Nonlinear Filters

For a nonlinear filtering problem, the most heuristic and
easiest approximation is to use the Taylor series expansion
and apply the conventional linear recursive Kalman filter
algorithm directly to the linearized nonlinear measurement
and transition equations. First, it is discussed that the
Taylor series expansion approach gives us the biased
estimators. Next, a Monte-Carlo simulation filter is
proposed, where each expectation of the nonlinear functions
is evaluated generating random draws. It is shown from
Monte-Carlo experiments that the Monte-Carlo simulation
filter yields the unbiased but inefficient estimator.
Anotherapproach to the nonlinear filtering problem is to
approximate the underlyingdensity functions of the state
vector. In this monograph, a nonlinear and nonnormal filter
is proposed by utilizing Monte-Carlo integration, in which a
recursive algorithm of the weighting functions is derived.
The densityapproximation approach gives us an
asymptotically unbiased estimator. Moreover, in terms of
programming and computational time, the nonlinear filter
using Monte-Carlo integration can be easily extended to
higher dimensional cases, compared with Kitagawa's nonlinear
filter using numericalintegration.