# [gmx-users] What is the autocorrelation time

Christopher Neale chris.neale at mail.utoronto.ca
Sat May 19 03:38:55 CEST 2012

```Thank you very much Patrick. This is exactly what I was looking for.  Based on what you sent, I'm going to go with the following text in my work:

The value of  tau_int is approximately half of the amount of time required, on average, to obtain a statistically independent (decorrelated) sample [ref Janke].

I really appreciate it!

Chris.

-- original message --

Hi Chris,I understand your question, this autocorrelation time puzzled me for a
long time as well. Not far from the interpretation you give, Scott
Feller defines it (http://dx.doi.org/10.1007/978-1-59745-519-0_7) as the
time a given observable takes to lose the memory of its previous state,
or in other words the time it takes to relax (that's why it's sometimes
called relaxation time). He also discusses it as a tool to choose the
block size for calculating an error estimate of an observable (one
single simulation can be used as independant samples if each block size
is >> autocorrelation time).
We also had a nice discussion some years ago on the mailing list on free
energy calculation and error estimate:
http://lists.gromacs.org/pipermail/gmx-users/2007-May/027281.html. John
Chodera pointed me to a useful article from Wolfhard Janke (the link in
the discussion is broken, here's the new one:
http://www2.fz-juelich.de/nic-series/volume10/janke2.pdf). There you'll
find a rigorous mathematical definition of autocorrelation time. Quoting
this paper "This shows more clearly that only every 2
tau_int
iterations the measurements are approximately uncorrelated and gives a
better idea of the relevant effective size of the statistical sample"
(tau_int is the integrated autocorrelation time; as you said the
autocorrelation function is usually a single exponential, but sometimes
it's more complex and one needs to evaluate it by integration of the
autocorrelation function).
After all these considerations, the autocorrelation time can be seen as
a tool to assess the time that is needed to have a good estimate of an
observable: the simulation must be many many times longer than the
autocorrelation time. And sometimes it's directly related to
experimental observables (i.e. NMR relaxation experiments).
Hope it's useful,

Patrick

Le 16/05/2012 23:39, Christopher Neale a écrit :
> Thank you Stephane.
>
> Unfortunately, neither of those links contains the information that I am
> seeking. Those links contain some example plots of autocorrelation
> functions including a discussion of time-spans over which the example
> time-series is autocorrelated and when it is not, but neither link
> defines the (exponential or integral) autocorrelation time except to
> show a plot and indicate when it is non-zero and when it fluctuates
>
> For example, I already know that the autocorrelation time describes the
> exponential decay of the correlation and that two values drawn from the
> same simulation are statistically independent if they are separated by a
> sufficient number of (accurate) autocorrelation times, but this
> information is not exactly a definition of the autocorrelation time.
>
> I am hoping to find a definition of the autocorrelation time in terms of
> the probability of drawing uncorrelated samples, although any complete
> definition will do.
>
> If anybody else has the time, I would appreciate it.
>
> Thank you,
> Chris.
>
> -- original message --
>
> Probably these links give you simple and clear response for your
> question
> http://idlastro.gsfc.nasa.gov/idl_html_help/Time-Series_Analysis.html
> and http://www.statsoft.com/textbook/time-series-analysis/ HTH Stephane
>
>

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