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Part-A- (Pure and Applied
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Annual
Forecasting Using a Hybrid Approach

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Qais Mustafa Abdulqader

Technical College of Petroleum and Mineral Sciences,
Duhok Polytechnic University, Zakho-Iraq

E-mail: [email protected]

 
Article
info

 

 
Abstract

Original:
Revised:
Accepted:
Published online: 

            
 

 

In
this paper, we used a hybrid method based on wavelet transforms and ARIMA
models and applied on the time series annual data of rain precipitation in
the Province of Erbil-Iraq in millimeters. A sample size of (45) values has
been taken during the period 1970 – 2014.We intended to obtain the ability to
explain how the hybrid method can be useful when making a forecast of time
series and how the quality of forecasting can be enhanced through applying it
on actual data and comparing the classical ARIMA method and our suggested
method depending on some statistical criteria. Results of the study proved an
advantage of the statistical hybrid method and showed that the forecast error
could be reduced when applying Wavelet-ARIMA technique and this helps to give
the enhancement of forecasting of the classical model. In addition, it was
found that out of wavelet families, Daubechies wavelet of order two using
fixed form thresholding with soft function is very suitable when de-noising
the data and performed better than the others. The annual rainfall in Erbil
in the coming years will be close to 370 millimeters.
 

Key Words:
Forecasting
Time series
ARIMA
Wavelet transforms
De-noising

 

Introduction

   
Rainfall forecasting is one of the most challenging objects. Many
algorithms have been developed and proposed but still an accurate prediction of
rainfall is very difficult. (Tantanee et al., 2005), presented in their study a
new procedure for predicting rainfall and depending on a combination of wavelet
analysis and conventional autoregressive AR model. The research showed that the
wavelet autoregressive model procedure gives a better prediction of annual
rainfall than the classical AR model. (Al-Safawi et al., 2009) have estimated
the autoregressive model using wave shrink. The results showed that the
suitable model using classical ARIMA method is AR(6) and this model has
improved when using wave shrink technique and especially when using Haar
wavelet with a soft threshold to forecast the quantity of the annual rainfall
in Erbil city for the period 1992-2007. (Al-Shakarchy, 2010) applied the factor
analysis for forecasting two series representing rain rates and relative
humidity in Mosul province. Results showed that the suitable model for the two
series is ARIMA(0,0,1) and ARIMA(1,0,0) respectively. (Ali, 2013) used ARIMA
method for analyzing and forecasting of Baghdad rainfall. It is found that the
seasonal model SARIMA(2,1,3)x(0,1,1) is the best model and according to this
model, rainfall forecast for the coming years was also prepared and presenting
and showing similar trend and range of the real data. (Venkata
Ramana et al., 2013) searched to obtain a good model for monthly prediction
of rainfall data by using hybrid technique consisting of the wavelet technique
with artificial neural network ANN. The results of the analysis showed that the
performances of the obtained models are more efficient than the ANN models.
(Shoba and Shobha, 2014) have made an analysis of various algorithms of data
mining used for rainfall prediction model. The study showed that sometimes when
certain algorithms are combined, they perform better and are more effective.
(Eni and Adeyeye, 2015) applied seasonal ARIMA method for building a suitable
model and to forecast the rainfall data in Warri Town, Nigeria. Results showed
that seasonal model ARIMA (1, 1, 1) (0, 1, 1) is adequate depending on some
statistical criteria.

    
Recently, (Shafaei et al., 2016) offered some techniques for testing
their capability of predicting the monthly precipitation such as wavelet
analysis WA, seasonally mixed model SARIMA and ANN method. The study concluded
that searching for the effect of decomposition level on model performance, it
was indicated that going from 2 to 3 decomposition levels increased the correlation
between observed and estimated data, but no significant difference was found
between predictions from 2 and 3 level models. (Ramesh Reddy et al., 2017)
applied ARIMA model to forecast the monthly mean rainfall of coastal Andhra
-India. They found that the best model for fitting data is ARIMA (5,0,0)(2,0,0)
depending on some performance criteria. (Ashley et al., 2017) applied DCT
presenting the discrete cosine transform and DWT presenting discrete wavelet
transform to make a reduction in the 5 dimensionalities of rainfall time series
observations. The conclusions of the research demonstrated that the DWT has the
superiority to the DCT and best preserves and characterizes the observed
rainfall records of the data.

    
From the above-suggested methods, we observe that most of these
approaches and methods are applying to forecast the short period. This paper
offers a new technique for forecasting the long-range of annual rainfall data.
In another word, it mainly deals with combining wavelet transformation with
classical ARIMA methodology for modeling of annual rain precipitation based on
the available data. The procedure of this paper is prepared as the following: First,
we provide brief explanations of ARIMA methodology and wavelet transformation
and then we offer the hybrid method. Next, we deal with application on real
data. Finally, we present some conclusions of the study.

 

ARIMA Methodology, Wavelet Transformation, and Hybrid
Method

 

ARIMA Methodology

     Box-Jenkins suggested an approach for analyzing
time series data including an identification of the model, parameters
estimation, diagnostic checking for the suggested model, and using the model
for forecasting. ARIMA model is a mixed model which depends on parameters p, d,
q representing a combination of autoregressive order part (AR); the degree of
difference involved and the moving average order part (MA) respectively. The
model becomes popular by (Box et al., 1970) and can be well explained through
the mathematical formula:

 

 

Here, p represents a non-seasonal autoregressive order, q is
a moving average order of the non-seasonal, are called coefficients of
autoregressive,  are coefficients of moving
average and is a random error. If the data are not stationary, then the difference
of first or second order has to be taken. For obtaining a convenient model, we
depend on two functions called ACF as Autocorrelation Function and PACF as
Partial Autocorrelation Function. The pattern of both functions plot provides
us an idea towards which one of the specified model could be the best for
fitting and appropriate for making a prediction and depending on some
statistical performance. Also, in this study, we will apply the Portmanteau
test statistic (i;e. Box-Pierce) for the purpose of randomness of time series.
We refer to (Makridakis et al., 1998) for more details.

 

Wavelet Transformation

 

     A
wavelet transformation is a proceeding subject, very efficient, and effective
in the field of processing the signal that has been very interest after
developing the theory of wavelet methodology (Grossman
and Morlet, 1984). applications of
wavelet analysis have increased in many fields such as in edge detection, image
compression, optical engineering, and the applications of time series as
alternate to the classical Fourier transformation in local maintain, not
involving cyclic and multi-scaled phenomenon. Wavelets can give the specific
locality of any changes in the dynamical patterns of the sequence, while the
transformations of Fourier focus essentially on their frequency and this is the
major difference between wavelets analysis and Fourier analysis. in addition,
the transformation of Fourier supposes unlimited length signals, while the transformation
of wavelet can be used to any form and any size of time series data, even when
these time series are not identically sampled (Antonios
and Constantine, 2003). Generally,
wavelet transforms can be applied for seeking, reducing the noise and filtering
time series data which help and also support forecasting and other analysis of
the experiment. The formula of wavelet transform can be presented as the
following:

 

 

Here, ?(t) represents the essential wavelet with efficient length (t)
that is commonly much shorter than the target time series f(t), ‘a’ represents
the scale factor or dilation that specifies the information of characteristic
frequency so that its variation yields increase to a spectrum and ‘b’
represents the translation of time information so, its difference displays the
‘sliding’ of the wavelet over f(t) (Burrus
et al., 1998).

 

Hybrid Method

 

     The
concept of the suggested method is based on combining ARIMA methodology with
wavelet transforms. As the wavelet approach can be easily used for signal
analysis, this study used the approach to decompose the details (which are
small differences) from the approximations (which represents the important
part) of data. In wavelet analysis, the approximations are the high-scale and
limited frequency components of the signal, and the details represent the
limited-scale and high-frequency components (Fugal,
2009). The process is
done by applying discrete wavelet transform DWT because the data of the study are
recorded in discrete time.  The procedure
of hybrid method can be expressed in figure1

 

Figure-1:
The process of hybrid method

 

Application

Information About Erbil City

     Erbil which is the Kurdish central is the capital city of Kurdistan Region in Iraq. The city is located between (36°12?17?N 44°20?33?E). It is located
about 350 kilometers north of Baghdad. The climate of Erbil is very hot in
summer and very cold and wet in winters. There is
more rainfall in the winter than in the summer in Erbil. The
average total of receiving rain of the city is between 300-400 millimeters
annually. The city represents the managerial center of Erbil province. It
is bounded from the north by Turkey and nearby Duhok Province, from the east by
Iran and near to Sulaymaniyah Province, from the south, is close to Kirkuk
province, and near to Mosul province from the west (Wahab and
Khayyat, 2014).

 

Application
Using ARIMA Methodology

   
The variable used in the analysis represents the annual data of rain
precipitation in Erbil province in Kurdistan Region of Iraq (in millimeters)
and represents taking (45) observations as sample size during the period 1970 –
2014 which is shown in table1. The data were obtained from the General
Directorate of Meteorology and Seismic Monitoring in Erbil province.

 

Table-1: Annually data on rain precipitation from 1970 to 2014

Year

Amount of Rain

 

         Year

Amount of Rain

 

 

 

 

 

 

 

 

 

 

 

 

1970

 

 

255.4

 

1993

601.6

 

1971

 

448.2

 

1994

583.0

 

1972

 

406.4

 

1995

494.4

 

1973

 

261.5

 

1996

418.9

 

1974

 

547.5

 

1997

441.6

 

1975

 

417.2

 

1998

337.2

 

1976

 

452.3

 

1999

229.2

 

1977

 

347.2

 

2000

272.3

 

1978

 

380.1

 

2001

330.9

 

1979

 

375.6

 

2002

361.5

 

1980

 

321.5

 

2003

587.7

 

1981

 

141.8

 

2004

255.6

 

1982

 

444.1

 

2005

297.5

 

1983

 

178.3

 

2006

514.6

 

1984

 

43.9

 

2007

273.4

 

1985

 

463.9

 

2008

410.7

 

1986

 

154.0

 

2009

411.0

 

1987

 

235.9

 

2010

359.6

 

1988

 

626.9

 

2011

301.6

 

1989

 

367.3

 

2012

366.4

 

1990

 

332.0

 

2013

345.2

 

1991

 

344.1

 

2014

385.2

 

1992

 

694.0

 

 

 

 

    Figure2 shows the plots of time
series of rain data for Erbil city. Depending on Box-Jenkins procedure, the
first step to do is identification through employing the ACF and PACF plots
which are clear in figure 3.

 

 

 

 

Figure-2: Time series plot of rain data in Erbil province from 1970
to 2014

Figure-3: Autocorrelation function and partial autocorrelation
function of rain data

 

     Depending on PACF and PACF
plots and checking for stationarity in mean and variance, the appropriate model
for the respected series is identified as ARIMA(2,1,0) after well consideration
of modelling and fitting and depending on two performance measures such as RMSE
as root mean square error and MAE as mean absolute error. The estimated model
is shown in table2.

Table 2: Estimation of ARIMA(2,1,0)

Parameter 

Estimates

Std. Error

 t-ratio

P-value

AR(1)

-0.72091

0.129125

-5.58304

0.000002

AR(2)

-0.540025

0.128616

-4.19875

0.000136

 

 

 

 

     After getting the estimation
of the ARIMA (2,1,0) model, we should look for getting randomness. Figure 4
offers the residuals pattern and stability of ACF and PACF inside the intervals
using classical ARIMA (2,1,0).

Figure-4: ACF and PACF of residuals using
ARIMA(2,1,0) on series data.

 

    From Figure 4, there is no
significant appear from the autocorrelations coefficients of ACF and PACF,
which concludes that the time series is random (i.e.; white noise). Concerning
the randomness of residuals, we did a test using a Portmanteau test (or
Box-Pierce test), which has been mentioned in theoretical part. The value of
the test was (7.326) comparing to the P-value (0.835) indicates that the
hypothesis cannot be rejected at the 95% or higher confidence level and
concluding that the series is random.

 

Application Using a Hybrid Method

    In this part, the conversion
of original data from time domain to frequency domain has been done to make
filtration. Figure 5 shows applying Daubechies wavelet
with multiresolution of five levels for the rain precipitation for 45
values as sequential observations, denoting s as a signal and it means the
summation of signal approximation and its details, a5 is an approximation at
level 5 and d5; d4; d3; d2; d1 is the details level from 1 to 5 respectively.

 

 

Figure-5:  Daubechies wavelet
of the rain precipitation using multiresolution of five levels.

 

     The real data of rain
precipitation were reduced from noise using wavelet denoising procedure (using
the software MATLAB, version 2013) with Daubechies wavelet family from order 2 to
order 5as shown in figure6. It should be noted that after
making many empirical experiments, it has been found that the performance of
Daubechies wavelet was better than others in terms of de-noising the rain data.
Figure 7 shows the real and de-noised signals by applying the Daubechies
wavelet with Fixed Form Threshold (Patil and Raskar, 2015).

Figure-6: Daubechies wavelet of order 2,3,4, and 5

 

Figure-7: The original and de-noised
signals using Daubechies wavelet with Fixed Form Threshold.

 

    The data were analyzed using
five levels of multiresolution for the selected wavelet, and then de-noised
using Fixed Form Threshold and depending on soft thresholding. After that, the
new series was modeled again using ARIMA methodology. Also, the values of
forecasting criteria were compared with those in the first method. Table 3
presents the performance values of the two indicators of selecting an optimal
model for the original data model using ARIMA method and hybrid method.

 

Table-3: The
values of the performance measures for the original data model using classical
ARIMA methodology and hybrid method.

Method

Kind

RMSE

MAE

Classical
ARIMA Method
Original (raw) data

ARIMA(2,1,0)

133.937

106.565

Hybrid Method
Fixed Form
De-noised data

Daubechies(2)

131.380

104.143

Daubechies(3)

131.555

104.553

Daubechies(4)

131.593

104.411

Daubechies(5)

131.706

104.546

 

    From Table 3, we observe that
the best model for the original data was ARIMA(2,1,0). However, when the hybrid
method applied to the original data the errors of the forecasting have
decreased for all wavelet orders and the new models have been enhanced
depending on the forecasting measures. To make a comparison of the two procedures,
we can see that the maximum reduction is when applying Fixed Form Thresholding
and using Daubechies wavelet of order 2 (i.e.; from the Table 3 the good
reduction in RMSE and MAE from 133.937 to 131.380 and from 106.565 to 104.143,
respectively). Figure 8 presents the original and filtered data using
Daubechies wavelet of order 2.

 

Figure-8: The original and filtered signals
using Daubechies wavelet of order 2

 

    The forecast values of our
hybrid method are presented in table 4 which shows the forecasting for the next
years starting from 2015 up to 2030 of the annual rain precipitation (in
millimeters) of Erbil province – Iraq.

 

Table-4:
Forecast values of the annual rain of Erbil province-Iraq using hybrid method

Forecast

Period

367.8

2015

360.3

2016

373.5

2017

368.1

2018

364.9

2019

370.1

2020

368.1

2021

366.7

2022

368.8

2023

368.0

2024

367.4

2025

368.3

2026

368.0

2027

367.7

2028

368.1

2029

368.0

2030

 

 

 

 

 

 

 

 

 

 

 

 

 

Conclusions

   
In this research, we offered a new technique as hybrid method for
enhancing the Box-Jenkins ARIMA analysis when forecasting time series data.
Indeed, we concluded that:

1- The appropriate
model for forecasting using classical Box – Jenkins method was ARIMA(2,1,0).

2- The classical
model has been enhanced and improved when making filtration of the data and
using Daubechies wavelets orders from 1 to 5 and among them, the Daubechies
wavelet of order 2 gave results better than others.

3- Depending on our hybrid method to
forecast for the coming years, the Erbil city will receive an average total
rainfall of 360-370 millimeters annually.

 

References

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1136-1142, (2013).

2 Al-Safawi S., Ali T., and Badal
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Scientific Conference of Mathematics – Statistics and Informatics, University
of Mosul, 274-299, (2009).

3 Al-Shakarchy DH., “Using
factor analysis to forecast of time series with an application on two series
rain rates and relative humidity in Mosul city”, Tikrit Journal of
Administrative and Economic Sciences, Vol.6, 93-108, (2010).

4 Antonios A., and Constantine E.V.,
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9 Fugal D., “Conceptual
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13 Ramesh Reddy J. C., Ganesh T.,
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14 Shafaei M., Adamowski J.,
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16 Tantanee S., Patamatammakul S.,
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17 Venkata Ramana R. Krishna S.,
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18 Wahab S., and
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