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.[1] Consequently, parameters such as mean and variance also do not change over time.
In other words, a line drawn through the middle of a stationary process — i.e. the trend line — is flat. It may have 'seasonal' cycles around this trend line, but overall it does not trend up nor down.
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).
^Gagniuc, Paul A. (2017). Markov Chains: From Theory to Implementation and Experimentation. USA, NJ: John Wiley & Sons. pp. 1–256. ISBN 978-1-119-38755-8.
and 24 Related for: Stationary process information
statistics, a stationaryprocess (or a strict/strictly stationaryprocess or strong/strongly stationaryprocess) is a stochastic process whose unconditional...
stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the random process will...
time series to be non-stationary, yet have no unit root and be trend-stationary. In both unit root and trend-stationaryprocesses, the mean can be growing...
distribution of a stationaryprocess or stationary time series The set of joint probability distributions of a stationaryprocess or stationary time series...
stochastic process. For a strongly stationaryprocess, the conditional entropy for latest random variable eventually tend towards this rate value. A process X...
time. If { X t } {\displaystyle \left\{X_{t}\right\}} is a weakly stationary (WSS) process, then the following are true:: p. 163 μ t 1 = μ t 2 ≜ μ {\displaystyle...
states; every stationaryprocess in N outcomes is a Bernoulli scheme, and vice versa. Bessel process Birth–death process Branching process Branching random...
Unit root processes, trend-stationaryprocesses, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation...
The term "stationary source" may refer to one of the following: A source of data produced by a stationaryprocess, in the mathematical theory of probability...
[-\log c(n,n,X)]} exist and are equal for any stationaryprocess including the stationary ergodic process X. Denote it as H. Argue that both c ( i , k...
n-fBm. n-fBm is a Gaussian, self-similar, non-stationaryprocess whose increments of order n are stationary. For n = 1, n-fBm is classical fBm. Like the...
covariance-stationary series. A time series is integrated of order d if ( 1 − L ) d X t {\displaystyle (1-L)^{d}X_{t}\ } is a stationaryprocess, where...
distribution of the stationary stochastic process remains the same. A sequence of random variables forms a stationary stochastic process only if the random...
various statistics of a stochastic process. For example, a wide-sense stationaryprocess X ( t ) {\displaystyle X(t)} has constant mean μ X = E [ X ( t ) ]...
hold the stationary phase in place. The separation process in CPC is governed solely by the partitioning of solutes between the stationary and mobile...
process the two concepts are equivalent.: p. 518 A Gaussian stochastic process is strict-sense stationary if and only if it is wide-sense stationary...
interleaved stationaryprocesses. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum...
the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second...
In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on...
time-reversed process is defined to be X ^ t = X T − t {\displaystyle {\hat {X}}_{t}=X_{T-t}} . By Kelly's lemma this process has the same stationary distribution...
and random processes. Academic Press. pp. 370–5. The Wiener–Khinchin theorem makes sense of this formula for any wide-sense stationaryprocess under weaker...
time series to be non-stationary, have no unit root yet be trend-stationary. In both unit root and trend-stationaryprocesses, the mean can be growing...
The Airy processes are a family of stationary stochastic processes that appear as limit processes in the theory of random growth models and random matrix...
frequently are non-stationary. They are typically modelled as either trend-stationary or difference stationary. A trend stationaryprocess {yt} evolves according...
Global Information