  The Kalman filter is an established tool for the analysis of dynamic systems
with normally distributed noise, and it has been successfully applied in
numerous application areas. It provides sequentially calculated estimates of
the system states along with a corresponding covariance matrix. For nonlinear
systems, the extended Kalman filter is often used which is derived from the
Kalman filter by linearization around the current estimate. A key issue in
metrology is the evaluation of the uncertainty associated with the Kalman
filter state estimates. The "Guide to the Expression of Uncertainty in
Measurements" (GUM) and its supplements serve as the de facto standard for
uncertainty evaluation in metrology. We explore the relationship between the
covariance matrix produced by the Kalman filter and a GUM-compliant uncertainty
analysis. In addition, also the results of a Bayesian analysis are considered.
For the case of linear systems with known system matrices, we show that all
three approaches are compatible. When the system matrices are not precisely
known, however, or when the system is nonlinear, this equivalence breaks down
and different results can be reached then. Though for precisely known nonlinear
systems the result of the extended Kalman filter still corresponds to the
linearized uncertainty propagation of GUM. The extended Kalman filter can
suffer from linearization and convergence errors. These disadvantages can be
avoided to some extent by applying Monte Carlo procedures, and we propose such
a method which is GUM-compliant and can also be applied online during the
estimation. We illustrate all procedures in terms of a two-dimensional dynamic
system and compare the results with those obtained by particle filtering, which
has been proposed for the approximate calculation of a Bayesian solution.
Finally, we give some recommendations based on our findings.
