/
basis.R
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basis.R
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check_dim = function(dim){
if(!is.numeric(dim)){
stop("Dimension should be a number", call. = FALSE)
}
if(!dim>0){
stop("dimension should be positive", call. = FALSE)
}
}
#' @title Coordinates basis
#'
#' @description
#' Obtain coordinates basis
#' @param H coordinates for which basis should be shown
#' @return basis used to create coordinates H
#' @export
basis = function(H){
if(is.null(attr(H, 'basis'))) return(message('No basis specified'))
attr(H, 'basis')
}
#' Isometric/Orthonormal log-ratio basis for log-transformed compositions.
#'
#' By default the basis of the clr-given by Egozcue et al., 2013
#' Build an isometric log-ratio basis for a composition with k+1 parts
#' \deqn{h_i = \sqrt{\frac{i}{i+1}} \log\frac{\sqrt[i]{\prod_{j=1}^i x_j}}{x_{i+1}}}{%
#' h[i] = \sqrt(i/(i+1)) ( log(x[1] \ldots x[i])/i - log(x[i+1]) )}
#' for \eqn{i \in 1\ldots k}.
#'
#'Modifying parameter type (pivot or cdp) other ilr/olr basis can be generated
#'
#' @param dim number of components
#' @param type if different than `pivot` (pivot balances) or `cdp` (codapack balances) default balances are returned, which computes a triangular Helmert matrix as defined by Egozcue et al., 2013.
#' @return matrix
#' @references
#' Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G. and Barceló-Vidal C. (2003).
#' \emph{Isometric logratio transformations for compositional data analysis}.
#' Mathematical Geology, \strong{35}(3) 279-300
#' @examples
#' ilr_basis(5)
#' @export
ilr_basis = function(dim, type = 'default'){
check_dim(dim)
if(type == 'cdp'){
B = cdp_basis_(dim)
}else{
B = ilr_basis_default(dim)
if(type == 'pivot'){
B = (-B)[,ncol(B):1, drop = FALSE][nrow(B):1,]
}
}
colnames(B) = sprintf("ilr%d", 1:ncol(B))
rownames(B) = sprintf("c%d", 1:nrow(B))
B
}
#' @rdname ilr_basis
#' @export
olr_basis = function(dim, type = 'default'){
check_dim(dim)
if(type == 'cdp'){
B = cdp_basis_(dim)
}else{
B = ilr_basis_default(dim)
if(type == 'pivot'){
B = (-B)[,ncol(B):1, drop = FALSE][nrow(B):1,]
}
}
colnames(B) = sprintf("olr%d", 1:ncol(B))
rownames(B) = sprintf("c%d", 1:nrow(B))
B
}
#' Centered log-ratio basis
#'
#' Compute the transformation matrix to express a composition using
#' the linearly dependant centered log-ratio coordinates.
#'
#' @param dim number of parts
#' @return matrix
#' @references
#' Aitchison, J. (1986)
#' \emph{The Statistical Analysis of Compositional Data}.
#' Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). 416p.
#' @examples
#' (B <- clr_basis(5))
#' # CLR coordinates are linearly dependant coordinates.
#' (clr_coordinates <- coordinates(c(1,2,3,4,5), B))
#' # The sum of all coordinates equal to zero
#' sum(clr_coordinates) < 1e-15
#' @export
clr_basis = function(dim){
check_dim(dim)
B = clr_basis_default(dim)
colnames(B) = sprintf("clr%d", 1:ncol(B))
rownames(B) = sprintf("c%d", 1:nrow(B))
B
}
#' Additive log-ratio basis
#'
#' Compute the transformation matrix to express a composition using the oblique additive log-ratio
#' coordinates.
#'
#' @param dim number of parts
#' @param denominator part used as denominator (default behaviour is to use last part)
#' @param numerator parts to be used as numerator. By default all except the denominator parts are chosen following original order.
#' @return matrix
#' @examples
#' alr_basis(5)
#' # Third part is used as denominator
#' alr_basis(5, 3)
#' # Third part is used as denominator, and
#' # other parts are rearranged
#' alr_basis(5, 3, c(1,5,2,4))
#' @references
#' Aitchison, J. (1986)
#' \emph{The Statistical Analysis of Compositional Data}.
#' Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). 416p.
#' @export
alr_basis = function(dim, denominator = dim, numerator = which(denominator != 1:dim)){
check_dim(dim)
res = alr_basis_default(dim)
res = cbind(res, 0)
if(dim != denominator){
res[c(denominator, dim),] = res[c(dim, denominator),, drop = FALSE]
res[,c(denominator, dim)] = res[,c(dim, denominator), drop = FALSE]
}
B = res[,numerator, drop = FALSE]
colnames(B) = sprintf("alr%d", 1:ncol(B))
rownames(B) = sprintf("c%d", 1:nrow(B))
B
}
#' Isometric log-ratio basis based on Principal Components.
#'
#' Different approximations to approximate the principal balances of a compositional dataset.
#'
#' @param X compositional dataset
#' @return matrix
#'
#' @export
pc_basis = function(X){
B = ilr_basis(ncol(X))
B = B %*% svd(scale(log(as.matrix(X)) %*% B, scale=FALSE))$v
parts = colnames(X)
if(is.null(parts)){
parts = paste0('c', 1:nrow(B))
}
rownames(B) = parts
colnames(B) = paste0('pc', 1:ncol(B))
as.matrix(B)
}
#' Isometric log-ratio basis based on canonical correlations
#'
#'
#' @param Y compositional dataset
#' @param X explanatory dataset
#' @return matrix
#'
cc_basis = function(Y, X){
Y = as.matrix(Y)
X = cbind(X)
B = ilr_basis_default(ncol(Y))
cc = stats::cancor(coordinates(Y), X)
B = B %*% cc$xcoef
parts = colnames(Y)
if(is.null(parts)){
parts = paste0('c', 1:nrow(B))
}
rownames(B) = parts
colnames(B) = paste0('cc', 1:ncol(B))
B
}
#' Balance generated from the first canonical correlation component
#'
#'
#' @param Y compositional dataset
#' @param X explanatory dataset
#' @return matrix
#'
#' @export
cbalance_approx = function(Y,X){
Y = as.matrix(Y)
X = cbind(X)
B = ilr_basis(ncol(Y))
cc1 = B %*% stats::cancor(coordinates(Y), X)$xcoef[,1,drop=F]
ord = order(abs(cc1))
cb1_ = sign(cc1)
cb1 = cb1_
cor1 = abs(suppressWarnings(stats::cancor(coordinates(Y,sbp_basis(cb1_)), X)$cor))
for(i in 1:(ncol(Y)-2)){
cb1_[ord[i]] = 0
cor1_ = abs(suppressWarnings(stats::cancor(coordinates(Y,sbp_basis(cb1_)), X)$cor))
if(cor1_ > cor1){
cb1 = cb1_
cor1 = cor1_
}
}
suppressWarnings(sbp_basis(cb1))
}
#' Isometric log-ratio basis based on Balances
#'
#' Build an \code{\link{ilr_basis}} using a sequential binary partition or
#' a generic coordinate system based on balances.
#'
#' @param sbp parts to consider in the numerator and the denominator. Can be
#' defined either using a list of formulas setting parts (see examples) or using
#' a matrix where each column define a balance. Positive values are parts in
#' the numerator, negative values are parts in the denominator, zeros are parts
#' not used to build the balance.
#' @param data composition from where name parts are extracted
#' @param fill should the balances be completed to become an orthonormal basis?
#' if the given balances are not orthonormal, the function will complete the
#' balance to become a basis.
#' @param silent inform about orthogonality
#' @return matrix
#' @examples
#' X = data.frame(a=1:2, b=2:3, c=4:5, d=5:6, e=10:11, f=100:101, g=1:2)
#' sbp_basis(list(b1 = a~b+c+d+e+f+g,
#' b2 = b~c+d+e+f+g,
#' b3 = c~d+e+f+g,
#' b4 = d~e+f+g,
#' b5 = e~f+g,
#' b6 = f~g), data = X)
#' sbp_basis(list(b1 = a~b,
#' b2 = b1~c,
#' b3 = b2~d,
#' b4 = b3~e,
#' b5 = b4~f,
#' b6 = b5~g), data = X)
#' # A non-orthogonal basis can also be calculated.
#' sbp_basis(list(b1 = a+b+c~e+f+g,
#' b2 = d~a+b+c,
#' b3 = d~e+g,
#' b4 = a~e+b,
#' b5 = b~f,
#' b6 = c~g), data = X)
#' @export
sbp_basis = function(sbp, data = NULL, fill = FALSE, silent=FALSE){
if(is.null(data) & is.matrix(sbp)){
# P = t(sbp)
df = as.data.frame(matrix(1, nrow(sbp), nrow = 1))
if(!is.null(rownames(sbp))){
colnames(df) = rownames(sbp)
}
str_to_frm = function(vec){
frm = paste(stats::aggregate(nm ~ vec, subset(data.frame(nm = paste0('`',names(df), '`'), vec = -1 * vec,
stringsAsFactors = FALSE), vec != 0),
FUN = paste, collapse= ' + ')[['nm']], collapse=' ~ ')
stats::as.formula(frm)
}
return(sbp_basis(apply(sbp, 2, str_to_frm),
data = df,
fill = fill,
silent = silent))
# return(do.call('sbp_basis', c(apply(P, 1, str_to_frm), list(data=df,
# fill = fill,
# silent = silent)))) #, envir = as.environment('package:coda.base')
}
if (!is.data.frame(data) && !is.environment(data) && ( (is.matrix(data) && !is.null(colnames(data))) | !is.null(attr(data, "class"))))
data <- as.data.frame(data)
else if (is.array(data))
stop("'data' must be a data.frame or a matrix with column names")
if(!all(unlist(lapply(sbp, all.vars)) %in% c(names(data), names(sbp)))){
stop("Balances should be columns of 'data'")
}
nms = setdiff(names(sbp), "")
if(length(nms) > 0){
substitutions = lapply(sbp, all.vars)
substitutions = substitutions[nms]
while(!all(is.na(substitutions)) &
!all(unlist(substitutions) %in% names(data))){
for(nm in nms){
substitutions = lapply(substitutions, function(subs){
I = match(nm, subs)
if(!is.na(I)){
c(subs[-I], substitutions[[nm]])
}else{
subs
}
})
}
}
}
sbp_split = function(part){
RIGHT = attr(stats::terms(part), 'term.labels')
LEFT = setdiff(all.vars(part), RIGHT)
if(length(nms) > 0){
for(nm in nms){
I = match(nm, RIGHT)
if(!is.na(I)){
RIGHT = c(RIGHT[-I], substitutions[[nm]])
}
I = match(nm, LEFT)
if(!is.na(I)){
LEFT = c(LEFT[-I], substitutions[[nm]])
}
}
}
list(LEFT, RIGHT)
}
sbp_clean = lapply(sbp, sbp_split)
RES = sapply(sbp_clean, function(balance){
I1 = length(balance[[1]])
I2 = length(balance[[2]])
l = +1/I1 * sqrt(I1*I2/(I1+I2))
r = -1/I2 * sqrt(I1*I2/(I1+I2))
bal = stats::setNames(rep(0, length(names(data))), names(data))
bal[balance[[1]]] = bal[balance[[1]]] + l
bal[balance[[2]]] = bal[balance[[2]]] + r
bal
})
if(fill){
return(Recall(fill_sbp(sign(RES))))
}
if(!silent){
if(qr(RES)$rank != NCOL(data)-1){
warning('Given partition is not a basis')
}else{
Z = t(RES) %*% RES
if( !all( Z - diag(diag(Z), nrow=NROW(Z), ncol=NCOL(Z)) < 10e-10 ) ){
warning('Given basis is not orthogonal')
}else{
if(!all( Z - diag(1, nrow=NROW(Z), ncol=NCOL(Z)) < 10e-10 )){
warning('Given basis is not orthonormal')
}
}
}
}
RES
}
#' Isometric log-ratio basis based on Principal Balances.
#'
#' Exact method to calculate the principal balances of a compositional dataset. Different methods to approximate the principal balances of a compositional dataset are also included.
#'
#' @param X compositional dataset
#' @param method method to be used with Principal Balances. Methods available are: 'exact', 'constrained' or 'cluster'.
#' @param constrained.complete_up When searching up, should the algorithm try to find possible siblings for the current balance (TRUE) or build a parent directly forcing current balance to be part of the next balance (default: FALSE). While the first is more exhaustive and given better results the second is faster and can be used with highe dimensional datasets.
#' @param cluster.method Method to be used with the hclust function (default: `ward.D2`) or any other method available in hclust function
#' @param ordering should the principal balances found be returned ordered? (first column, first
#' principal balance and so on)
#' @param ... parameters passed to hclust function
#' @return matrix
#' @references
#' Martín-Fernández, J.A., Pawlowsky-Glahn, V., Egozcue, J.J., Tolosana-Delgado R. (2018).
#' Advances in Principal Balances for Compositional Data.
#' \emph{Mathematical Geosciencies}, 50, 273-298.
#' @examples
#' set.seed(1)
#' X = matrix(exp(rnorm(5*100)), nrow=100, ncol=5)
#'
#' # Optimal variance obtained with Principal components
#' (v1 <- apply(coordinates(X, 'pc'), 2, var))
#' # Optimal variance obtained with Principal balances
#' (v2 <- apply(coordinates(X,pb_basis(X, method='exact')), 2, var))
#' # Solution obtained using constrained method
#' (v3 <- apply(coordinates(X,pb_basis(X, method='constrained')), 2, var))
#' # Solution obtained using Ward method
#' (v4 <- apply(coordinates(X,pb_basis(X, method='cluster')), 2, var))
#'
#' # Plotting the variances
#' barplot(rbind(v1,v2,v3,v4), beside = TRUE, ylim = c(0,2),
#' legend = c('Principal Components','PB (Exact method)',
#' 'PB (Constrained)','PB (Ward approximation)'),
#' names = paste0('Comp.', 1:4), args.legend = list(cex = 0.8), ylab = 'Variance')
#'
#' @export
pb_basis = function(X, method, constrained.complete_up = FALSE, cluster.method = 'ward.D2',
ordering = TRUE, ...){
X = as.matrix(X)
if(!(all(X > 0))){
stop("All components must be strictly positive.", call. = FALSE)
}
if(method %in% c('constrained', 'exact')){
if(method == 'exact'){
M = 'PB'
B = find_PB(X)
}
if(method == 'constrained'){
M = 'CS'
# B = t(fBalChip(X)$bal)
B = constrained_pb(as.matrix(X))
}
if(method == 'constrained2'){
M = 'CS'
B = find_PB_using_pc(X)
}
# if(method == 'lsearch'){
# if(rep == 0){
# B = find_PB_pc_local_search(X)
# }else{
# B = find_PB_rnd_local_search(stats::cov(log(X)), rep=rep)
# }
# }
}else if(method == 'cluster'){
M = 'CL'
# Passing arguments to hclust function
hh = stats::hclust(stats::as.dist(variation_array(X)), method=cluster.method, ...)
B = matrix(0, ncol = nrow(hh$merge), nrow = ncol(X))
for(i in 1:nrow(hh$merge)){
if(hh$merge[i,1] < 0 & hh$merge[i,2] < 0){
B[-hh$merge[i,],i] = c(-1,+1)
}else{
if(hh$merge[i,1] > 0){
B[B[,hh$merge[i,1]] != 0,i] = -1
}else{
B[-hh$merge[i,1],i] = -1
}
if(hh$merge[i,2] > 0){
B[B[,hh$merge[i,2]] != 0,i] = +1
}else{
B[-hh$merge[i,2],i] = +1
}
}
}
B = sbp_basis(B[,nrow(hh$merge):1, drop = FALSE])
} else{
stop(sprintf("Method %s does not exist", method))
}
if(ordering){
B = B[,order(apply(coordinates(X, B, basis_return = FALSE), 2, stats::var), decreasing = TRUE), drop = FALSE]
}
parts = colnames(X)
if(is.null(parts)){
parts = paste0('c', 1:nrow(B))
}
rownames(B) = parts
colnames(B) = paste0('pb', 1:ncol(B))
B
}
#' Isometric log-ratio basis based on Balances.
#'
#' The function return default balances used in CoDaPack software.
#'
#' @param dim dimension to build the ILR basis based on balanced balances
#' @return matrix
#' @export
cdp_basis = function(dim){
check_dim(dim)
B = cdp_basis_(dim)
rownames(B) = paste0("c", 1:dim)
colnames(B) = paste0("ilr", 1:ncol(B))
B
}
cdp_basis_ = function(dim, wR = 1:ceiling(dim/2), wL = ceiling(dim/2) + 1:floor(dim/2)){
R = length(wR)
L = length(wL)
D = R + L
v = rep(0, dim)
v[wR] = +sqrt(L/R/D)
v[wL] = -sqrt(R/L/D)
if(R == 1 & L == 1){
return(v)
}
if(R == 1){
return(cbind(v,
Recall(dim, wR = wL[1:ceiling(L/2)], wL = wL[ceiling(L/2) + 1:floor(L/2)])))
}
if(L == 1){
return(cbind(v,
Recall(dim, wR = wR[1:ceiling(R/2)], wL = wR[ceiling(R/2) + 1:floor(R/2)])))
}
cbind(v,
Recall(dim, wR = wR[1:ceiling(R/2)], wL = wR[ceiling(R/2) + 1:floor(R/2)]),
Recall(dim, wR = wL[1:ceiling(L/2)], wL = wL[ceiling(L/2) + 1:floor(L/2)]))
}
#' Pairwise log-ratio generator system
#'
#' The function returns all combinations of pairs of log-ratios.
#'
#' @param dim dimension to build the pairwise log-ratio generator system
#' @return matrix
#' @export
pairwise_basis = function(dim){
check_dim(dim)
I = utils::combn(dim,2)
B = apply(I, 2, function(i){
b = rep(0, dim)
b[i] = c(1,-1)
b
})
colnames(B) = paste0('pw', apply(I, 2, paste, collapse = '_'))
rownames(B) = paste0("c", 1:dim)
B
}