Parameters and mathematical forms of Arima

What is an Arima model

The full name of the Arima model is called the autoregressive moving average model, which is the full name (Arima, autoregressive Integrated moving Average model). Also known as Arima (P,D,Q), it is one of the most common models of statistical models (statistic model) used to predict time series. 1. Advantages and disadvantages of Arima

Pros: The model is simple and requires only endogenous variables without the need for other exogenous variables.

Disadvantages:
1. The timing data is required to be stable (stationary), or it is stable after differential differentiation (differencing).
2. Essentially, only linear relationships can be captured, not nonlinear relationships.
Note that using the Arima model to predict time series data must be stable, and if the data is unstable, it cannot be caught by law. For example, the reason for the unpredictability of stock data with Arima is that the stock data is non-stable and often fluctuates by the influence of policies and news. 2. Judging is the time series data is a stable method.

Rigorous definition: A random variable of a time series is stable when and only if all its statistical characteristics are independent of time (is a constant about time).
The method of judging: Stable data is no trend (trend), no periodicity (seasonality); That is, its mean value, which has a constant amplitude on the timeline, and its variance, tends to be the same stable value on the timeline. You can use Dickey-fuller test for hypothesis testing. (another article introduction) 3. Parameters and mathematical forms of Arima

Arima models have three parameters: P,d,q. p--represents the time lag (lags) of the time series data used in the predictive model, also called the ar/auto-regressive item d--, which is stable, also called the integrated term, which requires several order difference differentiation. q--represents the lag number (lags) of the predictive error used in the predictive model, also called the ma/moving average term

First explain the difference: suppose y represents the difference of Y at the T moment.
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−2

The predictive model of Arima can be expressed as:

The predicted value of y = constant C and/or the weighted sum of one or more recent time Y and and/or the prediction error of one or more recent time.

Assuming P,q,d is known,
Arima is expressed in mathematical form as:
Ytˆ=μ+ϕ1∗yt−1+...+ϕp∗yt−p+θ1∗et−1+...+θq∗et−q
Wherein, Φ represents the coefficient of AR