Open AccessEnsemble State Estimation for Nonlinear Systems Using Polynomial Expansions in the Innovation
Author(s) -
Daniel Hodyss
Publication year - 2011
Publication title -
monthly weather review
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.862
H-Index - 179
eISSN - 1520-0493
pISSN - 0027-0644
DOI - 10.1175/2011mwr3558.1
Subject(s) - nonlinear system , probability density function , scalar (mathematics) , polynomial , state vector , mathematics , state (computer science) , statistical physics , computer science , statistics , algorithm , mathematical analysis , physics , classical mechanics , quantum mechanics , geometry
A new framework is presented for understanding how a nonnormal probability density function (pdf) may affect a state estimate and how one might usefully exploit the nonnormal properties of the pdf when constructing a state estimate. A Bayesian framework is constructed that naturally leads to an expansion of the expected forecast error in a polynomial series consisting of powers of the innovation vector. This polynomial expansion in the innovation reveals a new view of the geometric nature of the state estimation problem. It is shown that this expansion in powers of the innovation provides a direct relationship between a nonnormal pdf describing the likely distribution of states and a normal pdf determined by powers of the forecast error. One implication of this perspective is that when state estimation is performed on a nonnormal pdf it leads to state estimates based on the mean to be nonlinear functions of the innovation. A direct relationship is shown between the degree to which the state estimate varies with the innovation and the moments of the distribution. These and other implications of this new view of ensemble state estimation in nonlinear systems are illustrated in simple scalar systems as well as on the Lorenz attractor.