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Researchers
Abstract
Key words
Introduction
Materials and Methods
Results
Conclusions
References
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RAZAN MAHMOUD AL ZIR 1, EHAB ABDULLAH 2
1Ph.D. Research scholar, water resources management and engineering department,
Al Baath University, Homs, Syria.
2Professor in water resources management and engineering department,
Al Baath University, Homs, Syria . |
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Abstract
The global mean warming and rising temperature, is one of the world's critical problems today. This briefing paper has been produced to provide prediction models of temperature due to the most distinguished climate change as the temperature changes that impact climate elements. It is vital to find good statistical prediction model for temperature. In this study, it was found that the series of the used data is unstable on average (there is an increasing general trend), the (Box-Jenkins) method was used in the analysis of time series, and seasonal autoregressive integrated moving average models S A R I M A ( p , d , q ) ( P , D , Q ) s were the appropriate models for the data used. Consequently, the model S A R I M A ( 4 , 1 , 1 ) ( 1 , 1 , 1 ) 12 is the prediction model for the data used between October 2005 to November 2009 we got from the Homs Climate Station.
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| Keywords: Temperature, Autoregression, moving average, partial autocorrelation, Statistical models. |
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Introduction
Global warming is an important issue in the world today. Rising average surface temperatures is one of many indicators of global warming [1].
The process of forecasting is a common issue in many fields of science, and given the importance of time series, many works can be observed in the literature on these topics, especially those based on statistical models, and there are many possible methods to describe temporal behavior, and the Box-Jenkins method is an attractive method in the analysis of time series, as it provides us with a comprehensive statistical modeling methodology and covers a wide variety of patterns, ranging from stability to instability and seasonality of time series [2].
The great interest in the subject of time sequences is attributed to the urgent need for a reliable prediction system, so that it can be relied upon to explain many phenomena in various areas of life, and this system requires the construction of accurate models called time-series models [3].
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2. Materials and Methods
2.1Time Series Analyses (Box-Jenkines) method:
This method depends on the conclusion of the expected changes of the observed values, and the time series is divided into three linear filters: the stationary data filter (integrated), the autoregression filter, and the moving average filter. The Box-Jenkins method can be considered a series of sieves through which time series data passes, and when the data passes through it some of the characteristic elements of the series remain [6].
2.2 Nonstationary Time Series Models:
The Nonstationary time series can be studied using Autoregression models and seasonal integrative moving averages SARIMA ( p , d , q ) ( P , D , Q ) s which have the form:[5]
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3.1 Temperature time series analysis:
The time series consists of monthly temperature values between October 2005 to November 2009 obtained from the Homs climate station and these values are:
Date Temperature
10/2005 18.75
11/2005 11.85
12/2005 9.75
1/2006 6.48
2/2006 9.56
3/2006 12.94
4/2006 15.99
5/2006 21.22
6/2006 25.25
7/2006 26.20
8/2006 27.79
9/2006 25.35
10/2006 20.03
11/2006 12.05
12/2006 7.42
1/2007 8.10
2/2007 9.80
3/2007 12.30
4/2007 14.80
5/2007 23.50
6/2007 27.00
7/2007 31.70
8/2007 27.40
9/2007 25.93
10/2007 20.90
11/2007 14.90
12/2007 7.30
1/2008 4.50
2/2008 8.30
3/2008 17.20
4/2008 21.70
5/2008 23.00
6/2008 28.10
7/2008 28.20
8/2008 31.80
9/2008 27.30
10/2008 21.70
11/2008 14.50
12/2008 8.70
1/2009 6.67
2/2009 9.89
3/2009 11.20
4/2009 15.69
5/2009 20.55
6/2009 25.95
7/2009 27.12
8/2009 27.75
9/2009 23.99
10/2009 22.29
11/2009 18.75
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| Figure 1 signfies the graph of the studied chain with the general direction of the chain and its equation, From Figure 1, we can see an increasing general trend, the series is unstable on average |
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Figure 1 : The general direction and equation of the time series of temperature values
The study test the normal distribution of the chain through the method (Kolmogorov-Smirnov) and as shown in Figure 2 that the series is subject to normal distribution (p-value>0.05). Then we find the Autocorrelation function ( A C F ) shown in Figure 3 and the Partial Autocorrelation Function (P A C F ) shown in Figure 4 [5].
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| Figure 2: Test of the normal distribution of the temperature time series |
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| Figure 3: Autocorrelation function for temperature time series |
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As shown in figures 3,4 this series is seasonally unstable where there is slow inertia in addition to cyclicality we make differentiation 12 and checked that the new time series is seasonally stable and ready to build the Statistical model.
3.2 The model building:
Several ARIMA models have been selected and tested by carrying out diagnostic tests.
Using the Akaiki Information Criteria, it is found that the appropriate model SARIMA ( 4 , 1 , 1 ) ( 1 , 1 , 1 ) 12 is of the form:
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| Figure 5 compares the actual temperature with the predicted values using the proposed model |
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| Figure 5: compare the actual values of temperature with the values of the model |
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Diagnose the model
A. Independence of residuals: By examining the shape of the autocorrelation and partial autocorrelation functions of the residuals, we notice that the autocorrelation coefficients of the residual series are mostly within confidence limits, and have the form of a white noise series, and that the values of the autocorrelation coefficients for most of the time gaps are close to Zero This means that the series is stable and that there is no interconnection between the elements of the series.
B. Residues of a normal distribution:
This was confirmed using the Columgrove-Smironov test, as shown in the figure
C. For residuals of zero mean: To show that the mean residuals of the model is equal to zero, the null hypothesis H0 : μ = 0 was tested against the alternative hypothesis H1 : μ ≠ 0. the test results shows that p =0.867>0.05 so the mean residuals can be considered equal to zero.
D. Residues of S A R I M A ( 4 , 1 , 1 ) ( 1 , 1 , 1 ) 12 random: the results of the test on the residuals of the model shows that the residuals of the model are random.
3.4 Forecasting the future series:
After the success of the tests on the residuals, the proposed model became suitable for predicting the temperature in the studied station and generating a future series. Forecasts of temperature values for ten months were generated using the proposed model and compared with the measured values. Figure (6) shows a comparison between the measured and predicted values of temperatures using the model.
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Figure 6: a comparison between the measured and predicted values
The percentage errors of the model were calculated in Table 1, and it was found that the highest value of the percentage error is 7%, which is less than the permissible value.
Table 1: The percentage errors of the model |
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4. Conclusions:
- The time series that represents the monthly temperatures in the city of Homs for the period (2005-2009) is unstable on the average due to the presence of a general trend in it (increasing).
-The stability of the time series is achieved after the first difference is taken.
- The S A R I M A ( 4 , 1 , 1) ( 1 , 1 , 1 ) 12 model is an appropriate data model that can be relied upon in making predictions of temperatures in the city of Homs.
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References
1. the Commonwealth Scientific and Industrial Research Organisation (CSIRO) and Bureau of Meteorology,2007-Observed Changes and Projections.
2. Berri, A,2002-Statistical prediction methods. Saud king university, Saudi Arabia,342.
3. MASO A,2008-Satistics using Minitab. Aleppo university, second audition,Syria,446.
4. KAUFMANN R, KAUPPI M. and Stock J, 2011-Reconciling anthropogenic climate change with observed temperature 1998–2008. Proceedings of the National Academy of Sciences, 108 (29),11,790.
5. ALI J, MAJEED A,2017-temperature analysis in Baghdad city using time series for the period (2015-2016),Alqadesiah university, Iraq,198p.
6. ABRAHAM B and BOX G.E.P, 1975- Linear Models, Time S e r i e s andO u t l i e r s 3: S t o c h a s t i c D i f f e r e n c e Equation Models. Department of Statistics, University of isc cons, Madison, 438.
7. BOX, G.E.P. and Jenkins G.M,1970- Time S e r i e s A n a l y s i s , Forecastingand C o n t r o l . Holden-Day, San Francisco, CA.
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