In
filtering theory the Zakai equation is a linear
stochastic partial differential equation
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
They hav ...
for the un-normalized density of a hidden state. In contrast, the
Kushner equation gives a non-linear stochastic partial differential equation for the normalized density of the hidden state. In principle either approach allows one to estimate a quantity function (the state of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
) from noisy measurements, even when the system is non-linear (thus generalizing the earlier results of Wiener and Kalman for linear systems and solving a central problem in
estimation theory
Estimation theory is a branch of statistics that deals with estimating the values of Statistical parameter, parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such ...
). The application of this approach to a specific
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
situation may be problematic however, as these equations are quite complex. The Zakai equation is a bilinear
stochastic partial differential equation
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
They hav ...
. It was named after
Moshe Zakai.
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Overview
Assume the state of the system evolves according to
:
and a noisy measurement of the system state is available:
:
where
are independent
Wiener process
In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one o ...
es. Then the unnormalized conditional probability density
of the state at time t is given by the Zakai equation:
:
where
:
is a Kolmogorov forward operator.
As previously mentioned,
is an unnormalized density and thus does not necessarily integrate to 1. After solving for
, integration and normalization can be done if desired (an extra step not required in the Kushner approach).
Note that if the last term on the right hand side is omitted (by choosing h identically zero), the result is a nonstochastic PDE: the familiar
Fokker–Planck equation
In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag (physi ...
, which describes the evolution of the state when no measurement information is available.
See also
*
Kushner equation
*
Kalman filter
In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unk ...
*
Wiener filter
References
Further reading
*
* {{cite book , first=Zeev , last=Schuss , chapter=Nonlinear Filtering and Smoothing of Diffusions , pages=85–106 , title=Nonlinear Filtering and Optimal Phase Tracking , location=Boston , publisher=Springer , year=2012 , isbn=978-1-4614-0486-6 , doi=10.1007/978-1-4614-0487-3_3
Signal estimation
Stochastic differential equations