
460 Data Analysis and Statistics for Geography, Environmental Science, and Engineering
In general, for m variables, the trace will be the sum of m variances and it is equal to the sum of
the m eigenvalues. Therefore, by eigen-decomposition, we allocate the total variance to the eigen-
values in such a way that the amount is sorted from the largest to smallest eigenvalue.
First, center each column vector on its sample mean. This is to say, subtract the sample mean of
Xi from all observations x
ij
, in the ith column of X,
x =
−−
−−
xX xX xX
xX xX xX
x
11 1212 31
12 1222 32
1112233nn
XxXx
X−−
(14.11)
We have used ...