In mathematics and statistics, a stationary process stationary process or strong(ly) stationary
process) is a stochastic process whose joint probability distribution does not change when
shifted in time. Consequently, parameters such as the mean and variance, if they are
present, also do not change over time and do not follow any trends.
Stationarity is used as a tool in time series analysis, where the raw data is often transformed
to become stationary; for example, economic data are often seasonal and/or dependent on
a non-stationary price level. An important type of non-stationary process that does not
include a trend-like behavior is the cyclostationary process.
Note that a "stationary process" is not the same thing as a "process with a stationary
distribution". Indeed there are further possibilities for confusion with the use of "stationary"
in the context of stochastic processes; for example a "time-homogeneous" Markov chain
is sometimes said to have "stationary transition probabilities". Besides, all stationary Markov
random processes are time-homogeneous.
Definition Formally, let be a stochastic process and
let represent the cumulative distribution function of the joint distribution of at times
. Then, is said to be stationary if, for all , for all , and for all ,
Since does not affect , is not a function of time.
Examples
As an example, white noise is stationary. The sound of a cymbal clashing, if hit only
once, is not stationary because the acoustic power of the clash diminishes with time. However,
it would be possible to invent a stochastic process describing when the cymbal is hit,
such that the overall response would form a stationary process.
An example of a discrete-time stationary process where the sample space is also discrete is
a Bernoulli scheme. Other examples of a discrete-time stationary process with continuous sample
space include some autoregressive and moving average processes which are both subsets of
the autoregressive moving average model. Models with a non-trivial autoregressive component
may be either stationary or non-stationary, depending on the parameter values, and important
non-stationary special cases are where unit roots exist in the model.
Let Y be any scalar random variable, and define a time-series { Xt }, by
. Then { Xt } is a stationary time series,
for which realisations consist of a series of constant values, with a different constant
value for each realisation. A law of large numbers does not apply on this case, as the
limiting value of an average from a single realisation takes the random value determined
by Y, rather than taking the expected value of Y.
As a further example of a stationary process for which any single realisation has an apparently
noise-free structure, let Y have a uniform distribution on (0,2π] and define the time
series { Xt } by
Then { Xt } is strictly stationary. Weaker forms of stationarity
Weak or wide-sense stationarity A weaker form of stationarity commonly employed
in signal processing is known as weak-sense stationarity, wide-sense stationarity, covariance
stationarity, or second-order stationarity. WSS random processes only require that 1st
moment and covariance do not vary with respect to time. Any strictly stationary process which
has a mean and a covariance is also WSS. So, a continuous-time random process x(t)
which is WSS has the following restrictions on its mean function
and autocovariance function
The first property implies that the mean function mx(t) must be constant. The second property
implies that the covariance function depends only on the difference between and and only
needs to be indexed by one variable rather than two variables. Thus, instead of writing,
the notation is often abbreviated and written as:
This also implies that the autocorrelation depends only on , since
The main advantage of wide-sense stationarity is that it places the time-series in the context
of Hilbert spaces. Let H be the Hilbert space generated by {x(t)}. By the positive definiteness
of the autocovariance function, it follows from Bochner's theorem that there exists a
positive measure μ on the real line such that H is isomorphic to the Hilbert subspace
of L2(μ) generated by {e-2πiξ⋅t}. This then gives the following Fourier-type decomposition
for continuous time stationary stochastic process: there exists a stochastic process
ωξ with orthogonal increments such that, for all t
where the integral on the right hand side is interpreted in a suitable sense. Same result
holds for a discrete-time stationary process, with the spectral measure now defined on the
unit circle. When processing WSS random signals with linear,
time-invariant filters, it is helpful to think of the correlation function as a linear operator.
Since it is a circulant operator, its eigenfunctions are the Fourier complex exponentials. Additionally,
since the eigenfunctions of LTI operators are also complex exponentials, LTI processing
of WSS random signals is highly tractable—all computations can be performed in the frequency
domain. Thus, the WSS assumption is widely employed in signal processing algorithms.
Other terminology The terminology used for types of stationarity
other than strict stationarity can be rather mixed. Some examples follow.
Priestley uses stationary up to order m if conditions similar to those given here for
wide sense stationarity apply relating to moments up to order m. Thus wide sense stationarity
would be equivalent to "stationary to order 2", which is different from the definition
of second-order stationarity given here.
Honarkhah and Caers also use the assumption of stationarity in the context of multiple-point
geostatistics, where higher n-point statistics are assumed to be stationary in the spatial
domain.
See also Lévy process
Stationary ergodic process Wiener–Khinchin theorem
Ergodicity References
Further reading Enders, Walter. Applied Econometric Time Series.
New York: Wiley. pp. 53–57. ISBN 978-0-470-50539-7. External links
Spectral decomposition of a random function