**ABSTRACT**

Water plays a significant role at the present time and receives more concern in future due to its necessity for the development of the life of people, because water is considered as the most important natural resources and depends on it different activities. This study presents the statistical models via the application of the theories of a time series to water variables of Thalweg of the Euphrates River between Al-Qadissiya Dam and Abu-Ghraib stream. Measurement were made and SPSS 7.5 Software was used for the analysis and the best model was determined by checking the explained variation.

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**How to cite this article**

*Journal of Applied Sciences, 6: 1929-1934.*

**DOI:**10.3923/jas.2006.1929.1934

**URL:**https://scialert.net/abstract/?doi=jas.2006.1929.1934

**INTRODUCTION**

The Thalweg of the cross section of the Euphrates river between Al-Qadisya Dam and Abu-Graib stream were recorded. Box-Jenkins method is used to estimate the Auto Correlation Factor (ACF) and Partial Auto Correlation Factor (PACF) for the notices above by using the SPSS software (Hussain, 2000). This system accounts the final estimates of the autocorrelation and the autoregressive and moving average in addition to the greatest log likelihood, AIC and SBC and the total of square and variance and residual difference and checking the whiteness after achieving the account of the ACF of the residuals by using Chi-square test and at the end we chose the best model to the ARIMA from the possible alternatives which gives the best check of variations for the results depending on the statistical tests (Al-Samawi and Abas, 1994).

**ANALYSIS**

It considers observed data across time periods and statistic methodology which deal with analysing these previews are called time series analysis (Muhsin, 1995), this analysis is distinguished among others by giving it a significant in keeping on previews sequences in time series (Pandit and Wu, 1983).

**METHOD**

Box-Jenkins method was used, this method is considered one of the common prediction methods because it is a general method use for dealing with all kinds of stationary and non stationary time series (Box and Jenkins, 1976). The theoretical basis of the method is explained below;

• | Linear Filter model |

(1) |

Ψ_{i}: | Values of coefficient weights |

μ : | Average of variable series |

• | Autoregressive Model (AR) |

(2) |

X_{t}: | Values of observations of time series during the period (t-p)t |

ξ : | Time series average. |

ε_{t}: | Random error combination that follows natural distribution with zero average and contrast is (σ^{2}_{t}) |

• | Moving Average Model (MA) |

(3) |

X_{t}: | The value of real observation in (t) |

μ : | Time series mean |

θ_{1}, θ_{2},...., θ_{q}: | Features of the model |

ε_{t-1}, ε_{t-2},...., ε_{t-q}: | Random changes |

• | Autoregressive Moving Average Model (ARMA) |

(4) |

X_{t}: | Values of real observation for time series in (t) |

ξ : | Time series mean |

φ_{1}, φ_{2},...., φ_{p} : | Coefficient of autoregressive limitations |

θ_{1}, θ_{2},...., θ_{q} : | Features of moving average limitations |

ε_{t}, ε_{t-1},...., ε_{t-q} : | Values of random changes |

Box and Jenkins (1976) presented a general way to indicate on model ARIMA the best for any time series then using this model for purposes of prediction.

This way is summarized with the following steps:-

• | Model Identification (Coulbeck and Orr, 1988). |

• | Estimation (Box and Jenkins, 1976). |

• | Diagnostic checking (Hanke and Reitsch, 1981). |

**Selection of parsimonious model: **The best degree of the model which has the smallest number of variables is used to give the best possible results. We account well the variable (a) standard error test (Donovan and Thomas, 1983). (b) variance ratio test (c) Akaike information criteria, which is used by Srinivas Edwin (1982).

**EUPHRATES THALWEG MODEL**

The Thalweg variable consists of (52) observations in the year 1999 between Al-Qadisya dam and Abu-Graib stream from the Euphrates. The Fig. 1 shows the drawing of the thalweg in the return of the distance which illustrates the most important series characteristics. The clear characteristics such as the direction, the discontinuities will be clear if it is founded in the series so the drawings illustrates the chosen test of stable series and the model definition.

The Fig. 1 shows wild observations or they are outliers which don’t appear organized with other information. The outlier observations form wild observations. That needs to their smooth to their expected value before the analysis of the series. The thalweg series has three outliers the smoothed of these wild observation is it taking equated to the average of the two observation which occurs before and after which is shown in Table 1.

Figure 2 represents relationship between standard deviation and mean transformation test using pearson method the correlation coefficient is found equal to (0.77).

Common logarithm function is used to transform the data to the normal distribution (Fig. 3) the correlation coefficient between the standard deviations and the means of the thalweg is reduced to (-0.016). This means that fluctuations of the data are reduced.

Table 1: | Thalweg outliers and their smooth points |

Fig. 1: | Thalweg time series |

Fig. 2: | Standard deviations vs means of thalweg |

Figure 4 represents (ACF) autocorrelation function for the variables the extend of values decreases gradually till it reach to zero.

The partial autocorrelation function (PACF) which is illustrated in the Fig. 5 is accounted. The cut is after the first lag so the addition is to the first real value. This value is bigger than the level of confidence limit error. From these Fig. 4 and 5 the model ARIMA(1,0,0) or we can define it AR_{(1)}.

Fig. 3: | Standard deviations vs. means of thalweg common log |

Fig. 4: | Auto correlation function of thalweg |

Figure 6 shows the model (AR_{(1)})all the variables of the model are new, because the values is more than of double of standard error.

The model equation:

(5) |

The statistic tests:

LLH = 76, AIC = -148, SBC = -144

Because of the value (φ_{1} = 0.99) is nearer to one it is possible to take first differencing.

Fig. 5: | Partial auto correlation function of thalweg |

Fig. 6: | Estimation of thalweg model AR_{(1)} |

The (ACF) illustrates for a model to the lowest level of the bottom thalweg after taking the first difference in Fig. 7 which has the real value to three of Autocorrelation these values are bigger than the confidence limit error at lag (1), lag (9) and lag (10).

Fig. 7: | Auto correlation function of thalweg difference (1) |

Fig. 8: | Partial auto correlation function of thalweg difference (1) |

Figure 8 shows the partial Auto correlation function of thalweg after taking difference (1), which has real values for two of partial auto correlation these values are bigger than confidence limit error at lag (1), lag (8).

From the two Fig. 7 and 8 we can know the model ARIMA (2, 1, 3).

Figure 9 represents estimation of Thalweg model ARIMA (2, 1, 3).

Fig. 9: | Estimation of thalweg model ARIMA (2,1,3) |

The model equation:

(6) |

the model didn’t achieve the standard error test because the variable of the first AR_{(1) }is (-0.003) and the variable of MA1, MA2, MA3 are (0.5), (0.9), (0.59) less than the double confidence limit (1.48), (3.78), (2.54), (0.22). So the model ARIMA (1,1,0) is identified to adjust the failure in the model.

Figure 10 shows estimation of thalweg model ARIMA (1,1,0)

The model equation is :

(7) |

the statistic to test are :

LLH = 85, AIC = 166, SBC = -163

The model variables of the two models are equal so we can not use variance ratio test. From the statistical test to the Table 2 the model ARIMA is the best chosen model because it has the largest value of LLH and the smallest values of AIC, SBC.

Fig. 10: | Estimation of thalweg model ARIMA (1,1,0) |

Fig. 11: | Auto correlation function of thalweg residuals model ARIMA (1,1,0) |

Figure 11 shows the accuracy of auto correlation function (AIC) of residual model ARIMA (1,1,0) FOR (lag 13) it is obvious that one of the auto correlation has bigger value than the double confidence limit error at lag (9). by using diagnostic process by finding the total squares for the values (ACF) and multiplying the total with the final number of the observation (N = 51), (Q) portmanteau coefficient was equal to (17.94) which is less than chi-square (χ^{2} = 19.68) at (11) degree of freedom and 5% of confidence limit error level (χ^{2}>Q) and more than the auto correlation which doesn’t have real values and so the checking of ARIMA (1,1,0) model validity is done concerning the residuals are whiteness.

Table 2: | Comparison between Thalweg models AR_{(1)} and ARIMA (1,1,0) |

Fig. 12: | Thalweg residuals histogram model ARIMA (1,1,0) |

Fig. 13: | Cumulative probability plot of a residuals form thalweg ARIMA (1,1,0) |

Table 3: | Chi-square test |

Figure 12 shows the thalweg residuals histogram model ARIMA (1,1,0). While Fig. 13 shows cumulative probability plot of a residuals from thalweg ARIMA (1,1,0).

Table 3 shows Chi-square test for the better application to common model.

**CONCLUSIONS**

• | The flexibility of Box-Jenkins method that flexibility comes of the numbering of the models of this method widely so there aspace to choose the suitable model to the series under research with bigger freedom. |

• | There are irregular values, which are substituted by the observations which are surrounded to it, because the deviations can not be completed or they have irregular values. |

• | From the variables drawings to the following study thalweg to the Euphrates between the Qadisya Dam abd Abu-Grabe stream. |

• | The thalweg model represented by the first order autoregressive model AR_{(1)} this refers that the memory of the variables, which are measured according. |

• | To the distance is short and the deviations, extences to one distance only the lowest level of the bottom thalweg has the smallest value of the memory (-0.394) which refer that the periodic value to the error component to this variables is large. |

In this research the formation of stochastic vocabularies which suits with the standard common distribution in addition the suitability test to these deviations for the common standard distribution by using probability plot for the common standard distribution which probability values spread to the deviations on the straight line whenever these points on the straight line that will a referring on the weakness of this suitability. The probability plot to models residual shows that the residual deviations to the model the thalweg is the closest for the common standard distribution. This is affirmed the value of Chi-square χ^{2} which is accounted for the residual model (χ^{2}_{(2)} = 4.73305) it shows confidence at the level of (0.0938063) function as in Table 3.