In practice, due to the lack of sample data, it is basically impossible to find the real random process of generating samples from sample data. Theoretical research shows that any stationary time series can be approximately represented by ARMA process (including AR process, MA process and mixed process), and it can be accurately predicted by ARMA model. Box-Jenkins modeling method is about how to analyze stationary time series, establish ARMA model and make prediction, and it is also a popular modeling method at present. The modeling process can be basically divided into the following three steps.
1) model identification: investigate the characteristics of time series, carry out model identification, and identify valuable model subclasses with simple parameters, such as AR(3) and ARMA (2,2).
2) Parameter estimation and diagnostic test: data fitting and parameter estimation are carried out for the identified model subclasses. Under appropriate conditions, the model parameters are effectively inferred and estimated by using sample data, and the model is diagnosed and tested. By testing the relationship between the fitting model and the data, the inadequacies of the model are revealed, so as to improve the model. Model identification, parameter estimation and diagnostic testing are continuous cycles and improvement processes, through which appropriate model expressions can be found.
3) Prediction: Using the fitted time series model to infer other statistical properties of the series or predict the future development of the series, it is usually required that the number of observation values used for modeling should be at least 50, preferably 100 or more. When 50 or more historical observations cannot be obtained, for example, when forecasting the demand of new products, the initial model can be obtained by using experience or historical demand information of similar products; The model can be updated at any time when more data is obtained.