Stationary process

Nth-order stationarity

In Eq.1, the distribution of ${\displaystyle n}$ samples of the stochastic process must be equal to the distribution of the samples shifted in time for all ⁠${\displaystyle n}$⁠. ⁠${\displaystyle N}$⁠th-order stationarity is a weaker form of stationarity where this is only requested for all ${\displaystyle n}$ up to a certain order ⁠${\displaystyle N}$⁠. A random process ${\displaystyle \left\{X_t\right\}}$ is said to be ⁠${\displaystyle N}$⁠th-order stationary if:

Weak or wide-sense stationarity

Definition

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 that has a finite mean and covariance is also WSS.

So, a continuous time random process ${\displaystyle \left\{X_t\right\}}$ that is WSS has the following restrictions on its mean function ${\displaystyle m_X(t)\triangleq \mathrm{E}[X_t]}$ and autocovariance function ⁠${\displaystyle K_{XX}(t_1,t_2)\triangleq \mathrm{E}[(X_{t_1}-m_X(t_1))(X_{t_2}-m_X(t_2))]}$⁠:

The first property implies that the mean function ${\displaystyle m_X(t)}$ must be constant. The second property implies that the autocovariance function depends only on the difference between ${\displaystyle t_1}$ and ${\displaystyle t_2}$ and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,

${\displaystyle K_{XX}(t_1-t_2,0)}$

the notation is often abbreviated by the substitution ⁠${\displaystyle \tau =t_1-t_2}$⁠:

${\displaystyle K_{XX}(\tau )\triangleq K_{XX}(t_1-t_2,0)}$

This also implies that the autocorrelation depends only on ⁠${\displaystyle \tau =t_1-t_2}$⁠, that is

${\displaystyle R_X(t_1,t_2)=R_X(t_1-t_2,0)\triangleq R_X(\tau ).}$

The third property says that the second moments must be finite for any time ⁠${\displaystyle t}$⁠.

Motivation

The main advantage of wide-sense stationarity is that it places the time-series in the context of Hilbert spaces. Let ⁠${\displaystyle H}$⁠ be the Hilbert space generated by ⁠${\displaystyle \{x(t)\}}$⁠ (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 ${\displaystyle \mu }$ on the real line such that ⁠${\displaystyle H}$⁠ is isomorphic to the Hilbert subspace of ⁠${\displaystyle L^2(\mu )}$⁠ generated by ⁠${\displaystyle \{e^{-2\pi i\xi \cdot t}\}}$⁠. This then gives the following Fourier-type decomposition for a continuous time stationary stochastic process: there exists a stochastic process ${\displaystyle {\omega }_{\xi }}$ with orthogonal increments such that, for all ⁠${\displaystyle t}$⁠:

${\displaystyle X_t=\int e^{-2\pi i\lambda \cdot t}d{\omega }_{\lambda },}$

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.

Definition for complex stochastic process

In the case where ${\displaystyle \left\{X_t\right\}}$ is a complex stochastic process the autocovariance function is defined as ${\displaystyle K_{XX}(t_1,t_2)=\mathrm{E}[(X_{t_1}-m_X(t_1))\overline{(X_{t_2}-m_X(t_2))}]}$ and, in addition to the requirements in Eq.3, it is required that the pseudo-autocovariance function ${\displaystyle J_{XX}(t_1,t_2)=\mathrm{E}[(X_{t_1}-m_X(t_1))(X_{t_2}-m_X(t_2))]}$ depends only on the time lag. In formulas, ${\displaystyle \left\{X_t\right\}}$ is WSS, if

Joint stationarity

The concept of stationarity may be extended to two stochastic processes.

Joint strict-sense stationarity

Two stochastic processes ${\displaystyle \left\{X_t\right\}}$ and ${\displaystyle \left\{Y_t\right\}}$ are called jointly strict-sense stationary if their joint cumulative distribution ${\displaystyle F_{XY}(x_{t_1},\dots ,x_{t_m},y_{t_1^{{}^{\prime }}},\dots ,y_{t_n^{{}^{\prime }}})}$ remains unchanged under time shifts, i.e. if

Joint (M + N)th-order stationarity

Two stochastic processes ${\displaystyle \left\{X_t\right\}}$ and ${\displaystyle \left\{Y_t\right\}}$ are said to be jointly (⁠${\displaystyle M+N}$⁠)th-order stationary if:

Joint weak or wide-sense stationarity

Two stochastic processes ${\displaystyle \left\{X_t\right\}}$ and ${\displaystyle \left\{Y_t\right\}}$ are called jointly wide-sense stationary if they are both wide-sense stationary and their cross-covariance function ${\displaystyle K_{XY}(t_1,t_2)=\mathrm{E}[(X_{t_1}-m_X(t_1))(Y_{t_2}-m_Y(t_2))]}$ depends only on the time difference ⁠${\displaystyle \tau =t_1-t_2}$⁠. This may be summarized as follows:

Relationships between types of stationarity

• If a stochastic process is ⁠${\displaystyle N}$⁠th-order stationary, then it is also ⁠${\displaystyle M}$⁠th-order stationary for all ⁠${\displaystyle M\le N}$⁠.
• If a stochastic process is second-order stationary (⁠${\displaystyle N=2}$⁠) and has finite second moments, then it is also wide-sense stationary.
• If a stochastic process is wide-sense stationary, it is not necessarily second-order stationary.
• If a stochastic process is strict-sense stationary and has finite second moments, it is wide-sense stationary.
• If two stochastic processes are jointly (⁠${\displaystyle M+N}$⁠)th-order stationary, this does not guarantee that the individual processes are ⁠${\displaystyle M}$⁠th- respectively ⁠${\displaystyle N}$⁠th-order stationary.

Other terminology

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

• Priestley uses stationary up to order ⁠${\displaystyle m}$⁠ if conditions similar to those given here for wide sense stationarity apply relating to moments up to order ⁠${\displaystyle m}$⁠. Thus wide sense stationarity would be equivalent to "stationary to order 2", which is different from the definition of second-order stationarity given here.
• Honarkhah and Caers also use the assumption of stationarity in the context of multiple-point geostatistics, where higher ⁠${\displaystyle n}$⁠-point statistics are assumed to be stationary in the spatial domain.

Techniques to stationarize a non-stationary process

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, non-stationary spectral components. Additionally, surrogate data methods can be used to construct strictly stationary versions of the original time series.

One of the ways of identifying non-stationary time series is the ACF (Autocorrelation Function) plot, which plots the autocorrelation against the lag. Sometimes, patterns related to non-stationarity will be more visible in the ACF plot than in the original time series; however, this is not always the case.

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.

Stationarization by means of differencing

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.

Stationarization by means of the surrogate method

The surrogate method for stationarization works by generating a new time series that preserves certain statistical properties of the original series while removing its nonstationary components. 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 ${\displaystyle [-\pi ,\pi ]}$ while enforcing conjugate symmetry to ensure a real-valued inverse. Applying the inverse Fourier Transform to the modified spectra yields a strictly stationary surrogate time series: one with the same power spectrum as the original but lacking the temporal structures that caused non-stationarity. This technique is often used in hypothesis tests for probing the stationarity property.