In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. More formally, the joint probability distribution of the process remains the same when shifted in time. This implies that the process is statistically consistent across different time periods. Because many statistical procedures in time series analysis assume stationarity, non-stationary data are frequently transformed to achieve stationarity before analysis.
A common cause of non-stationarity is a trend in the mean, which can be due to either a unit root or a deterministic trend. In the case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. With a deterministic trend, the process is called trend-stationary, and shocks have only transitory effects, with the variable tending towards a deterministically evolving mean. A trend-stationary process is not strictly stationary but can be made stationary by removing the trend. Similarly, processes with unit roots can be made stationary through differencing.
Another type of non-stationary process, distinct from those with trends, is a cyclostationary process, which exhibits cyclical variations over time.
Strict stationarity, as defined above, can be too restrictive for many applications. Therefore, other forms of stationarity, such as wide-sense stationarity or N-th-order stationarity, are often used. The definitions for different kinds of stationarity are not consistent among different authors (see Other terminology).
Formally, let
\left\{Xt\right\}
FX
| (x | |
| t1+\tau |
,\ldots,
| x | |
| tn+\tau |
)
\left\{Xt\right\}
t1+\tau,\ldots,tn+\tau
\left\{Xt\right\}
Since
\tau
FX( ⋅ )
FX
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.
Let
Y
\left\{Xt\right\}
Xt=Y forallt.
\left\{Xt\right\}
Y
Y
The time average of
Xt
As a further example of a stationary process for which any single realisation has an apparently noise-free structure, let
Y
[0,2\pi]
\left\{Xt\right\}
Xt=\cos(t+Y) fort\inR.
\left\{Xt\right\}
(t+Y)
2\pi
Y
t
Keep in mind that a weakly white noise is not necessarily strictly stationary. Let
\omega
(0,2\pi)
\left\{zt\right\}
zt=\cos(t\omega) (t=1,2,...)
Then
\begin{align} E(zt)&=
| 1 | |
| 2\pi |
| 2\pi | |
| \int | |
| 0 |
\cos(t\omega)d\omega=0,\\ \operatorname{Var}(zt)&=
| 1 | |
| 2\pi |
| 2\pi | |
| \int | |
| 0 |
\cos2(t\omega)d\omega=1/2,\\ \operatorname{Cov}(zt,zj)&=
| 1 | |
| 2\pi |
| 2\pi | |
| \int | |
| 0 |
\cos(t\omega)\cos(j\omega)d\omega=0 \forallt ≠ j. \end{align}
\{zt\}
In, the distribution of
n
n
n
N
\left\{Xt\right\}
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]
\left\{Xt\right\}
mX(t)\triangleq\operatornameE[Xt]
KXX(t1,t2)\triangleq\operatorname
| E[(X | |
| t1 |
-mX(t1))(X
| t2 |
-mX(t2))]
The first property implies that the mean function
mX(t)
t1
t2
KXX(t1-t2,0)
the notation is often abbreviated by the substitution
\tau=t1-t2
KXX(\tau)\triangleqKXX(t1-t2,0)
This also implies that the autocorrelation depends only on
\tau=t1-t2
RX(t1,t2)=RX(t1-t2,0)\triangleqRX(\tau).
The third property says that the second moments must be finite for any time
t
The main advantage of wide-sense stationarity is that it places the time-series in the context of Hilbert spaces. Let H be the Hilbert space generated by (that is, the closure of the set of all linear combinations of these random variables in the Hilbert space of all square-integrable random variables on the given probability space). By the positive definiteness of the autocovariance function, it follows from Bochner's theorem that there exists a positive measure
\mu
\omega\xi
t
Xt=\inte-d\omegaλ,
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.
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.
In the case where
\left\{Xt\right\}
KXX(t1,t2)=\operatorname
| E[(X | |
| t1 |
-mX(t1))\overline{(X
| t2 |
-mX(t2))}]
JXX(t1,t2)=\operatorname
| E[(X | |
| t1 |
-mX(t1))(X
| t2 |
-mX(t2))]
\left\{Xt\right\}
The concept of stationarity may be extended to two stochastic processes.
Two stochastic processes
\left\{Xt\right\}
\left\{Yt\right\}
FXY
| (x | |
| t1 |
,\ldots,
| x | |
| tm |
| ,y | |||||||
|
,\ldots,
| y | |||||||
|
)
Two random processes
\left\{Xt\right\}
\left\{Yt\right\}
Two stochastic processes
\left\{Xt\right\}
\left\{Yt\right\}
KXY(t1,t2)=\operatorname
| E[(X | |
| t1 |
-mX(t1))(Y
| t2 |
-mY(t2))]
\tau=t1-t2
N=2
The terminology used for types of stationarity other than strict stationarity can be rather mixed. Some examples follow.
In time series analysis and stochastic processes, stationarizing a time series is a crucial preprocessing step aimed at transforming a non-stationary process into a stationary one. Several techniques exist for achieving this, depending on the type and order of non-stationarity present. For first-order non-stationarity, where the mean of the process varies over time, differencing is a common and effective method: it transforms the series by subtracting each value from its predecessor, thus stabilizing the mean. For non-stationarities up to the second order, time-frequency analysis (e.g., Wavelet transform, Wigner distribution function, or Short-time Fourier transform) can be employed to isolate and suppress time-localized, nonstationary spectral components. Additionally, surrogate data methods can be used to construct strictly stationary versions of the original 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]
The choice of method for time series stationarization depends on the nature of the non-stationarity and the goals of the analysis, especially when building models that require strict stationarity assumptions, such as ARMA or spectral-based techniques. More details on some time series stationarization methods are presented below.
One way to make some time series first-order 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.
The surrogate method for stationarization [7] works by generating a new time series that preserves certain statistical properties of the original series while removing its nonstationary components.[8] [9] [10] A common approach is to apply the Fourier Transform to the original time series to obtain its magnitude and phase spectra. The magnitude spectrum, which determines the power distribution across frequencies, is retained to preserve the global autocorrelation structure. The phase spectrum, which encodes the temporal alignment of frequency components and is often responsible for time-dependent dynamics in the time series (like non-stationarities), is then randomized, typically by replacing it with a set of random phases drawn uniformly from
[-\pi,\pi]
