Filtering problem (stochastic processes)

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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 non-stationary case) was solved by Ruslan L. Stratonovich (1959^[1], 1960^[2]). The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well-understood: for example, the linear filter is optimal for Gaussian random variables, and is known as the Kalman-Bucy filter.

If the separation principle applies, then filtering also arises as part of the solution of a optimal control problem i.e. the Kalman filter is the estimation part of the optimal control solution to the Linear-quadratic-Gaussian control problem.

[edit] The mathematical formalism

Consider a probability space (Ω, Σ, P) and suppose that the (random) state Y_t in n-dimensional Euclidean space Rⁿ of a system of interest at time t is a random variable Y_t : Ω → Rⁿ given by the solution to an Itō stochastic differential equation of the form

$\mathrm{d} Y_{t} = b(t, Y_{t}) \, \mathrm{d} t + \sigma (t, Y_{t}) \, \mathrm{d} B_{t},$

where B denotes standard p-dimensional Brownian motion, b : [0, +∞) × Rⁿ → Rⁿ is the drift field, and σ : [0, +∞) × Rⁿ → R^n×p is the diffusion field. It is assumed that observations H_t in R^m (note that m and n may, in general, be unequal) are taken for each time t according to

$H_{t} = c(t, Y_{t}) + \gamma (t, Y_{t}) \cdot \mbox{noise}.$

Adopting the Itō interpretation of the stochastic differential and setting

$Z_{t} = \int_{0}^{t} H_{s} \, \mathrm{d} s,$

this gives the following stochastic integral representation for the observations Z_t:

$\mathrm{d} Z_{t} = c(t, Y_{t}) \, \mathrm{d} t + \gamma (t, Y_{t}) \, \mathrm{d} W_{t},$

where W denotes standard r-dimensional Brownian motion, independent of B and the initial condition X₀, and c : [0, +∞) × Rⁿ → Rⁿ and γ : [0, +∞) × Rⁿ → R^n×r satisfy

$\big| c (t, x) \big| + \big| \gamma (t, x) \big| \leq C \big( 1 + | x | \big)$

for all t and x and some constant C.

The filtering problem is the following: given observations Z_s for 0 ≤ s ≤ t, what is the best estimate Ŷ_t of the true state Y_t of the system based on those observations?

By "based on those observations" it is meant that Ŷ_t is measurable with respect to the σ-algebra G_t generated by the observations Z_s, 0 ≤ s ≤ t. Denote by K = K(Z, t) be collection of all Rⁿ-valued random variables Y that are square-integrable and G_t-measurable:

$K = K(Z, t) = L^{2} (\Omega, G_{t}, \mathbf{P}; \mathbf{R}^{n}).$

By "best estimate", it is meant that Ŷ_t minimizes the mean-square distance between Y_t and all candidates in K:

$\mathbf{E} \left[ \big| Y_{t} - \hat{Y}_{t} \big|^{2} \right] = \inf_{Y \in K} \mathbf{E} \left[ \big| Y_{t} - \hat{Y} \big|^{2} \right]. \qquad \mbox{(M)}$

[edit] Basic result: orthogonal projection

The space K(Z, t) of candidates is a Hilbert space, and the general theory of Hilbert spaces implies that the solution Ŷ_t of the minimization problem (M) is given by

$\hat{Y}_{t} = P_{K(Z, t)} \big( X_{t} \big),$

where P_K(Z,t) denotes the orthogonal projection of L²(Ω, Σ, P; Rⁿ) onto the linear subspace K(Z, t) = L²(Ω, G_t, P; Rⁿ). Furthermore, it is a general fact about conditional expectations that if F is any sub-σ-algebra of Σ then the orthogonal projection

$P_{F} : L^{2} (\Omega, \Sigma, \mathbf{P}; \mathbf{R}^{n}) \to L^{2} (\Omega, F, \mathbf{P}; \mathbf{R}^{n})$

is exactly the conditional expectation operator E[·|F], i.e.,

$P_{F} (X) = \mathbf{E} \big[ X \big | F \big].$

Hence,

$\hat{Y}_{t} = P_{K(Z, t)} \big( X_{t} \big) = \mathbf{E} \big[ X_{t} \big | G_{t} \big].$

This elementary result is the basis for the general Fujisaki-Kallianpur-Kunita equation of filtering theory.

[edit] References

Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications, Sixth edition, Berlin: Springer. ISBN 3-540-04758-1. (See Section 6.1)

^ 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.
^ Stratonovich, R.L. (1960). Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp.1-19.