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Multiple binormal restraint

The polymodal pdf for a geometric feature $(f_1, f_2)$ (, a pair of dihedral angles) is


$\displaystyle p$ $\textstyle =$ $\displaystyle \sum_{i=1}^n \omega_i p_i \; = \; \sum_{i=1}^n \omega_i
\frac{1}{2 \pi \sigma_{1i} \sigma_{2i} \sqrt{(1-\rho_i^2)}} \; \cdot$  
    $\displaystyle \exp \left\{-\frac{1}{2 (1-\rho_i^2)}
\left[
\left(\frac{f_1-\bar...
...i}} +
\left(\frac{f_2-\bar{f}_{2i}}{\sigma_{2i}}\right)^2
\right]
\right\} \; .$ (7.50)

where $\rho < 1$. $\rho$ is the correlation coefficient between $f_1$ and $f_2$. 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} (7.51)

The first derivatives with respect to features $f_1$ and $f_2$ are:

$\displaystyle \frac{\partial c}{\partial f_1}$ $\textstyle =$ $\displaystyle \frac{1}{p}
\sum_{i=1}^n \left[ \omega_i p_i \cdot
\frac{1}{\sigm...
...{1i}}{\sigma_{1i}} -
\rho_i\frac{f_2-\bar{f}_{2i}}{\sigma_{2i}}
\right)
\right]$ (7.52)
$\displaystyle \frac{\partial c}{\partial f_2}$ $\textstyle =$ $\displaystyle \frac{1}{p}
\sum_{i=1}^n \left[ \omega_i p_i \cdot
\frac{1}{\sigm...
...{\sigma_{2i}} -
\rho_i\frac{f_1-\bar{f}_{1i}}{\sigma_{1i}}
\right)
\right] \; .$ (7.53)



Bozidar BJ Jerkovic 2001-12-21