ARMA vs. AR and then what?

Question

Sorry if this sounds elementary but I am struggling to grasp the physical idea behind ARMA (auto-regressive, moving average) process. The "AR" part is intuitive and so is "MA", but put together?

If I model my time series using "AR", I can predict the next sample using linear terms of the previous samples and the difference between the predicted and the real one should then have a normal distribution provided the noise was guassian white (the unpredictable). But what does it mean to say that my time series can be ARMA modelled?

thanks

Answer 1

The arma approach is to model the current output of the system as the sum of past outputs and past inputs explicitly. The assumption of gaussian model for the noise statistics still can be used for the unpredictable signal which cannot be modeled as arma.

From system's frequency response spectrum modeling aspect: an ar model is able to model only the spectrum peaks as it fits the peaks better (using poles) and the ma model is able to model the valleys better (using zeros) in the spectrum. Hence, by using an arma model we will be better enable both peaks and valleys using a reasonable order for both ar and ma.

However, it is always possible to model a zero (in z-domain) using a infinite number of poles so an approximation to an arma model (whose parameter estimation is relatively involved) is to use a high order ar model (whose paramter estimation is relatively easy).