Gauss–Markov process

{{distinguish|text=the Gauss–Markov theorem of mathematical statistics}}

Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes.{{cite book|last=C. E. Rasmussen & C. K. I. Williams|title=Gaussian Processes for Machine Learning|date=2006|publisher=MIT Press|isbn=026218253X|page=Appendix B|url=http://www.gaussianprocess.org/gpml/chapters/RWB.pdf}}{{cite book|last=Lamon|first=Pierre|title=3D-Position Tracking and Control for All-Terrain Robots|url=https://archive.org/details/dpositiontrackin00lamo|url-access=limited|date=2008|publisher=Springer|isbn=978-3-540-78286-5|pages=[https://archive.org/details/dpositiontrackin00lamo/page/n99 93]–95}} A stationary Gauss–Markov process is unique{{Citation needed|reason=here also some assumption is missing: a process with iid Gaussian values is Gauss-Markov|date=February 2019}} up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.

Gauss–Markov processes obey Langevin equations.{{cite book|author=Bob Schutz, Byron Tapley, George H. Born |title=Statistical Orbit Determination |date=2004-06-26 |isbn=978-0-08-054173-0 |pages=230|url=https://books.google.com/books?id=Ct3qN1VCHewC&q=Gauss%E2%80%93Markov+process+%22langevin+equation%22&pg=PA230}}

Basic properties

Every Gauss–Markov process X(t) possesses the three following properties: C. B. Mehr and J. A. McFadden. Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 3(1965), pp. 505-522

If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process
If f(t) is a non-decreasing scalar function of t, then Z(t) = X(f(t)) is also a Gauss–Markov process
If the process is non-degenerate and mean-square continuous, then there exists a non-zero scalar function h(t) and a strictly increasing scalar function f(t) such that X(t) = h(t)W(f(t)), where W(t) is the standard Wiener process.

Property (3) means that every non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

Other properties

A stationary Gauss–Markov process with variance $\textbf{E}(X^{2}(t)) = \sigma^{2}$ and time constant $\beta^{-1}$ has the following properties.

Exponential autocorrelation: $\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}.$
A power spectral density (PSD) function that has the same shape as the Cauchy distribution: $\textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.$ (Note that the Cauchy distribution and this spectrum differ by scale factors.)
The above yields the following spectral factorization: $\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}}$

= \frac{\sqrt{2\beta}\,\sigma}{(s + \beta)} \cdot\frac{\sqrt{2\beta}\,\sigma}{(-s + \beta)}. which is important in Wiener filtering and other areas.

There are also some trivial exceptions to all of the above.{{clarify|date=April 2018}}

References

Category:Markov processes