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, 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 caus ...
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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
with the help of often incomplete measurements marred by
distortion.
[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 – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
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 determini ...
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. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
(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
* Hidden Markov model
A hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it X — with unobservable ("''hidden''") states. As part of the definition, HMM requires that there be an ob ...
* Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
* 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 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 expressing it through a set o ...
* Nyquist–Shannon sampling theorem
References
{{DEFAULTSORT:Masreliez's theorem
Control theory
Signal processing
Control engineering