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Multiple Gaussian restraint

The polymodal pdf for a geometric feature $f$ (, distance, angle, dihedral angle) is

\begin{displaymath}
p = \sum_{i=1}^n \omega_i p_i = \sum_{i=1}^n \omega_i
\fr...
...1}{2}
\left(\frac{f-\bar{f}_i}{\sigma_i}\right)^2\right] \; .
\end{displaymath} (7.41)

A corresponding restraint $c$ in the sum that defines the objective function $F$ is
\begin{displaymath}
c = -\ln p = -\ln \sum_{i=1}^n \omega_i p_i
\end{displaymath} (7.42)

The first derivatives with respect to feature $f$ are:

$\displaystyle \frac{ \; {d}c}{ \; {d}f}$ $\textstyle =$ $\displaystyle \frac{1}{p} \sum_{i=1}^n \omega_i p_i \cdot
\left[\frac{f-\bar{f}_i}{\sigma_i}\frac{1}{\sigma_i}\right]
\; .$ (7.43)

When any of the normalized deviations $v_i = (f-\bar{f}_i) / \sigma_i$ is large, there are numerical instabilities in calculating the derivatives because $v_i$ are arguments to the exp function. Robustness is ensured as follows. The `effective' normalized deviation is used in all the equations above when the magnitude of normalized violation $v$ is larger than cutoff rgauss1 (10 for double precision). This scheme works up to rgauss2 (200 for double precision); violations larger than that are ignored. This trick is equivalent to increasing the standard deviation $\sigma_i$. A slight disadvantage is that there is a discontinuity in the first derivatives at rgauss1. However, if continuity were imposed, the range would not be extended (this is equivalent to linearizing the Gaussian, but since it is already linear for large deviations, a linearization with derivatives smoothness would not introduce much change at all).


$\displaystyle M$ $\textstyle =$ $\displaystyle 37 \ \ \ ; \ \ \ \
\mbox{$M^2/2$\ has to be smaller than the largest argument to $\exp$}$ (7.44)
$\displaystyle A$ $\textstyle =$ $\displaystyle \frac{1}{M} \frac{\mbox{\tt rgauss2}-M}
{\mbox{\tt rgauss2} -\mbox{\tt rgauss1}}$ (7.45)
$\displaystyle B$ $\textstyle =$ $\displaystyle \frac{\mbox{\tt rgauss2}}{M} \frac{M-\mbox{\tt rgauss1}}
{\mbox{\tt rgauss2}-\mbox{\tt rgauss1}}$ (7.46)
$\displaystyle v$ $\textstyle =$ $\displaystyle \frac{f - \bar{f}_i}{\sigma_i}$ (7.47)
$\displaystyle F$ $\textstyle =$ $\displaystyle A \left\vert v \right\vert + B$ (7.48)
$\displaystyle v'$ $\textstyle =$ $\displaystyle v / F$ (7.49)

Now, Eqs. 5.41-5.43 are used with $v'$ instead of $v$. For single precision, $M = 12$, rgauss1 = 4, rgauss2 = 100.


next up previous contents index
Next: Multiple binormal restraint Up: Restraints and their derivatives Previous: Single Gaussian restraint   Contents   Index
Bozidar BJ Jerkovic 2001-12-21