**Auto correlation** is a special case of cross correlation
with only one time series. The single series is cross correlated
with itself. Lag 0 has a correlation coefficient of 1, and any
other lags with high correlation coefficients suggest an
underlying periodicity in the data. Positive and negative lags
will be identical, because to the mathematics computing the
correlation coefficient the two cases look identical:

lag 2

```
ABCDEFGHIJKLMNOPQRSTUVWXYZ
ABCDEFGHIJKLMNOPQRSTUVWXYZ
```

lag -2

```
ABCDEFGHIJKLMNOPQRSTUVWXYZ
ABCDEFGHIJKLMNOPQRSTUVWXYZ
```

Series used for autocorrelation. It is not obvious if there is any periodicity to the data. |

Correlogram for the time series above. Note the symmetry about lag 0. |

Time series in the two graphs above,
plotted with a lag of 2000 which has a correlation coefficient of
0.35. The highest correlation coefficient away from 0 lag (which
will always has a correlation coefficient of 1) occurs at a lag
of 1991 where the value of r=0.35.
Note that the peaks and troughs generally line up, but the relationship is far from perfect although this is the best value beyond the lags around 0. |

*Last revised 2/2/2009*