SSA-based hybrid forecasting models and applications

Bulletin of Electrical Engineering and Informatics
Vol. 9, No. 5, October 2020, pp. 2178~2188
ISSN: 2302-9285, DOI: 10.11591/eei.v9i5.1950  2178
Journal homepage: http://beei.org
SSA-based hybrid forecasting models and applications
Winita Sulandari1
, Subanar2
, Suhartono3
, Herni Utami4
, Muhammad Hisyam Lee5
Paulo Canas Rodrigues6
1,2,4
Department of Matematics, Universitas Gadjah Mada, Indonesia
1
Study Program of Statistics, Universitas Sebelas Maret, Indonesia
3
Department of Statistics, Institut Teknologi Sepuluh Nopember, Indonesia
5
Department of Mathematical Sciences, Universiti Teknologi Malaysia, Malaysia
6
Department of Statistics, Federal University of Bahia, Brazil
Article Info ABSTRACT
Article history:
Received Sep 24, 2019
Revised Dec 11, 2019
Accepted May 5, 2020
This study attempted to combine SSA (singular spectrum analysis) with other
methods to improve the performance of forecasting model for time series
with a complex pattern. This work discussed two modifications of TLSAR
(two-level seasonal autoregressive) modeling by considering the SSA
decomposition results, namely TLSNN (two-level seasonal neural network)
and TLCSNN (two-level complex seasonal neural network). TLSAR
consisted of a linear trend, harmonic, and autoregressive component. In
contrast, the two proposed hybrid approaches consisted of flexible trend
function, harmonic, and neural networks. Trend and harmonic function were
considered as the deterministic part identified based on SSA decomposition.
Meanwhile, NN was intended to handle the nonlinearity relationship in
the stochastic part. These two SSA-based hybrid models were contemplated
to be more flexible than TLSAR and more applicable to the series with
an intricate pattern. The experimental studies to the monthly accidental
deaths in USA and daily electricity load Jawa-Bali showed that the proposed
SSA-based hybrid model reduced RMSE for the testing data from that
obtained by TLSAR model up to 95%.
Keywords:
Hybrid
SSA
TLCSNN
TLSAR
TLSNN
This is an open access article under the CC BY-SA license.
Corresponding Author:
Winita Sulandari,
Study Program of Statistics,
Universitas Sebelas Maret,
Jl. Ir. Sutami 36 A Kampus Kentingan, Surakarta, Central of Java, Indonesia 57126.
Email: winita@mipa.uns.ac.id
1. INTRODUCTION
Decomposition is the basis for modeling complex seasonal time series. A time series is said to have
a complex seasonal pattern when it has trend and multiple seasonal patterns, perhaps with non-integer period.
Recently, this kind of series has been intriguing researchers to study and develop methodologies to improve
the forecast accuracy. Soares and Medeiros [1] have discussed TLSAR (two level seasonal autoregressive)
model that consists of two parts, deterministic and stochastic component. This method is a simple statistical
model that combines linear trend, trigonometry and the autoregressive model. Meanwhile, [2-7] developed
an innovation state space exponential smoothing model. Taylor [2] added the second seasonal component
to the Holt-Winter exponential smoothing, namely HWT, and in the following year, [6] presented BATS
(Box-Cox transform, ARMA errors, trend, and seasonal component) and TBATS (trigonometric BATS) to
reduce the parameters of HWT model. Different from other previous methods, TBATS has accommodated
the fractional seasonality in the data. Still, it unable to take into account the time varying component.

Bulletin of Electr Eng & Inf ISSN: 2302-9285 
SSA-based hybrid forecasting models and applications (Winita Sulandari)
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Meanwhile, other researchers [8-11] considered SSA method in complex time series forecasting.
The development of the SSA method was inspired by [12-15]. Much of subsequent works refer to their
works. Later, [16-23] continued to develop SSA as a tool of time series forecasting. Vautard et al. [16]
reviewed the capability of SSA in extracting information from short and noisy time series without any prior
knowledge about the data. Yiou et al. [17] regarded SSA as a method to identifying and extracting oscillatory
component from the original series. Vahabie et al. [18] combined SSA with autoregressive (AR) model to
predict the electricity demand. In the following year, [19, 20] reviewed and demonstrated the success of SSA
in economic and financial time series forecasting. Golyandina and Korobeynikov [21] described the RSSA
package and showed how the methodology of SSA can be implemented for analyzing, estimating parameter
and forecasting. Hassani et al. [22] studied the effect of outliers and showed that the vector SSA forecasts
were more robust than the recurrent SSA forecast. Khan and Poskitt [23] discussed the properties of forecast
obtained by SSA in theory and practice. Recently, [24, 25] implemented SSA method to decompose a series
into several components and modeled the components by the Box-Jenkins method. However, some data
exhibit a deterministic behavior rather than stochastic. Thus, there will be parts that have not been
accommodated by the Box-Jenkins model. Recently, [26] to combine SSA-LRF with NN (neural network)
and fuzzy time series to forecast hourly electricity load in Indonesia. The success of NN in dealing the
nonlinearity can also be seen in [27-31]. Moreover, the method proposed by [24-26] did not take into account
the seasonal with non-integer period, time varying amplitude, and calendar variation. Hence, some
modifications are needed to accommodate the features of the series instead.
This work proposes two SSA-based hybrid model, namely TLSNN (two level seasonal neural
networks) and TLCSNN (two level complex seasonal neural networks) model, modifications of TLSAR
model. TLSAR consists of deterministic and stochastic component. The deterministic is expressed as
a combination of linear, harmonic, and dummy variables to take into account the linear trend, seasonal, and
holiday influences. Meanwhile, the two proposed models provide flexibility in determining deterministic
component, especially in describing nonlinear trends. Moreover, one of the proposed models can capture
the oscillation pattern with time varying amplitude. In this study, the two models are constructed by
considering the results of SSA decomposition. We take advantage of the capability of SSA in decomposing
time series so that we get several separable and interpretable components including the information on how
much each component contributes to the series. Further, SSA method makes the identifying process of
the component patterns easier. Our methodology is as simple as that in [1]. The main difference is that we
take into account the time-varying components of the series that are not included in TLSAR model or other
methods mentioned in the literature. Further, we can show that our methodology can improve the multi-step
forecasting accuracy.
2. THE PROPOSED METHOD
2.1. Mathematical definition for the proposed SSA-based hybrid model
The two proposed models are called the two-level seasonal neural networks (TLSNN) and the two-level
complex seasonal neural networks (TLCSNN). Both models are modifications of the TLSAR model.
The higher order of polynomial trend and the time-varying sinusoidal function are included in
the deterministic component to capture a more complex pattern in the time series. In this case, the results
from the SSA decomposition make the identification and determination of the proper deterministic function
for each component of TLSNN and TLCSNN easier. Generally, TLSNN and TLCSNN are expressed in
𝑌𝑡 = 𝑆𝑡 + 𝑍𝑡.
(1)
Where 𝑌𝑡 is observation at time t, 𝑆𝑡 is the deterministic component and 𝑍𝑡 is the stochastic component.
The difference between the two is in the form of 𝑆𝑡. Meanwhile, 𝑍𝑡, is approximated by neural network (NN).
These two models are explained further in definition 1 and definition 2.
Definition 1. A time series {𝑌𝑡, 𝑡 = 1, … , 𝑁} with the sample size N and which has a complex pattern follows
TLSNN if the deterministic component in (1) can be presented as
𝑆𝑡 = ∑ 𝑎𝑖𝑡𝑖
𝑛𝑝𝑜𝑙
𝑖=0
+ ∑ [𝛼0ℎ cos(2πℎ𝑡/𝑇) + 𝛽0ℎ sin(2𝜋ℎ𝑡/𝑇)]
𝑛ℎ
ℎ=1 + ∑ 𝛿𝑘𝐷𝑘𝑡
𝑛𝐷
𝑘 (2)
and the stochastic component is
𝑍𝑡 = ∑ 𝑏𝑙𝑓𝑙
𝑛ℎ𝑖𝑑
𝑙 (∑ 𝑐𝑗𝑍𝑡−𝑗
𝑛𝑖𝑛𝑝
𝑗
) + 𝑒𝑡 (3)

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where
𝑓𝑙 (∑ 𝑐𝑗𝑍𝑡−𝑗
𝑛𝑖𝑛𝑝
𝑗
) =
2
1−exp(−2 ∑ 𝑐𝑗𝑍𝑡−𝑗
𝑛𝑖𝑛𝑝
𝑗
)
− 1 (4)
The notation 𝑎𝑖, 𝛼0ℎ, 𝛽0ℎ, 𝛿𝑘 for 𝑖 = 1, … , 𝑛𝑝𝑜𝑙, ℎ = 1, … , 𝑛ℎ and 𝑘 = 1, 2, … , 𝑛𝐷 are the unknown parameters
of the model, 𝑇 is the seasonal period, 𝑛ℎ is the number of harmonics, and 𝑛𝐷 is the number of dummy
variables included in the model. 𝐷𝑘𝑡 notates the effect of the 𝑘th category at time 𝑡.
Definition 2. A time series {𝑌𝑡, 𝑡 = 1, … , 𝑁} with a complex pattern follows TLCSNN if the deterministic
component in (1) can be presented as
𝑆𝑡 = ∑ 𝑎𝑖𝑡𝑖
𝑛𝑝𝑜𝑙
𝑖=0
+ ∑ ∑ [𝛼𝑣𝑞𝑡𝑞
cos(𝜔𝑣𝑡) + 𝛽𝑣𝑞𝑡𝑞
sin(𝜔𝑣𝑡)]
𝑛𝑝𝑜𝑙𝑣
𝑞=0
𝑛𝑜𝑠
𝑣=1
+ ∑ [𝛼0ℎ cos(2πℎ𝑡/𝑇) + 𝛽0ℎ sin(2𝜋ℎ𝑡/𝑇)]
𝑛ℎ
𝑛𝐷
𝑘
(5)
and the stochastic component is as in (3) and (4).
The notation 𝑎𝑖, 𝛼𝑣𝑞, 𝛽𝑣𝑞, 𝛼0ℎ, 𝛽0ℎ, 𝛿𝑘 for 𝑖 = 1, … , 𝑛𝑝𝑜𝑙, 𝑣 = 1, … , 𝑛𝑜𝑠, ℎ = 1, … , 𝑛ℎ, 𝑞 = 1, … , 𝑛𝑝𝑜𝑙𝑣
, and 𝑘 =
1, 2, … , 𝑛𝐷 are the unknown parameters of the model, 𝑇 is the seasonal period, 𝑛ℎ is the number of
harmonics, and 𝑛𝐷 is the number of dummy variables included in the model. 𝐷𝑘𝑡 notates the effect of the 𝑘th
category at time 𝑡.
The TLSNN and TLCSNN models accommodate, not only trend linear, but also quadratic or
another polynomial with higher order to capture other trend behaviors. The differences between TLSNN and
TLCSNN are in their harmonic terms. We add the time-varying amplitude sinusoid component, do estimate
its parameters separately using iterative ordinary least square to estimate the parameter of the amplitude
modulated sinusoid model (see [32, 33]) and then combine its best function with the deterministic component
of TLSAR model. Meanwhile, the parameters of the deterministic model in TLSNN are estimated
simultaneously using householder transformation. This estimation method was discussed detail in [34-36].
2.2. Modeling strategy for the two proposed SSA-based hybrid model
2.2.1. TLSNN model
The steps of TLSNN modeling procedure are explained below.
Level 0: the series is decomposed by SSA method into several separable components
Level 1: obtaining the deterministic model (2) based on the results of SSA decomposition
a. identifying the trend function and the dummy variables to accommodate calendar effects.
b. estimating the parameters of the deterministic model by using least square based on
householder method [34-36]
c. defining the fitted value based on the deterministic model, 𝑆
̂𝑡
Level 2: obtaining the stochastic model
a. determining the stochastic component, 𝑍𝑡 = 𝑌𝑡 − 𝑆
̂𝑡
b. testing the linearity in the series by using Terasvirta method [37]
c. modeling by NN when the linearity test detecting the nonlinearity relationship in the data
evaluating the accuracy of the forecast values
2.2.2. TLCSNN model
The procedure of TLCSNN modeling are explained in the following steps.
Level 0: the series is decomposed by SSA method into several separable components
Level 1: obtaining the deterministic model (5) based on the resuts of SSA decomposition
a. .identifying and estimating each component function
− estimating parameters of the trend function by ordinary least square (OLS) method
− estimating parameters of the oscillatory function by iterative OLS [33]
− defining the appropriate function for representing the irregular component. This function
may involve Fourier function with the dummy variables. The number of harmonics can be
obtained based on AIC value.
b. defining the fitted value based on the deterministic model, 𝑆
̂𝑡
Level 2: obtaining stochastic model
a. determining the stochastic component, 𝑍𝑡 = 𝑌𝑡 − 𝑆
̂𝑡

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b. testing the linearity in the series by using Terasvirta method [37]
c. modeling by NN when the linearity test detecting the nonlinearity relationship in the data
evaluating the accuracy of the forecast values
3. RESEARCH METHOD
The SSA-based hybrid model is defined by the method that combines deterministic and stochastic
component, where the deterministic component is obtained based on the SSA decomposition result.
3.1. The benchmark model
The TLSAR is used as the benchmark model in this research. TLSAR model is presented as
a combination of a trend linear and trigonometric function of time and the autoregressive model. The TLSAR
model is expressed as the combination of deterministic and stochastic component as in (1), where
the deterministic component is written as
𝑆𝑡 = 𝑎0 + 𝑎1𝑡 + ∑ [𝛼0ℎ cos(2πℎ𝑡/𝑃) + 𝛽0ℎ sin(2𝜋ℎ𝑡/𝑃)]
𝑛ℎ
𝑛𝐷
𝑘 . (6)
The notation 𝑎0, 𝑎1, 𝛼0ℎ, 𝛽0ℎ, 𝛿𝑘 for ℎ = 1, … , 𝑛ℎ and 𝑘 = 1, 2, … , 𝑛𝐷 are the unknown parameters of
the model, 𝑃 is the seasonal period, 𝑛ℎ is the number of harmonics, and 𝑛𝐷 is the number of dummy
variables included in the model. The number of harmonics 𝑛ℎ in (6) is determined by minimizing the Akaike
Information Criteria (AIC),
AIC = 𝑛 ln(𝜎
̂2
) + 2𝑀, (7)
Where 𝜎
̂2
=
1
𝑛
∑ (𝑌𝑡 − 𝑌
̂𝑡)2
𝑛
𝑡=1 is the estimate of the residual variance. 𝑀 is the number of parameters fitted in
the model, and 𝑛 is the number of residuals that can be calculated from the series (see [34, 38, 39]). The
dummy variable 𝐷𝑘𝑡 identifies the effect of the 𝑘th category at time 𝑡, for example, the 𝑘th hours of
the day to 𝑌𝑡 or the 𝑘th days of the weeks to 𝑌𝑡. This value of 𝐷𝑘𝑡 is one if the 𝑘th category is observed at
time 𝑡 and zero for otherwise. In the original TLSAR model, 𝑍𝑡 is the stochastic component that follows the
AR model while in this study, 𝑍𝑡 follows the SAR model. The autoregressive model of order 𝑝, denoted by
AR(𝑝) is given by
𝜙𝑝(𝐵)𝑍𝑡 = 𝑒𝑡, (8)
Where 𝜙𝑝(𝐵) = 1 − 𝜙𝑝𝐵 − ⋯ − 𝜙𝑝𝐵𝑝
and 𝑒𝑡 is zero mean white noise process. This model can be applied
to the series in which the preceding values and the random shock influence its present value.
The process needs to be stationary so that the roots of 𝜙𝑝(𝐵)=0 must lie outside the unit circle.
When the seasonal part is stochastic and cannot be handled by the deterministic model, (8) can be
extended as
𝜙𝑝(𝐵)𝜑𝑃(𝐵𝑠
)𝑍𝑡 = 𝑒𝑡, (9)
Where 𝜙𝑝(𝐵) = 1 − 𝜙𝑝𝐵 − ⋯ − 𝜙𝑝𝐵𝑝
, 𝜑𝑃(𝐵𝑠
) = 1 − 𝜑1(𝐵𝑠
) − ⋯ − 𝜑𝑃(𝐵𝑃𝑠
), 𝑠 is the seasonal period
and 𝑃 is the order of the seasonal component. The model presented in (9) denoted by SAR(𝑝, 0,0)(𝑃, 0,0)𝑠
3.2. SSA method in decomposing a time series
SSA method decomposes an original time series into trend, several oscillatory components, and a
noise. The major references related to this method can be found in [11, 40-43]. This method consists of four
steps, embedding, singular value decomposition (SVD), grouping and diagonal averaging. The time series
{𝑌𝑡, 𝑡 = 1, 2, … , 𝑁} with the sample size 𝑁 is transformed into a Hankel matrix Y with 𝐿- dimension vectors,
that is
𝐘 = [
𝑌1 𝑌2
𝑌2 𝑌3
⋯
⋯
𝑌𝐾
𝑌𝐾+1
⋮ ⋮ ⋱ ⋮
𝑌𝐿 𝑌𝐿+1 ⋯ 𝑌𝑁
] (10)

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Where 𝐿 is a window length and K=N – L + 1. This transformation process is called by
the embedding step. The embedding step is then followed by calculating 𝐘𝐘′ and applying SVD to determine
its eigentriples (√𝜆𝑖, 𝑈𝑖, 𝑉𝑖) for 𝑖=1, 2, …, 𝐿. The 𝜆𝑗 represent the i-th eigenvalue of matrix 𝐘𝐘′ and √𝜆𝑖
denotes the i-th singular value of matrix Y. The 𝑈𝑖 and 𝑉𝑖 are the left and the right singular values of matrix
Y. SVD of the Hankel matrix in (10) can be written as
𝐘 = 𝐘𝟏 + 𝐘𝟐 + ⋯ + 𝐘𝒅 (11)
where d is the rank of matrix Y and 𝐘𝑗 = √𝜆𝑗 𝑈𝑗𝑉
𝑗′ for 𝑗=1, 2, …, 𝑑. The contribution of the matrix 𝐘𝑗 in (11)
can be determined by calculating the ratio 𝜆𝑗/ ∑ 𝜆𝑗.
𝑑
𝑗=1 The matrices (𝐘𝟏, 𝐘𝟐, … , 𝐘𝒅) are arranged in 𝑚 groups,
where 𝑚 ≤ 𝑑 such that (11) can be written as
𝐘 = 𝐘𝑰𝟏
+ 𝐘𝑰𝟐
+ ⋯ + 𝒀𝑰𝒎
. (12)
In this case, each matrix 𝐘𝑰𝒊
in (12) may be reconstructed from one or more eigentriples (√𝜆𝑗, 𝑈𝑗, 𝑉
𝑗) and the
contribution of this matrix can be measured by the ratio ∑ 𝜆𝑗/
𝑚
𝑗∈𝐼𝑖
∑ 𝜆𝑗
𝑑
𝑗=1 .
Each matrix 𝐘𝑰𝒊
is then retransformed into a time series by the diagonal averaging procedure.
The diagonal averaging is a Hankelization process that transforms the elements of matrix 𝐘𝑰𝒊
, say 𝑦𝑖𝑗 into the
𝑡-th observation of the 𝑖-th (𝑖=1, 2, …, 𝑚) component series by averaging 𝑦𝑖𝑗 for all 𝑖, 𝑗 such that 𝑖 + 𝑗 =
𝑡 + 2. Thus the original series {𝑌𝑡, 𝑡 = 1, 2, … , 𝑁} can be represented as the sum of 𝑚 series,
𝑌𝑡 = ∑ 𝑌𝑡
(𝑖)
𝑚
𝑖=1 (13)
where 𝑌𝑡
(𝑖)
is the series reconstructed from the matrix 𝐘𝑰𝒊
.
The separability of all component series, 𝑌𝑡
(𝑖)
, 𝑖 = 1, . . , 𝑚, in (13) can be measured by calculating w-
correlation values, which are in the range of [-1,1] and visualize them in the form of a w-correlation matrix
diagram with different level of color from the brightest (white means no correlation or separable) to
the darkest (black means perfect correlation). The strongest separability is represented by the white color and
the weakest separability is shown by black color (see [40]). Other alternatives to achieve separability to
obtain a more parsimonious forecasting model were proposed by [44, 45].
3.3. Diagnostic checking and evaluating the performance of forecast accuracy
Model diagnostic checking is needed to check whether the model assumption that is zero mean
white noise residual series is satisfied. To see whether the residuals are uncorrelated random shock with zero
mean and constant variance, we can examine both the plot of residuals and its sample ACF. We can also
verify the results by using the Ljung-Box test [46, 47]. In this work, we use RMSE and MAPE to measure
the performance of the forecast accuracy.
RMSE = √
1
𝑛
∑ (𝑌𝑡 − 𝑌
̂𝑡)2
𝑛
𝑡=1 , (14)
MAPE =
1
𝑛
∑ |
𝑌𝑡−𝑌
̂𝑡
𝑌𝑡
|
𝑛
𝑡=1 . (15)
where 𝑛 is the number of observations involved in the calculation. 𝑌𝑡 and 𝑌
̂𝑡 are the real value at time 𝑡 and
the forecast value at time 𝑡, respectively. 𝑌
̂ is the mean of the in sample data. In (14) and (15) are used for
evaluating the training data and the testing data.
4. RESULTS AND DISCUSSION
The well-known monthly accidental deaths in the USA and the daily load series of Jawa-Bali for
hour 1.00 am to hour 4.00 am were used in this empirical study. In this section, we show how the two
proposed method implemented to the complex series and compare the results with TLSAR as the benchmark.
4.1. Monthly accidental deaths in USA
The monthly accidental deaths series depicted in Figure 1(a) is quite popular and has been discussed
in several studies. Brockwell and Davis [48] have used this data as an example of the application of

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the ARIMA, Holt-Winter seasonal, ARAR model, while [11] discussed the application of the SSA-LRF
model. The data are divided into two parts, the training and the testing data set. We used the data from
January 1973 to December 1978 as the training data and January 1979 to June 1979 as the testing data.
As in [11], window length L used in this study is 24, proportional to the period. Based on the w-matrix
correlation, the time series can be decomposed into 4 components (see, Figure 1(b)), namely trend, two oscillation
components, and noise.
(a) (b)
Figure 1. The monthly accidental deaths series from January 1973 to June 1979, (a) The original series,
(b) Its decomposition
In TLCSNN model, the quadratic function is the most suitable for modeling trend components,
while the first and the second oscillation components can be approximated by quadratic amplitude modulated
sinusoidal function and linear amplitude modulated sinusoidal function, respectively. Since Terasvirta test
indicates that there is a nonlinearity relationship in the stochastic component, NN must be more appropriate
to model the stochastic component than the AR or other linear models. The best NN is determined based on
the RMSE value. The number of input variables are chosen between 6 or 12 regarding the seasonal periods
while the number of hidden units are chosen between 1 to 10 so that the network produces the smallest
RMSE value with random residuals. Based on the experimental results, NN with 6 input units and 8 hidden
units, denoted NN (6-8-1) is the best fit NN model for the stochastic component.
Further, in TLSNN model, we also choose the quadratic trend in the deterministic component and
NN(6-8-1) for the stochastic component. In this case, the number of harmonic is 5 with period 11.98.
The comparison results of RMSE and MAPE for the three models are summarized in Table 1. It can be seen
that TLSNN and TLCSNN produce smaller RMSE and MAPE than TLSAR both for the training and testing
data. TLSNN and TLCSNN respectively reduce the RMSE of TLSAR by 85.26% and 82.44% for
the training data and by 82.73% and 95.00% for the testing data. Meanwhile, sequentially TLSNN and
TLCSNN reduce the MAPE of TLSAR by 86.69% and 87.15% for the training data and by 82.52% and 95.58%
for the testing data.
Table 1. Comparison of RMSE and MAPE obtained by TLSAR, TLSNN, and TLCSNN
model for the death series
Model
Training Testing
RMSE MAPE RMSE MAPE
TLSAR (h=2, P=12, SARIMA(0,1,1)(0,1,1)12 270.4575 2.4997 334.7116 3.1314
TLSNN, npol=2, nh=5, NN(6-8-1) 39.8595 0.3327 57.7906 0.5474
TLCSNN, npol=2, nos=2, nh=0, NN(6-8-1) 47.4913 0.3213 16.7392 0.1384

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4.2. Daily electricity load of Jawa-Bali in the specific hours
The electricity load of Jawa-Bali, in megawatt per hour (MWh), obtained from a power utility
company in Indonesia, Jawa-Bali control center (JCC), that were observed hourly are considered in this
work. However, the data discussed is only the electricity load at 01.00 am, 02.00 am, 03.00 am, and 04.00
am. These data have a unique pattern compared to other hours due to the influence of the habits of Indonesian
citizens in the month of Ramadan. These four periods are considered to represent complex seasonal patterns,
which have seasonal patterns, trends, and calendar influences. The data pattern of time series 01.00, 02.00,
03.00, and 04.00 for the period of January 1, 2009, to December 31, 2011, is presented in Figure 2.
Each series consists of 1095 observations. It is divided into two parts, the first 1088 observations are the
estimation sample (the training data) and the last 7 observations (a week) are the test sample.
(a) (b)
(c) (d)
Figure 2. Daily load series from 1 January 2009 to 31 December 2011 for, (a) 01.00, (b) 02.00,
(c) 03.00, and (d) 04.00
There was a noticeable surge in several days and repeated from year to year. Higher surges occur at
03.00 and 04.00. This surge occurred during the sahur hours in the month of Ramadan and eventually
dropped dramatically before and after the Eid al-Fitr. Each series exhibits trend and seasonal. The demand for
electricity that increases from time to time is related to population growth and society needs. This causes
a positive trend in the series and can be modeled by the function of time. Due to the similarity results in
the component patterns of SSA decomposition for 01.00 to 04.00, we only report 01.00 in detail. Nevertheles,
the overall results will be dicussed. The results of SSA decomposition for time series 04.00 is shown in
Figures 3(a) and 3(b).
Based on Figure 3(c), the dummy variables in the TLSAR, TLSNN, and TLCSNN models can be
described in Table 2. In this case, the electricity load tends to fall on the next day (the day after a holiday).
Especially for the month of Ramadan, the electricity load increases. The k dummy variable {k=1, 2,. . . , 10}
is one if time t shows the event as described in Table 2 and has a value of 0 for the other. All the models
described in Table 3 have harmonic components with the number of harmonics 𝑛ℎ=3 and dummy variables,
as stated in Table 2. The results of the comparison of RMSE and MAPE are presented in Table 3.

2185
(a)
(b) (c)
Figure 3. The components of series 04.00 obtained by SSA decomposition with L=490,
(a) Trend component, (b) Oscillation component, (c) Irregular component
Table 2. Types of days presented as dummy variables in the TLSAR, TLSNN, and TLCSNN model
Code Description
1 The second day of Ramadan up to five days before the first Eid Al-Fitr
2 The two days of Eid Al-Fitr and the two days after the second Eid Al-Fitr
3 The third day after the second Eid Al-Fitr
4 A day before the first Eid Al-Fitr=and the fourth day after the second Eid Al-Fitr
5 The second day before the first Eid Al-Fitr, and the fifth and sixth day after the second Eid Al-Fitr
6 The third day before the first Eid Al-Fitr, a day before the first Eid Al-Fitr, and the seventh day after the second
Eid Al-Fitr
7 A day after the independent day, new year, and holidays related new year.
8 Eid al Adha and a day after Eid Al Adha
9 A day after Chrismas, a day after Waisya, a day after Nyepi, a day after Imlek, a day after the good Friday, a day after the
ascension day of Jesus, a day after New Hijr.=
10 A day after presidential and legislative elections
Table 3. RMSE and MAPE calculated from the three models for the training and testing data
Ho
ur
Method
RMSE MAPE
Training Testing Training
Testing (25-31 December 2011)
1 2 3 4 5 6 7
1
TLSAR 258.1585 604.5183 1.5269 3.0608 2.2952 2.6805 2.7826 2.7215 3.3189 3.8516
TLSNN 225.0217 210.5599 1.3292 1.7816 1.1340 1.6498 1.4100 1.4120 1.3194 1.3108
TLCSNN 219.6763 223.3640 1.2854 1.9569 1.8204 2.0931 1.6147 1.3931 1.2624 1.3573
2
TLSAR 258.4460 598.5510 1.5406 1.9241 1.5728 2.2643 2.5783 2.7039 3.3912 3.8528
TLSNN 226.9396 291.9397 1.3586 1.5159 1.8073 1.8611 1.9852 1.9391 1.6678 1.9340
TLCSNN 226.5588 284.2629 1.3412 0.7976 0.8453 1.3011 1.9835 2.0730 1.7419 1.5428
3
TLSAR 258.0119 594.6462 1.5426 2.6900 1.8375 2.4459 3.0090 3.0498 3.6538 3.9846
TLSNN 228.1649 364.7621 1.3492 3.0535 1.6601 2.0184 2.9127 2.3784 2.2030 1.9297
TLCSNN 216.0240 282.1139 1.2868 1.1977 1.1545 2.0739 2.3221 2.1597 1.8226 1.5671
4
TLSAR 166.6101 674.3093 1.0052 3.2022 1.6647 2.1011 2.8500 3.1066 3.7152 4.3406
TLSNN 230.2454 431.8189 1.3081 2.6571 1.6434 2.0005 2.7185 2.4458 2.7858 2.7745
TLCSNN 218.3380 229.3032 1.2888 1.6704 1.6563 1.3914 1.9303 1.7635 1.4701 1.2627
Based on Table 3, it can be concluded that the TLSNN and TLCSNN models provide a more
accurate forecasting value than the TLSAR model. The modulated oscillation component in the TLCSNN
model increases the forecast accuracy of the TLSNN model, especially at 03.00 and 04.00. The TLSNN
model is able to reduce the value of the RMSE of TLSAR between 35.96% and 65.17% for the testing data,

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while the TLCSNN model is able to reduce the RMSE of TLSAR between 52.51% and 65.99% for
the testing data. For the case of electricity load forecasting at 01.00, 02.00, 03.00, and at 04.00, the TLSNN
and TLCSNN models are more acceptable than the TLSAR model. Furthermore, the results of the TLSNN
model forecasting for 02.00 and the TLCSNN model for 04.00 provide MAPE values that meet PLN
standards, which are below 2%.
5. CONCLUSION
In this study, we proposed two SSA-based hybrid forecasting methods for complex seasonal time
series and provided five data series to show the application examples. The proposed methods named TLSNN
and TLCSNN were developed from TLSAR by considering components of SSA decomposition results.
The proposed hybrid models consist of deterministic and stochastic components. The deterministic
components of TLSNN and TLCSNN models constructed based on the SSA decomposition results
accommodated not only trend linear but also quadratic or another polynomial with higher order to capture
other trend behaviors. The differences between TLSNN and TLCSNN were in their harmonic term.
Parameters of the deterministic model in TLSNN were estimated simultaneously using householder
transformation. Meanwhile in TLCSNN, we added the time-varying amplitude sinusoid components to
the trend and harmonic or stationary sinusoids. The parameters for each functions were then estimated
separately. Finally, the stochastic component of TLSNN and TLCSNN were modeled by NN to overcome
nonlinearity relationship problem in the data. Being modified, the models tend to produce better performance
for the multi-steps forecast values than TLSAR for the case of the monthly accidental death series and Jawa-
Bali load series. The findings of this study have several implications for future practice. Another possible
future research would be to develop the methods for multivariate problems.
ACKNOWLEDGEMENTS
We are grateful to M. Reza Pulungan (Universitas Gadjah Mada) for his constructive remarks and to
the anonymous reviewers for their valuable comments which helped the authors to improve the quality of
the manuscript. The authors would also like to thank to PLN for its support by providing Indonesia electricity load.
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BIOGRAPHIES OF AUTHORS
Winita Sulandari currently is an Assistant Professor in the Study Program of Statistics, Faculty
of Mathematics and Natural Sciences, Universitas Sebelas Maret, Surakarta. She got Master of
Science and Doctorate degree in Mathematics from Universitas Gadjah Mada. She published
several scientific papers in reputed journals or conferences. Her research interests include time
series forecasting, neural networks, and fuzzy time series.
Subanar currently is a professor of Department of Mathematics, Universitas Gadjah Mada. He
holds PhD degree from University of Wisconsin-Madison. He has been the Centre Director of
SEAMEO Regional Centre for QITEP in Mathematics (SEAQiM) for two periods, from 2011 to
2016, and was instrumental in the progress for mathematics education in Indonesia. He has more
than 30 years of teaching and research experience. His research interests include mathematical
statistics, time series forecasting, wavelet neural networks, and fuzzy time series
Suhartono currently is the chief of forum pendidikan tinggi statistika (statistics higher education
forum), Indonesia. He is a faculty at Institut Teknologi Sepuluh Nopember, Surabaya. He got
Master of Science in Statistics from University of Manchester Institute of Science and
Technology. He holds PhD degree in mathematics from Univervitas Gadjah Mada. He has
published more than 100 research papers in journals or conferences. His research interests
include time series forecasting, neural networks, fuzzy time series, and spatio-temporal model.
Herni Utami currently is a researcher and lecturer at Universitas Gadjah Mada, Yogyakarta. She
got Master of Science and Doctorate degree in Mathematics from Universitas Gadjah Mada. Her
research interests include time series analysis, and regression analysis.
Muhammad Hisyam Lee currently is a professor of Department of Mathematical Sciences,
Universiti Teknoogi Malaysia (UTM). He got Master of Science (Statistics) from Universiti
Kebangsaan Malaysia and PhD (Mathematics) from Universiti Teknologi Malaysia. He has
published more than 100 research papers in reputed journals or conferences. His research
interests include high dimensional data modeling, time series analysis, neuro fuzzy, and
regression model.
Paulo Canas Rodrigues is an Assistant Professor in the Department of Statistics, Federal
University of Bahia, Brazil. He is the current President of the Brazilian Region of the
International Biometric Society, and Member of the Executive Committee of the International
Association for Statistical Computing. He published more than 45 scientific papers with more
than 70 co-authors, and is member of the editorial board several international scientific journals.

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