## 2 Materials and Methods
### Compositional Data Analysis  
Compositional data (CoDa) refer to vectors of positive components representing proportions of some whole [44]. All components, also called parts, are assumed positive,  
Figure 1: Scheme of the system designed to extract aromatic compounds present in wine sparkling samples onto desorption tubes using a TD method. The red arrow indicates the direction of the intake air. and the only relevant information is contained in the ratios between them. A simplified manner to represent CoDa is using the closed form, which is a positive vector where the parts sum up to a positive constant, i.e., \(\mathbf{x}=\left(x_{1},x_{2},\ldots,\ x_{D}\right)\) with \(x_{i}>0\) and \(x_{1}+\cdots+x_{D}=k\). Typically, \(k=1\) for parts per unit or \(k=100\) for percentages. The sample space of CoDa is called the simplex, and it is denoted as \(S^{D}\). In the context of our study, the volatile compound data obtained using the method described in the previous section and through the normalizing approach align with this definition. In general, standard statistical methods presuppose an absolute scale for data, which conflicts with the inherent relativity of CoDa. Therefore, we will require statistical methods that acknowledge this relative nature, and this implies working with ratios or, more specifically, with log-ratios. In the literature, we can find different possible log-ratios introduced as one-to-one transformation from \(S^{D}\) to the real space [44]. For the purpose of this paper, we will consider the so-called centered log-ratio transformation (clr), where the composition \(\mathbf{x}\) is expressed as vector of the score. \[\mathrm{clr}(\mathbf{x})=\left(\ln\frac{x_{1}}{g(\mathbf{x})},\ldots,\ln\frac {x_{D}}{g(\mathbf{x})}\right)\]  
where \(g(\mathbf{x})\) is the geometric mean of the parts of \(\mathbf{x}\); that is, \(g(\mathbf{x})=\sqrt[D]{x_{1}\cdots x_{D}}\). When interpreting the components of the clr-vector, it is essential to consider that they represent part \(x_{i}\) relative to the average of all parts. Moreover, some simple log-ratios, including \(\ln\bigl{(}x_{i}/x_{j}\bigr{)}\), will also be used. This log-ratio approach is already justified and applied in some works using data obtained from chromatography [45]. In the case of descriptive statistics, there are some alternatives to the standard arithmetic mean and variance called the center, variation matrix and total variance [46]. Let us consider a CoDa set represented as a matrix \(\mathbf{X}=\left(x_{ij}\right),\ i=1,\ldots,n;j=1,\ldots,D\), with n rows (samples) and D columns (volatile compounds). The center of the data set \(\mathbf{X}\) is \(\mathit{cen}(\mathbf{X})=C(g_{1},\ldots,g_{D})\), where \(g_{j}\) is the geometric mean of the column \(j\) and \(C(\cdot)\) represents the closure operator applied to rescale the resulting vector to the constant sum \(k\). Note that here, the geometric mean is considered column-wise. In addition, the dispersion of a CoDa set is usually quantified using the variation matrix, a \(D\times D\) matrix formed by the usual variance of the simple log-ratios, and \(\mathit{var}\left(\ln\bigl{(}x_{i}/x_{j}\bigr{)}\right),\ i,j=1,\ldots,D\). Note here that variances close to zero indicate two redundant parts. A global measure of relative dispersion, called total variance, is defined as the sum of all the variances of the variation matrix divided by \(2D\). The total variance of \(\mathbf{X}\) equals the trace of the covariance matrix of the clr-scores data set. A compositional MANOVA test [47] can be applied to determine whether some significant difference among group means exists. The non-parametric compositional MANOVA test and a homoscedasticity study are recommended when the multivariant normality on \(S^{D}\)[48] cannot be accepted. When we have two or more groups in our data set, we can use the geometric-mean bar plot to graphically compare the centers. For each group, we initially compute the ratio between the overall geometric mean and the group's geometric mean.