Copyright © 2009 I. A. Boguslavsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new multipolynomial approximations algorithm (the MPA algorithm) is proposed for estimating the state vector θ of virtually any dynamical (evolutionary) system. The input of the algorithm consists of discrete-time observations Y. An adjustment of the algorithm is required to the generation of
arrays of random sequences of state vectors and observations scalars
corresponding to a given sequence of time instants.
The distributions of the random factors (vectors of the initial
states and random perturbations of the system, scalars of random observational errors)
can be arbitrary but have to be prescribed beforehand.
The output of the algorithm is a vector polynomial series
with respect to products of nonnegative integer powers of the results of real observations or
some functions of these results. The sum of the powers does not exceed some
given integer d. The series is a vector polynomial approximation of the vector E(θ∣Y), which is the conditional expectation of the vector under evaluation
(or given functions of the components of that vector). The vector coefficients of the
polynomial series are constructed in such a way that the approximation errors uniformly
tend to zero as the integer d increases. These coefficients are found by the Monte-Carlo method and a process
of recurrent calculations that do not require
matrix inversion.