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 (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[edit]

A stationary Gauss–Markov process with variance 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:

which is important in Wiener filtering and other areas.

There are also some trivial exceptions to all of the above.

See also[edit]

Ornstein–Uhlenbeck process

References[edit]

  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.