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The observation function in Equation 5.7
does not encapsulate the most general
form of observation, since it assumes that the observation vector
can be separated as a function of the state . It is sometimes therefore
necessary to introduce a generalised observation of the form

where again represents a random noise vector having covariance .
However with some manipulation and extra computation we can effectively
convert the linearised version of the -type function into an
-type function, allowing it to be incorporated in the same way.
We linearise with respect to and around the
estimated state and observation , assuming
that the noise is small:

where here represents the true value of the state vector, and
is the true observation vector (as opposed to the actually measured vector
), so that
.
We identify the following quantities with their equivalents for an -type
observation:
- The
**innovation** vector is
.
- The
**Jacobian** matrix is
.
- The
**noise vector** is
.
- The
**noise covariance** matrix is
.

Extra computation is therefore needed to convert the observation covariance
from to . The innovation vector
, Jacobian matrix and
observation covariance are substituted into the Levenberg-Marquardt
algorithm in place of their equivalents for the -type observation.
There is no reason why there should not be a robust version of the -type
observation, but currently it is not implemented.

** Next:** Levenberg-Marquardt software
** Up:** Levenberg-Marquardt minimisation
** Previous:** Robust observations
** Contents**
2006-03-17