# Working with log-ratio coordinates in coda.base

#### 2019-04-15

In this vignette we show how to define log-ratio coordinates using coda.base package and its function coordinates with parameters X, a composition, and basis, defining the independent log-contrasts for building the coordinates.

In this vignette we work with a subcomposition of the results obtained in different regions of Catalonia in 2017’s parliament elections:

library(coda.base)
# By default basis is not shown, in this vignette we turn on basis showing.
options('coda.base.basis' = TRUE)
data('catalan_elections_2017')
X = parliament2017[,c('erc','jxcat','psc','cs')]

## Defining log-ratio coordinates with coda.base

### The additive logratio (alr) coordinates

The alr coordinates are accessible by setting parameter basis='alr' or by using the building function alr_basis.

The easiest way to define an alr-coordinates is to set basis='alr'.

H1.alr = coordinates(X, basis = 'alr')
head(H1.alr)
#>          alr1        alr2       alr3
#> 1  0.23864536 0.446503630 -0.7201917
#> 2 -0.10388120 0.216858085 -1.0473730
#> 3  0.36723896 0.542010167 -0.5320675
#> 4  0.53209369 0.798479995 -0.4799141
#> 5  0.54918649 0.477309280 -0.1028807
#> 6 -0.09742133 0.002856425 -0.6858265
#>  Basis:
#>    alr1 alr2 alr3
#> P1    1    0    0
#> P2    0    1    0
#> P3    0    0    1
#> P4   -1   -1   -1

It defines an alr-coordinates were the first parts are used for the numerator of the log-quotient and the last part for the denominator.

The basis can be reproduced using the function alr_basis:

alr_basis(dim = 4)
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> [4,]   -1   -1   -1

Function alr_basis allows to define other alr-coordinates by defining the numerator and the denominator.

B.alr = alr_basis(dim = 4, numerator = c(4,2,3), denominator = 1)
B.alr
#>      [,1] [,2] [,3]
#> [1,]   -1   -1   -1
#> [2,]    0    1    0
#> [3,]    0    0    1
#> [4,]    1    0    0

The log-contrast matrix defining the basis can be used in basis parameter:

H2.alr = coordinates(X, basis = B.alr)
head(H2.alr)
#>            x1          x2         x3
#> 1 -0.23864536  0.20785827 -0.9588371
#> 2  0.10388120  0.32073928 -0.9434918
#> 3 -0.36723896  0.17477121 -0.8993065
#> 4 -0.53209369  0.26638630 -1.0120078
#> 5 -0.54918649 -0.07187721 -0.6520672
#> 6  0.09742133  0.10027776 -0.5884051
#>  Basis:
#>    x1 x2 x3
#> P1 -1 -1 -1
#> P2  0  1  0
#> P3  0  0  1
#> P4  1  0  0

### The centered logratio (clr) coordinates

Building centred log-ratio coordinates can be accomplished by setting parameter basis=TRUE:

H.clr = coordinates(X, basis = 'clr')
head(H.clr, basis = TRUE)
#>         clr1      clr2       clr3         clr4
#> 1 0.24740605 0.4552643 -0.7114311  0.008760689
#> 2 0.12971783 0.4504571 -0.8137740  0.233599031
#> 3 0.27294355 0.4477148 -0.6263629 -0.094295406
#> 4 0.31942879 0.5858151 -0.6925790 -0.212664904
#> 5 0.31828271 0.2464055 -0.3337844 -0.230903777
#> 6 0.09767651 0.1979543 -0.4907286  0.195097842
#>  Basis:
#>     clr1  clr2  clr3  clr4
#> P1  0.75 -0.25 -0.25 -0.25
#> P2 -0.25  0.75 -0.25 -0.25
#> P3 -0.25 -0.25  0.75 -0.25
#> P4 -0.25 -0.25 -0.25  0.75

### The isometric logratio (ilr) coordinates

coda.base allows to define a wide variety of ilr coordinates: principal components (pc) coordinates, specific user balances coordinates, principal balances (pb) coordinates, balanced coordinates (default’s CoDaPack’s coordinates).

The default ilr coordinates used by coda.base are accessible by simply calling function coordinates without parameters.

H1.ilr = coordinates(X)
head(H1.ilr)
#>          ilr1      ilr2        ilr3
#> 1 -0.14697799 0.8677450 -0.01011597
#> 2 -0.22679692 0.9012991 -0.26973693
#> 3 -0.12358191 0.8056307  0.10888296
#> 4 -0.18836356 0.9350526  0.24556428
#> 5  0.05082486 0.5030669  0.26662472
#> 6 -0.07090708 0.5213690 -0.22527958
#>  Basis:
#>          ilr1       ilr2       ilr3
#> P1  0.7071068  0.4082483  0.2886751
#> P2 -0.7071068  0.4082483  0.2886751
#> P3  0.0000000 -0.8164966  0.2886751
#> P4  0.0000000  0.0000000 -0.8660254

Parameter basis is set to ilr by default:

all.equal( coordinates(X, basis = 'ilr'),
H1.ilr )
#> [1] TRUE

Other easily accessible coordinate is the pc coordinates. pc coordinates define the first coordinate as the log-contrast with highest variance, the second the one independent from the first and with highest variance and so on:

H2.ilr = coordinates(X, basis = 'pc')
head(H2.ilr)
#>         pc1         pc2        pc3
#> 1 0.6787536  0.35694598 -0.4319368
#> 2 0.5581520  0.57775877 -0.5396259
#> 3 0.7013616  0.25302877 -0.3467523
#> 4 0.8973701  0.25915667 -0.3125234
#> 5 0.5362270 -0.05527103 -0.1901418
#> 6 0.2676101  0.32802497 -0.3852126
#>  Basis:
#>           pc1        pc2        pc3
#> P1  0.3469512 -0.5978990 -0.5216720
#> P2  0.6300769  0.4877904  0.3392104
#> P3 -0.4368610 -0.3913286  0.6371926
#> P4 -0.5401671  0.5014372 -0.4547309
barplot(apply(H2.ilr, 2, var))

The pb coordinates are similar to pc coordinates but with the restriction that the log contrast are balances

H3.ilr = coordinates(X, basis = 'pb')
head(H3.ilr)
#>          pb1         pb2         pb3
#> 1 -0.7026704 -0.14697799 -0.50925247
#> 2 -0.5801749 -0.22679692 -0.74060456
#> 3 -0.7206583 -0.12358191 -0.37622854
#> 4 -0.9052439 -0.18836356 -0.33935049
#> 5 -0.5646882  0.05082486 -0.07274761
#> 6 -0.2956308 -0.07090708 -0.48495254
#>  Basis:
#>     pb1        pb2        pb3
#> P1 -0.5  0.7071068  0.0000000
#> P2 -0.5 -0.7071068  0.0000000
#> P3  0.5  0.0000000  0.7071068
#> P4  0.5  0.0000000 -0.7071068
barplot(apply(H3.ilr, 2, var))

Finally, coda.base allows to define the default CoDaPack basis which consists in defining well balanced balances, i.e. equal number of branches in each balance.

H4.ilr = coordinates(X, basis = 'cdp')
head(H4.ilr)
#>        cdp1        cdp2        cdp3
#> 1 0.7026704 -0.14697799 -0.50925247
#> 2 0.5801749 -0.22679692 -0.74060456
#> 3 0.7206583 -0.12358191 -0.37622854
#> 4 0.9052439 -0.18836356 -0.33935049
#> 5 0.5646882  0.05082486 -0.07274761
#> 6 0.2956308 -0.07090708 -0.48495254
#>  Basis:
#>    cdp1       cdp2       cdp3
#> P1  0.5  0.7071068  0.0000000
#> P2  0.5 -0.7071068  0.0000000
#> P3 -0.5  0.0000000  0.7071068
#> P4 -0.5  0.0000000 -0.7071068

### Defining coordinates manually

We can define the coordinates directly by providing the log-contrast matrix.

B = matrix(c(1,-1,2,0,
1,0,-0.5,-0.5,
-0.5,0.5,0,0), ncol = 3)
H1.man = coordinates(X, basis = B)
head(H1.man)
#>         x1        x2          x3
#> 1 15.45379 0.5987412  0.10392914
#> 2 17.02629 0.4198053  0.16036964
#> 3 17.44436 0.6332727  0.08738560
#> 4 13.77042 0.7720507  0.13319315
#> 5 11.62718 0.6006268 -0.03593861
#> 6 18.10523 0.2454919  0.05013888
#>  Basis:
#>    x1   x2   x3
#> P1  1  1.0 -0.5
#> P2 -1  0.0  0.5
#> P3  2 -0.5  0.0
#> P4  0 -0.5  0.0

We can also define balances using formula numerator~denominator:

B.man = sbp_basis(b1 = erc~jxcat,
b2 = psc~cs,
b3 = erc+jxcat~psc+cs,
data=X)
H2.man = coordinates(X, basis = B.man)
head(H2.man)

With sbp_basis we do not need to define neither a basis nor a system generator

B = sbp_basis(b1 = erc+jxcat~psc+cs,
data=X)
#> Warning in sbp_basis(b1 = erc + jxcat ~ psc + cs, data = X): Given
#> partition is not a basis
H3.man = coordinates(X, basis = B)
head(H3.man)
#>          x1
#> 1 0.7026704
#> 2 0.5801749
#> 3 0.7206583
#> 4 0.9052439
#> 5 0.5646882
#> 6 0.2956308
#>  Basis:
#>      x1
#> P1  0.5
#> P2  0.5
#> P3 -0.5
#> P4 -0.5

or

B = sbp_basis(b1 = erc~jxcat+psc~cs,
b2 = jxcat~erc+psc+cs,
b3 = psc~erc+jxcat+cs,
b4 = cs~erc+jxcat+psc,
data=X)
#> Warning in sbp_basis(b1 = erc ~ jxcat + psc ~ cs, b2 = jxcat ~ erc + psc
#> + : Given basis is not orthogonal
H4.man = coordinates(X, basis = B)
head(H4.man)
#>            x1        x2         x3          x4
#> 1 -0.01011597 0.5256940 -0.8214898  0.01011597
#> 2 -0.26973693 0.5201431 -0.9396653  0.26973693
#> 3  0.10888296 0.5169765 -0.7232616 -0.10888296
#> 4  0.24556428 0.6764410 -0.7997213 -0.24556428
#> 5  0.26662472 0.2845246 -0.3854211 -0.26662472
#> 6 -0.22527958 0.2285779 -0.5666446  0.22527958
#>  Basis:
#>            x1         x2         x3         x4
#> P1  0.2886751 -0.2886751 -0.2886751 -0.2886751
#> P2  0.2886751  0.8660254 -0.2886751 -0.2886751
#> P3  0.2886751 -0.2886751  0.8660254 -0.2886751
#> P4 -0.8660254 -0.2886751 -0.2886751  0.8660254

We can also define sequential binary partition using a matrix.

P =  matrix(c(1, 1,-1,-1,
1,-1, 0, 0,
0, 0, 1,-1), ncol= 3)
B = sbp_basis(P)
H5.man = coordinates(X, basis = B)
head(H5.man)
#>          x1          x2          x3
#> 1 0.7026704 -0.14697799 -0.50925247
#> 2 0.5801749 -0.22679692 -0.74060456
#> 3 0.7206583 -0.12358191 -0.37622854
#> 4 0.9052439 -0.18836356 -0.33935049
#> 5 0.5646882  0.05082486 -0.07274761
#> 6 0.2956308 -0.07090708 -0.48495254
#>  Basis:
#>      x1         x2         x3
#> P1  0.5  0.7071068  0.0000000
#> P2  0.5 -0.7071068  0.0000000
#> P3 -0.5  0.0000000  0.7071068
#> P4 -0.5  0.0000000 -0.7071068