ARMA/ARIMA MODEL
Anuj Singh
Student, Data Science
NMIMS, Indore
ARMA and ARIMA models are used to forecast the observation at (t+1) based on the historical data of previous time spots recorded for the same observation. However, it is necessary to make sure that the time series is stationary over the historical data of observation overtime period. If the time series is not stationary then we could apply the differencing factor on the records and see if the graph of the time series is a stationary overtime period.
ACF (Auto Correlation Function)
Auto Correlation function takes into consideration of all the past observations irrespective of its effect on the future or present time period. It calculates the correlation between the t and (t-k) time period. It includes all the lags or intervals between t and (t-k) time periods. Correlation is always calculated using the Pearson Correlation formula.
PACF (Partial Correlation Function)
The PACF determines the partial correlation between time period t and t-k. It doesn’t take into consideration all the time lags between t and t-k. For e.g. let’s assume that today’s stock price may be dependent on 3 days prior stock price but it might not take into consideration yesterday’s stock price closure. Hence we consider only the time lags having a direct impact on future time period by neglecting the insignificant time lags in between the two-time slots t and t-k.
ARMA (Auto Regressive Moving Average) Model
This is a model that is combined from the AR and MA models. In this model, the impact of previous lags along with the residuals is considered for forecasting the future values of the time series. Here β represents the coefficients of the AR model and α represents the coefficients of the MA model.
Yt = β₁* yₜ-₁ + α₁* Ɛₜ-₁ + β₂* yₜ-₂ + α₂ * Ɛₜ-₂ + β₃ * yₜ-₃ + α₃ * Ɛₜ-₃ +………… + βₖ * yₜ-ₖ + αₖ * Ɛₜ-ₖ
Consider the above graphs where the MA and AR values are plotted with their respective significant values. Let’s assume that we consider only 1 significant value from the AR model and likewise 1 significant value from the MA model. So the ARMA model will be obtained from the combined values of the other two models will be of the order of ARMA(1,1).
ARIMA (Auto-Regressive Integrated Moving Average) Model
We know that in order to apply the various models we must in the beginning convert the series into Stationary Time Series. In order to achieve the same, we apply the differencing or Integrated method where we subtract the t-1 value from t values of time series. After applying the first differencing if we are still unable to get the Stationary time series then we again apply the second-order differencing.
The ARIMA model is quite similar to the ARMA model other than the fact that it includes one more factor known as Integrated (I) i.e. differencing which stands for I in the ARIMA model. So in short ARIMA model is a combination of a number of differences already applied on the model in order to make it stationary, the number of previous lags along with residuals errors in order to forecast future values.
Also, the graph was initially non-stationary and we had to perform differencing operation once in order to convert into a stationary set. Hence the ARIMA model which will be obtained from the combined values of the other two models AR and MA along with the Integral operator can be displayed as ARIMA(1,1,1).