2011 International Conference on Alternative Energy in Developing Countries and Emerging Economies
- 345 -
A. Box-Jenkins Method
The stochastic model using Box-Jenkins has been
applied in several applications [11]. The general Box-
Jenkins model of order (p, P, q, Q) is the seasonal
autoregressive integrated moving average model, which
is defined by notation SARIMA(p, d, q)(P, D, Q)
s
, can be
expressed as follows [3],
D
d
s
s
s
p
P
t
q
Q
t
B B 1 B 1 B Y
B B
I )
G T 4 H
(1)
where
t
Y
is the wind speed series,
t
H
is the series of
independently and normally distributed with zero mean
and constant variance
2
V
for time
t = 1, 2, …, n
, where n
is the number of observations, s is the number of seasons,
d and D are the degrees of non-seasonal and seasonal
differencing used, respectively, B is the backshift
operator where
s
t
t s
B Y Y
,
G
=
s
p
P
B B
PI )
is a constant term, where
P
is the true mean of the
stationary time series being modeled,
2
p
p
1
2
p
B 1 B B
B
I I I
I
is the non-seasonal autoregressive (AR) operator of order
p,
s
s
2s
Ps
P
1
2
P
B 1 B B
B
) ) )
)
is the seasonal autoregressive (SAR) operator of order P,
2
q
q
1
2
q
B 1 B B
B
T T T
T
is the non-seasonal moving average (MA) operator of
order q,
s
s
2s
Qs
Q
1
2
Q
B 1 B B
B
4 4 4
4
is the seasonal moving average (SMA) operator of order
Q.
The different SARIMA models can be obtained by a
combination of AR, SAR, MA, and SMA components. A
model with minimum value of BIC [12] and insignificant
of Ljung-Box Q statistics [13] has been identified as a
good model. Model parameters of SARIMA model in this
study were estimated using the SPSS V.17 by applying
four steps: tentative identification, estimation, diagnostic
checking, and forecasting.
B. Decomposition Method
Decomposition method is a procedure to separate the
time series into linear trend and seasonal components, as
well as error, and provide forecasts. The seasonal
component may be in the form additive or multiplicative
with the trend. The multiplicative model will be used
when the size of the seasonal pattern in the data depends
on the level of the data that is the data increase, so does
the season pattern; apart from this case is an additive
model [14]. In this study, the forecasting model of the
decomposition method is constructed by using the
Minitab V.15. The manual of this program shows the
additive and multiplicative decomposition models as
following forms.
Additive Decomposition Model:
t
Y
= Trend + Seasonal + Error.
(2)
Multiplicative Decomposition Model:
t
Y
= Trend
u
Seasonal + Error.
(3)
C. Forecast Performance Measure
A criterion to compare the performance of forecast
values from the SARIMA and decomposition models in
this study is mean square error (MSE). MSE can be
considered as the mean of square values of the forecast
errors and can be expressed with the following formula,
n
n
2
2
t
t
t
t 1
t 1
1
1
ˆ
MSE
e
Y Y
n
n
¦ ¦
(4)
where
t
Y
is the realized value at time t,
t
ˆ
Y
is the
forecast value pertaining to that time period, and n is the
number of observations.
III. R
ESULT AND
D
ISCUSSION
The result of this study is divided into three parts. First
part is to show the algorithms to construct the SARIMA
model. The decomposition of the wind speed into various
components to create the model is shown in the second
part. And the last part is devoted to compare of both and
to select the best model which has minimum MSE.
The step to create SARIMA model is starting by
depicting the run plot of the 3-h wind speed data set of
736 observations as shown in Fig. 3 and then constructs
the autocorrelation function (ACF) and partial
autocorrelation function (PACF) of the wind speed data
as shown in Fig. 4 and 5, respectively. A visual
inspection of Fig. 3 indicates that the data seem to
fluctuate around a constant mean and in Fig. 4 ACF dies
down in damped sine-wave fashion with a period is
around eight lags. Then, it is reasonable to believe that
this time series is non-stationary, we can sometimes
transform it to be a stationary time series values by taking
the first seasonal differences that is to take 8
th
differences
of the raw data before estimation, D = 1 and s = 8, the
results of ACF and PACF are shown in Fig. 6 and 7,
respectively. The tentative SARIMA models chosen
based on Fig. 6 and 7 are shown in Table 1 and 2. All
estimated parameters of the SARIMA(1, 0, 1)(1, 1, 1)
8
model with no constant term in Table 2 are significant at
the level of significant 0.05 and in Table 3 this model has
the minimum value of BIC and is insignificant of Ljung-
Box Q statistics. As the results, this model can be
identified as the best model. So, the fitted SARIMA
model for predicting the wind speed is:
t
t 1
t 8
t 9
t 16
t 17
t 1
t 8
t 9
ˆ
Y 0.370Y 1.106Y 0.40922Y 0.106Y 0.03922Y
0.165 0.905 0.149325
(5)
H
H
H
