Factor Analysis and Principal Components Analysis 253
matrix ΣX using the eig function of MATLAB. We obtain the following
matrices:
=
V
–0. 3563 0.4623–0. 1593 0.0885–0. 0951–0. 3155–0. 6800 0.2347
0.2721–0. 0733–0. 1822 0.4334–0. 0359 0.7056–0. 3014 0.3311
–0. 1703–0. 2009–0. 5150 0.0241 0.6772–0. 1653 0.1890 0.3784
–0. 6236–0. 3148 0.0961–0. 3693–0. 3385 0.2945 0.0895 0.3963
0.5319–0. 4428 0.0909–0. 3693–0. 0972–0. 3284–0. 3586 0.3611
–0. 0088–0. 1331 0.0672 0.6355–0. 4031–0. 4157 0.3484 0.3422
0.0075 0.2697 0.7580 0.0421 0.4326 0.0557 0.0579 0.3966
0.3131 0.5957 –0. 2823–0. 3540–0. 2395 0.0693 0.3869 0.3610
and
=
D
0.0381 0000000
00.0541 000000
00 0.1131 00000
000 0.1563 0000
0000 0.2221 000
00000 0.4529 00
000000 1.3474 0
0000000 5.6160
Note that the product VV' is a diagonal matrix with ones on the main diag-
onal. This means that the eigenvectors are mutually orthogonal vectors; that is,
their scalar product is 0 if the vectors are different, with length equal to 1.
Step 2: Construct principal Components by Multiplying data by the eigenvectors If
we multiply the data X by any of the eigenvectors (i.e., any of the columns
of the matrix V), we obtain a new time series formed by a linear combina-
tion of the data. For example, suppose we multiply the data X by the first
eigenvector; we obtain our first principal component (denoted by PC 1 ):
PC XV
XXXX
11
1234
=
=− ×+0.3563 ×−0.2721 ×−0.1703 ×0..6236
+×XX 5 0.5319−× 678 0.0088+×XX0.0075+×0. 31131
We can therefore construct a new 20 × 8 matrix
PC=XV (12.13)
