Can autocorrelation be greater than one?

[Rib5]

I have to find the autocorrelation of a random variable. When I compute the theoretical autocorrelation I get the result where it is [1 -2 3 -2 1] centered around zero, and zero everywhere else.

I tried estimating the autocorrelation of the random variable using

ryy_est = xcorr(Y,20, 'unbiased');

in Octave, and I get the exactly same distribution. Can anyone explain to me what is going on? I was under the impression that a correlation can't be greater than 1. I've though about just dividing everything by 3, since it is equal to 3 at m = 0, however I don't know if this would be right, and I still don't know why I am getting values like that.

The random variable is a gaussian random variable of zero mean and unit variance, put through a filter y[n] = x[n] - x[n-1] + x[n -2].

 

[jrlaguna]

Hm... can't help if you don't tell me how you're computing the autocorrelation...

 

[morphemera]

Usually, the autocorrelation function is defined to be normalized so that its value should be in the -1 and 1. The things that you should check is the exact definition of the function xcorr. Don't blindly guess the meaning of a function, cos they usually do not use the standard definition with some reason such as efficient.

 

[thrillhouse86]

I think the problem is that the autocorrelation function in most definitions is normalised by dividing through by the variance (or the product of the standard deviations at the times of interest if its not a covariant stationary process).

The function xcorr doesn't normalize the resulting autocorrelation function by this variance. If you want it to do that you need to go:
ryy_est = xcorr(Y,20, 'unbiased','coeff')

Can someone verify that this is true ?

Regards,
Thrillhouse

 

[DeeNos]

Autocorrelation is a statistical measure that quantifies the similarity between a signal or time series and a lagged version of itself. It is a measure of how correlated a signal is with itself at different points in time. Autocorrelation values can range from -1 to 1, with a value of 1 indicating a perfect positive correlation, 0 indicating no correlation, and -1 indicating a perfect negative correlation.

In theory, autocorrelation values cannot exceed 1. However, in practice, due to the limitations of data and measurement errors, it is possible to obtain values slightly higher than 1. This is especially true when the data is noisy or when the autocorrelation is estimated from a finite sample size.

In your case, it is possible that the autocorrelation values you are obtaining are slightly higher than 1 due to the noise in the data or the finite sample size. However, it is important to note that even small deviations from 1 can have significant effects on the results and should be carefully considered.

In order to ensure accurate results, it is recommended to use a larger sample size and to carefully evaluate the data for any potential sources of noise. Additionally, dividing the values by 3 to bring them closer to 1 may not be appropriate as it could distort the results. It is important to consult with a statistician or an expert in the field to determine the best course of action in this situation.

Overall, while it is possible to obtain autocorrelation values slightly higher than 1, it is important to carefully evaluate the data and consider potential sources of error to ensure accurate and meaningful results.