Gauss–Markov process explained
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.[1] [2] A stationary Gauss–Markov process is unique up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.
Gauss–Markov processes obey Langevin equations.[3]
Basic properties
Every Gauss–Markov process X(t) possesses the three following properties:[4]
- 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
and
time constant
has the following properties.
- Exponential autocorrelation:
- A power spectral density (PSD) function that has the same shape as the Cauchy distribution: (Note that the Cauchy distribution and this spectrum differ by scale factors.)
- The above yields the following spectral factorization:
= \frac \cdot\frac. which is important in Wiener filtering and other areas.
There are also some trivial exceptions to all of the above.
Notes and References
- Book: C. E. Rasmussen & C. K. I. Williams. Gaussian Processes for Machine Learning. 2006. MIT Press. 026218253X. Appendix B.
- Book: Lamon, Pierre. 3D-Position Tracking and Control for All-Terrain Robots. limited. 2008. Springer. 978-3-540-78286-5. 93–95.
- Book: Bob Schutz, Byron Tapley, George H. Born . Statistical Orbit Determination . 2004-06-26 . 978-0-08-054173-0 . 230.
- 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