Autocorrelation and partial autocorrelation functions

Autocorrelation and partial autocorrelation are measures of association between current and past series values and indicate which past series values are most useful in predicting future values. With this knowledge, you can determine the order of processes in an ARIMA model. More specifically,

  • Autocorrelation function (ACF). At lag k, this is the correlation between series values that are k intervals apart.
  • Partial autocorrelation function (PACF). At lag k, this is the correlation between series values that are k intervals apart, accounting for the values of the intervals between.
Figure 1. ACF plot for a series
ACF plot for a series

The x axis of the ACF plot indicates the lag at which the autocorrelation is computed; the y axis indicates the value of the correlation (between −1 and 1). For example, a spike at lag 1 in an ACF plot indicates a strong correlation between each series value and the preceding value, a spike at lag 2 indicates a strong correlation between each value and the value occurring two points previously, and so on.

  • A positive correlation indicates that large current values correspond with large values at the specified lag; a negative correlation indicates that large current values correspond with small values at the specified lag.
  • The absolute value of a correlation is a measure of the strength of the association, with larger absolute values indicating stronger relationships.