Stochastic extension and functional restrictions of ill-posed estimation problems

Abstract

In Theorem 1 it is shown that under mild conditions the minimum-variance smoothed-estimation of a Gaussian process y(t) with covariance Ry(t,τ) from noise-corrupted measurements z(t)=y(t)+n(t), t∈T, with n(t) Gaussian with covariance R_n(t,τ), is equivalent to a purely deterministic optimization problem, namely, findingy ∈ H (R_y) that minimizes the functional

$$L \left( {y\left| z \right.} \right) = \left\| y \right\|^2 _{R_n } - 2 Re\left[ {\left\langle {z, y} \right\rangle _{R_n } } \right] + \left\| y \right\|^2 _{Ry} .$$

Here 〈·,·〉_R denotes inner-product of the reproducing kernel Hilbert space (RKHS)H(R). Theorems 2 and 3 deal with the more general situation where y is a set of linear measurements on an unknown function w(α), α ∈ A.

The above stochastic-deterministic equivalence provides valuable insight and important consequences for modelling. An example shows how this equivalence allows one to make use of Kalman-Bucy filtering in a purely deterministic problem of smoothing.