Sequentially Best Filter

Abstract

The noisy-state noisy-observation filtering problem is broached in the language of stochastic differential equations. The state and observation equations are nonlinear and time varying, \[\begin{gathered} (1)\qquad dx(t) = m(x(t),t)dt + \sigma (x(t),t)d\xi (t), \hfill \\ (2)\qquad dy(t) = n(x(t),t)dt + d\eta (t). \hfill \\ \end{gathered} \] By taking $\xi (t)$ and $\eta (t)$ to be independent Brownian motions, (1) and (2) define a two-dimensional diffusion process. The pair $\dot \xi $ and $\dot \eta $ simulate white Gaussian noise processes which drive the state variable $x(t)$ and corrupt the observation $y(t)$, respectively. Only recursive estimators will be considered, so only filters whose dynamics can be put into the form \[(3)\qquad dz(t) = g(z(t),t)dt + f(z(t),t)dy(t)\] will be admissible. ($z(t)$ is the estimation of $x(t)$, the instantaneous value of the state variable.) Even though the square loss function is assumed, ambiguity persists with respect to the time t at which $E(z(t) - x(t))^2 $ should be minimized. A precise criterion is provided with the definition of the sequentially best filter. A filter is sequentially best if every other dynamic scheme which has a smaller error at some instant does so by accruing a larger error beforehand. A dividend of the above criterion is that the point by point optimization lends itself to an algorithm for the generation of the sequentially best $f( \cdot , \cdot )$ and $g( \cdot , \cdot )$ in (3). To demonstrate the usefulness of this philosophy, an analytic (linear case) and a computerized (a nonlinear case) example are described. A heuristic proof of the coincidence of the sequentially best filter and the uniformly best (Kalman–Bucy) filter in the time-invariant case is offered.