Is autocorrelation function a copy of original function?

Answer 1

No.

You probably know what a sample correlation is. This statistic is often used to measure how well a linear function of one variable predicts the value of another variable. The statistic can assume any value from -1 to 1, and the extreme values show the strongest (linear) relationship.

Calculating the autocorrelation function for a time series involves doing a series of calculations that are the same as those done to obtain a sample correlation coefficient. Since these values must always be between -1 and 1 they cannot in general form a copy of the original function.

Here is where the idea of copying appears. Suppose you want to calculate the 1st autocorrelation coefficient from the series v0, v1, v2, v3, v4, ... .

Then calculate the sample correlation for the pairs (v1, v0), (v2, v1), (v3, v2), (v4, v3), ... Notice that it is as if you were to write down the original time series on one line and then copy it on a second line shifting it one item to the right so that the pairs needed to compute the sample correlation could be read from the columns of the two lines.

The 2nd autocorrelation would be computed as if by copying the second line shifting it two places to the right and so on.