Statistics Toolbox

The Principal Components (First Output)

The first output of the princomp function, pcs, contains the nine principal components. These are the linear combinations of the original variables that generate the new variables.

Let's look at the first three principal component vectors.

• p3 = pcs(:,1:3)
  p3 =
      0.2064    0.2178   -0.6900
      0.3565    0.2506   -0.2082
      0.4602   -0.2995   -0.0073
      0.2813    0.3553    0.1851
      0.3512   -0.1796    0.1464
      0.2753   -0.4834    0.2297
      0.4631   -0.1948   -0.0265
      0.3279    0.3845   -0.0509
      0.1354    0.4713    0.6073
  

The largest weights in the first column (first principal component) are the third and seventh elements, corresponding to the variables health and arts. All the elements of the first principal component are the same sign, making it a weighted average of all the variables.

To show the orthogonality of the principal components, note that premultiplying them by their transpose yields the identity matrix.

• I = p3'*p3
  I =
      1.0000   -0.0000   -0.0000
     -0.0000    1.0000   -0.0000
     -0.0000   -0.0000    1.0000
  

Example: Principal Components Analysis

The Component Scores (Second Output)