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| '''Time reversibility''' is an attribute of some [[stochastic process]]es and some [[deterministic]] processes.
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| If a [[stochastic process]] is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which state arrived later.
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| If a [[deterministic]] process is time reversible, then the time-reversed process satisfies the same dynamical equations as the original process (see [[reversible dynamics]]); in other words, the equations are invariant or symmetric under a change in the sign of time. [[Classical mechanics]] and [[optics]] are both time-reversible. Modern physics is not quite time-reversible; instead it exhibits a broader symmetry, [[CPT symmetry]].{{fact|date=December 2013}}
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| Time reversibility generally occurs when, within a process, it can be broken up into sub-processes which undo the effects of each other. For example, in [[phylogenetics]], a time-reversible nucleotide substitution model such as the [[Substitution model#GTR: Generalised time reversible|generalised time reversible]] model has the total overall rate into a certain nucleotide equal to the total rate out of that same nucleotide.
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| [[Time Reversal Signal Processing|Time Reversal]], specifically in the field of [[acoustics]], is a process by which the linearity of sound waves is used to reverse a received signal; this signal is then re-emitted and a temporal compression occurs, resulting in a reverse of the initial excitation waveform being played at the initial source. Mathias Fink is credited with proving Acoustic Time Reversal by experiment.
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| ==Stochastic processes==
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| A formal definition of time-reversibility is stated by Tong<ref>Tong(1990), Section 4.4</ref> in the context of time-series. In general, a [[Gaussian process]] is time-reversible. The process defined by a time-series model which represents values as a linear combination of past values and of present and past innovations (see [[Autoregressive moving average model]]) is, except for limited special cases, not time-reversible unless the innovations have a [[normal distribution]] (in which case the model is a Gaussian process).
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| A stationary [[Markov Chain]] is reversible if the transition matrix {''p<sub>ij</sub>''} and the stationary distribution {''π<sub>j</sub>''} satisfy
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| :<math>\pi_i p_{ij} =\pi_j p_{ji}, \,</math> | |
| for all ''i'' and ''j''.<ref>Isham (1991), p 186</ref> Such Markov Chains provide examples of stochastic processes which are time-reversible but non-Gaussian.
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| Time reversal of numerous classes of stochastic processes have been studied including [[Lévy processes]]<ref>{{cite doi|10.1214/aop/1176991776}}</ref> [[stochastic network]]s ([[Kelly's lemma]])<ref>{{cite jstor|1425912}}</ref> [[birth and death processes]] <ref>{{cite doi|10.3836/tjm/1270133555}}</ref> [[Markov chain]]s<ref>{{cite book | title = Markov Chains | first = J. R. | last= Norris | authorlink = James R. Norris | publisher = Cambridge University Press | year =1998 | isbn = 0521633966}}</ref> and [[piecewise deterministic Markov processes]].<ref>{{cite doi|10.1214/EJP.v18-1958}}</ref>
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| ==See also==
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| *[[Memorylessness]]
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| *[[Kolmogorov’s criterion]]
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| *[[Reversible cellular automaton]]
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| *[[Reversible dynamics]]
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| *[[T-symmetry]]
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| ==Notes==
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| <references/>
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| ==References==
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| :*Isham, V. (1991) "Modelling stochastic phenomena". In: ''Stochastic Theory and Modelling'', Hinkley, DV., Reid, N., Snell, E.J. (Eds). Chapman and Hall. ISBN 0-412-30390-9 {{Please check ISBN|reason=Check digit (9) does not correspond to calculated figure.}}.
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| :*Tong, H. (1990) ''Non-linear Time Series: A Dynamical System Approach''. Oxford UP. ISBN 0-19-852300-9
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| [[Category:Stochastic processes]]
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| [[Category:Time series analysis]]
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