Gauss–Markov process

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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 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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  2. Lamon, Pierre (2008). 3D-Position Tracking and Control for All-Terrain Robots. Springer. pp. 93–95. ISBN 978-3-540-78286-5.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>