# Auto Correlation

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