Stationary process

Stationarity called in mathematics a property of time-ordered, random or lawful processes ( stochastic processes or physical sizes), as well as of ordinary differential equations. This article deals with temporal processes. Stationarity is given with this if can specify an arithmetic mean and a statement of the deviations from the mean, which do not depend even on the time.

Definition

A stochastic process is called strongly stationary if the distribution of does not depend on the shift.

A stochastic process is called weakly stationary (rarely covariance stationary ) if

Here, the expectation value. stands for an arbitrary index set ( on which a binary operation is explained ), usually the integers, sometimes the natural numbers or the real numbers. Is often modeled with the time. denotes the variance covariance.

Interpretation

Stationarity is one of the most important properties of stochastic processes in time series analysis. With the stationarity properties that apply not only for individual time points, but invariants are obtained over time are. The time series is at all times the same mean and the same variance. ( The most important class of non-stationary processes are integrated processes. )

With the first property you can move to a new process, then for the. This process is also called centered process. One can therefore without loss of generality assume a stationary stochastic process has the mean 0

The second property simply says that each of the random variables has a finite variance and therefore belongs to the Hilbert space. From this, it also follows that the expected value exists.

The third requirement establishes a relationship between the different time points, making it the most significant feature. It indicates that the covariance between the points in time does not depend on the two points in time itself, but only on the distance of the two points in time to each other. The condition can also be formulated so that a function is only one variable. This has, among others, the consequence that an infinite block Toeplitz matrix.

Geometrical Meaning

The geometric interpretation of the univariate case ( ) relies on the Hilbert space whose elements are the individual random variables of the process. The geometric interpretation supports a deeper understanding of the concept of stationarity.

Then says the above interpretation following that for observation all the same angle. Is increased by one, then still further rotated by the same angle.

Claim ( ii ) means nothing more than, ie the angle between the unit and each process variable is constant. Here, a degree of latitude is cut out of the unit sphere.

Station linearization

To make a non-stationary time series stationary is an important first task in time series analysis. Widespread methods here are the formation of differences, the rescaling or the logarithm of the time series. Generally, one can try to obtain a stationary time series by using a suitable seasonal trend model.

Examples

The most important (weakly ) stationary process is the white noise. Furthermore, certain Gaussian processes and ARMA models stationary. From theoretical significance are also harmonic processes that are stationary under certain conditions.