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Autocorrelation and autocorrelation time

  1. Jun 17, 2009 #1
    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]?
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  3. Jun 17, 2009 #2


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  4. Jun 18, 2009 #3
    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?
  5. Jun 18, 2009 #4


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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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