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| In [[mathematics]], 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. Consequently, parameters such as the [[mean]] and [[variance]], if they are present, also do not change over time and do not follow any trends.
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| Stationarity is used as a tool in [[time series analysis]], where the raw data is often transformed to become stationary; for example, [[economics|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]].
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| Note that a "stationary process" is not the same thing as a "process with a [[stationary distribution]]".{{clarification needed|date=January 2013|see talk page of 11 January 2013}} 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.
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| ==Definition==
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| Formally, let <math>\left\{X_t\right\}</math> be a [[stochastic process]] and let <math>F_{X}(x_{t_1 + \tau}, \ldots, x_{t_k + \tau})</math> represent the [[cumulative distribution function]] of the [[joint distribution]] of <math>\left\{X_t\right\}</math> at times <math>t_1 + \tau, \ldots, t_k + \tau</math>. Then, <math>\left\{X_t\right\}</math> is said to be stationary if, for all <math>k</math>, for all <math>\tau</math>, and for all <math>t_1, \ldots, t_k</math>,
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| :<math> F_{X}(x_{t_1+\tau} ,\ldots, x_{t_k+\tau}) = F_{X}(x_{t_1},\ldots, x_{t_k}).</math>
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| Since <math>\tau</math> does not affect <math>F_X(.)</math>, <math> F_{X}</math> is not a function of time.
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| ==Examples==
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| [[File:Stationarycomparison.png|thumb|right|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.]]
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| 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.
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| An example of a [[Discrete-time stochastic process|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 model|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 root]]s exist in the model.
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| Let ''Y'' be any scalar [[random variable]], and define a time-series { ''X<sub>t</sub>'' }, by
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| :<math>X_t=Y \qquad \text{ for all } t.</math>.
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| Then { ''X<sub>t</sub>'' } 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.''
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| 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 (continuous)|uniform distribution]] on (0,2π] and define the time series { ''X<sub>t</sub>'' } by
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| :<math>X_t=\cos (t+Y) \quad \text{ for } t \in \mathbb{R}. </math>
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| Then { ''X<sub>t</sub>'' } is strictly stationary.
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| ==Weaker forms of stationarity==
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| ===Weak or wide-sense stationarity===<!-- Wide-sense stationary process redirects here -->
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| 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 (mathematics)|moment]] and [[covariance]] do not vary with respect to time. Any strictly stationary process which has a [[mean]] and a [[covariance]] is also WSS.
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| So, a [[continuous function|continuous]]-time [[random process]] ''x''(''t'') which is WSS has the following restrictions on its mean function | |
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| :<math>\mathbb{E}[x(t)] = m_x(t) = m_x(t + \tau) \,\, \text{ for all } \, \tau \in \mathbb{R}</math> | |
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| and [[autocovariance]] function
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| :<math>\mathbb{E}[(x(t_1)-m_x(t_1))(x(t_2)-m_x(t_2))] = C_x(t_1, t_2) = C_x(t_1 + (-t_2), t_2 + (-t_2)) = C_x(t_1 - t_2, 0).</math>
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| The first property implies that the mean function ''m''<sub>''x''</sub>(''t'') must be constant. The second property implies that the covariance function depends only on the ''difference'' between <math>t_1</math> and <math>t_2</math> and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,
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| :<math>\,\!C_x(t_1 - t_2, 0)\,</math>
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| the notation is often abbreviated and written as:
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| :<math>C_x(\tau) \,\! \mbox{ where } \tau = t_1 - t_2.</math> | |
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| This also implies that the [[autocorrelation]] depends only on <math>\tau = t_1 - t_2</math>, since
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| :<math>\,\! R_x(t_1,t_2) = R_x(t_1-t_2).</math>
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| The main advantage of wide-sense stationarity is that it places the time-series in the context of [[Hilbert space]]s. 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 ''L''<sup>2</sup>(''μ'') generated by {''e<sup>-2πiξ⋅t</sup>''}. This then gives the following Fourier-type decomposition for continuous time stationary stochastic process: there exists a stochastic process ''ω<sub>ξ</sub>'' with [[orthogonal increments]] such that, for all ''t''
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| :<math>x(t) = \int e^{- 2 \pi i \lambda \cdot t} d \omega_{\lambda},</math>
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| where the integral on the right hand side is interpreted in a suitable (Riemann) sense. Same result holds for a discrete-time stationary process, with the spectral measure now defined on the unit circle.
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| When processing WSS random signals with [[linear]], [[time-invariant]] ([[LTI system theory|LTI]]) [[filter (signal processing)|filter]]s, it is helpful to think of the correlation function as a [[linear operator]]. Since it is a [[circulant matrix|circulant]] operator (depends only on the difference between the two arguments), its eigenfunctions are the [[Fourier series|Fourier]] complex exponentials. Additionally, since the [[eigenfunction]]s of LTI operators are also [[exponential function|complex exponential]]s, 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 [[algorithm]]s.
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| ===Second-order stationarity===
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| 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 ''N''th order (for finite ''N'') and strict stationary means stationary of all orders.
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| A process is second order stationary if the first and second order density functions satisfy
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| :<math> f_X(x_1 : t_1 ) = f_X(x_1 : t_1 + \Delta), \, </math>
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| :<math> f_X(x_1 ,x_2 : t_1, t_2 ) = f_X(x_1 ,x_2 : t_1 + \Delta, t_2 +\Delta ), \, </math>
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| for all <math> t_1 </math>, <math>t_2</math>, and <math>\Delta</math>. 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.{{Citation needed|reason=There is no reference given for this section on second-order stationarity; specifically, there is no reference for the difference between wide sense and second order stationarity when the first and second order statistics are not finite.|date=April 2013}}
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| ===Other terminology===
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| The terminology used for types of stationarity other than strict stationarity can be rather mixed. Some examples follow.
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| :*Priestley<ref>{{cite book |last=Priestley |first=M. B. |year=1981 |title=Spectral Analysis and Time Series |publisher=Academic Press |isbn=0-12-564922-3 }}</ref><ref>{{cite book |last=Priestley |first=M. B. |year=1988 |title=Non-linear and Non-stationary Time Series Analysis |publisher=Academic Press |isbn=0-12-564911-8 }}</ref> 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.
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| :* 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.<ref>{{cite journal |last=Honarkhah |first=M. |last2=Caers |first2=J. |year=2010 |doi=10.1007/s11004-010-9276-7 |title=Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling |journal=Mathematical Geosciences |volume=42 |issue=5 |pages=487–517 }}</ref>
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| ==See also==
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| * [[Lévy process]]
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| * [[Stationary ergodic process]]
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| * [[Wiener–Khinchin theorem]]
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| * [[Ergodicity]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://eom.springer.de/s/s086360.htm Spectral decomposition of a random function (Springer)]
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| {{Statistics|analysis}}
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| [[Category:Stochastic processes]]
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| [[Category:Signal processing]]
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