# Why is ARMA preferred instead of AR?

ARMA basically is an AR model which considers past inputs to the filter as well. But what is the benefit of taking past inputs to the filter if we are considering real-time processes which are all stochastic and we don't know the input signal among the noise that is coming to the input of filter?

Auto-regressive (AR) and moving average (MA) models, or the combination of said models (ARMA) are linear models that work off of an assumption of a stationary input. Under this assumption, they can be used to predict a future occurrence based on previous observations if suitably defined (e.g., of a sufficient order). Note that other models exist such as the Auto-regressive Integrated Moving Average (ARIMA) which performs an initial operation to make the input stationary before operating like a typical ARMA model. A disadvantage of auto-regressive models is that they are unable to apply information about correlated noise structures of those previous observations to the prediction. By incorporating that structure into the model, one can improve the performance of the model's predictability.

An AR model takes the input $$x[n]$$ and generates the output $$y[n]$$ using only the present input and past computed outputs: $$y[n] = \sum_{k=1}^{K} a_k y[n-k] + b_0 x[n]$$

An ARMA model allows the present output $$y[n]$$ to also depend on past inputs: $$y[n] = \sum_{k=1}^{K} a_k y[n-k] + \sum_{m=1}^M b_m x[n-m] + b_0 x[n]$$

what is the benefit of taking past inputs to the filter if we are considering real-time processes which are all stochastic and we don't know the input signal among the noise that is coming to the input of filter?

If we don't know the input signal, $$x[n]$$, then surely neither representation is any good? Both are defined in terms of it.

From a purely theoretical viewpoint, there is no particular reason to use an ARMA over an AR, because we can choose the order of the AR filter to be as high as we want in order to approximate the signal we are seeing.

However, from a practical viewpoint, high order filters cause numerical computation problems. What the ARMA approach allows is, if the signal $$y[n]$$ has spectral zeros (the frequency response is close to zero), then the ARMA model makes it easy (in terms of numbers of coefficients) to model this.

AR models work only with "poles".

MA models work only with "zeros".

ARMA models use both poles and zeros.

So, the benefit is that we can use much lower order (values of $$K$$ and $$M$$) models to accurately model processes using ARMA models than using AR alone.

The downside is that ARMA model estimation is a much harder problem than AR model estimation.