In
filtering theory
In the theory of stochastic processes, filtering describes the problem of determining the State (controls), state of a system from an incomplete and potentially Noise (signal processing), noisy set of observations. While originally motivated by pro ...
the Kushner equation (after
Harold Kushner
Harold Samuel Kushner (born April 3, 1935) is a prominent American rabbi and author. He is a member of the Rabbinical Assembly of Conservative Judaism and served as the congregational rabbi of Temple Israel of Natick, in Natick, Massachusetts, ...
) is an equation for the
conditional probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of the state of a
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
non-linear
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
, given noisy measurements of the state. It therefore provides the solution of the
nonlinear filtering problem in
estimation theory
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their val ...
. The equation is sometimes referred to as the Stratonovich–Kushner
[ Stratonovich, R.L. (1960). ''Conditional Markov Processes''. Theory of Probability and Its Applications, 5, pp. 156–178.] (or Kushner–Stratonovich) equation.
Overview
Assume the state of the system evolves according to
:
and a noisy measurement of the system state is available:
:
where ''w'', ''v'' are independent
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It i ...
es. Then the conditional probability density ''p''(''x'', ''t'') of the state at time ''t'' is given by the Kushner equation:
:
where
is the Kolmogorov Forward operator and
is the variation of the conditional probability.
The term
is the
innovation
Innovation is the practical implementation of ideas that result in the introduction of new goods or service (economics), services or improvement in offering goods or services. ISO TC 279 in the standard ISO 56000:2020 defines innovation as "a ...
i.e. the difference between the measurement and its expected value.
Kalman–Bucy filter
One can simply use the Kushner equation to derive the
Kalman–Bucy filter for a linear diffusion process. Suppose we have
and
. The Kushner equation will be given by
:
where
is the mean of the conditional probability at time
. Multiplying by
and integrating over it, we obtain the variation of the mean
:
Likewise, the variation of the variance
is given by
:
The conditional probability is then given at every instant by a normal distribution
.
See also
*
Zakai equation
References
{{Reflist
Signal estimation
Nonlinear filters