  In this paper is proposed a novel incremental iterative
Gauss-Newton-Markov-Kalman filter method for state estimation of dynamic models
given noisy measurements. The mathematical formulation of the proposed filter
is based on the construction of an optimal nonlinear map between the observable
and parameter (state) spaces via a convergent sequence of linear maps obtained
by successive linearisation of the observation operator in a Gauss-Newton-like
form. To allow automatic linearisation of the dynamical system in a sparse
form, the smoother is designed in a hierarchical setting such that the forward
map and its linearised counterpart are estimated in a Bayesian manner given a
forecasted data set. To improve the algorithm convergence, the smoother is
further reformulated in its incremental form in which the current and
intermediate states are assimilated before the initial one, and the
corresponding posterior estimates are taken as pseudo-measurements. As the
latter ones are random variables, and not deterministic any more, the novel
stochastic iterative filter is designed to take this into account. To correct
the bias in the posterior outcome, the procedure is built in a
predictor-corrector form in which the predictor phase is used to assimilate
noisy measurement data, whereas the corrector phase is constructed to correct
the mean bias. The resulting filter is further discretised via time-adapting
sparse polynomial chaos expansions obtained either via modified Gram-Schmidt
orthogonalisation or by a carefully chosen nonlinear mapping, both of which are
estimated in a Bayesian manner by promoting the sparsity of the outcomes. The
time adaptive basis with non-Gaussian arguments is further mapped to the
polynomial chaos one by a suitably chosen isoprobabilistic transformation.
Finally, the proposed method is tested on a chaotic nonlinear Lorenz 1984
system.
