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E ( X ) = ,
with,
XXX X
aab
(
,
,...,
)
T
1
2
n
T
(,,)
0
and
ª
º
1cos
ZZ
sin
«
»
«
»
1cos2
ZZ
sin2
A =
«
»
.
«
»
...
....
....
«
»
«
»
1cos
n
ZZ
sin
n
¬
¼
Minimizing the least-squares
n
(
X
aa
cos
Z
t
b
sin
Z
t
)
2
¦
t
0
t
1
the ș can be estimated using the pseudo inverse relation
ˆ
) T T
1
T
(
A AAX
.
In order to use the spectral expansion technique for forecasting purposes, we
need first to observe the given time series carefully to check whether it contains
any trend and/or seasonality. This can, for instance, be identified by visual
inspection of the graph of the given series.
Trend removal from the time series can be carried out in two ways:
x by taking the first or the second difference of the given time series data
x by fitting a polynomial to it and then by subtracting the fitted polynomial
from the given time series data.
The coefficients of the fitted polynomial can be estimated by a least-squares fitting
method which minimizes the sum of squares of the difference between actual data
and the polynomial data to be fitted.
The first difference is taken as ( x t - x t+ 1 ) for t ranging from 1 to N -1, and where
x t is the actual time series data at time instant t and N is the total number of
observations in the given time series. Similarly, the first difference applied to the
resulting first difference will give rise to the second difference. Once the series is
de-trended, we have to check for various frequency components present in the
residual of the time series. This is accomplished by first transforming the signals
from the time domain into the frequency domain using a fast Fourier transform
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