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EE
{}
, for
j
= 1, 2, … ,
n
.
j
j kk
It is usually supposed that the mean value of expectation is
E
H , for all
j
{} 0
j
and the condition that the covariance matrix of
H
2
E
{
HH
}
¦
H
, for
k
= 0.
j
j
k
H
Finally
x
DD
x
2 2
...
x
D
x
QQ
,
t
11
t
t
t
where the values of parameters
D
to
D
and those of the covariance matrix should
be estimated. This is rather mathematically complicated and requires computer
support.
For
dimensionally reduced modelling
of multivariable time series, the method
of
principal components analysis
is used (Jolliffe, 1986). The analysis helps to
reduce the initial number of correlated variables to a small number of variables,
i.e
.
to the
principal factors
that still contain (with minimal loss) the essential
information of the initial number of variables. This reduces the computational
effort needed for further time series data processing.
However, the reduction in the number of initial variables is not a process of
simple elimination of some non-relevant variables, as the eliminated variables still
have an influence, or “echo”, on the remaining variables. This is because the
principal components are first determined using a smaller number of linear
combinations of the initial variables that are still able to reproduce the entire
collection of observed variables within a relatively good accuracy. Applying
principal component analysis, the optimal number of linear combinations can be
found that are best predictors of the entire set of variables. The prediction accuracy
achieved is considered as the best performance measure. It is also to be noted that -
after transforming the initial variables to the reduced number of variables using
linear combinations - the
back-transformation
of the reduced variables to the
initial variables is not possible.
Consider now the five observations of each of three variables
123
x
,,
xx
presented in matrix form
ª º
«
« »
« »
x
xx
11
21
31
x
xx
12
22
32
x
«
»
«
»
x
xx
¬
¼
15
25
35
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