The additive logratio (alr) coordinates

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
#>  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 in order and the last part is used in 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

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(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 can be used in basis parameter:

H2.alr = coordinates(X, basis = B.alr)
head(H2.alr)
#>            h1          h2         h3
#> 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:
#>    h1 h2 h3
#> P1 -1 -1 -1
#> P2  0  1  0
#> P3  0  0  1
#> P4  1  0  0

The centered logratio (clr) coordinates

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

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
#>  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 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
#>  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 ilr-coordinates: Principal Components and Principal balances

Other easily accessible coordinates are the Principal Component (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))

Note that the PC coordinates are independent:

cov(H2.ilr)
#>               pc1           pc2           pc3
#> pc1  4.475083e-01 -5.791265e-16 -2.995221e-16
#> pc2 -5.791265e-16  3.650673e-02 -1.781210e-18
#> pc3 -2.995221e-16 -1.781210e-18  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
#>  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))

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 hard to compute when the number of components is very high. coda.base allows to build 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 they 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 
#> 1.686637 1.334032 1.532599

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 with an specific basis

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)
#>          h1        h2          h3
#> 1 -2.125532 0.5987412  0.10392914
#> 2 -2.207723 0.4198053  0.16036964
#> 3 -1.973384 0.6332727  0.08738560
#> 4 -2.290402 0.7720507  0.13319315
#> 5 -1.232257 0.6006268 -0.03593861
#> 6 -1.277088 0.2454919  0.05013888
#>  Basis:
#>    h1   h2   h3
#> P1 -1  1.0 -0.5
#> P2 -1  0.0  0.5
#> P3  2 -0.5  0.0
#> P4  0 -0.5  0.0

Defining coordinates using balances

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)
#>          h1
#> 1 0.7026704
#> 2 0.5801749
#> 3 0.7206583
#> 4 0.9052439
#> 5 0.5646882
#> 6 0.2956308
#>  Basis:
#>      h1
#> 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)
#>            h1        h2         h3          h4
#> 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:
#>            h1         h2         h3         h4
#> 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)
#>          h1          h2          h3
#> 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:
#>      h1         h2         h3
#> 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