ARIMA components

First, the series of the nitrate concentrations within the storage reservoir is made stationary in order to obtain the parameters d and sd for the trend and the seasonal ARIMA model. With one-time differencing at the differences 1, the series becomes stationary and the parameter d is set to unity (Fig. 6-24), but seasonal fluctuations are present. With one-time differencing of the original nitrate series at the difference 12, the seasonal fluctuations disappear, but the trend is present (Fig. 6-25). It is, therefore, necessary to include the seasonal ARIMA component in the model, the parameter sd is set to zero. The deduced possible model is ARIMA ( ,1, )( ,0, ). 

These were the general conclusions from ACF and PACF of the time series for modeling ARIMA (p,d,q) sp,sd,sq). Now the second term of the multiplicative ARIMA model - the seasonal ARIMA component, ARIMA (()S)S))(sp,sd,sq) - must be estimated. 

Last but not least, it is necessary to check the correlation between the ARIMA components the autoregression component and the seasonal autoregression component using the correlation matrix. In an adequate model, there should be no significant correlations between the single components. 

For composed trend and seasonal processes, the ARIMA trend and the ARIMA seasonal model were multiplied, e.g. both AR components as well as both integration components and both MA components. Then the notation is ... 

Improving the Noncorrelation between the ARIMA Model Components... 

The PACF illustrates the order 1 for the AR component, but at this stage of estimation of the model it is unknown if the trend or the seasonal model follow the autoregression with the order of one. No moving average component can be found from the PACF. Deduced possible models are ARIMA (1,1,0)( 1,0,0), ARIMA (0,1,0)( 1,0,0), or ARIMA (1,1,0)(0,0,0). 

One of the single seasonal models deduced from the previous conclusion is ARIMA (0,0,0)(1,0,0). This model will be proved for its significance relating to the seasonal fluctuations of the time series. The model ARIMA (0,0,0)(1,0,0) (Tab. 6-4) confirms the high significance of the seasonal AR component. Therefore, sp is set to unity. The resulting standard error of the model is 6.35 mg L 1 NO . The resulting fit and the errors from ARIMA (0,0,0)(1,0,0) are demonstrated in Fig. 6-26. 

The ACF and PACF of the resulting errors from ARIMA (1,1,0)(1,0,0) do not show spikes (Figs. 6-30 and 6-31). This means they do not have significant autoregression or moving average components. 

In the example with the ARIMA model (1,1,0)(1,0,0), the correlation coefficient between the autoregression and the seasonal autoregression component is -0.19. This means they are not correlated with an error probability of 0.002. Therefore, this model is valid. 

In ARIMA modeling, the order of the autoregressive component is frequently zero, one or sometimes two. Therefore, only short forecasting intervals are of any use. Disturbances in future values, normally smoothed by the moving average, were set to zero. The following example demonstrates this fact ... 

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