Masreliez theorem
describes a
recursive algorithm
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves ...
within the technology of extended
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 ...
, named after the Swedish-American
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
John Masreliez, who is its author. The algorithm estimates the state of a
dynamic system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock ...
with the help of often incomplete measurements marred by
distortion
In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...
.
[T. Cipra & A. Rubio]
''Kalman filter with a non-linear non-Gaussian observation relation''
Springer (1991).
Masreliez's theorem produces estimates that are quite good approximations to the exact
conditional mean
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on ...
in
non-Gaussian additive outlier (AO) situations. Some evidence for this can be had by
Monte Carlo simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.
[ (PDF 1465 kB)]
The key approximation property used to construct these filters is that the state prediction density is approximately
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. Masreliez discovered in 1975
that this approximation yields an intuitively appealing non-Gaussian filter recursions, with
data dependent covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
(unlike the Gaussian case) this derivation also provides one of the nicest ways of establishing the standard Kalman filter recursions. Some theoretical justification for use of the Masreliez approximation is provided by the "continuity of state prediction densities" theorem in Martin (1979).
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See also
* Control engineering
Control engineering, also known as control systems engineering and, in some European countries, automation engineering, is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with d ...
* Hidden Markov model
A hidden Markov model (HMM) is a Markov model in which the observations are dependent on a latent (or ''hidden'') Markov process (referred to as X). An HMM requires that there be an observable process Y whose outcomes depend on the outcomes of X ...
* Bayes' theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For exampl ...
* Robust optimization
Robust optimization is a field of mathematical optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the ...
* Probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
* Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample r ...
References
{{DEFAULTSORT:Masreliez's theorem
Control theory
Signal processing
Control engineering