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...a represents the p = 6-part colourscompositions of n = 22 paintings created for teaching purposes. Here a so-called form biplot (the default)shas been produced. This preserves distances between rows (=-=Aitchison and Greenacre, 2002-=-). Bysspecifying additional input options we can just as easily produce a covariance biplot that preserves thescovariance structure between log-ratios (as in Figure 3 of Aitchison and Greenacre, 2002)...

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...because the closure may lead to unstable or erroneous results with regard to the correlations [14]. A mathematical approach consists of first applying a centered log-ratio transformation, clr(.), to the compositions thereby avoiding the problems of spurious correlation and negative bias as described by Chayes [32]. The centered log-ratio transformation is defined by Aitchinson [30]: clr xð Þ ln x1 g xð Þ ; ln x2 g xð Þ ;…; ln xD g xð Þ ; where g(x) denotes the geometric mean of the parts: g(x)= (x1,.., xD)1/D. Then, the classical biplots can be made from this new set of coordinates (see [33] for further detail and how to read and interpret a bi-plot graph; [34]). For our purposes, the interesting feature of this modified version of the biplot is that it is equivalent to the analysis of all of the pairwise log-ratios [35]. As a consequence, the data are no longer treated Table 6 Confusion matrix for the control data (i.e. 21 rock samples from series B considered as belonging to the same groups compared to their analogs belonging to series A). Calculations are made using individual spectrum of each sample. The prior probabilities are 1/3, 1/3, and 1/3 for each of the three groups. ...

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...is possible to calculate with CoDaPack, following Aitchison methodology (1986), the center and total variance (Fig. 4). The main part of this data set corresponds to K which has the highest value on the center (0.83) followed by Ti (0.11) while the rest of the values are very small, specially Au (2.1x10-7). The variation array (Fig. 4) shows that Mn is the part with largest relative variability in the clrtransformed data set, followed by Cu and As. It is possible to see a two dimensional graphical display of both parts and observations of the clrtransformed data set. The compositional biplot (Aitchison and Greenacre, 2002) displayed on Figure 6 explains an 82% of total variability of the data set. It verifies that largest variability appears on clr(Mn), followed by clr(Cu), clr(As) and clr(W). As the link between two vertices is roughly proportional to the logratio variance var[ln(Xi/ Xj)], and the corresponding vertices of Pb, K and W are very close, the variation between those parts is expected to be very low. Figure 6. Compositional biplot of the data set. Also it is possible to see that both groups, metavulcanitas and granitoides, have differences on compositions. Metavulcanitas have larger values on Mn, Cu...

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...variables in it are expressed in the same relative scale, and their variations are thus comparable. Once this homogeneous system is obtained, we could directly do compositional statistics with it. For instance, it is well known that a singular value decomposition (SVD, Eckart and Young, 1936) would give us a series of maximal variant directions as linear combinations of components. In an Aitchison geometry of the simplex, these linear combinations will be weighted products of components, and the graphical representation of the two first dimensions is the compositional biplot (Aitchison, 1997; Aitchison and Greenacre, 2002). Figure 3A contains such a biplot, including isotope data, and Figure 3B contains the biplot without isotope data. The two biplots are nearly identical, and both [32SO4 =] and [34SO4 =] lie exactly in the same arrow thus indicating that differences between them are masked by the variability in sulphate concentration. The biplot with [18OSO4] and [ 16OSO4] is identical to that of Figure 2B. The same results are obtained if water is replaced by [18OH2O] and [ 16OH2O], or [D] and [H]. Mg Ca Sr BaHCO3 H Na K NH4 Cl NO3 PO4 TOC ppm 34SSO4 ppm 32SSO4 Sr NO3 H Na K Mg Ca Ba NH4 Cl PO4 SO4 HCO3 TOC A...

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...centred log-ratio) to avoid spurious correlations generated by the intrinsically closednature of the concentration data (Aitchison, 1986), as recommended by Van den Boogaart and Tolosana-Delgado (2013): clrðxÞ ln x1 gmðxÞ; ln x2 gmðxÞ;…; ln xD gmðxÞ where gm(x) is the geometric mean of the parts: gm(x) (x1… xD)1/ D. Covariance biplots of compositional data represent the relationships among log-ratios of the variables (trace elements) and sites (peatlands) on the same plane (Borgheresi et al., 2013). Covariance biplots have properties that must be addressed for a correct interpretation (Aitchison and Greenacre, 2002; Van den Boogaart and Tolosana-Delgado, 2013): (i) the lengths of links (lines joining two arrowheads) are approximations of the standard deviation of the corresponding log-ratio, and (ii) the angle cosines between links estimate correlations between log-ratios (e.g., a link perpendicular to another indicates that log-ratios among the first set of variables have correlations near zero with the log-ratios among the second set). This modified version of the biplot for compositional data has the interesting feature of being equivalent to the analysis of all pairwise log-ratios (Daunis-i-Estadell...