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Working with log-ratio coordinates in coda.base

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

If you don’t want the last part in the denominator, 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

It defines an alr-coordinates were the last part is used in the denominator. The basis used to build H1.alr can be obtained with the function alr_bases():

alr_basis(X)
#>       alr1 alr2 alr3
#> erc      1    0    0
#> jxcat    0    1    0
#> psc      0    0    1
#> cs      -1   -1   -1

In fact, function alr_basis allows to define any type of alr-like coordinate by defining the numerator and the denominator:

B.alr = alr_basis(X, numerator = c(4,2,3), denominator = 1)
B.alr
#>       alr1 alr2 alr3
#> erc     -1   -1   -1
#> jxcat    0    1    0
#> psc      0    0    1
#> cs       1    0    0

The log-contrast matrix can be used as basis parameter in coordinates() function:

H2.alr = coordinates(X, basis = B.alr)
head(H2.alr)
#>          alr1        alr2       alr3
#> 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

Building centered log-ratio coordinates can be accomplished by setting parameter basis='clr' or

H.clr = coordinates(X, basis = 'clr')
head(H.clr)
#>         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

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 coordinate 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

Parameter basis is set to ilr by default:

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

Other easily accessible coordinates are the Principal Component (PC) coordinates. PC coordinates define the first coordinate as the log-contrast with the highest variance, the second the one independent of the first and with the 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
barplot(apply(H2.ilr, 2, var))

Note that the PC coordinates are independent:

cov(H2.ilr)
#>               pc1           pc2           pc3
#> pc1  4.475083e-01 -6.329939e-17  3.397482e-17
#> pc2 -6.329939e-17  3.650673e-02 -9.452139e-19
#> pc3  3.397482e-17 -9.452139e-19  1.257989e-02

The Principal Balance 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
barplot(apply(H3.ilr, 2, var))

Moreover, they are not independent:

cor(H3.ilr)
#>            pb1       pb2        pb3
#> pb1  1.0000000 0.6043786 -0.3197742
#> pb2  0.6043786 1.0000000  0.1594538
#> pb3 -0.3197742 0.1594538  1.0000000

Principal Balances are challenging to compute when the number of components is very high. coda.base allows building PB approximations using different algorithms.

X100 = exp(matrix(rnorm(1000*100), ncol = 100))

• Hierarchical clustering based algorithm.

PB1.ward = pb_basis(X100, method = 'cluster')

• Constrained search algorithm

PB1.constrained = pb_basis(X100, method = 'constrained')

We can compare their performance (variance explained by the first balance) with respect to the principal components.

PC_approx = coordinates(X100, cbind(pc_basis(X100)[,1], PB1.ward[,1], PB1.constrained[,1]))
names(PC_approx) = c('PC', 'Ward', 'Constrained')
apply(PC_approx, 2, var)
#>       h1       h2       h3 
#> 1.707414 1.317988 1.567546

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)
#>        ilr1        ilr2        ilr3
#> 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

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