Autocorrelation and autocorrelation time

In summary, to determine the integrated autocorrelation time using the Metropolis-Markov algorithm, you need to find the autocorrelation by taking measurements at each sweep after equilibration. The autocorrelation is calculated using the formula A_O(t) = (O_i O_{i+t} - \langle O_i\rangle^2) / (\langle O_i^2\rangle - \langle O_i\rangle^2), where the averages are taken over the data sequence indexed by i. The longer the duration of your data relative to the lag, the higher the level of statistical confidence in your results.
  • #1
tomkeus
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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]?
 
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  • #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?
 
  • #4
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.
 

1. What is autocorrelation?

Autocorrelation is a statistical measure that quantifies the degree of similarity between a signal and a delayed version of itself. It is commonly used in time series analysis to identify patterns and trends in data.

2. Why is autocorrelation important?

Autocorrelation helps us understand the relationship between a variable and its past values. It can indicate whether a variable follows a predictable pattern or if it is random. This information is useful in forecasting future values and making predictions.

3. How is autocorrelation calculated?

Autocorrelation is calculated by taking the correlation coefficient between a signal and a lagged version of itself. This can be done using statistical software or manually by calculating the covariance and variances of the two variables.

4. What is autocorrelation time?

Autocorrelation time is the amount of time it takes for a signal to become uncorrelated with its past values. It is also known as the lag time or memory time. A shorter autocorrelation time indicates a faster decay of correlation between a signal and its past values.

5. How is autocorrelation time used in data analysis?

Autocorrelation time is used to estimate the number of data points needed in a time series to capture the underlying patterns and trends. It is also used to identify the appropriate time window for analysis and to determine the significance of any observed autocorrelation.

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