Stationary process

In the mathematical sciences, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time or space. Consequently, parameters such as the mean and variance, if they exist, also do not change over time or position.

Stationarity is used as a tool in time series analysis, where the raw data are 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 (which condition is sometimes called by the name "stationary Markov chain") is sometimes said to have "stationary transition probabilities".

Definition

Formally, let $\left\{X_t\right\}$ be a stochastic process and let $F_{X}(x_{t_1 + \tau}, \ldots, x_{t_k + \tau})$ represent the cumulative distribution function of the joint distribution of $\left\{X_t\right\}$ at times $t_1 + \tau, \ldots, t_k + \tau$. Then, $\left\{X_t\right\}$ is said to be stationary if, for all k, for all τ, and for all $t_1, \ldots, t_k$,

$F_{X}(x_{t_1+\tau} ,\ldots, x_{t_k+\tau}) = F_{X}(x_{t_1},\ldots, x_{t_k}).$

Since τ does not affect F_X(.), F_X is not a function of time.

Examples

Two simulated time series processes, one stationary the other non-stationary. The Augmented Dickey–Fuller test is reported for each process and non-stationarity cannot be rejected for the second process.

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 (and hence its variance) 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 (so that the random variable may take one of N possible values) 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 { X_t }, by

$X_t=Y \qquad \text{ for all } t.$.

Then { X_t } 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 { X_t } by

$X_t=\cos (t+Y) \quad \text{ for } t \in \mathbb{R}.$

Then { X_t } 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 (WSS) or covariance stationarity. WSS random processes only require that 1st and 2nd moments 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

$\mathbb{E}\{x(t)\} = m_x(t) = m_x(t + \tau) \,\, \text{ for all } \, \tau \in \mathbb{R}$

and autocorrelation function

$\mathbb{E}\{x(t_1)x(t_2)\} = R_x(t_1, t_2) = R_x(t_1 + \tau, t_2 + \tau) = R_x(t_1 - t_2, 0) \,\, \text{ for all } \, \tau \in \mathbb{R}.$

The first property implies that the mean function m_x(t) must be constant. The second property implies that the correlation function depends only on the difference between t₁ and t₂ and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,

$\,\!R_x(t_1 - t_2, 0)\,$

the notation is often abbreviated and written as:

$R_x(\tau) \,\! \mbox{ where } \tau = t_1 - t_2.$

This also implies that the autocovariance depends only on τ = t₁ − t₂, since

$\,\! C_x(t_1,t_2) = C_x(t_1-t_2,0) = C_x(\tau).$

When processing WSS random signals with linear, time-invariant (LTI) filters, it is helpful to think of the correlation function as a linear operator. Since it is a circulant operator (depends only on the difference between the two arguments), 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.

Second-order stationarity

The case of second-order stationarity arises when the requirements of strict stationarity are only applied to pairs of random variables from the time-series. The definition of second order stationarity can be generalized to Nth order (for finite N) and strict stationary means stationary of all orders.

A process is second order stationary if the first and second order density functions satisfy

$f_X(x_1 : t_1 ) = f_X(x_1 : t_1 + \Delta), \,$

$f_X(x_1 ,x_2 : t_1, t_2 ) = f_X(x_1 ,x_2 : t_1 + \Delta, t_2 +\Delta ), \,$

for all t₁, t₂, and Δ. Such a process will be wide sense stationary if the mean and correlation functions are finite. A process can be wide sense stationary without being second order stationary.

Other terminology

The terminology used for types of stationarity other than strict stationarity can be rather mixed. Some examples follow.

• Priestley^[1][2] 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^[3] also uses 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

References

1. ^ Priestley, M.B. (1981) Spectral Analysis and Time Series, Academic Press. ISBN 0-12-564922-3
2. ^ Priestley, M.B. (1988) Non-linear and Non-stationary Time Series Analysis, Academic Press. ISBN 0-12-564911-8
3. ^ Honarkhah, M and Caers, J, 2010, Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling, Mathematical Geosciences, 42: 487 - 517

External links

• Spectral decomposition of a random function (Springer)