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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. [1] In this regime, any collection of random samples from a process must represent the average statistical properties of the entire regime.
The mathematical definition of ergodicity aims to capture ordinary every-day ideas about randomness.This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as diffusion and Brownian motion, as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, the ...
In probability theory, a 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 ...
Ergodic theory is often concerned with ergodic transformations.The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not ...
Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible. Systems that are ergodic are said to have the property of ergodicity; a broad range of systems in geometry, physics, and probability are ergodic. Ergodic systems are studied in ergodic theory.
In ergodic theory, a measure-preserving dynamical system is called "ergodic" iff any measurable subset such that () = implies = or (up to a null set). The terminology is inconsistent. Given a Markov chain with a stationary distribution that is strictly positive on all states, the Markov chain is irreducible iff its corresponding measure ...
The TRM reverses and retransmits the signal as usual, and a more focused wave travels toward the target. As the process is repeated, the waves become more and more focused on the target. Yet another variation is to use a single transducer and an ergodic cavity. Intuitively, an ergodic cavity is one that will allow a wave originating at any ...
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.