Stationary process

In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over time.

Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become stationary. The most common cause of violation of stationarity is a trend in the mean, which can be due either to the presence of a unit root or of a deterministic trend. 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.

A trend stationary process is not strictly stationary, but can easily be transformed into a stationary process by removing the underlying trend, which is solely a function of time. Similarly, processes with one or more unit roots can be made stationary through differencing. An important type of non-stationary process that does not include a trend-like behavior is a cyclostationary process, which is a stochastic process that varies cyclically with time.

For many applications strict-sense stationarity is too restrictive. Other forms of stationarity such as wide-sense stationarity or N-th-order stationarity are then employed. The definitions for different kinds of stationarity are not consistent among different authors (see Other terminology).

Strict-sense stationarity[edit]

Definition[edit]

Formally, let $\left\{X_{t}\right\}$ be a stochastic process and let $F_{X}(x_{t_{1}+\tau },\ldots ,x_{t_{n}+\tau })$ represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of $\left\{X_{t}\right\}$ at times $t_{1}+\tau ,\ldots ,t_{n}+\tau$ . Then, $\left\{X_{t}\right\}$ is said to be strictly stationary, strongly stationary or strict-sense stationary if^[1]^{: p. 155}

Weak or wide-sense stationarity[edit]

Definition[edit]

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 moment (i.e. the mean) and autocovariance do not vary with respect to time and that the 2nd moment is finite for all times. Any strictly stationary process which has a finite mean and covariance is also WSS.^[2]^{: p. 299}

So, a continuous time random process $\left\{X_{t}\right\}$ which is WSS has the following restrictions on its mean function $m_{X}(t)\triangleq \operatorname {E} [X_{t}]$ and autocovariance function $K_{XX}(t_{1},t_{2})\triangleq \operatorname {E} [(X_{t_{1}}-m_{X}(t_{1}))(X_{t_{2}}-m_{X}(t_{2}))]$ :

If a stochastic process is N-th-order stationary, then it is also M-th-order stationary for all $M\leq N$ .

If a stochastic process is second order stationary ( $N=2$ ) and has finite second moments, then it is also wide-sense stationary.^{: p. 159}

[1]

If a stochastic process is wide-sense stationary, it is not necessarily second-order stationary.^{: p. 159}

[1]

If a stochastic process is strict-sense stationary and has finite second moments, it is wide-sense stationary.^{: p. 299}

[2]

If two stochastic processes are jointly (M + N)-th-order stationary, this does not guarantee that the individual processes are M-th- respectively N-th-order stationary.^{: p. 159}

[1]

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.^[3]^[4] Thus wide sense stationarity would be equivalent to "stationary to order 2", which is different from the definition of second-order stationarity given here.

Priestley

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.^[5]

Honarkhah

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

Differencing[edit]

One way to make some time series stationary is to compute the differences between consecutive observations. This is known as differencing. Differencing can help stabilize the mean of a time series by removing changes in the level of a time series, and so eliminating trends. This can also remove seasonality, if differences are taken appropriately (e.g. differencing observations 1 year apart to remove a yearly trend).

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). Related techniques from signal analysis such as the wavelet transform and Fourier transform may also be helpful.

Lévy process

Stationary ergodic process

Wiener–Khinchin theorem

Ergodicity

Statistical regularity

Autocorrelation

Whittle likelihood

Enders, Walter (2010). Applied Econometric Time Series (Third ed.). New York: Wiley. pp. 53–57. 978-0-470-50539-7.

ISBN

Jestrovic, I.; Coyle, J. L.; Sejdic, E (2015). . Brain Research. 1589: 45–53. doi:10.1016/j.brainres.2014.09.035. PMC 4253861. PMID 25245522.

"The effects of increased fluid viscosity on stationary characteristics of EEG signal in healthy adults"

Hyndman, Athanasopoulos (2013). Forecasting: Principles and Practice. Otexts.

Stationary process

Strict-sense stationarity[edit]

Definition[edit]

Weak or wide-sense stationarity[edit]

Definition[edit]

[1]

[1]

[2]

[1]

Priestley

Honarkhah

Differencing[edit]

Lévy process

Stationary ergodic process

Wiener–Khinchin theorem

Ergodicity

Statistical regularity

Autocorrelation

Whittle likelihood

ISBN

"The effects of increased fluid viscosity on stationary characteristics of EEG signal in healthy adults"

https://www.otexts.org/fpp/8/1

Spectral decomposition of a random function (Springer)