Comments (12)
Just as an addition: A dynamical system does not need to be hyperbolic for the Pesin's identity to be fulfilled, but that is indeed a sufficient condition.
Pesin's identity is fulfilled if and only if the dynamical system is endowed with an ergodic SRB-measure, which means briefly that the attractor has smooth densities along the unstable manifolds. This was shown in this paper:
https://www.math.nyu.edu/~lsy/papers/metric-entropy-part1.pdf
Here is a review on SRB-measures and which dynamical systems have them:
https://cims.nyu.edu/~lsy/papers/SRBsurvey.ps
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Are you talking about the Pesin's identity/inequality? I think in practice people just assume hyperbolicity and calculate the sum of positive Lyapunov exponents.
http://www.scholarpedia.org/article/Pesin_entropy_formula
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I am talking about the Kolmogorov Sinai entropy: http://www.scholarpedia.org/article/Kolmogorov-Sinai_entropy
From what I recall from the lectures I have attended:
You define a partition size ε and get the Shannon entropy at orbit of length n. Then you get the shannon entropy at orbit of length n+1 and define the KS entropy as:
KS = lim(ε -> 0), lim(N -> oo) (1/N)*Sum(Hn+1 - Hn), the sum being from n=0 to N-1.
I wish github would allow latex, god damn. This is from the book of Schuster "Deterministic Chaos".
I looked at the page you cited, but the word "Kolmogorov" is not even mentioned on the entire page, so I am sceptical as to whether there is some straight-forward connection.
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Alright after some search apparently they are connect, but I still don't see a straightforward way of computing them.
Using just the sum of positive lyapunovs is not enough of course :P people can already do it with the lyapunovs
function.
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Oh, I didn't realize that the Scholarpedia entry of the Pesin's formula didn't mention KS entropy by name. Not a good reference indeed.
Using just the sum of positive lyapunovs is not enough of course :P
Why do you think so? I don't think you want to directly estimate KS entropy, because of combinatorial explosion of the pre-images of the partitions you have to track.
Checkout, e.g., Eq (4.4) and the equation next to it from Eckmann & Ruelle (1985) and (A3) from Hunt & Ott (2015). I think this is also how people compute in practice: e.g., Monteforte Wolf (2010).
- Eckmann, J., & Ruelle, D. (1985, July). Ergodic theory of chaos and strange attractors. Reviews of Modern Physics. http://doi.org/10.1103/RevModPhys.57.617
- Hunt, B. R., & Ott, E. (2015). Defining chaos. Chaos (Woodbury, N.Y.), 25(9), 97618. http://doi.org/10.1063/1.4922973
- Monteforte, M., & Wolf, F. (2010). Dynamical Entropy Production in Spiking Neuron Networks in the Balanced State. Physical Review Letters, 105(26). http://doi.org/10.1103/PhysRevLett.105.268104
PS: Yeah, github should support equations...
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Hmm... But not impossible even without the Lyapunov exponents?
http://iopscience.iop.org/article/10.1209/epl/i2005-10515-2/meta
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@RainerEngelken Cool! Thanks for filling the important detail. (BTW, I like your JuliaCon talk!)
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Hi people! First of all let me just say that I am very glad that anybody else besides myself wants to improve these libraries!
A bit off-topic, but let me also say that unfortunately in the coming months I won't have any time to make any improvements because I need to focus on my phd... :( However, I promise that if anybody wants to make a PR I will definitely review it and merge it asap; I can also make you collaborators.
Alrighty, now on-topic. I am not familiar with SRB and the papers Rainer cited, but it is much appreciated! If at somebody somebody wants to contribute something for this issue they can take advice.
I think we can then say that this method is a "low-priority"? Since Takafumi pointed out that in research people often use sum of maximum λs? @RainerEngelken how did you do it for the neural system you had? Computed the sum as well or did a method to compute the entropy directly?
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In addition, a warning/heads-up:
If you want to do a PR, please consider doing it after this issue is closed: JuliaDynamics/DynamicalSystemsBase.jl#18 so that you don't have to rewrite anything again and again. It will be a massive change of the internals so pretty much all code that doesn't use Dataset
will be affected.
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Yeah, I think this issue is low-priority, too. The breaking changes propagated from DifferentialEquations.jl sound much more important.
PS: Good luck on your PhD!
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This paper:
P. Grassberger and I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A, vol. 28, no. 4, pp. 2591–2593, 1983
has a straight-forward numeric algorithm on how to calculate an approximately equal quantity to KS-entropy (that they call K_2)
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solution for this issue exists in the book by Schuster and Just, equation 6.107
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