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'''Time reversibility''' is an attribute of some [[stochastic process]]es and some [[deterministic]] processes.
 
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.  
 
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}}
 
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.
 
[[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.
 
==Stochastic processes==
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).
 
A stationary [[Markov Chain]] is reversible if the transition matrix {''p<sub>ij</sub>''} and the stationary distribution {''π<sub>j</sub>''} satisfy
:<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.
 
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>
 
==See also==
*[[Memorylessness]]
*[[Kolmogorov’s criterion]]
*[[Reversible cellular automaton]]
*[[Reversible dynamics]]
*[[T-symmetry]]
 
==Notes==
<references/>
 
==References==
 
:*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.}}.
:*Tong, H. (1990) ''Non-linear Time Series: A Dynamical System Approach''. Oxford UP. ISBN 0-19-852300-9
 
 
[[Category:Stochastic processes]]
[[Category:Time series analysis]]

Revision as of 12:50, 31 January 2014

Time reversibility is an attribute of some stochastic processes and some deterministic processes.

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.

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.Template:Fact

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 generalised time reversible model has the total overall rate into a certain nucleotide equal to the total rate out of that same nucleotide.

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.

Stochastic processes

A formal definition of time-reversibility is stated by Tong[1] 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).

A stationary Markov Chain is reversible if the transition matrix {pij} and the stationary distribution {πj} satisfy

for all i and j.[2] Such Markov Chains provide examples of stochastic processes which are time-reversible but non-Gaussian.

Time reversal of numerous classes of stochastic processes have been studied including Lévy processes[3] stochastic networks (Kelly's lemma)[4] birth and death processes [5] Markov chains[6] and piecewise deterministic Markov processes.[7]

See also

Notes

  1. ↑ Tong(1990), Section 4.4
  2. ↑ Isham (1991), p 186
  3. ↑ Template:Cite doi
  4. ↑ Template:Cite jstor
  5. ↑ Template:Cite doi
  6. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  7. ↑ Template:Cite doi

References

  • 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 Template:Please check ISBN.
  • Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford UP. ISBN 0-19-852300-9
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