Differencing seasonal

ARIMA modeling in contrast with the ARMA model, includes trend or seasonality of time series. For such series, the trend can be removed by one-step differencing. Seasonality may be removed by 12-step differencing. After differencing the time series one gets a stationary time series which can be described as an ARMA process. 

The remaining series from seasonal differencing is a good starting point for trend detection methods (Fig. 6-9). 

A significant slope signifies a trend. For the appropriate test see Section 2.4. After seasonal differencing the example (Fig. 6-11) shows a significant trend with a slope of -0.0618, i.e. a small negative trend or a decreasing nitrate concentration over the total time range of the observations. 

This differencing is continued until the time series is stationary (mean and variance are not dependent on time). Frequently, single time differencing, i.e. first order for the seasonal ARIMA model, is sufficient. Second order differencing is necessary for quadratic trends. Please note the loss of values after differencing (e.g. after first-order seasonal differencing twelve values will be lost). 

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, ). 

It can be noted that the spring and summer temperature series have a similar undifferenced behaviour, which suggests that an integrator could be present. Differencing the summer temperature series reveals the potential of seasonal components at 3, 4, and 8 years. On the other hand, the winter temperature series has a different behaviour with a single peak at 3.3 years/ cycle and no suggestions of an integrator. 

Based on the above discussion, different models were fit, including seasonal differencing of 3 and 4 years, differencing of 1 year, and model orders between 1 and 3 parameters for both the seasonal and nonseasonal components. After trying different models, the final model was determined to be... 

---