4.4 Chemical Features of Wine. Table 4.13 shows the PCA output on data (nonnormalized) in which the variables represent chemical characteristics of wine, and each case is a different wine.

a. The data are in the file Wine.csv. Consider the rows labeled "Proportion of Variance." Explain why the value for PC1 is so much greater than that of any other column.

b. Comment on the use of normalization (standardization) in part (a).

TABLE 4.13 PRINCIPAL COMPONENTS OF NON-NORMALIZED WINE DATA

code for running PCA on the wine data

wine.df <- read.csv("Wine.csv") pcs.cor <- prcomp(wine.df[,-1]) summary(pcs.cor) pcs.cor$rot[,1:4] Output

> summary(pcs.cor)

importance of components:

PC1 PC2 PC3 PC4 PC5 Standard deviation 314.9632 13.13527 3.07215 2.23409 1.10853 Proportion of Variance 0.9981 0.00174 0.00009 0.00005 0.00001 Cumulative Proportion 0.9981 0.99983 0.99992 0.99997 0.99998 PC6 PC7 PC8 PC9 PC10 Standard deviation 0.91710 0.5282 0.3891 0.3348 0.2678 Proportion of Variance 0.00001 0.0000 0.0000 0.0000 0.0000 Cumulative Proportion 0.99999 1.0000 1.0000 1.0000 1.0000 PC11 PC12 PC13 Standard deviation 0.1938 0.1452 0.09057 Proportion of Variance 0.0000 0.0000 0.00000 Cumulative Proportion 1.0000 1.0000 1.00000

> pcs.cor$rot[,1:4]

PC1 PC2 PC3 PC4 Alcohol -0.0016592647 -1.203406e-03 -0.016873809 0.141446778 Malic_Acid 0.0006810156 -2.154982e-03 -0.122003373 0.160389543 Ash -0.0001949057 -4.593693e-03 -0.051987430 -0.009772810 Ash_Alcalinity 0.0046713006 -2.645039e-02 -0.938593003 -0.330965260 Magnesium -0.0178680075 -9.993442e-01 0.029780248 -0.005393756 Total_Phenols -0.0009898297 -8.779622e-04 0.040484644 -0.074584656 Flavanoids -0.0015672883 5.185073e-05 0.085443339 -0.169086724 Nonflavanoid_Phenols 0.0001230867 1.354479e-03 -0.013510780 0.010805561 Proanthocyanins -0.0006006078 -5.004400e-03 0.024659382 -0.050120952 Color_Intensity -0.0023271432 -1.510035e-02 -0.291398464 0.878893693 Hue -0.0001713800 7.626731e-04 0.025977662 -0.060034945 OD280_OD315 -0.0007049316 3.495364e-03 0.070323969 -0.178200254 Proline -0.9998229365 1.777381e-02 -0.004528682 -0.003112916