Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

58 3. Properties of Sample Principal ComponentsTable 3.4. Correlation matrix for ten variables measuring reflexes.V1 V2 V3 V4 V5 V6 V7 V8 V9 V10V1 1.00V2 0.98 1.00V3 0.60 0.62 1.00V4 0.71 0.73 0.88 1.00V5 0.55 0.57 0.61 0.68 1.00V6 0.55 0.57 0.56 0.68 0.97 1.00V7 0.38 0.40 0.48 0.53 0.33 0.33 1.00V8 0.25 0.28 0.42 0.47 0.27 0.27 0.90 1.00V9 0.22 0.21 0.19 0.23 0.16 0.19 0.40 0.41 1.00V10 0.20 0.19 0.18 0.21 0.13 0.16 0.39 0.40 0.94 1.00The correlation matrix for these data is given in Table 3.4, and thecoefficients of, and the variation accounted for by, the corresponding PCsare presented in Table 3.5. It should first be noted that the ten variablesfall into five pairs. Thus, V1, V2, respectively, denote strength of reflexesfor right and left triceps, with {V3, V4}, {V5, V6}, {V7, V8}, {V9, V10}similarly defined for right and left biceps, right and left wrists, right and leftknees, and right and left ankles. The correlations between variables withineach pair are large, so that the differences between variables in each pairhave small variances. This is reflected in the last five PCs, which are mainlywithin-pair contrasts, with the more highly correlated pairs correspondingto the later components.Turning to the first two PCs, there is a suggestion in the correlationmatrix that, although all correlations are positive, the variables can bedivided into two groups {V1–V6}, {V7–V10}. These correspond to sites inthe arms and legs, respectively. Reflecting this group structure, the first andsecond PCs have their largest coefficients on the first and second groups ofvariables, respectively. Because the group structure is not clear-cut, thesetwo PCs also have contributions from the less dominant group, and thefirst PC is a weighted average of variables from both groups, whereas thesecond PC is a weighted contrast between the groups.The third, fourth and fifth PCs reinforce the idea of the two groups. Thethird PC is a contrast between the two pairs of variables in the second(smaller) group and the fourth and fifth PCs both give contrasts betweenthe three pairs of variables in the first group.It is relatively rare for examples with as many as ten variables to havesuch a nicely defined structure as in the present case for all their PCs.However, as will be seen in the examples of subsequent chapters, it is notunusual to be able to deduce the structure of at least a few PCs in thismanner.