Gauss–Markov process

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] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions.

Every Gauss–Markov process X(t) possesses the three following properties:

1. If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process
2. If f(t) is a non-decreasing scalar function of t, then Z(t) = X(f(t)) is also a Gauss–Markov process
3. There exists a non-zero scalar function h(t) and a non-decreasing 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 Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

Properties of the Stationary Gauss-Markov Processes

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.

See also

Ornstein–Uhlenbeck process

References

1. C. E. Rasmussen & C. K. I. Williams, (2006). Gaussian Processes for Machine Learning (PDF). MIT Press. p. Appendix B. ISBN 026218253X.
2. Lamon, Pierre (2008). 3D-Position Tracking and Control for All-Terrain Robots. Springer. pp. 93–95. ISBN 978-3-540-78286-5.