Re: Quantitative Test for Equilibration of Data Time Series

From: Dave Rogers (
Date: Wed Oct 15 2008 - 13:27:23 CDT

  In stating that the system has reached equilibrium, there is a
fundamental assumption that the system dynamics does not have a
correlation time longer than the actual time observed (i.e. a
persistent long-time correlation). A first approach, as you suggest
is to "look" at the computed autocorrelation function for a quantity
of interest and make sure it approaches zero relatively rapidly. To
make this more quantitative you could fit to an exponential to get a
correlation time to compare to total sim. time.
  Allen and Tildesley (Computer Simulation of Liquids) have a
discussion on using block averages of different sizes to estimate a
"statistical inefficiency" on pp.191-194 -- which is another way of
getting at the correlation time. Jaynes has a more abstract
discussion of time averages and ensemble averages on pp. 64-68 of
"Where Do We Stand on Maximum Entropy?"
(, no 36.) and in his book
Probability Theory, The Logic of Science, presents a Markov-chain
model for estimating correlations in dynamic processes which may be
worth further investigation.
  There are also a host of advocates for re-doing everything several
times from random starting points in the literature. I am interested
in seeing what other answers come up for this question.

~ David Rogers
Dr. Thomas Beck Lab
University of Cincinnati

On Wed, Oct 15, 2008 at 1:06 PM, Joshua Adelman <> wrote:
> Does anyone know of a quantitative test or analysis that can be applied to a
> data time series to test to see when the data has equilibrated? I have a
> large number of simulations that seem to be equilibrating on slightly
> different time scales and am trying to find an automated way of figuring out
> how much of the initial sampling needs to be discarded in order to calculate
> the average value of the time series. Most of the suggestions on this list
> seem to imply visual inspection of the data, which is sub-optimal for this
> application.
> Any suggestions would be appreciated.
> Josh

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