## 2 Processing of HS-SPME-GC-MS Data on Wine Volatiles by Multivariate Statistics
### Linear Discriminant Analysis  
Linear Discriminant Analysis (LDA) is one of the most used methods to perform supervised pattern recognition. This method is usually based on identifying linear combinations of independent variables (called Linear Discriminant Functions, LDF) maximizing the between-class variance and minimizing the within-class variance. In particular, if k is the number of classes, and if the number of independent variables is larger than k, then k-1 LDFs are identified. The resulting LDF may be used as a linear classifier, or for dimensionality reduction before later classification. LDF can be used as a dimensionality reduction technique as it determines a hyperspace with less dimensions than the original data, on which these can be projected to achieve the best possible linear separation among the given classes. The LDA is based on fairly strong assumptions, i.e., that the classes are linearly separable in the multidimensional space of the independent variables and that the variance-covariance matrices are equal for each class. These assumptions do not allow this type of analysis to be applied without a careful validation. A further condition for the application of the LDA is that the number of samples in the training dataset must be greater than the number of independent variables. If the number of independent variables is greater, \"feature selection\" techniques, i.e., determining a limited but representativesubset of independent variables, or \"feature reduction\" techniques, that enable a reduction of the size of data without significant loss of information, can be adopted. As previously mentioned, the PCA can be used for this purpose.  
Turning to applications of LDA to wine volatiles' data, the combination of LDA and PCA analysis was successfully applied for the varietal differentiation of _Loureira_, _Dona Branca_, and _Triexadura_ wines [28], starting from their volatile profiles. Thirty-four _Sauwignon Blanc_ wine samples from three different countries and six regions were analyzed by HS-SPME-GC-MS by [PERSON] et al. [65] and LDA was applied to the resulting data, showing three distinct clusters or classes of wines with different aroma profiles. In particular, wines from the Loire Region in France and wines from Tasmania and Western Australia were found to have similar aroma patterns. In a research on the effect of time and storage conditions on major volatile compounds of _Zalema_ white wine, LDA was exploited to distinguish among wines with different conditions and times of storage [66]. [PERSON] et al. [67] adopted the synergy between PCA and LDA to find the differences in the volatile profiles among Chilean sparkling wines obtained with different production methods. In addition, LDA, coupled with PCA and multilayer perceptrons' neural networks (MLP-NN) were used as chemometric tools to differentiate several Spanish white wines according to their geographical origin, using selected volatile compounds as input variables [68]. Similarly, linear discriminant analysis (LDA) was successfully applied to obtain an appropriate classification of white and red wines of various geographical origins using the volatolomic approach [8].  
Recently, [PERSON] et al. [42] used LDA to study the volatile profile of different crosses of white wines obtained from _Monastrell_ and other varieties, showing that the white crosses obtained from red varieties were aromatically more similar to the white wine than to the respective parental.