_exposures_list=aggregate_exposures_list))
}


res=function(x,b,in_matrix){
  b - in_matrix %*% x
}


norm_res=function(x,b,in_matrix){
  norm(as.matrix(b - in_matrix %*% x),"F")
}


#' @importFrom lsei lsei
#' 
LCD_SMC <- function(in_mutation_sub_catalogue_list,
                    in_signatures_df,in_F_df=NULL){
  ## find general properties
  number_of_strata <- length(in_mutation_sub_catalogue_list)
  number_of_sigs <- dim(in_signatures_df)[2]
  number_of_PIDs <- dim(in_mutation_sub_catalogue_list[[1]])[2]
  number_of_features <- dim(in_mutation_sub_catalogue_list[[1]])[1]
  ## 1. construct composite signatures_matrix
  signatures_matrix_element <- as.matrix(in_signatures_df)
  zero_element <- matrix(
    rep(0,dim(signatures_matrix_element)[1]*dim(signatures_matrix_element)[2]),
    ncol=dim(signatures_matrix_element)[2])
  temp_matrix <- NULL
  for (i in seq_len(number_of_strata)) {
    temp_row <- NULL
    for (j in seq_len(number_of_strata)) {
      if (i==j) {
        temp_row <- cbind(temp_row,signatures_matrix_element)
      } else {
        temp_row <- cbind(temp_row,zero_element)
      }
    }
    temp_matrix <- rbind(temp_matrix,temp_row)
  }
  signatures_matrix <- temp_matrix
  ## 2. construct composite mutation catalogue
  temp_matrix <- NULL
  for (i in seq_len(number_of_strata)) {
    temp_matrix <- rbind(temp_matrix,in_mutation_sub_catalogue_list[[i]])
  }
  pasted_mutation_catalogue_df <- temp_matrix
  ## 3. account for boundary conditions
  ## 3.a) account for equality boundary condition
  if (!is.null(in_F_df)) {
    # This condition is fulfilled when an exposures file has been supplied.
    F_df <- in_F_df
  } else {
    # This is the standard case when no exposures file has been supplied and 
    # the exposures have to be computed by LCD.
    sum_df <- data.frame(matrix(rep(0,number_of_PIDs*number_of_features),
                                ncol=number_of_PIDs))
    for (i in seq_len(number_of_strata)) {
      sum_df <- sum_df + in_mutation_sub_catalogue_list[[i]]
    }
    all_mutation_catalogue_df <- sum_df
    F_df <- LCD(all_mutation_catalogue_df,in_signatures_df)    
  }
  diagonal_element <- diag(number_of_sigs)
  temp_matrix <- NULL
  for (i in seq_len(number_of_strata)) {
    temp_matrix <- cbind(temp_matrix,diagonal_element)
  }
  E <- temp_matrix
  ## 3.b) account for inequality boundary condition
  G <- diag(dim(signatures_matrix)[2])
  H <- rep(0,dim(signatures_matrix)[2])
  out_exposures_df <- data.frame()
  for (i in seq_len(ncol(pasted_mutation_catalogue_df))) {
    # temp_fractions <- limSolve::lsei(A = signatures_matrix, 
    #                                  B = pasted_mutation_catalogue_df[,i], 
    #                                  E=E, F=F_df[,i], G=G, H=H)
    temp_fractions <- lsei::lsei(a = signatures_matrix, 
                                 b = pasted_mutation_catalogue_df[,i], 
                                 c=E, d=F_df[,i], e=G, f=H)
    temp_exposures_vector <- round(temp_fractions,digits = 6)
    names(temp_exposures_vector) <- names(in_signatures_df)
    out_exposures_df[seq(1,dim(signatures_matrix)[2],1),i] <- 
      #as.vector(temp_fractions$X)
      as.vector(temp_exposures_vector)
    rm(temp_fractions)
  }
  out_list <- list()
  for (i in seq_len(number_of_strata)) {
    out_list[[i]] <- as.data.frame(
      out_exposures_df[seq((number_of_sigs*(i-1)+1),(number_of_sigs*i),1),])
    colnames(out_list[[i]]) <- colnames(in_mutation_sub_catalogue_list[[1]])
    rownames(out_list[[i]]) <- colnames(in_signatures_df)
  }
  colnames(F_df) <- colnames(in_mutation_sub_catalogue_list[[1]])
  rownames(F_df) <- colnames(in_signatures_df)
  return(list(exposures_all_df=F_df,sub_exposures_list=out_list))
}