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In the mathematical sciences
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process
Stochastic process

A stochastic process, or sometimes random process, is the counterpart to a deterministic process in probability theory. Instead of dealing with only one possible 'reality' of how the process might evolve under time , in a stochastic or random process there is some indeterminacy in its future evolution described by probability distribu...
 whose joint probability distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
 does not change when shifted in time or space. As a result, parameters such as the mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
 and variance
Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
, if they exist, also do not change over time or position.

Stationarity is used as a tool in time series analysis, where the raw data are often transformed to become stationary, for example, economic
Economics

File:Ballard Farmers' Market - vegetables.jpgEconomics is the Social sciences that studies the Production theory basics, Distribution , and Consumption of Good and Service ....
 data are often seasonal and/or dependent on the price level.






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In the mathematical sciences
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process
Stochastic process

A stochastic process, or sometimes random process, is the counterpart to a deterministic process in probability theory. Instead of dealing with only one possible 'reality' of how the process might evolve under time , in a stochastic or random process there is some indeterminacy in its future evolution described by probability distribu...
 whose joint probability distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
 does not change when shifted in time or space. As a result, parameters such as the mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
 and variance
Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
, if they exist, also do not change over time or position.

Stationarity is used as a tool in time series analysis, where the raw data are often transformed to become stationary, for example, economic
Economics

File:Ballard Farmers' Market - vegetables.jpgEconomics is the Social sciences that studies the Production theory basics, Distribution , and Consumption of Good and Service ....
 data are often seasonal and/or dependent on the price level. Processes are described as trend stationary if they are a linear combination of a stationary process and one or more processes exhibiting a trend
Trend estimation

When a series of measurements of a process is treated as a time series, trend estimation is the application of statistics techniques to make and justify statements about trends in the data....
. Transforming this data to leave a stationary data set for analysis is referred to as de-trending.

Definition


Formally, let be a stochastic process and let represent the cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 of the joint distribution of at times . Then, is said to be stationary if, for all , for all , and for all ,

Examples


As an example, white noise
White noise

White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency....
 is stationary. However, the sound of a cymbal
Cymbal

Cymbals are a modern percussion instrument. Cymbals consist of thin, normally round plates of various cymbal alloys; see cymbal making for a discussion of their manufacture....
 crashing is not stationary because the acoustic power of the crash (and hence its variance) diminishes with time.

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
Bernoulli scheme

In mathematics, the Bernoulli scheme is a generalization of the Bernoulli process to more than two possible outcomes. That is, it is a discrete-time stochastic process where each statistical independence random variable may take on one of N distinct possible values, with the outcome i occurring with probability , with , and...
. Other examples of a discrete-time stationary process with continuous sample space include autoregressive and moving average
Moving average model

In time series analysis, the moving average model is common approach for modeling univariate time series models. The notation MA refers to the moving average model of order q:...
 processes which are both subsets of the autoregressive moving average model
Autoregressive moving average model

In statistics and signal processing, autoregressive moving average models, sometimes called Box-Jenkins models after the iterative Box-Jenkins methodology usually used to estimate them, are typically applied to time series data....
.

Weaker forms of stationarity


Weak or wide-sense stationarity


A weaker form of stationarity commonly employed in signal processing
Signal processing

Signal processing is the analysis, interpretation, and manipulation of signal . Signals of interest include: audio signal processing, , time-varying measurement values and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals such as radio signals, and many others....
 is known as weak-sense stationarity, wide-sense stationarity (WSS) or covariance stationarity. WSS random processes only require that 1st and 2nd moments
Moment (mathematics)

The concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f of a real variable about a value c is...
 do not vary with respect to time. Any strictly stationary process which has a mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
 and a covariance
Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together .If two variables tend to vary together , then the covariance between the two variables will be positive....
 is also WSS.

So, a continuous
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
-time random process x(t) which is WSS has the following restrictions on its mean function

and autocorrelation
Autocorrelation

Autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies....
 function

The first property implies that the mean function mx(t) must be constant. The second property implies that the correlation function depends only on the difference between and and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,

we usually abbreviate the notation and write

This also implies that the autocovariance
Autocovariance

In statistics, given a real stochastic process X, the autocovariance is simply the covariance of the signal against a time-shifted version of itself....
 depends only on , since

When processing WSS random signals with linear
Linear

The word linear comes from the Latin word linearis, which means created by lines.In mathematics, a linear map or function f is a function which satisfies the following two properties......
, time-invariant (LTI
LTI system theory

Linear time-invariant system theory, most commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, electrical networks, signal processing, control theory, and other technical areas....
) filters, it is helpful to think of the correlation function as a linear operator. Since it is a circulant
Circulant matrix

In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector....
 operator (depends only on the difference between the two arguments), its eigenfunction
Eigenfunction

In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor....
s are the Fourier
Fourier series

In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sine wave . The study of Fourier series is a branch of Fourier analysis....
 complex exponential
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
s. Additionally, since the eigenfunction
Eigenfunction

In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor....
s of LTI operators are also complex exponential
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
s, LTI processing of WSS random signals is highly tractable—all computations can be performed in the frequency domain
Frequency domain

In electronics and control systems engineering, frequency domain is a term used to describe the analysis of mathematical functions or Signal with respect to frequency, rather than time....
. Thus, the WSS assumption is widely employed in signal processing
Signal processing

Signal processing is the analysis, interpretation, and manipulation of signal . Signals of interest include: audio signal processing, , time-varying measurement values and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals such as radio signals, and many others....
 algorithm
Algorithm

In mathematics, computing, linguistics and related subjects, an algorithm is a sequence of finite instructions, often used for calculation and data processing....
s.

Second-order stationarity


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 generalised to Nth order (for finite N) and strict stationary means stationary of all orders.

A process is second order stationary if the first and second order density functions satisfy for all , , and . Such a process will be wide sense stationary if the mean and corelation functions are finite. A process can be wide sense stationary without being second 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 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.

See also

  • Cyclostationary process
  • Stationary ergodic process
    Stationary ergodic process

    In probability theory, stationary ergodic process is a stochastic process which exhibits both stationary process and ergodic process. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties can be deduced from a single, sufficiently long sample of the process....
  • Time-invariant system
    Time-invariant system

    A time-invariant system is one whose output does not depend explicitly on time.Formally, if is the shifting operator ,then the operator is called time-invariant, if...