Filtering problem (stochastic processes)

In the theory of <a href="/facts/Stochastic_processes/Ng1n1dnB">stochastic processes</a>, filtering describes the problem of determining the <a href="/facts/State_(controls)/Jf4EanCP">state</a> of a system from an incomplete and potentially <a href="/facts/Noise_(signal_processing)/JT3lGj8v">noisy</a> set of observations. For example, in GPS navigation, filtering helps estimate a car’s true position (the state) from noisy satellite signals (the observations). While originally motivated by problems in engineering, filtering found applications in many fields from signal processing to finance.
The problem of optimal non-linear filtering (even for the non-stationary case) was solved by <a href="/facts/Ruslan_L._Stratonovich/4xUGIjrm">Ruslan L. Stratonovich</a> (1959, 1960), see also <a href="/facts/Harold_J._Kushner/70RDFkQQ">Harold J. Kushner</a>'s work and <a href="/facts/Moshe_Zakai/tHuCokcT">Moshe Zakai</a>'s, who introduced a simplified dynamics for the unnormalized conditional law of the filter known as the <a href="/facts/Zakai_equation/q7bV3Qmk">Zakai equation</a>. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the <a href="/facts/Wiener_filter/oGWdBTYa">Wiener filter</a> and the <a href="/facts/Kalman-Bucy_filter/wNK7rnbk">Kalman-Bucy filter</a>. 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 <a href="/facts/Nonlinear_filter/sgt9wmvb">nonlinear filter</a> may be more based on heuristics, such as the <a href="/facts/Extended_Kalman_filter/b08h7X3x">extended Kalman filter</a> or the assumed density filters, or more methodologically oriented such as for example the <a href="/facts/Projection_filters/VTmB9nKW">projection filters</a>, some sub-families of which are shown to coincide with the Assumed Density Filters. 
<a href="/facts/Particle_filter/192bI79g">Particle filters</a> are another option to attack the infinite dimensional filtering problem and are based on sequential Monte Carlo methods.
In general, if the <a href="/facts/Separation_principle/tMHqgCHS">separation principle</a> applies, then filtering also arises as part of the solution of an <a href="/facts/Optimal_control/9SmcQHjJ">optimal control</a> problem. For example, the <a href="/facts/Kalman_filter/wNK7rnbk">Kalman filter</a> is the estimation part of the optimal control solution to the <a href="/facts/Linear-quadratic-Gaussian_control/3sI4R7zA">linear-quadratic-Gaussian control</a> problem.

Filtering problem (stochastic processes) open-in-new

Filtering problem (stochastic processes)