These are all classical models of forecasting based on time series data, with slight differences. This post focuses on the conceptual differences to gain an intuition and overview.

## Autoregressive model

Autoregression refers to a regression model based on itself (“auto”). Basically, it takes p number of time steps before the one to be forecast and fits a regression model based on that. Or, in other words, predict the current value based on p previous values. Time steps before the one to be forecast are referred to as lags, so one could also say, predict the current value based on p lags. And the number of lags is called the order of the model.

$X_{t}=c+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}\,$
• $X_{t}$: value to be forecast at time $t$.
• $c$: some constant value
• $\varepsilon_{t}$: error value at time $t$
• $p$ number of lags, or order of the model
• $\varphi_{i}$: model parameter at index $i$

## Moving-average regression model

The moving-average regression model takes a slightly different approach. It takes the average of the time series, and then predicts the current value based on the error term of n previous time steps. Similar to above, the n is referred to as the order.

How does this compare to the autoregressive approach above?

Pros:

• The error terms of the lags are modeled to affect the forecasted value directly, rather than indirectly.
• The error influence from lags is limited to the order, and not infinitely like the autoregressive approach.

Cons:

• Fitting the model can be more complicated, as the error terms are not directly observable (they are included within the lag total values).
$X_{t}=\mu +\varepsilon _{t}+\theta _{1}\varepsilon _{t-1}+\cdots +\theta _{q}\varepsilon _{t-q}\,$
• $X_{t}$: value to be forecast at time $t$.
• $\mu$: the average of the series
• $\varepsilon_{t}$: error value at time $t$
• $q$: the number of lags to use, or order of the model
• $\theta_{t}$: the parameter of the model at time t

This is alternatively written with the summmation operator like so:

$X_{t}=\mu +\varepsilon _{t}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.\,$

Note that this model name sounds similar to the moving average concept from statistics, but is conceptually different.

## Autoregressive moving average model (ARMA)

After reviewing the above, ARMA becomes much easier to understand, as it’s simply a combination of the above two.

$X_{t}=c+\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.\,$

ARMA forecasts the targeted value as a combination of both p previous lag values and q previous error values.

## Autoregressive integrated moving average (ARIMA)

The ARIMA approach extends ARMA by using difference values as opposed to the lag values themselves. This is the integrated part.

What is a difference value? A difference value for time step t would be the differnce between it and the previous value at time step t-1. Or, formulaically:

$X_{t} - X_{t-1}$

Using the differences as opposed to the values themselves is an effort to eliminate non-stationarity, or differences in the data when shifted in time. Difference values may be calculated repeatedly on a series, also giving them an order.
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