whereby the covariance matrix of Y, i.e. is equal to the correlation coefficient matrix of
, i.e..CY
X X
The second step is to transform the vector of standardized basic random variables Y, into a
vector of uncorrelated basic random variables U. This last transformation may be performed
in several ways. The approach described in the following utilises the Cholesky factorisation
from matrix algebra and is efficient for both hand calculations and for implementation in
computer programs.
The desired transformation may be written as:
Y=TU (6.23)where is a lower triangular matrix such that T Tij 0 for j&i. It is then seen that the
covariance matrix CY can be written as:
EE TTTTT E
CY YY TUU T T UU T T×T XT (6.24)
where denotes the transpose of a matrix. It is seen from Equation T (6.24) that the
components of T may be determined as:
11
21 12
31 13
2
22 2132 23 31 21
22
22
33 31 321
1
1
T
T
T
TT
T TT
T
TT
1
1
1
T
(6.25)
Considering the example from before but now with the additional information that the random
variables and A Rare correlated with correlation coefficient matrix:
1
1
1AR AS
RA RS
SA SR11
1
11
^8
8
(^89)
X 1
9
9 (6.26)
with 0.1 (^1) AR (^1) RA
T
and all other correlation coefficients equal to zero. The transformation
matrix can now be calculated as:
100
0.1 0.995 0
001
^8
8
(^89)
T^99 (6.27)
The components of the vector Y may then be calculated as: