Formulae for averaging
Note: this section was taken from ref 179.
When analyzing a MD trajectory averages \(\left<x\right>\) and
fluctuations
(1)\[\left<(\Delta x)^2\right>^{{\frac{1}{2}}} ~=~ \left<[x-\left<x\right>]^2\right>^{{\frac{1}{2}}}\]
of a quantity \(x\) are to be computed. The variance
\(\sigma_x\) of a series of N\(_x\) values, {x:math:_i}, can
be computed from
(2)\[\sigma_x~=~ \sum_{i=1}^{N_x} x_i^2 ~-~ \frac{1}{N_x}\left(\sum_{i=1}^{N_x}x_i\right)^2\]
Unfortunately this formula is numerically not very accurate, especially
when \(\sigma_x^{{\frac{1}{2}}}\) is small compared to the values of
\(x_i\). The following (equivalent) expression is numerically more
accurate
\[ \begin{align}\begin{aligned}\sigma_x ~=~ \sum_{i=1}^{N_x} [x_i - \left<x\right>]^2\\with\end{aligned}\end{align} \]
(3)\[\left<x\right> ~=~ \frac{1}{N_x} \sum_{i=1}^{N_x} x_i\]
Using (2) and
(3) one has to go through the series of
\(x_i\) values twice, once to determine \(\left<x\right>\) and
again to compute \(\sigma_x\), whereas
(1) requires only one sequential scan of
the series {x:math:_i}. However, one may cast
(2) in another form, containing partial
sums, which allows for a sequential update algorithm. Define the partial
sum
\[X_{n,m} ~=~ \sum_{i=n}^{m} x_i\]
and the partial variance
(4)\[\sigma_{n,m} ~=~ \sum_{i=n}^{m} \left[x_i - \frac{X_{n,m}}{m-n+1}\right]^2\]
It can be shown that
(5)\[X_{n,m+k} ~=~ X_{n,m} + X_{m+1,m+k}\]
and
(6)\[\begin{split}\begin{aligned}
\sigma_{n,m+k} &=& \sigma_{n,m} + \sigma_{m+1,m+k} + \left[~\frac {X_{n,m}}{m-n+1} - \frac{X_{n,m+k}}{m+k-n+1}~\right]^2~* \nonumber\\
&& ~\frac{(m-n+1)(m+k-n+1)}{k}
\end{aligned}\end{split}\]
For \(n=1\) one finds
(7)\[\sigma_{1,m+k} ~=~ \sigma_{1,m} + \sigma_{m+1,m+k}~+~
\left[~\frac{X_{1,m}}{m} - \frac{X_{1,m+k}}{m+k}~\right]^2~ \frac{m(m+k)}{k}\]
and for \(n=1\) and \(k=1\)
((6)) becomes
(8)\[\begin{split}\begin{aligned}
\sigma_{1,m+1} &=& \sigma_{1,m} +
\left[\frac{X_{1,m}}{m} - \frac{X_{1,m+1}}{m+1}\right]^2 m(m+1)\\
&=& \sigma_{1,m} +
\frac {[~X_{1,m} - m x_{m+1}~]^2}{m(m+1)}
\end{aligned}\end{split}\]
where we have used the relation
(9)\[X_{1,m+1} ~=~ X_{1,m} + x_{m+1}\]
Using formulae ((8)) and
((9)) the average
\[\left<x\right> ~=~ \frac{X_{1,N_x}}{N_x}\]
and the fluctuation
\[\left<(\Delta x)^2\right>^{{\frac{1}{2}}} = \left[\frac {\sigma_{1,N_x}}{N_x}\right]^{{\frac{1}{2}}}\]
can be obtained by one sweep through the data.
Implementation
In GROMACS the instantaneous energies \(E(m)\) are stored in the
energy file, along with the values of \(\sigma_{1,m}\) and
\(X_{1,m}\). Although the steps are counted from 0, for the energy
and fluctuations steps are counted from 1. This means that the equations
presented here are the ones that are implemented. We give somewhat
lengthy derivations in this section to simplify checking of code and
equations later on.
Part of a Simulation
It is not uncommon to perform a simulation where the first part, e.g.
100 ps, is taken as equilibration. However, the averages and
fluctuations as printed in the log file are computed over the whole
simulation. The equilibration time, which is now part of the simulation,
may in such a case invalidate the averages and fluctuations, because
these numbers are now dominated by the initial drift towards
equilibrium.
Using (5) and
(6) the average and standard deviation
over part of the trajectory can be computed as:
\[\begin{split}\begin{aligned}
X_{m+1,m+k} &=& X_{1,m+k} - X_{1,m} \\
\sigma_{m+1,m+k} &=& \sigma_{1,m+k}-\sigma_{1,m} - \left[~\frac{X_{1,m}}{m} - \frac{X_{1,m+k}}{m+k}~\right]^{2}~ \frac{m(m+k)}{k}\end{aligned}\end{split}\]
or, more generally (with \(p \geq 1\) and \(q \geq p\)):
\[\begin{split}\begin{aligned}
X_{p,q} &=& X_{1,q} - X_{1,p-1} \\
\sigma_{p,q} &=& \sigma_{1,q}-\sigma_{1,p-1} - \left[~\frac{X_{1,p-1}}{p-1} - \frac{X_{1,q}}{q}~\right]^{2}~ \frac{(p-1)q}{q-p+1}\end{aligned}\end{split}\]
Note that implementation of this is not entirely trivial, since
energies are not stored every time step of the simulation. We therefore
have to construct \(X_{1,p-1}\) and \(\sigma_{1,p-1}\) from the
information at time \(p\) using
(8) and
(9):
\[\begin{split}\begin{aligned}
X_{1,p-1} &=& X_{1,p} - x_p \\
\sigma_{1,p-1} &=& \sigma_{1,p} - \frac {[~X_{1,p-1} - (p-1) x_{p}~]^2}{(p-1)p}\end{aligned}\end{split}\]
Combining two simulations
Another frequently occurring problem is, that the fluctuations of two
simulations must be combined. Consider the following example: we have
two simulations (A) of \(n\) and (B) of \(m\) steps, in which
the second simulation is a continuation of the first. However, the
second simulation starts numbering from 1 instead of from \(n+1\).
For the partial sum this is no problem, we have to add \(X_{1,n}^A\)
from run A:
(10)\[X_{1,n+m}^{AB} ~=~ X_{1,n}^A + X_{1,m}^B\]
When we want to compute the partial variance from the two components we
have to make a correction \(\Delta\sigma\):
\[\sigma_{1,n+m}^{AB} ~=~ \sigma_{1,n}^A + \sigma_{1,m}^B +\Delta\sigma\]
if we define \(x_i^{AB}\) as the combined and renumbered set of
data points we can write:
\[\sigma_{1,n+m}^{AB} ~=~ \sum_{i=1}^{n+m} \left[x_i^{AB} - \frac{X_{1,n+m}^{AB}}{n+m}\right]^2\]
and thus
\[\sum_{i=1}^{n+m} \left[x_i^{AB} - \frac{X_{1,n+m}^{AB}}{n+m}\right]^2 ~=~
\sum_{i=1}^{n} \left[x_i^{A} - \frac{X_{1,n}^{A}}{n}\right]^2 +
\sum_{i=1}^{m} \left[x_i^{B} - \frac{X_{1,m}^{B}}{m}\right]^2 +\Delta\sigma\]
or
\[\begin{split}\begin{aligned}
\sum_{i=1}^{n+m} \left[(x_i^{AB})^2 - 2 x_i^{AB}\frac{X^{AB}_{1,n+m}}{n+m} + \left(\frac{X^{AB}_{1,n+m}}{n+m}\right)^2 \right] &-& \nonumber \\
\sum_{i=1}^{n} \left[(x_i^{A})^2 - 2 x_i^{A}\frac{X^A_{1,n}}{n} + \left(\frac{X^A_{1,n}}{n}\right)^2 \right] &-& \nonumber \\
\sum_{i=1}^{m} \left[(x_i^{B})^2 - 2 x_i^{B}\frac{X^B_{1,m}}{m} + \left(\frac{X^B_{1,m}}{m}\right)^2 \right] &=& \Delta\sigma\end{aligned}\end{split}\]
all the \(x_i^2\) terms drop out, and the terms independent of the
summation counter \(i\) can be simplified:
\[\begin{split}\begin{aligned}
\frac{\left(X^{AB}_{1,n+m}\right)^2}{n+m} \,-\,
\frac{\left(X^A_{1,n}\right)^2}{n} \,-\,
\frac{\left(X^B_{1,m}\right)^2}{m} &-& \nonumber \\
2\,\frac{X^{AB}_{1,n+m}}{n+m}\sum_{i=1}^{n+m}x_i^{AB} \,+\,
2\,\frac{X^{A}_{1,n}}{n}\sum_{i=1}^{n}x_i^{A} \,+\,
2\,\frac{X^{B}_{1,m}}{m}\sum_{i=1}^{m}x_i^{B} &=& \Delta\sigma\end{aligned}\end{split}\]
we recognize the three partial sums on the second line and use
(10) to obtain:
\[\Delta\sigma ~=~ \frac{\left(mX^A_{1,n} - nX^B_{1,m}\right)^2}{nm(n+m)}\]
if we check this by inserting \(m=1\) we get back
(8)
Summing energy terms
The gmx energy program
can also sum energy terms into one, e.g. potential + kinetic = total.
For the partial averages this is again easy if we have \(S\) energy
components \(s\):
(11)\[X_{m,n}^S ~=~ \sum_{i=m}^n \sum_{s=1}^S x_i^s ~=~ \sum_{s=1}^S \sum_{i=m}^n x_i^s ~=~ \sum_{s=1}^S X_{m,n}^s\]
For the fluctuations it is less trivial again, considering for example
that the fluctuation in potential and kinetic energy should cancel.
Nevertheless we can try the same approach as before by writing:
\[\sigma_{m,n}^S ~=~ \sum_{s=1}^S \sigma_{m,n}^s + \Delta\sigma\]
if we fill in (4):
(12)\[\sum_{i=m}^n \left[\left(\sum_{s=1}^S x_i^s\right) - \frac{X_{m,n}^S}{m-n+1}\right]^2 ~=~
\sum_{s=1}^S \sum_{i=m}^n \left[\left(x_i^s\right) - \frac{X_{m,n}^s}{m-n+1}\right]^2 + \Delta\sigma\]
which we can expand to:
\[\begin{split}\begin{aligned}
&~&\sum_{i=m}^n \left[\sum_{s=1}^S (x_i^s)^2 + \left(\frac{X_{m,n}^S}{m-n+1}\right)^2 -2\left(\frac{X_{m,n}^S}{m-n+1}\sum_{s=1}^S x_i^s + \sum_{s=1}^S \sum_{s'=s+1}^S x_i^s x_i^{s'} \right)\right] \nonumber \\
&-&\sum_{s=1}^S \sum_{i=m}^n \left[(x_i^s)^2 - 2\,\frac{X_{m,n}^s}{m-n+1}\,x_i^s + \left(\frac{X_{m,n}^s}{m-n+1}\right)^2\right] ~=~\Delta\sigma \end{aligned}\end{split}\]
the terms with \((x_i^s)^2\) cancel, so that we can simplify to:
\[\begin{split}\begin{aligned}
&~&\frac{\left(X_{m,n}^S\right)^2}{m-n+1} -2 \frac{X_{m,n}^S}{m-n+1}\sum_{i=m}^n\sum_{s=1}^S x_i^s -2\sum_{i=m}^n\sum_{s=1}^S \sum_{s'=s+1}^S x_i^s x_i^{s'}\, - \nonumber \\
&~&\sum_{s=1}^S \sum_{i=m}^n \left[- 2\,\frac{X_{m,n}^s}{m-n+1}\,x_i^s + \left(\frac{X_{m,n}^s}{m-n+1}\right)^2\right] ~=~\Delta\sigma \end{aligned}\end{split}\]
or
\[-\frac{\left(X_{m,n}^S\right)^2}{m-n+1} -2\sum_{i=m}^n\sum_{s=1}^S \sum_{s'=s+1}^S x_i^s x_i^{s'}\, + \sum_{s=1}^S \frac{\left(X_{m,n}^s\right)^2}{m-n+1} ~=~\Delta\sigma\]
If we now expand the first term using
(11) we obtain:
\[-\frac{\left(\sum_{s=1}^SX_{m,n}^s\right)^2}{m-n+1} -2\sum_{i=m}^n\sum_{s=1}^S \sum_{s'=s+1}^S x_i^s x_i^{s'}\, + \sum_{s=1}^S \frac{\left(X_{m,n}^s\right)^2}{m-n+1} ~=~\Delta\sigma\]
which we can reformulate to:
\[-2\left[\sum_{s=1}^S \sum_{s'=s+1}^S X_{m,n}^s X_{m,n}^{s'}\,+\sum_{i=m}^n\sum_{s=1}^S \sum_{s'=s+1}^S x_i^s x_i^{s'}\right] ~=~\Delta\sigma\]
or
\[-2\left[\sum_{s=1}^S X_{m,n}^s \sum_{s'=s+1}^S X_{m,n}^{s'}\,+\,\sum_{s=1}^S \sum_{i=m}^nx_i^s \sum_{s'=s+1}^S x_i^{s'}\right] ~=~\Delta\sigma\]
which gives
\[-2\sum_{s=1}^S \left[X_{m,n}^s \sum_{s'=s+1}^S \sum_{i=m}^n x_i^{s'}\,+\,\sum_{i=m}^n x_i^s \sum_{s'=s+1}^S x_i^{s'}\right] ~=~\Delta\sigma\]
Since we need all data points \(i\) to evaluate this, in general
this is not possible. We can then make an estimate of
\(\sigma_{m,n}^S\) using only the data points that are available
using the left hand side of (12).
While the average can be computed using all time steps in the
simulation, the accuracy of the fluctuations is thus limited by the
frequency with which energies are saved. Since this can be easily done
with a program such as xmgr this is not
built-in in GROMACS.