Autocorrelation and autocorrelation time

  • Thread starter tomkeus
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  • #1
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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 [tex]O_i,i=1...N[/tex], for some observable.

Definition of autocorrelation says

[tex]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}[/tex]

What are those averages over? Should I average over [tex]i[/tex]?
 

Answers and Replies

  • #3
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One more question. I have obtained N sweeps, so basically time goes from [tex]i=0[/tex] to [tex]i=N[/tex]. When I want to obtain autocorrelation for lag [tex]t=N-1[/tex] i just have

[tex]A_O(t=N-1)=\frac{O_1 O_N-\langle O\rangle^2}{\sigma_O^2}[/tex]

so I don't have any averaging for term [tex]O_1 O_N[/tex]. Is that right or duration of my data should always be longer than maximum lag I'm calculating autocorrelation for?
 
  • #4
EnumaElish
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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.
 

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