Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Autocorrelation Function

The autocorrelation is calculated in the same way as the correlation between two variables (just using the same variable twice). If we consider time signals, the values of the autocorrelation may be calculated by shifting one of the two formal variables by a certain amount  Δt. If we plot the results of these calculations against the time shift, we obtain the autocorrelation function, or autocorrelogram. Here's an  interactive example  which shows the principal function of an autocorrelogram.

The idea is that a time-series yt (with the measurements x1, x2, x3, ..., xn) is correlated with itself. In the first step, the pairs to be correlated are:
First Series x1 x2 x3 ...  xn
Second Series x1 x2 x3 ...  xn

The correlation coefficient is obviously 1. In the next step, the second series is shifted one time-step to the right:
First Series x1 x2 ...  xn-1
Second Series x2 x3 ...  xn
and so on.

Here's another  interactive example  which shows the difference between an uncorrelated signal and a signal exhibiting strong autocorrelation.

Last Update: 2012-10-08