The effect of differencing in stationary and accuracy of time series in predicting of lake level

One of the most important assumptions in the modeling of time series, it is to be stationary. The amount of stationary can be various, so that different definitions exist such as first order and second order stationary and strong and strict stationary. Therefore, this study cover the effect of differencing on the stationary value as well as the precision of the ARMA, ARIMA and SARIMA models in the modeling and monthly prediction of time series.
For this purpose, 96 years data of lake level, which is monthly measurement related to Michigan-Huron Lake on the border of United States and Canada, are used. The 76-years of primary utilized for training and the rest of 20-years are used for validation. Firstly, the existing of the trend and period components in the time-series were assessed using Fischer and man-Kendal tests. These two components are the main factors in the appearance of non-stationary in time series. Therefore seasonal differencing, non-seasonal differencing and both of them at same time were measured and their results were compared by non- differencing data. To assessment of achieved time-series differencing, the ACF diagram and generalized Dicky-Fouler test were utilized. The type and amount of required parameters in different models were determined by ACF diagram. Then, each of series was modeled and predicted using appropriate model. The results indicated that there is not a certain trend and period in series. However, the using of seasonal differencing increased the stationary but non-seasonal differencing lead to non-stationary of these time series. The most increasing in stationary was indicated by using of seasonal and non-seasonal differencing. Due to ACF diagram, using both of differencing results in use of seasonal parameters in model. Therefore, series without differencing with ARMA model and series with seasonal differencing with SARIMA are modeled.
The investigations showed the concurrent using of seasonal and non-seasonal differencing has the most impact on the rate of getting stationary alignment of the Lake in compare with other methods. As a result, the numbers of model needed to achieve the most accurate predictions were reduced in large scale. In such a way in non-differencing situation, 1444 model of ARMA were needed that this amount in situation of seasonal differencing and non-seasonal differencing were reduced in 64 models of SARIMA. On the other hand, by reducing much more number of parameters (two parameters) in SARIMA model, similar result is even better than ARMA model with 21 parameters.
The results showed that the more making stationary of monthly lake level which itself is stationary, reduces the number of models and the number of model's parameters needed to achieve the best outcome too much. For this purpose, combined differencing made the series stationary more than the other methods.