Applied Statistics Using SPSS, STATISTICA, MATLAB and R

344 8 Data Structure Analysis
In order to analyse this issue in further detail, let us consider the simple dataset
shown in Figure 8.9a, consisting of normally distributed bivariate data generated
with (true) mean µo = [3 3] ’ and the following (true) covariance matrix:
⎡5
3⎤
Σ o = ⎢ ⎥ .
⎣3
2⎦
Figure 8.9b shows this dataset after standardisation (subtraction of the mean and
division by the standard deviation) with the new covariance matrix:
⎡ 1 0.
9478⎤
Σ = ⎢
⎥ .
⎣0.
9478 1 ⎦
The standardised data has unit variance along all variables with the new
covariance: σ12 = σ21 = 3/( 5 2 ) = 0.9487. The eigenvalues and eigenvectors of Σ
(computed with MATLAB function eig), are:
⎡1.
9487 0 ⎤
Λ = ⎢
⎥ ;
⎣ 0 0.
0513⎦
⎡−1
/ 2 1/
2⎤
U = ⎢
⎥ .
⎢⎣
1/
2 1/
2⎥⎦
Note that tr(Λ) = 2, the total variance, and that the first principal component
explains 97% of the total variance.
Figure 8.9c shows the standardised data projected onto the new system of
variables F1 and F2.
Let us now consider a group of data with mean mo = [4 4] ’ and a one-standarddeviation
boundary corresponding to the ellipsis shown in Figure 8.9a, with sx
= 5 /2 and sy = 2 /2, respectively. The mean vector maps onto m = mo – µo =
[1 1] ’ ; given the values of the standard deviation, the ellipsis maps onto a circle of
radius 0.5 (Figure 8.9b). This same group of data is shown in the F1-F2 plane
(Figure 8.9c) with mean:
m p
⎡−1
/ 2
= U’
m = ⎢
⎢⎣
1/
2
1/
1/
2⎤
⎡1⎤
⎥ ⎢ ⎥
2⎥⎦
⎣1⎦
⎡
= ⎢
⎣
0 ⎤
2
⎥ .
⎦
Figure 8.9d shows the correlations of the principal components with the original
variables, computed with formula 8.9:
rF1 X = rF1 Y = 0.987; rF2 X = − rF2 Y = 0.16 .
These correlations always lie inside a unit-radius circle. Equal magnitude
correlations occur when the original variables are perfectly correlated with
λ1 = λ2 = 1. The correlations are then rF1 X = rF1 Y =1/ 2 (apply formula 8.9).