3
$\begingroup$

Given a set of data from a generic Montecarlo simulation $x_i$, $(1=1,...,N)$, autocorrelation is expected to happen between data points within relaxation time $\tau$ (correlation time) distance between each other.

Now, I know that a possibile approach to reduce/avoid correlation is to set up bins much greater than the relaxation time and compute an average from each bin as well as an error.

What if I just consider an initial $x_i$ and then hop to the next one $x_{i+\tau}$ and so on and so forth for $x_{i+2\tau}$ ,$x_{i+3\tau}$ ...essentially considering these new samples as uncorrelated? In this case I'm not computing averages and errors, I'm just skipping enough points to make the remaining ones sort of uncorrelated.

I've read about both approaches in literature, but I'm actually not sure if they are both viable.

$\endgroup$
6
$\begingroup$

Both the approaches you describe are kind-of viable, and should give similar results. Obviously, in the case of just sampling data points at intervals of $\tau$, you are throwing away the intermediate data points; but you may reasonably expect that these do not contain significantly more information, since they are highly correlated with the data points that you do sample, given that $\tau$ is of the order of the correlation time. But I would think that there is nothing to lose by adopting the binning method.

I say "kind-of" because both methods presuppose that you know the correlation time before you start. It is (in my opinion) far more important to choose a data analysis method that essentially determines the correlation time as part of the error estimation. The classical paper on this is Flyvbjerg and Petersen J Chem Phys, 91, 461 (1989), and this uses the binning approach. In brief, you start with a variance, calculated from every individual data point. Then you average each successive pair of data points, giving you half as many data points, each representing a bin of length 2, and you calculate the variance of those data points. The process continues recursively, with bins of length 4, 8, 16 etc., and can be programmed fairly economically. Their analysis, based on renormalization ideas, shows how the variances may be used in a formula that converges towards the best estimate of the error in the mean (provided the simulation run is long enough). The method is described in most textbooks on simulation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.