Autocorrelation and autocorrelation time

**tomkeus** · Jun17-09, 11:02 AM

I have Metropolis-Markov algorithm and I need to determine integrated autocorrelation time. In order to do that i have to find autocorrelation and I don't quite get what to do.

For example, after equilibration I did N sweeps, took measurements at each one and I obtained N results

, for some observable.

Definition of autocorrelation says

$LaTeX Code: A_O (t)=\\frac{\\langle O_i O_{i+t}\\rangle-\\langle O_i\\rangle^2}{\\langle O_i^2\\rangle-\\langle O_i\\rangle^2}$

What are those averages over? Should I average over

?

**EnumaElish** · Jun17-09, 12:46 PM

Yes. The denominator is the variance of your data (sequence indexed by i). The numerator is the covariance between O(i) and O(i+t).

http://en.wikipedia.org/wiki/Autocorrelation#Statistics

**tomkeus** · Jun18-09, 06:47 AM

One more question. I have obtained N sweeps, so basically time goes from

to

. When I want to obtain autocorrelation for lag

i just have

$LaTeX Code: A_O(t=N-1)=\\frac{O_1 O_N-\\langle O\\rangle^2}{\\sigma_O^2}$

so I don't have any averaging for term

. Is that right or duration of my data should always be longer than maximum lag I'm calculating autocorrelation for?

**EnumaElish** · Jun18-09, 11:09 AM

If you have a single observation, the formula will still work in the arithmetical sense, but your results will not have a high level of confidence because you'd be making an inference about the population based on a single individual observation, O(1) x O(N). The longer the duration (relative to the lag), the higher will be the level of statistical confidence.