Kushner equation

In filtering theory the Kushner equation (after Harold Kushner) is an equation for the conditional probability density of the state of a stochastic non-linear dynamical system, given noisy measurements of the state.[1] It therefore provides the solution of the nonlinear filtering problem in estimation theory. The equation is sometimes referred to as the Stratonovich–Kushner[2][3][4][5] (or Kushner–Stratonovich) equation. However, the correct equation in terms of Itō calculus was first derived by Kushner although a more heuristic Stratonovich version of it appeared already in Stratonovich's works in late fifties. However, the derivation in terms of Itō calculus is due to Richard Bucy.[6][clarification needed]

Overview

Assume the state of the system evolves according to

d x = f ( x , t ) d t + σ d w {\displaystyle dx=f(x,t)\,dt+\sigma \,dw}

and a noisy measurement of the system state is available:

d z = h ( x , t ) d t + η d v {\displaystyle dz=h(x,t)\,dt+\eta \,dv}

where w, v are independent Wiener processes. Then the conditional probability density p(x, t) of the state at time t is given by the Kushner equation:

d p ( x , t ) = L [ p ( x , t ) ] d t + p ( x , t ) ( h ( x , t ) − E t h ( x , t ) ) ⊤ η − ⊤ η − 1 ( d z − E t h ( x , t ) d t ) . {\displaystyle dp(x,t)=L[p(x,t)]dt+p(x,t){\big (}h(x,t)-E_{t}h(x,t){\big )}^{\top }\eta ^{-\top }\eta ^{-1}{\big (}dz-E_{t}h(x,t)dt{\big )}.}

where

L [ p ] := − ∑ ∂ ( f i p ) ∂ x i + 1 2 ∑ ( σ σ ⊤ ) i , j ∂ 2 p ∂ x i ∂ x j {\displaystyle L[p]:=-\sum {\frac {\partial (f_{i}p)}{\partial x_{i}}}+{\frac {1}{2}}\sum (\sigma \sigma ^{\top })_{i,j}{\frac {\partial ^{2}p}{\partial x_{i}\partial x_{j}}}}

is the Kolmogorov forward operator and

d p ( x , t ) = p ( x , t + d t ) − p ( x , t ) {\displaystyle dp(x,t)=p(x,t+dt)-p(x,t)}

is the variation of the conditional probability.

The term d z − E t h ( x , t ) d t {\displaystyle dz-E_{t}h(x,t)dt} is the innovation, i.e. the difference between the measurement and its expected value.

Kalman–Bucy filter

One can use the Kushner equation to derive the Kalman–Bucy filter for a linear diffusion process. Suppose we have f ( x , t ) = A x {\displaystyle f(x,t)=Ax} and h ( x , t ) = C x {\displaystyle h(x,t)=Cx} . The Kushner equation will be given by

d p ( x , t ) = L [ p ( x , t ) ] d t + p ( x , t ) ( C x − C μ ( t ) ) ⊤ η − ⊤ η − 1 ( d z − C μ ( t ) d t ) , {\displaystyle dp(x,t)=L[p(x,t)]dt+p(x,t){\big (}Cx-C\mu (t){\big )}^{\top }\eta ^{-\top }\eta ^{-1}{\big (}dz-C\mu (t)dt{\big )},}

where μ ( t ) {\displaystyle \mu (t)} is the mean of the conditional probability at time t {\displaystyle t} . Multiplying by x {\displaystyle x} and integrating over it, we obtain the variation of the mean

d μ ( t ) = A μ ( t ) d t + Σ ( t ) C ⊤ η − ⊤ η − 1 ( d z − C μ ( t ) d t ) . {\displaystyle d\mu (t)=A\mu (t)dt+\Sigma (t)C^{\top }\eta ^{-\top }\eta ^{-1}{\big (}dz-C\mu (t)dt{\big )}.}

Likewise, the variation of the variance Σ ( t ) {\displaystyle \Sigma (t)} is given by

d d t Σ ( t ) = A Σ ( t ) + Σ ( t ) A ⊤ + σ ⊤ σ − Σ ( t ) C ⊤ η − ⊤ η − 1 C Σ ( t ) . {\displaystyle {\tfrac {d}{dt}}\Sigma (t)=A\Sigma (t)+\Sigma (t)A^{\top }+\sigma ^{\top }\sigma -\Sigma (t)C^{\top }\eta ^{-\top }\eta ^{-1}C\,\Sigma (t).}

The conditional probability is then given at every instant by a normal distribution N ( μ ( t ) , Σ ( t ) ) {\displaystyle {\mathcal {N}}(\mu (t),\Sigma (t))} .

See also

  • Zakai equation

References

  1. ^ Kushner, H. J. (1964). "On the differential equations satisfied by conditional probability densities of Markov processes, with applications". J. SIAM Control Ser. A. 2 (1): 106–119. doi:10.1137/0302009.
  2. ^ 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.
  3. ^ Stratonovich, R.L. (1959). On the theory of optimal non-linear filtering of random functions. Theory of Probability and Its Applications, 4, pp. 223–225.
  4. ^ Stratonovich, R.L. (1960) Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp. 1–19.
  5. ^ Stratonovich, R.L. (1960). Conditional Markov Processes. Theory of Probability and Its Applications, 5, pp. 156–178.
  6. ^ Bucy, R. S. (1965). "Nonlinear filtering theory". IEEE Transactions on Automatic Control. 10 (2): 198. doi:10.1109/TAC.1965.1098109.