Overview www.wiley.com/wires/compstats TABLE 8 PCA Wine Characteristics.Loadings (i.e.,Q matrix)of the Variables on the First Two Components For For Hedonic Meat Dessert Price Sugar Alcohol Acidity PC1 -0.40 -0.45 -0.26 0.42 -0.05 -0.44 -0.45 PC2 0.11 -0.11 -0.59 -0.31 -0.72 0.06 0.09 TABLE 9 PCA Wine Characteristics.Correlation of the Variables with the First Two Components For For Hedonic meat dessert Price Sugar Alcohol Acidity PC1-0.87 -0.97-0.58 0.91 -0.11 -0.96 -0.99 PC2 0.15 -0.15 -0.79 -0.42 -0.97 0.07 0.12 PC2 spent on another item,we want to keep the same unit of measurement for the complete space.Therefore we will perform a covariance PCA,rather than a correlation PCA.The data are shown in Table 11. A PCA of the data table extracts seven compo- Hedonic nents (with eigenvalues of 3,023,141.24,290,575.84, ●Acidity 68,795.23,25,298.95,22,992.25,3,722.32,and 723.92,respectively).The first two components PC1 extract 96%of the inertia of the data table,and For meat we will keep only these two components for further Price● consideration (see also Table 14 for the choice of the number of components to keep).The factor scores for For dessert the first two components are given in Table 12 and the corresponding map is displayed in Figure 8. Sugar We can see from Figure 8 that the first component separates the different social classes,while FIGURE 6PCA wine characteristics.Correlation(and circle of the second component reflects the number of children correlations)of the Variables with Components 1 and 2.=4.76, per family.This shows that buying patterns differ t1=68%;2=1.81,t2=26%. both by social class and by number of children per family.The contributions and cosines,given in of 15(corresponding to a cosine of 0.97).This gives Table 12,confirm this interpretation.The values of the the new set of rotated loadings shown in Table 10. contributions of the observations to the components The rotation procedure is illustrated in Figure 7.The indicate that Component 1 contrasts blue collar improvement in the simplicity of the interpretation families with three children to upper class families is marginal,maybe because the component structure with three or more children whereas Component of such a small data set is already very simple.The 2 contrasts blue and white collar families with first dimension remains linked to price and the second five children to upper class families with three and dimension now appears more clearly as the dimension four children.In addition,the cosines between the of sweetness. components and the variables show that Component 1 contributes to the pattern of food spending seen by the blue collar and white collar families with two Covariance PCA and three children and to the upper class families with Here we use data from a survey performed in the three or more children while Component 2 contributes 1950s in France [data from Ref 45].The data table to the pattern of food spending by blue collar families gives the average number of Francs spent on several with five children. categories of food products according to social class To find the variables that account for these and the number of children per family.Because a differences,we refer to the squared loadings of the Franc spent on one item has the same value as a Franc variables on the two components (Table 13)and to 444 2010 John Wiley Sons,Inc. Volume 2,July/August 2010Overview www.wiley.com/wires/compstats TABLE 8 PCA Wine Characteristics. Loadings (i.e., Q matrix) of the Variables on the First Two Components For For Hedonic Meat Dessert Price Sugar Alcohol Acidity PC 1 −0.40 −0.45 −0.26 0.42 −0.05 −0.44 −0.45 PC 2 0.11 −0.11 −0.59 −0.31 −0.72 0.06 0.09 TABLE 9 PCA Wine Characteristics. Correlation of the Variables with the First Two Components For For Hedonic meat dessert Price Sugar Alcohol Acidity PC 1 −0.87 −0.97 −0.58 0.91 −0.11 −0.96 −0.99 PC 2 0.15 −0.15 −0.79 −0.42 −0.97 0.07 0.12 Price Sugar Alcohol Acidity Hedonic For meat For dessert PC1 PC2 FIGURE 6 | PCA wine characteristics. Correlation (and circle of correlations) of the Variables with Components 1 and 2. λ1 = 4.76, τ 1 = 68%; λ2 = 1.81, τ 2 = 26%. of 15◦ (corresponding to a cosine of 0.97). This gives the new set of rotated loadings shown in Table 10. The rotation procedure is illustrated in Figure 7. The improvement in the simplicity of the interpretation is marginal, maybe because the component structure of such a small data set is already very simple. The first dimension remains linked to price and the second dimension now appears more clearly as the dimension of sweetness. Covariance PCA Here we use data from a survey performed in the 1950s in France [data from Ref 45]. The data table gives the average number of Francs spent on several categories of food products according to social class and the number of children per family. Because a Franc spent on one item has the same value as a Franc spent on another item, we want to keep the same unit of measurement for the complete space. Therefore we will perform a covariance PCA, rather than a correlation PCA. The data are shown in Table 11. A PCA of the data table extracts seven compo￾nents (with eigenvalues of 3,023,141.24, 290,575.84, 68,795.23, 25,298.95, 22,992.25, 3,722.32, and 723.92, respectively). The first two components extract 96% of the inertia of the data table, and we will keep only these two components for further consideration (see also Table 14 for the choice of the number of components to keep). The factor scores for the first two components are given in Table 12 and the corresponding map is displayed in Figure 8. We can see from Figure 8 that the first component separates the different social classes, while the second component reflects the number of children per family. This shows that buying patterns differ both by social class and by number of children per family. The contributions and cosines, given in Table 12, confirm this interpretation. The values of the contributions of the observations to the components indicate that Component 1 contrasts blue collar families with three children to upper class families with three or more children whereas Component 2 contrasts blue and white collar families with five children to upper class families with three and four children. In addition, the cosines between the components and the variables show that Component 1 contributes to the pattern of food spending seen by the blue collar and white collar families with two and three children and to the upper class families with three or more children while Component 2 contributes to the pattern of food spending by blue collar families with five children. To find the variables that account for these differences, we refer to the squared loadings of the variables on the two components (Table 13) and to 444  2010 John Wiley & Son s, In c. Volume 2, July/Augu st 2010