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--- chaosbook:noise [2009/03/08 17:24]
predrag
+++ chaosbook:noise [2010/08/04 15:51] (current)
predrag Goldobin and Zaks noisy preprint
@@ Line 1: / Line 1: @@
+<- [[:chaosbook]]
 ====== Chapter: Noise ======
 (ChaosBook.org blog)
+<html>
+<span style="color:green;font-size:80%;text-align:right;">to find recent edits: click on [Recent changes], select changed page, then [Old revision], compare versions</span>
+</html>
+===== Noise blog =====
+<html>
+<span style="color:green;font-size:80%;text-align:right;">enter the latest posts at the top of this section</span>
+</html>
+**2010-08-03 PC to Domenico** [[http://arxiv.org/abs/1008.0073|Noise reduces disorder in chaotic dynamics]] by Denis S. Goldobin and Michael A. Zaks. They say: "We evoke the idea of representation of the chaotic attractor by the set of unstable periodic orbits and discover a novel noise-induced ordering phenomenon. For long unstable periodic orbits the weights (or natural measure) appear to be highly inhomogeneous over the set either diminishing or enhancing their contribution into system dynamics. We show analytically a weak noise to reduce this inhomogeneity and, additionally to obvious perturbing effect, make a regularizing influence on the chaotic dynamics. We demonstrate this universal effect rooted into the nature of deterministic chaos for the Lorenz system. The effect can be observed as shrinking of the distribution of averages over finite segments of the chaotic trajectory and lead to significant enhancement of the coherence of chaotic oscillations."
+**2010-07-02 PC to Sara**
+They probably exist - Google returns lots of hits of type "Rayleigh–Rice distribution"..."Lagruerre polynomials" etc. Check it out. There exist also orthogonal Rice polynomials (hypergeometric functions) but I think they are not related to the Rice distribution.
+**2010-07-01 Sara** What would be really useful is a family of orthogonal PDFs. If you coma across such thing ...
+**2010-07-01 PC to Sara**
+Rice distribution (see [[http://en.wikipedia.org/wiki/Rice_distribution|wikipedia]] and MathWorld [[|]]
+might be the right distribution to use in your work on linear-nonlinear models. I was told that it describes the distribution of distances between two random events. There is also a [[http://mathworld.wolfram.com/RayleighDistribution.html|Rayleigh Distribution]]. The original paper is [[http://www.cramster.com/reference/wiki.aspx?wiki_name=Stephen_O._Rice|Stephen O. Rice]], "Statistical properties of a sine wave plus
+random noise," //Bell Syst. Tech. J//. **22** 109–57 (1948 ).
+[7] Nakagami M 1960 The m-distribution—a general formula
+**2010-04-07 PC** In Goettingen Mirko ... and David Hofmann have started an interesting [[http://www.nld.ds.mpg.de/~langevin/|Stochastic Processes Seminar]].
+**2009-12-08 PC to Domenico** [[http://arxiv.org/abs/0912.1581|State and parameter estimation using Monte Carlo evaluation of path integrals]] by John C. Quinn and Henry D.I. Abarbanel might be of interest for your thesis preparation. They say:
+"
+ Transferring information from observations of a dynamical system to estimate
+the fixed parameters and unobserved states of a system model can be formulated
+as the evaluation of a discrete time path integral in model state space. The
+observations serve as a guiding potential working with the dynamical rules of
+the model to direct system orbits in state space. The path integral
+representation permits direct numerical evaluation of the conditional mean path
+through the state space as well as conditional moments about this mean.
+"
+They use (if not in this paper, than in the earlier ones ) the same convolution of noise variance and trajectory variance as we, as a part of the Kalman filter procedure. I think I reported on some of that in my notes on the Paris Lyapunov vectors conference, svn siminos/blog, and there is a mention of Kalman filters in stoch/flotsam.tex. We also heard Ott talk about use of Kalman filters by the Maryland group weather prediction work.
+As far as is plumbing is concerned, I think we might need something like that to marry our statespace picture of turbulent flows (computed in the full 3-dimensional Navier-Stokes) with partial information obtained in experiments (typically a full 3-dimensional velocity field, fully resolved in time, but measured only on a 2-dimensional disk section across the pipe). The challenge is to match this measurement of the turbulent flow with a statespace point in our 100K ODE representation, and then track the experimental observation to improve our theoretical prediction for the trajectory in the time ahead. That would be the absolutely best "weather prediction" attainable for a turbulent pipe flow, limited by a combination of Lyapunov time and observational noise. In our parlance, the "optimal partition of statespace".
+**2009-10-23 PC to Domenico** [[http://arxiv.org/abs/0910.4370|Random fluctuation leads to forbidden escape of particles]]
+by C. S. Rodrigues, A. P. S. de Moura, and C. Grebogi might be of interest to you. They say:
+"We show that, under finitely small random
+fluctuations of the field, trajectories starting inside KAM islands escape
+within finite time. The non-hyperbolic dynamics gains then hyperbolic
+characteristics due to the effect of the random perturbed field. As a
+consequence, trajectories which are started inside KAM curves escape with
+hyperbolic-like time decay distribution. We show a universal quadratic power
+law relating the exponential decay to the amplitude of noise."
 ===== How well can one resolve the state space of a chaotic flow? =====
-[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi07|pdf paper]]
+[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi08|the latest pdf version of DasArtikel]]
-\\
 \\
+(append posts at the end of a section, in sections named after people)
-{{gtspring2009:pc.jpg }} now on ''[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi07|arXiv:0902.4269]]'' and submitted to Phys Rev Letts  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-08 17:18//
+{{gtspring2009:pc.jpg }} Accepted for publication in Phys Rev Letters  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-12-16 00:10//
-{{gtspring2009:pc.jpg }}  Guys, if you have time and inclination, have a look at
+{{gtspring2009:pc.jpg }} now on ''[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi08|arXiv:0902.4269]]'' and submitted to Phys Rev Letters  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-08 17:18//
-[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi07|ChaosBook.org/~predrag/papers/pubs.html]].
-We have not put it on arXiv yet, but it is the latest crack on the
+{{gtspring2009:pc.jpg }} If you have time and inclination, have a look at
+[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi08|ChaosBook.org/~predrag/papers/pubs.html]]. It is our latest crack on the
 noise problem, differs quite a bit from papers 1-3. Word `quantum'
-is not mentioned, but between us - something like that should also
+is not mentioned, but something like that should also
 perhaps replace Berry-Keating cycle expansion cutoff, and Hermite
 basis should be also a natural basis for semi-classical corrections
 calculations.
-Grateful for any criticisms, references etc. - Domenico & Predrag
+Grateful for any criticisms, references etc.
- --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-01-14 00:36//
+ --- //Domenico & Predrag 2009-01-14//
+==== Domenico & Predrag ====
+----
+{{gtspring2009:pc.jpg }} it would be good to check a few things before the referee reports arrive:
+  - Continuing on [[chaosbook:noise#carl_p.dettmann|discussion with Carl]], text from me to you next to numerical estimates of escape rates: "make sure that the noiseless Fredholm determinant is converging super-exponentially, as it should, and that the noisy cycle expansions are not converging to the noiseless answer, as this graph seems to indicate:"  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-10 09:16//
+  - Further down you say: "The fact that adding 0110 and 0111 does not change the determinant at all is probably an accident. In the case of the quartic map, the determinant of the optimal partition differs greatly from the one given by the partition immediately simpler, but their leading zeroes only differ by 10^{-5}"
+    * Can you enter your transition graph, determinant into stoch.tex, isolate the terms that differ from the partition we are currently using, look at their values and geometrical (state space) placement carefully so we understand why they "shadow"?  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-10//
+    * I have rewritten ChaosBook chapters [[chaosbook:Markov|"Walkabout: Transition graphs"]] and [[chaosbook:count|"Counting"]] to include the discussion needed for your thesis, you might have a look at them, see whether they are more helpful now --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-20//
 ==== Marcos Saraceno ====
@@ Line 171: / Line 236: @@
 ----
-It seems I should read 	[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi07|DasArtikel]]
+It seems I should read 	[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#LipCvi08|DasArtikel]]
  --- //[[http://complex.elte.hu/~vattay| Vattay Gábor]] 2009-01-28//

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