The basic idea of ARIMA model is to treat the data series formed by the predicted object over time as random series, and then use autoregressive plus moving average to fit it after differential integration, and then predict the past and future values of this time series.
The ARIMA model is generally expressed as ARIMA(p, d, q), and its mathematical expression is
φp(B)( 1-B)dyt=θq(B)εt，(7-9)
Where: φ p (b) =1-φ1b-…-φ PBP, θ q (b) =1-θ1b-…-θ qbq;
AR is autoregressive, P is autoregressive term, MA is moving average, Q is moving average term, and D is difference; Yt is a time series, b is a backward shift operator, φ 1, …, φp is an autoregressive coefficient, θ 1, …, θq is a moving regression coefficient, and {ε t} is a white noise sequence.
2.2 Basic procedures. ARIMA model prediction
(1) stationary identification
Autocorrelation function and partial autocorrelation function diagram are used to judge whether the sequence is stable or not.
(2) Smooth nonstationary sequence.
There is a trend of increasing or decreasing, and differential processing is needed until the autocorrelation function value and partial correlation function value of the processed data are significantly equal to zero.
(3) Establish the corresponding model according to the identification rules of time series model.
According to the autocorrelation and partial correlation function diagram of the sequence, the type of model and the order of P and Q are determined.
On autocorrelation and partial correlation function graphs, the function is zero after a certain step, which is called truncation; It can't be zero after a certain step, but it appears in the form of exponential decay or positive and negative decline, which is called tailing.
According to whether the autocorrelation function and partial correlation function are truncated or tailed and their periods, the model can be judged. See Table 7-8 for standards.
Table 7-8 Criteria for Distinguishing Model Parameters by ACF-PACF Diagram
(4) Hypothesis test to diagnose whether the residual sequence is white noise.
Use χ2 test to detect the white noise residual of the estimation model, and the residual should be a random sequence, otherwise, carry out residual analysis and re-determine the model if necessary.
(5) Forecast and analysis
The predicted value of x(t) after t+ 1 period, that is, 1 period, is obtained by using the tested model. Let this forecast value be x(t+ 1), and call it the forecast value of future 1 period.