---
title: "Principal Balances"
author: "Marc Comas-CufÃ"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{Principal Balances}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
references:
- id: fernandez2017
title: Advances in Principal Balances for Compositional Data
author:
- family: MartÃn-Fernández
given: Josep A.
- family: Pawlowsky-Glahn
given: Vera
- family: Egozcue
given: Juan J.
- family: Tolosan-Delgado
given: Raimon
container-title: Mathematical Geosciences
volume: 50
URL: 'https://link.springer.com/article/10.1007/s11004-017-9712-z'
DOI: 10.1007/s11004-017-9712-z
page: 273–298
type: article-journal
issued:
year: 2017
- id: ruskey1993
title: Simple Combinatorial Gray Codes Constructed by Reversing Sublists
author:
- family: Ruskey
given: Frank
container-title: Lecture Notes in Computer Science
volume: 762
DOI: 10.1007/3-540-57568-5_250
page: 201-208
type: article-journal
issued:
year: 1993
- id: pawlowsky2011
title: Principal balances
author:
- family: Pawlowsky-Glahn
given: Vera
- family: Egozcue
given: Juan J.
- family: Tolosan-Delgado
given: Raimon
container-title: In Egozcue JJ, Tolosana-Delgado R, Ortego M (eds) Proceedings of the 4th international workshop on compositional data analysis, Girona, Spain
type: article-journal
page: 1-10
issued:
year: 2011
---
```{r setup, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
options(digits = 4)
```
_Principal balances_ are defined as follows [@pawlowsky2011]:
> Given an $n$-sample of a $D$-part random composition, the set of Principal Balances (PB) is a set of $D-1$ balances satisfying the following conditions:
>
> * Each sample PB is obtained as the projection of a sample composition on a unitary composition or balancing element associated to the PB.
> * The first PB is the balance with maximum sample variance.
> * The $i$-th PB has maximum variance conditional to its balancing element being orthogonal to the previous 1st, 2nd, ..., $(i-1)$-th balancing elements.
>
Because of the large number of possibilities, for a given compositional sample, finding the PB is a computational demanding task. @fernandez2017 proposed different approaches to deal with the calculation and approximation of such type of basis. `coda.base` implements all methods presented in [@fernandez2017].
To illustrate `coda.base` functionalities, we use the following dataset:
```{r}
library(coda.base)
X = parliament2017[,-1]
```
consisting of parties in the 2017 Catalan Parliament Elections.
## Exact Principal Balances
`coda.base` can calculate the PB with the function `pb_basis()`. Function `pb_basis()` needs the parameter `method` to be set. To obtain the PB users needs to set `method = "exact"`.
```{r}
B1 = pb_basis(X, method = "exact")
```
Where the obtained sequential binary partition can be summarized with the following sequential binary tree:
```{r, fig.width=7.5, fig.height=4.5}
plot_balance(B1)
```
```{r}
apply(coordinates(X, B1), 2, var)
```
When `method` is set to `"exact"`, exhaustive search is performed to obtain the PB. The time needed to calculate the PB grows exponentially regarding the number of parts of `X`. To iterate through all the possibilities, CoDaPack uses the algorithm @ruskey1993. Currently, exhaustive search can find the PB of a compositional data set with 20 parts in a reasonable amount of time (5 minutes approximately).
```{r, echo=FALSE, fig.width=7.5, fig.height=4.5, warning=FALSE}
library(ggplot2)
D = 10
# time_ = sapply(5:20, function(D){
# print(D)
# x = matrix(rlnorm(100*D), ncol=D)
# a = tic()
# r = pb_basis(x, method = 'exact')
# b = toc()
# b$toc - b$tic
# })
# dplot = data.frame(
# parts = tail(5:20, 11),
# time = tail(time_, 11)
# )
dplot = structure(list(parts = 10:20, time = c(0.00499999999999545, 0.0119999999999436,
0.0260000000000673, 0.0749999999999318, 0.243000000000052, 0.831999999999994,
2.64400000000001, 9.56200000000001, 28.787, 95.4560000000001,
311.133)), class = "data.frame", row.names = c(NA, -11L))
ggplot(data=dplot) +
geom_point(aes(x=parts, y=time)) +
geom_segment(aes(x=parts, xend =parts, y = 0, yend=time)) +
labs(x = 'Number of parts', y = 'Time in seconds (logarithmic scale)', title = 'Time needed to calculate Principal Balances',
caption = "Times measured in a MacBook Pro (13-inch, 2017) \n2,3 GHz Dual-Core Intel Core i5.16 GB 2133 MHz LPDDR3") +
scale_x_continuous(breaks = tail(5:20, 11)) +
scale_y_continuous(trans = 'log', breaks = c(0.004, 0.04, 0.4, 4, 40, 400), labels = c("0.004", "0.04", "0.4", 4, 40, 400)) +
theme_minimal()
```
When the number of part is higher, the exact method should be discarded in favor of other alternatives. Different alternatives are available in @pawlowsky2011 and @fernandez2017. Between these alternatives, the once that can be implemented are agglomerate partition approach based on the Ward method [@pawlowsky2011, @fernandez2017] and the approximation based on a reduced approximation to the principal components [@fernandez2017].
## Ward Method for Parts
For a composition $X$, @pawlowsky2011 proposed to build the variation array and use it to produce a matrix distance. The variation array calculates the variance between a pair of logratios.
```{r}
D = as.dist(variation_array(X))
D
```
Combining pairs with lower variance, it is expected to get a final merging with high variability.
```{r, fig.width=7.5, fig.height=4.5}
B2 = pb_basis(X, method = 'cluster')
plot_balance(B2)
```
```{r}
apply(coordinates(X, B2), 2, var)
```
## Constrained PCs Algorithm
```{r, fig.width=7.5, fig.height=4.5}
B3 = pb_basis(X, method = 'constrained')
plot_balance(B3)
```
```{r}
apply(coordinates(X, B3), 2, var)
```
```{r, eval=FALSE, echo=FALSE}
hc = hclust_dendrogram(B1)
hcd = as.dendrogram(hc)
dd = dendro_data(hcd)
ggdendrogram(dd)
dd$segments = dd$segments %>%
mutate(
end_node = yend == 0
)
p <- ggplot(dd$segments) +
geom_segment(aes(x = x, y = y, xend = xend, yend = yend, linetype = end_node))+
geom_label(data = dd$labels, aes(x, y, label = label),
hjust = 0.5, angle = 90, size = 4) +
theme_void() + scale_linetype_discrete(guide=FALSE)
p
## Build tree
build_tree_order = function(B, ibalance){
balance = B[,ibalance]
if(sum(balance != 0) == 2){
return(ibalance)
}
L = NULL; R = NULL
left_ = balance < 0
if(sum(left_) > 1){
L = Recall(B, which(apply((B != 0 & left_) | (B == 0 & !left_), 2, all)))
}
right_ = balance > 0
if(sum(right_) > 1){
R = Recall(B, which(apply((B != 0 & right_) | (B == 0 & !right_), 2, all)))
}
return(unname(c(L, R, ibalance)))
}
ord_ = build_tree_order(B, which(apply(B != 0, 2, all)))
TREE = setNames(lapply(1:ncol(X), function(i) list('c' = i)), names(X)[ORDER])
for(i in ord_){
balance = B[,i]
name_ = paste(sort(names(balance)[balance != 0]), collapse = '_')
name_left = paste(sort(names(balance)[balance < 0]), collapse = '_')
name_right = paste(sort(names(balance)[balance > 0]), collapse = '_')
l_node = list(
list(
name = name_,
left = name_left,
right = name_right,
l = TREE[[name_left]]$c,
r = TREE[[name_right]]$c,
c = (TREE[[name_left]]$c+TREE[[name_right]]$c)/2
))
names(l_node) = name_
TREE = c(TREE, l_node)
}
nodes = sapply(ord_, function(i){
balance = B[,i]
paste(sort(names(balance)[balance != 0]), collapse = '_')
})
TREE = TREE[nodes]
lapply(TREE, as_tibble) %>% bind_rows()
for(node in rev(nodes)){
TREE[[node]]
}
# lapply(nodes, function(name_){
# tibble(
# name = name_,
# name_l = TREE[[name_]]$left,
# name_r = TREE[[name_]]$right,
# l = TREE[[name_]]$l,
# r = TREE[[name_]]$r,
# c = TREE[[name_]]$c,
# var = var(H[,i])
# )
# }) %>% bind_rows()
H = coordinates(X, B)
H_mean = colMeans(H)
H_var = apply(H, 2, var)
names(X)[ORDER]
l_nodes = lapply(rev(ord_), function(i){
list(
balance = B[,i],
mean = boot::inv.logit(H_mean[i]),
var = H_var[i]
)
})
names(l_nodes) = sapply(l_nodes, function(x) paste(sort(names(x$balance)[x$balance != 0]), collapse = '_'))
l_nodes
```
```{r, eval=FALSE, include=FALSE}
library(coda.base)
X = iris[,1:4]
B = pb_basis(X, method = 'exact')
rownames(B) = names(X)
H = coordinates(X, B)
apply(H, 2, var)
```
```{r, eval = FALSE, include=FALSE}
plot(H[,1], H[,2])
```
```{r, eval=FALSE, include=FALSE}
lX = split(X, iris$Species)
m_ = lapply(lX, colMeans)
s_ = lapply(lX, cov)
S = cov(X)
S_ = replicate(3, S, simplify = FALSE)
```
```{r, eval=FALSE, include=FALSE}
Prob = mapply(function(m,s){
mvtnorm::dmvnorm(X, m, s)
}, m_, S_)
```
```{r, eval=FALSE, include=FALSE}
B_1 = pb_basis(Prob, method = 'exact')
H_1 = coordinates(Prob, B_1)
plot(H_1, col = iris$Species)
```
```{r, eval=FALSE, include=FALSE}
B_2 = pc_basis(Prob)
H_2 = coordinates(Prob, B_2)
plot(-H_2, col = iris$Species)
```
```{r, eval=FALSE, include=FALSE}
library(fpc)
dp = discrproj(X, iris$Species)
plot(dp$proj[,1:2], col = iris$Species)
```
# References
```{r, eval=FALSE, echo=FALSE}
@article{cite-key,
Date-Added = {2020-06-13 08:44:39 +0000},
Date-Modified = {2020-06-13 08:44:39 +0000},
Id = {ref25},
Title = {Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2011) Principal balances. In Egozcue JJ, Tolosana-Delgado R, Ortego M (eds) Proceedings of the 4th international workshop on compositional data analysis, Girona, Spain, pp 1--10},
Ty = {STD}}
@article{cite-key,
Author = {Mart{\'\i}n-Fern{\'a}ndez, J. A. and Pawlowsky-Glahn, V. and Egozcue, J. J. and Tolosona-Delgado, R.},
Da = {2018/04/01},
Date-Added = {2020-06-13 08:42:47 +0000},
Date-Modified = {2020-06-13 08:42:47 +0000},
Doi = {10.1007/s11004-017-9712-z},
Id = {Mart{\'\i}n-Fern{\'a}ndez2018},
Isbn = {1874-8953},
Journal = {Mathematical Geosciences},
Number = {3},
Pages = {273--298},
Title = {Advances in Principal Balances for Compositional Data},
Ty = {JOUR},
Url = {https://doi.org/10.1007/s11004-017-9712-z},
Volume = {50},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1007/s11004-017-9712-z}}
Frank Ruskey LEcture note computer science 762 (1993) 205-206
@String{j-LECT-NOTES-COMP-SCI = "Lecture Notes in Computer Science"}
@String{ack-nhfb = "Nelson H. F. Beebe,
University of Utah,
Department of Mathematics, 110 LCB,
155 S 1400 E RM 233,
Salt Lake City, UT 84112-0090, USA,
Tel: +1 801 581 5254,
FAX: +1 801 581 4148,
e-mail: \path|beebe@math.utah.edu|,
\path|beebe@acm.org|,
\path|beebe@computer.org| (Internet),
URL: \path|http://www.math.utah.edu/~beebe/|"}
@Article{Ruskey:1993:SCG,
author = "F. Ruskey",
title = "Simple Combinatorial Gray Codes Constructed by
Reversing Sublists",
journal = j-LECT-NOTES-COMP-SCI,
volume = "762",
pages = "201--208",
year = "1993",
CODEN = "LNCSD9",
ISSN = "0302-9743 (print), 1611-3349 (electronic)",
ISSN-L = "0302-9743",
bibdate = "Wed Sep 15 10:01:31 MDT 1999",
bibsource = "http://www.math.utah.edu/pub/tex/bib/lncs1993.bib",
acknowledgement = ack-nhfb,
keywords = "algorithms; computation; ISAAC",
}
```