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. 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.
Basic properties
Every Gauss–Markov process X(t) possesses the three following properties:
- 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
E
(
X
2
(
t
)
)
=
σ
2
{\displaystyle {\textbf {E}}(X^{2}(t))=\sigma ^{2}}
and time constant
β
−
1
{\displaystyle \beta ^{-1}}
has the following properties.
- Exponential autocorrelation:
R
x
(
τ
)
=
σ
2
e
−
β
|
τ
|
.
{\displaystyle {\textbf {R}}_{x}(\tau )=\sigma ^{2}e^{-\beta |\tau |}.}
- A power spectral density (PSD) function that has the same shape as the Cauchy distribution:
S
x
(
j
ω
)
=
2
σ
2
β
ω
2
+
β
2
.
{\displaystyle {\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:
S
x
(
s
)
=
2
σ
2
β
−
s
2
+
β
2
=
2
β
σ
(
s
+
β
)
⋅
2
β
σ
(
−
s
+
β
)
.
{\displaystyle {\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.
References
- ^ C. E. Rasmussen & C. K. I. Williams (2006). Gaussian Processes for Machine Learning (PDF). MIT Press. p. Appendix B. ISBN 026218253X.
- ^ Lamon, Pierre (2008). 3D-Position Tracking and Control for All-Terrain Robots. Springer. pp. 93–95. ISBN 978-3-540-78286-5.
- ^ Bob Schutz, Byron Tapley, George H. Born (2004-06-26). Statistical Orbit Determination. p. 230. ISBN 978-0-08-054173-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
- ^ 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