|
In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some (potentially noisy) observations of that system. The problem of optimal non-linear filtering (even for the non-stationary case) was solved by Ruslan L. Stratonovich (1959,〔Stratonovich, R. L. (1959). ''Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise''. Radiofizika, 2:6, pp. 892-901.〕 1960〔Stratonovich, R.L. (1960). ''Application of the Markov processes theory to optimal filtering''. Radio Engineering and Electronic Physics, 5:11, pp.1-19.〕), see also Harold J. Kushner's work 〔Kushner, Harold. (1967). Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode. Automatic Control, IEEE Transactions on Volume 12, Issue 3, Jun 1967 Page(s): 262 - 267〕 and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter〔Zakai, Moshe (1969), On the optimal filtering of diffusion processes. Zeit. Wahrsch. 11 230–243. , , 〕 known as Zakai equation. The solution, however, is infinite-dimensional in the general case.〔Mireille Chaleyat-Maurel and Dominique Michel. Des resultats de non existence de filtre de dimension finie. Stochastics, 13(1+2):83-102, 1984.〕 Certain approximations and special cases are well-understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters,〔Maybeck, Peter S., Stochastic models, estimation, and control, Volume 141, Series Mathematics in Science and Engineering, 1979, Academic Press〕 or more methodologically oriented such as for example the Projection Filters,〔Damiano Brigo, Bernard Hanzon and François LeGland, A Differential Geometric approach to nonlinear filtering: the Projection Filter, I.E.E.E. Transactions on Automatic Control Vol. 43, 2 (1998), pp 247--252.〕 some sub-families of which are shown to coincide with the Assumed Density Filters.〔Damiano Brigo, Bernard Hanzon and François Le Gland, Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities, Bernoulli, Vol. 5, N. 3 (1999), pp. 495--534〕 In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the Linear-quadratic-Gaussian control problem. ==The mathematical formalism== Consider a probability space (Ω, Σ, P) and suppose that the (random) state ''Y''''t'' in ''n''-dimensional Euclidean space R''n'' of a system of interest at time ''t'' is a random variable ''Y''''t'' : Ω → R''n'' given by the solution to an Itō stochastic differential equation of the form : where ''B'' denotes standard ''p''-dimensional Brownian motion, ''b'' : . \qquad \mbox 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Filtering problem (stochastic processes)」の詳細全文を読む スポンサード リンク
|