Partial Derivatives of the CGF.

To obtain the SPA to the PDF of $ {\bf z}$, we need the joint cumulant generating function (CGF) $ c_z($$ \lambda$$ )$ of $ {\bf z}$ and its partial derivatives. The joint CGF is defined by

$\displaystyle c_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle )= \log( g_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle )) =
- {\rm sum}\left(\mbox{\bf log}({\underline{\bf 1}}- {\bf P} \mbox{\boldmath$\lambda$})
\right)
$

where log is the vector $ \log$ function which operates on each element of its argument and $ {\rm sum}(\;)$ is the vector sum, which adds up all the elements of the argument. If we define $ \phi$$ ^{-1}($$ \lambda$$ )$ as the element-by-element reciprocal of $ {\underline{\bf 1}}- {\bf P}$   $ \lambda$, and $ \Phi$$ ($$ \lambda$$ )$ as the diagonal $ N$-by-$ N$ matrix with elements equal to the elements of $ {\underline{\bf 1}}- {\bf P}$   $ \lambda$, it is straight-forward to show that the gradient vector of $ c_z($$ \lambda$$ )$ is the $ M$-by-1 vector

   $\displaystyle \mbox{\boldmath$\delta$}$$\displaystyle ($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) \stackrel{\mbox{\tiny$\Delta$}}{=}
{\partial \over \partial \m...
...)
= {\bf P}^\prime \; \mbox{\boldmath$\phi$}^{-1}(\mbox{\boldmath$\lambda$}),
$

and the $ M$-by-$ M$ Hessian matrix of $ c_z($$ \lambda$$ )$ is

$\displaystyle {\bf C}_z($$\displaystyle \mbox{\boldmath$\lambda$}$$\displaystyle ) \stackrel{\mbox{\tiny$\Delta$}}{=}{\partial^2 \over \partial
\...
...P}^\prime \; \mbox{\boldmath$\Phi$}^{-2}(\mbox{\boldmath$\lambda$})\; {\bf P},
$



Baggenstoss 2017-05-19