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Similarly, the Holt-Winter exponential smoothing technique has been applied
to generate the second forecasts of the same temperature series
zt
ˆ ()
cz cz
cz
...
1
0
t
1
t
1
2
t
2
where
) i
c
, where i = 0, 1, 2, …
(1
D
and D is a constant value within the interval 0
D
d. This results in
1
z
ˆ
(1)
D
z
(1
D
)
z
ˆ
(1).
t
t
t
1
The two forecast series are then arranged as columns 1 and 2 and the actual
temperature series as column 3 of an HBXIO matrix
ª
º
f
f
d
B
1
H
1
1
«
»
f
f
d
«
»
HBXIO
«
B
2
H
2
2
»
...
...
...
«
»
«
»
f
f
d
¬
¼
Bm
Hm
m
The sum squared error (SSE) of the generated forecast has been also computed
as SSE = 0.5 EE , where E is the column vector of errors e i = ( f i - d i ), with f i , d i
representing the forecast at i th instant and actual value of the time series at i th
instant and E T is the transposition of E . Consequently, the sum squared error for the
Box-Jenkins forecast is 2.0080 and that of the Holt-Winter forecast is 1.1688,
computed for the entire forecast series (Palit, 1999).
It is important to note that in the above example of Holt-Winter's smoothing
technique the smoothing constant D = 1.6 has been selected because that gave the
minimum value of SSE for generated forecasts, which is quite unusual.
2.10.2 Quality Prediction of Crude Oil
In the following example, time series analysis is applied to crude oil physical and
chemical qualities prediction (Debska and Ivasczek, 2001). The observation data
are collected from oil fields within time period of 5 years and first analyzed
statistically for estimation of values of the most relevant chemical physical
parameters, such as specific gravity, density, colour, viscosity, relative and
kinematic viscosity, drip and set point, etc . The statistical methods used for these
purposes are: preprocessing and smoothing of data, partial and autocorrelation
calculation, seasonality and trend-analysis, decomposition, etc . For decomposition
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