Why Are the ARIMA and SARIMA not Sufficient
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The autoregressive moving average (ARMA) model and its variants like autoregressive integrated moving average (ARIMA), seasonal ARIMA (SARIMA) take the significant position in the time series analysis community. The ARMA model could describe a rational-spectra wide-sense stationary stochastic process and make use of the past information to approximate the underlying dynamics of the interested stochastic process so that we can make predictions of the future. As its supplementary, the ARIMA and SARIMA, collectively referred to as S-ARIMA for briefness, aim to transform the interested non-stationary and irrational-spectra wide-sense stationary stochastic process to be stationary and rational by difference and seasonal difference operator with proper order so that they can follow the philosophy of the ARMA model. However, such difference operators are merely empirical experience without exploring the exact philosophy behind. Therefore, we never know why they work well somewhere and ineffectively elsewhere. In this paper, we investigate the power of the (seasonal) difference operator from the perspective of spectral analysis, linear system theory and digital filtering, and point out the characteristics and limitations of (seasonal) difference operator and S-ARIMA. Besides, the general operator that transforms a non-stationary and/or irrational-spectra wide-sense stationary stochastic process to be stationary and/or rational will be presented. In the end, we show the overall methodology, ARMA-SIN, for predicting a non-stationary and/or wide-sense stationary stochastic process. As a closing note, we will also present the nature, philosophy, effectiveness, insufficiency, and improvement of the canonical time series decomposition (like STL, X11) and time series smoothing (like exponential smoothing, Holt's, and moving average) methods, and demonstrate that they are special cases of the ARMA-SIN.
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