ARIMA seasonality

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 ... 

ARIMA connects both autoregressive and moving average models and includes integrating effects, e.g. trends or seasonal effects. 

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. 

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, ). 

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). 

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. 

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 patterns of both ACF (Fig. 6-27) and PACF (Fig. 6-28) of the errors from the previous model are used to find the complete multiplicative trend and the seasonal ARIMA model. The following conclusions can be drawn ... 

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. 

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. 

The empirical analysis of ARIMA multiplicative seasonal model to forecast the total number of coal mine accidents... 

ABSTRACT This paper is to research an application of the multiplicative seasonal model to forecast the total number of China s coal mine accidents. By the empirical analysis on the data of coal mine accidents from January 2006 to December 2010, an accepted multiplicative seasonal forecasting model ARIMA(4,1,1)(1,1,1) is built up after differing the series to be stationary and estimating the order and parameters of the model. Furthermore, the test of this multiplicative seasonal model shows that it has a desirable fitting effect on the data of coal mine accidents. At last, this model is applied to forecast the number of national coal mine accidents from January 2010 to December 2010, and the forecasted values have a high accuracy when compared to the actual data. 

Also, a time series changing seasonally couldn t be modeled by ARMA immediately. Firstly, it should be differed to remove the seasonality of the series, and the length of one difference step is one seasonal cycle. Generally, if the series is processed by a D-th order difference the cycle length of which is s, the seasonality will almost be removed. Furthermore, if the series needs to be turned stationary by a d-th order difference before it s impacted by the seasonal difference, a model called the Multiplicative Seasonal Mod-el ARIMA(p, d, q) (P, D, Q) could analyze the original series, and the model is described as below ... 

Compared to ordinary time series models, the Multiplicative Seasonal Model needs more historical data, and the Multiplicative Seasonal Model can be applied to a wider field because data in daily life always have an obvious trend and seasonal features. Therefore, the Multiplicative Seasonal Model can well solve such problems that involve some issues about forecasting, and as well as reach a high precision. The model in this paper, ARIMA (4,1,1)(1,1,1) well matches the monthly changing number of national coal mine accidents. Moreover, the more historical data, the more accurate the forecasted result is. AH above, the Multiplicative Seasonal Model is a practical tool for us to forecast or to apply in many other fields. 

Zhang hui Liui Jia-kun, 2005. Modeling and Prediction of Freeway Traffic Flow using seasonal ARIMA modles. Journal of Tianjin University, 28(9) 838-841 (in Chinese). 

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