Stationary process
Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become stationary.
In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting.
In the latter case of a deterministic trend, the process is called a trend-stationary process, and stochastic shocks have only transitory effects after which the variable tends toward a deterministically evolving (non-constant) mean.
Similarly, processes with one or more unit roots can be made stationary through differencing.
White noise is the simplest example of 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.
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
As a further example of a stationary process for which any single realisation has an apparently noise-free structure, let
Keep in mind that a weakly white noise is not necessarily strictly stationary.
is a white noise in the weak sense (the mean and cross-covariances are zero, and the variances are all the same), however it is not strictly stationary.
Any strictly stationary process which has a finite mean and covariance is also WSS.[2]: p.
The second property implies that the autocovariance function depends only on the difference between
The main advantage of wide-sense stationarity is that it places the time-series in the context of Hilbert spaces.
on the real line such that H is isomorphic to the Hilbert subspace of L2(μ) generated by {e−2πiξ⋅t}.
where the integral on the right-hand side is interpreted in a suitable (Riemann) sense.
The same result holds for a discrete-time stationary process, with the spectral measure now defined on the unit circle.
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.
is a complex stochastic process the autocovariance function is defined as
is WSS, if The concept of stationarity may be extended to two stochastic processes.
remains unchanged under time shifts, i.e. if Two random processes
One way to make some time series stationary is to compute the differences between consecutive observations.
Transformations such as logarithms can help to stabilize the variance of a time series.
One of the ways for identifying non-stationary times series is the ACF plot.
Sometimes, patterns will be more visible in the ACF plot than in the original time series; however, this is not always the case.
[6] Another approach to identifying non-stationarity is to look at the Laplace transform of a series, which will identify both exponential trends and sinusoidal seasonality (complex exponential trends).
