# Functional Characterization of _Saccharomyces_ Yeasts from Cider Produced in Hardanger
## 3 Results and Discussion
### Relationships between Aroma Compounds and Variables
#### 3.6.2 Partial Least Squares (PLS) Regression Analysis  
In this section, the fermentents/ciders produced with 19 different yeast strains were further analyzed. Since this is a high-dimensional data set, to further determine the associations between the aroma groups of attributes (y variables, n = 5) and the other physicochemical parameters (x variables, n = 28) in ciders from 19 tested strains, we used partial least squares (PLS) regression--a multivariate statistical analysis used for both regression and classification tasks. To prepare for the PLS regression, we first had to distinguish between the response variables (y variables) and the predictor variables (x variables). Therefore, the five chemical classes/groups of aroma compounds (ethyl esters of fatty acids, ethyl esters of branched acids, acetate esters, C6 alcohols, and volatile phenols) were considered as the response variables that formed the Y-block in the PLS regression. The amino acids, sugars, ethanol, organic acids, etc., were entered and considered as predictor variables used to develop the X-block. Thus, the goal was to find out the relationship between two blocks, the Y-block and the X-block (i.e., the set of response variables and the predictor variables of interest). PLS regression was used to identify (1) the underlying factors responsible for the variation in the data; (2) the contributions of 28 traits/predictors, including amino acids, sugars, ethanol, acids, etc. (i.e., the variables forming the X-block) and to correlate and discover the five groups of cider aroma attributes (i.e., the variables forming the five groups in the Y-block) from 19 yeast strains used for fermentation (Table 3). The method works by finding the linear combinations of the predictor variables X (also called latent variables (LVs)) that are most strongly related to the response variables Y. This is done by maximizing the covariance between the X-block and the Y-block while ensuring that the predictor variables are orthogonal (i.e., uncorrelated) to each other. Subsequently, the PLS regression analysis was used to determine which variables contributed most to the Y-block and the X-block contributed most to the variation in the data. Finally, PLS regression analysis indicated how the variables were correlated with each other and then lumped the variables into a new latent variable (LV). Our results show that our X-block variables mentioned above are strongly correlated with each of the groups of aroma compounds, for which large correlations were always found in different numbers of LVs (Table 3). The results of our PLS regression model show how the amino acids, sugars, ethanol, acids, etc., contributed to the fermentations of the yeasts as well as to the chemical classes of the aroma compounds, and they also lead to some important findings. Table 3 summarizes the results of this analysis and shows that our PLS regression model (built with 10 LVs) includes 89% of the variation in the X-block data and 93% in the Y-block data. Although the optimal number of latent variables (LVs) was analytically set at ten by the PLS regression method, even with eight LVs, our PLS model can well explain the variation in the X-blockand Y-block data, that is, 82% of the variation in the X-block and 88% in the Y-block. PLS regression analysis achieved a correlation coefficient of R\({}^{2}\geq\) 0.85 for all aroma classes of compounds and captured R\({}^{2}\)X \(\geq\) 85% of the variation in the X-block data (i.e., predictor variables amino acids, sugars, ethanol, acids, etc.) and R\({}^{2}\)Y \(\geq\) 85% of the variation in the Y-block data (i.e., aroma classes of compounds), implying a robust linear relationship between the measured aroma classes of compounds and the predictors in all 19 yeast strains we tested in this study. Besides much useful information that Figure 6 can provide, it shows the correlation coefficients between each predictor variable and the response variable, as well as the loading weights that indicate the importance of each predictor variable in the model.