## 3 Results and Discussion
### Relating Samples to VOCs by CoDa Exploratory Analysis  
#### 3.5.1 Variation Array and Biplot  
Usually, the evolution of components is measured by the central tendency and variance. However, in the compositional analysis, the mean and variance are meaningless. Instead, the analysis is focused on the log-ratios [11]. The variability of the VOCs is estimated, column-wise, as the log-ratio variance of every component over each other. High variances are associated with differences between spoiled and normal fermentation samples. Their values are presented in the so-called variation array (Supplementary Table S3), which includes log-ratio variances (upper half) and log-ratio means (bottom half). The highest variance values are observed for methyl butanoate (V13), ethyl butanoate (V18), propanoic acid (V58), or butanoic acid (V62) with different levels according to the VOC in the denominator. However, when the original data are _clr_ transformed (coefficients), the individual variances (within, between, or regardless of treatment) of each _clr_ transformed VOC (_clr coefficients_) are estimated by the standard procedure. It is named the _clr_ variance and represents the _clr coefficients_' variability over treatments (Supplementary Table S3, last column at the right). The VOCs more affected by the spoilage (Supplementary Table S2) will show the highest values. Among them (Supplementary Table S3): ethyl butanoate (V18) (related to butyric spoilage), butanoic acid (V62) (butyric), methyl butanoate (V13) (butyric), 2-pentanone (V11) (sulfidic and putrid), and pentanoic acid (V68) (butyric). The tetrahedral plot of the samples as a function of the first four variables shows differences in contents between F282 and F283 (butyric samples, industry A) and F309 and F285 (normal process, industry A) or between F202 (industry B) and most of the other samples (regardless of industry), which are grouped in the c(V55) vertex. The close position of samples (regardless of industry) means that they have similar VOCs levels (Figure 4). Therefore, the plot was helpful for visualising the differences between samples, although the geometry only allowed for comparison as a function of four components. The CoDa biplot is one of the most efficient exploratory tools in CoDa analysis [26]. The PC1, PC2, and PC3 explained 42.36%, 18.77%, and 10.36% of the total variance (Figure 5). The biplots showed numerous variables following similar trends (redundant information). Besides, many of them present short arrows, which indicates a poor representation on the PC1 vs PC2 plane (Figure 5A) or PC1 vs PC3 (Figure 5B). Notably, on the PC1 vs PC2 plane, spoiled buryric samples were situated on the left, normal fermentation samples from industry A on the fourth quadrant, and all samples from industry B on the first (Figure 5A). The segregation of butyric samples from the rest was based on log-ratios between compounds on the left divided by those on the right, or vice versa, (e.g., In[methyl butanoate (V13)/2-pentanol (V28)] or In[ethanol (V10)/(Z)-3-hexen-1-ol (V48)] which, in turn, were associated to PC1 (Figure 5A). The plot also shows that the highest variance (distance between arrows) also corresponded to the same log-ratios. However, log-ratios along PC2 were considerably lower (e.g., In[1,4-dimethoxybenzene (V67)/ethyl lactate (V44)], although enough to separate samples from industry A and B and even the two samples affected by the butyric spoilage. Their variances are also low (short distance between arrows). In the presentation on PC1 vs PC3 (Figure 5B), the log-ratio (benzaldehyde \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**From\textbackslash{}To** & **1** & **2** & **3** & **4** & **5** & **Total** & **\% Correct** \\ \hline
1 & 4 & 0 & 0 & 0 & 0 & 4 & 100.0 \\
2 & 0 & 8 & 0 & 0 & 0 & 8 & 100.0 \\
3 & 0 & 0 & 0 & 2 & 0 & 2 & 0.0 \\
4 & 0 & 0 & 0 & 2 & 0 & 2 & 100.0 \\
5 & 0 & 0 & 0 & 0 & 4 & 4 & 100.0 \\ Total & 4 & 8 & 0 & 4 & 4 & 20 & 90.0 \\ \hline \end{tabular}
\end{table} 
Table 2: Classification tree analysis.