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Concept
Ergodic process
Summary
In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. In this regime, any collection of random samples from a process must represent the average statistical properties of the entire regime. Conversely, a process that is not in ergodic regime is said to be in non-ergodic regime.
One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process has constant mean
and autocovariance
that depends only on the lag and not on time . The properties and
are ensemble averages (calculated over all possible sample functions ), not time averages.
The process is said to be mean-ergodic or mean-square ergodic in the first moment
if the time average estimate
converges in squared mean to the ensemble average as .
Likewise,
the process is said to be autocovariance-ergodic or d moment
if the time average estimate
converges in squared mean to the ensemble average , as .
A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.
The notion of ergodicity also applies to discrete-time random processes
for integer .
A discrete-time random process is ergodic in mean if
converges in squared mean
to the ensemble average ,
as .
Ergodicity means the ensemble average equals the time average. Following are examples to illustrate this principle.
Each operator in a call centre spends time alternately speaking and listening on the telephone, as well as taking breaks between calls. Each break and each call are of different length, as are the durations of each 'burst' of speaking and listening, and indeed so is the rapidity of speech at any given moment, which could each be modelled as a random process.
Take N call centre operators (N should be a very large integer) and plot the number of words spoken per minute for each operator over a long period (several shifts). For each operator you will have a series of points, which could be joined with lines to create a 'waveform'.
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Ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process.
Ergodic theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc.
Stationary process
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over time. If you draw a line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles around the trend line, but overall it does not trend up nor down.