Autoregressive Integrated Moving Average (differential autoregressive moving average model)

Arima:autoregressive Integrated Moving Average (differential autoregressive moving average model)

The I-Finger integrated (differential) inside the Arima.  ARIMA (P,D,Q) represents P-order ar,d differential, Q-Order MA. Why the difference? The premise of Arima is that the data is stationary, that is, the statistical characteristics (mean,variance,correlation, etc.) do not vary with the time window. The mathematical representation is the same as the Union distribution:

Of course, many times do not meet this requirement, such as the airline passenger data here. There are many ways to transform the original data to make it stationary:

(1) differential, i.e. integrated. For example, first order difference is the value of subtracting the previous item from each item in the original sequence. The second order difference is the first difference based on the difference. This is the most recommended practice.


(2) The original data is roughly fitted with some function, and the remaining quantity is processed by Arima. For example, the trend of airline passenger is first fitted with a straight line, so the original data becomes the offset of each data point from the line. Then use the Arima to fit these offsets.


(3) Log or open the root of the original data. This is very effective for variance that are not constants.


How to look at the data is not stationary it. Two very common quantities are used here: ACF (auto correlation function) and PACF (patial Auto correlation function). For non-stationary data, the ACF graph does not tend to be 0, or the trend 0 is slow. Here are three ACF graphs, corresponding to the original data, first-order differential raw data, to remove periodic first-order differential data: