## 2 Processing of HS-SPME-GC-MS Data on Wine Volatiles by Multivariate Statistics
### Partial Least Squares  
If the number of independent variables is greater than the number of samples, it is still possible to use multivariate statistical techniques, such as Partial Least Square (PLS), that are not subject to this constraint. The PLS approach is particularly suited for data containing correlated independent variables, since it constructs new predictor variables, called latent variables or components, as linear combinations of the original ones. PLS is designed to evaluate these components while considering the observed response values. The working principle of PLS is to find a finite number of linear combinations of the independent variables describing its variance as much as possible and at the same time having the largest correlation with the dependent variables. This is obtained by algorithms capable of maximizing the covariance between independent and dependent variables.  
[PERSON] et al. [69], illustrated a study on the possibility of obtaining regression models by means of Partial Least Squares (PLS) and Multiple Linear Regression (MLR) to correlate a series of parameters, such as the concentration of short-chain organic acids, higher alcohols, and phenolic compounds, to the age of vintage _Sherry_ wines. PLS was also successfully applied in a study focused on the correlations between volatile compounds of _Albarrino_ wines and sensory descriptors [51]. As demonstrated by other authors, PLS regression is, among multivariate techniques, the best approach to highlight the correlations between chemical data, obtained by HS-SPME-GC-MS, and sensory descriptors [70; 71; 72].  
If the independent variable is categorical, the PLS technique can be used as a supervised pattern recognition technique. In this case, reference is made to the Partial Least Square Discriminant Analysis (PLS-DA) variant. In this variant, the operating principle remains unchanged, and the dependent categorical variable (the class) is replaced by a suitable \"dummy\" multivariate variable. This consists of as many single variables as classes. The value of dummy variables is set to 1 if the sample belongs to its corresponding class, and to 0 otherwise. PLS-DA is a compromise between the usual discriminant analysis and a discriminant analysis based on the principal components of the predictor variables. PLS-DA can provide a good insight into the factors leading to effective discrimination between samples by the analysis of the components (the loading vectors) and their related sample scores, which gives it a relevant role in exploratory data analysis. In order to discriminate between selected wines with different geographical origin (Azores, Canary and Madeira Islands) and of different types (white wine, red wine and fortified wine), the volatile profiles were characterized by [PERSON] et al. [73]. The authors applied the PLS-DA to the dataset to obtain a predictive model for classification of examined wine samples according to their geographical origin and type. This information is crucial to prevent fraud and, therefore, to guarantee wine authenticity. In a study on the volatiles of _Chardonnay_ wine, PLS-DA was adopted to find the key volatile metabolites able to discriminate wines fermented by different yeast strains [74].  
Recently, [PERSON] et al. [75] used PLS-DA to successfully study the discrimination in the volatile composition of a 48-month old bottle-stored white wine closed with either cork, micro-agglomerated or synthetic stoppers, revealing the most discriminating volatiles. PLS-DA was also applied, along with PCA, by [PERSON] et al. [10] to discriminate white wines produced in the _Friulano Collio_ area in the region of _Friuli_ (North-East of Italy) from those produced in other areas of the same region. In a paper by [PERSON] et al. [76], PLS-DA based on VOCs recognized by HS-SPME-GC-MS analysis was adopted to assess the differences between dry and semi-sec white wines produced from eight different Greek grape varieties.