Main functions
- Simulated data set:
nP = 6
nO = 8
set.seed(1)
X = matrix(rlnorm(nP * nO), ncol = nP, nrow = nO)
colnames(X) = paste0('P', 1:nP)
rownames(X) = paste0('O', 1:nO)
X
#> P1 P2 P3 P4 P5 P6
#> O1 0.5344838 1.7785196 0.9839401 1.8586041 1.4735458 0.8482977
#> O2 1.2015872 0.7368371 2.5698209 0.9454174 0.9476168 0.7761871
#> O3 0.4336018 4.5348008 2.2732743 0.8557342 0.2523194 2.0076470
#> O4 4.9297132 1.4767493 1.8110401 0.2297526 0.6603439 1.7448406
#> O5 1.3902836 0.5372775 2.5067256 0.6199292 0.6741586 0.5022006
#> O6 0.4402254 0.1091863 2.1861375 1.5188319 0.9424114 0.4928772
#> O7 1.6281250 3.0800041 1.0774154 3.8910520 3.0042422 1.4399119
#> O8 2.0924271 0.9560610 0.1367841 0.9023185 2.1450776 2.1566000Descriptive statistics
- Center:
center(X)
#> P1 P2 P3 P4 P5 P6
#> 0.1732294 0.1599997 0.1982046 0.1544340 0.1509712 0.1631611- Variation array:
variation_array(X)
#> P1 P2 P3 P4 P5 P6
#> P1 0.00000000 1.78073540 2.1542290 2.14577559 0.96385810 0.6955516
#> P2 0.07944508 0.00000000 2.6142898 1.90084736 2.05938765 0.6793622
#> P3 -0.13468257 -0.21412765 0.0000000 1.91662090 2.49665679 1.9518749
#> P4 0.11485030 0.03540522 0.2495329 0.00000000 0.52964238 1.2179905
#> P5 0.13752808 0.05808300 0.2722107 0.02267778 0.00000000 0.9414218
#> P6 0.05987869 -0.01956639 0.1945613 -0.05497161 -0.07764939 0.0000000Aitchison distances between compositions
stats::dist() is rewritten to include the Aitchison
distance between compositions:
dist(X, method = 'aitchison')
#> O1 O2 O3 O4 O5 O6 O7
#> O2 1.7312698
#> O3 2.4651454 2.6482563
#> O4 3.2924718 2.2982518 3.1363020
#> O5 2.1967501 0.5310050 2.9124924 2.1147115
#> O6 2.6637001 1.8840274 4.0189064 3.9336894 2.0813815
#> O7 0.7482808 1.9679763 2.9204289 3.2563331 2.3587517 2.8831297
#> O8 2.7579161 3.2673090 4.1643719 3.2700093 3.5174190 4.0863918 2.2159481Log-ratio coordinates. coordinates()
coordinates(X)
#> ilr1 ilr2 ilr3 ilr4 ilr5
#> O1 -0.8501086 -0.00746765 -0.5560864 -0.2230977 0.3219244
#> O2 0.3457976 -0.82034125 0.2859262 0.2193991 0.3613081
#> O3 -1.6598694 -0.39448617 0.5671774 1.5316655 -0.6427156
#> O4 0.8523731 0.32550510 2.0182094 0.6189998 -0.3815869
#> O5 0.6722806 -0.86944130 0.5951600 0.3860022 0.5839787
#> O6 0.9858706 -1.87771387 -1.0123423 -0.3572872 0.2999825
#> O7 -0.4507819 0.59736115 -0.6896777 -0.3028747 0.4240687
#> O8 0.5538473 1.90737459 -0.2850948 -0.9953748 -0.8176105
#> attr(,"class")
#> [1] "matrix" "array"By default, coda.base uses the isometric log-ratio
coordinates defined in Egozcue et al. 2003.
coordinates(X, 'ilr'): isometric log-ratio coordinates (Egozcue et al. 2003, defaults)coordinates(X, 'alr'): additive log-ratio coordinatescoordinates(X, 'clr'): centered log-ratio coordinatescoordinates(X, 'pw'): pairwise log-ratio coordinatescoordinates(X, 'pc'): principal component coordinatescoordinates(X, 'pb'): principal balance coordinatescoordinates(X, 'cdp'): balanced isometric log-ratio coordinates
coordinates(X, B) accepts a log-contrast matrix \(B\) to build the log-ratio coordinates.
Different log-contrast matrices \(B\)
can be constructed (following section).
Functions to build log-contrast matrices
ilr_basis(nP)
#> ilr1 ilr2 ilr3 ilr4 ilr5
#> c1 0.7071068 0.4082483 0.2886751 0.2236068 0.1825742
#> c2 -0.7071068 0.4082483 0.2886751 0.2236068 0.1825742
#> c3 0.0000000 -0.8164966 0.2886751 0.2236068 0.1825742
#> c4 0.0000000 0.0000000 -0.8660254 0.2236068 0.1825742
#> c5 0.0000000 0.0000000 0.0000000 -0.8944272 0.1825742
#> c6 0.0000000 0.0000000 0.0000000 0.0000000 -0.9128709all.equal(as.numeric(coordinates(X, 'ilr')),
as.numeric(log(X) %*% ilr_basis(nP)))
#> [1] TRUELog-ratio matrix transformations:
ilr_basis(nP)(Egozcue et al. 2003, defaults)ilr_basis(nP, type = 'pivot')to pivot log-ratio coordinatesilr_basis(nP, type = 'cdp')to balanced log-ratio coordinates (CoDaPack’s default)alr_basis(nP)to additive-log ratio coordinates. Numerator order and denominator can be modified. For example,alr_basis(nP, denominator = 1, numerator = nP:2).clr_basis(nP)to centered log-ratio coordinates.pc_basis(X)to principal components log-ratio coordinates.cc_basis(X, X2)to canonical correlations log-ratio coordinates.pb_basis(X, method = 'exact')to principal balances using the exact algorithm.pb_basis(X, method = 'constrained')to principal balances using pca constrained algorithm.pb_basis(X, method = 'cluster')to principal balances obtained using parts clustering algorithm.pairwise_basis(nP)to pairwise log-ratio coordinates.sbp_basis(b0 = b1~b2, b1 = P1~P2+P3, b2 = P4~P5+P6, data=X)