ARIMA (p,d,q) forecasting equation: Auto-RegressiveIntegrated Moving Average models are efficient class of models used for forecasting timeseries data which can be transformed to be “stationary” by differencing. Arandom variable is said to be stationary if the statistical properties of thatvariable remains constant over the whole time series. A series that isstationary in nature has a negligible trend and the fluctuations around itsmean have similar constant amplitude and it squirm in a regular patternsi.
e. its short term random patterns of time always look similar. The stationaryseries Autocorrelation factor also remain constant over that particular periodof time and the power spectrum remain constant over the entire series. A randomvariable composed of Autocorrelation factor and power spectrum series andcontains both signal and noise, the signal pattern could be fast or slowreversion or a sinusoid oscillation or it could be a rapid alteration in signcontaining a seasonal component. ARIMA model is considered as “filter” thatsegregates the signal from noise to gain valuable information for the futuremovements.ARIMA forecastingmathematical equation for stationary time series is linear like regressionequation in that the predictors/variables consists of lags of dependentvariables and the lags of forecast errors. The mathematical formulation ofmodel is,Forecasted value of Y = constant/weighted sum of single or morevalues of Y and/ weighted sum of single or more values of errors.
Non-Seasonal ARIMA modelis defined as ARIMA (p,q,d) where, p is the number of autoregressive terms, q is the number of lagged forecast errors in the prediction equation d is the number of non-seasonal differences needed for stationary, and The forecasting equationis constructed as follows. First, let y denotethe dth difference of Y, which means:If d=0: yt = YtIf d=1: yt = Yt – Yt-1If d=2: yt = (Yt – Yt-1) – (Yt-1 – Yt-2) = Yt -2Yt-1 + Yt-2Note that the seconddifference of Y (the d=2 case) is not the difference from two periodsago. Rather, it is the first-difference-of-the-firstdifference, which is the discrete analog of a second derivative, i.e., thelocal acceleration of the series rather than its local trend.In order to identify theappropriate ARIMA model for Y, the first step involves the determining of thedifferencing order (d) needed to stationary the series and also to remove thegross features of seasonality in conjunction with a variance-stabilizingtransformation such as logging or deflating the stationary series also maycontains auto correlated errors representing some number of AR terms (p>1)and some number MA terms (q>1) would also be need to formulate theforecasting equation.