Section 6 Transformation of the particle-size data

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Sand, silt and clay contents are reported as proportions that sum to 100%. However, models formulated for each of these fractions do not guarantee that their individual predictions sum to 100%. To avoid this compositional constraint, the particle-size data (\(V = {clay, silt, sand}\)) for both depths (\(l = {A: 0–0.2 m, B: 0.8–1.0 m}\)) were transformed using the additive log-ratio (\(alr\)) transformation (Odeh et al., 2003): \[Y_{l,i} = \frac{V_{l,i}}{V_{l,r}} \quad \forall \quad i = 1,2,.. (r -1) \quad \forall \quad l \in (A,B)\]
where \(Y_{l,i}\) is the resulting transformed variable, \(V_{l,i}\) is the ith variable of the set of compositional variables (silt and clay contents) at depth \(l\), \(V_{l,r}\) designates a reference compositional variable at depth \(l\) and \(r\) is the total number of compositional variables. In ou paper, we used the sand content as reference (\(V_{l,r}\)).

## The above equation can be simply applied in two lines of code

## Calibration dataset
## alr for silt contnet
train$alr_Silt <- log(train$Silt/train$Sand)
## alr for clay contnet
train$alr_Clay <- log(train$Clay/train$Sand)

## Validation dataset
## alr for silt contnet
valida$alr_Silt <- log(valida$Silt/valida$Sand)
## alr for clay contnet
valida$alr_Clay <- log(valida$Clay/valida$Sand)

## Prediction dataset
## alr for silt contnet
pred$alr_Silt <- log(pred$Silt/pred$Sand)
## alr for clay contnet
pred$alr_Clay <- log(pred$Clay/pred$Sand)