# Functional Characterization of _Saccharomyces_ Yeasts from Cider Produced in Hardanger
## 3 Results and Discussion
### Relationships between Aroma Compounds and Variables  
#### 3.6.1 Principal Component Analysis (PCA)  
To perceive an initial configuration of our data/variables and to simplify the dataset by identifying possible patterns and relationships between all variables, a principal component analysis (PCA) was performed and validated using the correlation matrix in IBM SPSS Statistics 27 and GraphPad Prism 9.5.1 (see Appendix A for additional description). In the PCA analysis, ethyl esters of fatty acids, ethyl esters of branched acids, acetate esters, C6 alcohols, and volatile phenols were the five chemical classes/groups of aroma compounds, and the other 28 features, including amino acids, sugars, ethanol, organic acids, etc., were entered as other important variables. From the eigenanalysis of the correlation matrix related to PCA, nine principal components (represented by PC or F) were extracted, and 54.0% of the variance in the data set was explained by the first three components (F1 = PC1 = 27.9%, F2 = PC2 = 13.8%, and F3 = PC3 = 12.4%). When we refer to explained variance in terms of the PCs, we are referring to the proportion of variance in the entire collection of response and predictor variables that is explained by the PCs. In our results, the first nine principal components have eigenvalues greater than 1 (see the scree plot in Figure 4). These nine components explained approximately 89% of the variation in the data (Figure 4). However, since the cumulative variance of 54.1% in the first three components does not report the adequate amount of variation we expected, we performed another statistical analysis, the partial least squares (PLS) regression analysis. The Component Plot in Rotated Space (CPRS) shown in Figure 4 displays the scores of the first three PCs, which capture most of the variance in our data set. This plot is a graphical representation of the results of PCA with orthogonal rotation. Each point on the plot represents a variable in our data set, and the position of the point in the new rotated space is determined by the scores of the observation on the PCs. In other words, the CPRS shows how the variables in our dataset are related based on the underlying patterns identified by the PCA method. Variables that are close to each other on the graph (e.g., lysine and putrescine) have similar principal component values, indicating that they share similar underlying patterns. Observations that are far apart on the graph (e.g., lysine and serine) have different values on the PCs, indicating that they have different underlying patterns. The CPRS can be useful in identifying clusters or groups of variables in the data set that have similar patterns. The problem of clustering yeast strains can also be exploredin the PCA Bootstrap hulls (implemented in XLSTAT 2023) in Figure 5 and the partial least squares (PLS) regression analysis (implemented in Minitab 21 and XLSTAT 2023) in Figure 6.  
Figure 4: The scree plot (**a**) and the component 3D plot in rotated space (**b**) allied with the PCA. The scree plot orders the eigenvalues from largest to smallest. Figure 5: The Bootstrap hulls: the graphical method for assessing the significance of the yeast strains in the PCA method and observing the possible clusters in the fermented yeast strains in our dataset. Figure 6: The 2D configuration, correlations, loading, clustering, and projection of observations (yeast strains) and all attributes and aroma groups of compounds (\(p<0.05\)) of all 19 different yeasts on the first two components (**top**), and the VIP values associated with the 28 predictor variables from the PLS regression model with 10 LVs (**bottom**). Legend: Ethyl esters from branched acids—EEBA; Ethyl esters from fatty acids—EEFA; Acetate esters—AE; Volatile phenols—VP; C6-alcohols—C6-OH. Bootstrap hulls involve generating a series of bootstrap samples from the original dataset and then computing the PCs for each bootstrap sample.