which the series is observed. Specifically, for a wide-sense stationary time series, the mean and the variance/autocovariance keep constant over time. Differencing in statistics is a transformation applied to a non-stationary time-series in order to make it stationary in the mean sense (viz., to remove the non-constant trend), but having nothing to do with the non-stationarity of the variance or autocovariance. Likewise, the seasonal differencing is applied to a seasonal time-series to remove the seasonal component. From the perspective of signal processing, especially the Fourier spectral analysis theory, the trend is the low-frequency part in the spectrum of a non-stationary time series, while the season is the periodic-frequency part in the spectrum of it. Therefore, the differencing works as a high-pass (i.e., low-stop) filter and the seasonal-differencing as a comb filter to suppress the low-frequency trend and the periodic-frequency season in the spectrum domain (rather than directly in the time domain), respectively. To difference the data, the difference between consecutive observations is computed. Mathematically, this is shown as Differencing removes the changes in the level of a time series, eliminating trend and seasonality and consequently stabilizing the mean of the time series. Sometimes it may be necessary to difference the data a second time to obtain a stationary time series, which is referred to as second-order differencing: Another method of differencing data is seasonal differencing, which involves computing the difference between an observation and the corresponding observation in the previous season e.g a year. This is shown as: The differenced data are then used for the estimation of an ARMA model. Examples Some well-known special cases arise naturally or are mathematically equivalent to other popular forecasting models. For example: An ARIMA(0, 1, 0) model (or model) is given by — which is simply a random walk. An ARIMA(0, 1, 0) with a constant, given by — which is a random walk with drift. An ARIMA(0, 0, 0) model is a white noise model. An ARIMA(0, 1, 2) model is a Damped Holt's model. An ARIMA(0, 1, 1) model without constant is a basic exponential smoothing model. An ARIMA(0, 2, 2) model is given by — which is equivalent to Holt's linear method with additive errors, or double exponential smoothing. Choosing the order The order p and q can be determined using the sample autocorrelation function (ACF), partial autocorrelation function (PACF), and/or extended autocorrelation function (EACF) method. Other alternative methods include