Stationary ergodic process
Encyclopedia
In probability theory
, stationary ergodic process is a stochastic process
which exhibits both stationarity
and ergodicity
. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the process.
Stationarity is the property of a random process which guarantees that its statistical properties, such as the mean value, its moments
and variance
, will not change over time. A stationary process is one whose probability distribution
is the same at all times. For more information see stationary process
.
Several sub-types of stationarity are defined: first-order, second-order, nth-order, wide-sense and strict-sense.
For details please see the reference below.
An ergodic
process is one which conforms to the ergodic theorem. The theorem allows the time average of a conforming process to equal the ensemble average. In practice this means that statistical sampling can be performed at one instant across a group of identical processes or sampled over time on a single process with no change in the measured result. Also see ergodic theory
and ergodic process
.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, stationary ergodic process is a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
which exhibits both stationarity
Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
and ergodicity
Ergodic process
In signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long sample of the process.- Specific definitions :...
. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the process.
Stationarity is the property of a random process which guarantees that its statistical properties, such as the mean value, its moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
, will not change over time. A stationary process is one whose probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
is the same at all times. For more information see stationary process
Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
.
Several sub-types of stationarity are defined: first-order, second-order, nth-order, wide-sense and strict-sense.
For details please see the reference below.
An ergodic
Ergodic (adjective)
In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.-Etymology:The word ergodic is...
process is one which conforms to the ergodic theorem. The theorem allows the time average of a conforming process to equal the ensemble average. In practice this means that statistical sampling can be performed at one instant across a group of identical processes or sampled over time on a single process with no change in the measured result. Also see ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
and ergodic process
Ergodic process
In signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long sample of the process.- Specific definitions :...
.