8.6 ARIMA Overview

Arima performs well when working with a time series where the the data is directly related to the time stamp: clear growth and seasonality based on time

ARIMA is a generalization of a ARMA model

It is fitted to time series data

ARIMA models are applied in some cases where data show evidence of non-stationarity, where an initial differencing step (corresponding to the “integrated” part of the model) can be applied one or more times to eliminate non-stationarity.

Components of ARIMA

AutoRegression: AR($p$)

It is a regression model that utilizes the dependent relationship between a current observation and observations over a preivous period (see definition)

Integrated: I($d$)

It is the differencing of observations in order to make the time series stationary. This means how many times did we have difference the data to get it stationary so that the AR and MA components could work.

It indicates that the data values have been replaced with the difference between their values and previous values.

Moving Average: MA($q$)

It is a model that uses the dependency between an observation and a residual error from a moving average model applied to lagged observations (see definition)

Stationarity

The key assumption to use ARIMA is stationarity (see stationary). A stationary data set will allow our model to predict that the mean and variance will be the same in future periods. We can use stationary test to check if the time series is stationary.

If the data is not stationary, we need to transform it to be stationary ir evaluate it and what type of ARIMA terms we’ll use.

For seasonal data, we can difference by a season. For instance, if we have monthly data with yearly seasonality, we could difference by a time unit of 12, instead of just 1.

Choosing $p$, $d$ and $q$

1. Using AutoCorrelation Plots and Partial AutoCorrelation Plots

This method is hard, the parameters $p$, $d$ and $q$ are chosen based on viewing the decay in the plot.

2. Grid Search

It takes more time but works best. Run ARIMA based on different combinations of $p$, $d$ and $q$ and compare the models based on some evaluation metric.