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 y_t (with the measurements x₁, x₂, x₃, ..., x_n) 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.