[gmx-users] g_energy averages

Berk Hess gmx3 at hotmail.com
Thu Jul 18 11:48:21 CEST 2002

>To get an accurate estimate of the standard error you will have to take the 
>autocorrelation time into account. Load the curve into xmgrace, calculate 
>the autocorrelation of the fluctuations, and fit e.g. a single exponential 
>A*exp(-t/tau) to it - 'tau' is your correlation time.
>Assuming the total length of your simulation is T, we can then estimate the 
>number of independent observations as T/tau, so the standard error
>    <rmsd fluctiations>
>           T/tau
>In the limit where the observations are completely uncorrelated the
>correlation time is the stepsize, so in that case the denominator would
>be the total number of steps.
>The alternative is to split the trajectory into parts that you *know* are 
>longer than the correlation time, calculate an average from each of
>these, and then the standard error as the fluctuations of these averages
>divided by the number of parts. This will only be an upper estimate of
>the standard error though, and you will still need an estimate of the
>correlation time.

Although tau is your correlation time, blocks of length tau are not 
at all. The correlation over length tau is still 1/e, which is quite large.
(and there is a square root missing in the denominator).
The exact formula is:

  standard deviation
    sqrt(T/(2 tau))

For error estimation I programmed the -ee option into g_analyze. For almost 
quantity you don't loose much accuracy when using the direct values in the 
file instead of the partial sums, since the correlation times are on the 
order of
or longer than the simulation time (unless you have written only 100 times 
the energy file). g_analyze -ee has the advantage that it estimates the 
error from
integral properties and uses the sum of two exponentials, so also a very 
fluctuation or stablization time of the system is taken into account, this 
can have a huge effect on the error. See ref.:
B. Hess
Determining the shear viscosity of model liquids from molecular dynamics
J. Chem. Phys. 116 209-217 (2002)

PS use the RMSD value from g_energy, the fluctuation value is the rmsd after
   subtracting a fitted straight line.


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