Is there any way to find out that how many data points are needed for reconstructing state-space of a scalar time-series? Beside Chaos Data analyzer and TISEAN package, are there any other software tools for doing nonlinear time-series analysis?
I think it could be useful for you this very recent paper by Hirata et. al
http://scitation.aip.org/content/aip/journal/chaos/25/1/10.1063/1.4906746
In addition you can consider this paper by Aguirre and Letellier
http://www.hindawi.com/journals/mpe/2009/238960/ref/
This is a very good summary of the main challenges in state space reconstruction by means of analytical methods (Hirata's paper addresses the matter on the grounds of a topological procedure).
As it is highlighted above, the number of points you need to characterize a given dynamics is determined by the metric and topological properties of the underlying attractor. However, and also accordingly to the previous comment, it is not easy to give a close figure for the number of points needed to perform such a reconstruction.
Takens' theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. For ex.
http://cnx.org/contents/939eff33-c4f0-43e4-ae2b-ca6d5b58e267@2/Takens%27_Embedding_Theorem
You need enough datapoint N to capture the dynamics of interest + (N - lag * number of surrogate dimensions)
Use the CRP toolbox (Matlab) available here: http://tocsy.pik-potsdam.de/CRPtoolbox/
Standard procedure:
1. Find an embedding delay using mutual information mi(), take the first local minimum. This is an optimization, in principle you could use any lag for embedding.
2. Use false nearest neighbours analysis fnn() to decide how many surrogate dimensions you need.
While you are at it, run a Recurrence Quantification Analysis: crqa()
Other issues... Need to rescale your data? Try rescaling to either the mean or max distance in reconstructed phase space, the function pss() gives you both distances.
There is also a package for R discussed here http://journal.frontiersin.org/Journal/10.3389/fpsyg.2014.00510/abstract
Here is a submitted paper in which I discuss embedding and RQA:
https://osf.io/bqayd/
Here are some examples from my group:
http://link.springer.com/article/10.1007%2Fs11881-012-0067-3
http://journal.frontiersin.org/Journal/10.3389/fphys.2012.00116/full
http://fredhasselman.com/main/wp-content/papercite-data/pdf/wijnants2009.pdf
Lichtwarck-Aschoff, A., Hasselman, F., Cox, R., Pepler, D., & Granic, I.. (2012). A characteristic destabilization profile in parent-child interactions associated with treatment efficacy for aggressive children. Nonlinear dynamics-psychology and life sciences, 16(3), 353.
The paper by Ji et al, "Influence of sampling length and sampling interval on calculating the fractal dimension of chaotic attractors", might also provide you some interesting hints.
http://www.worldscientific.com/doi/abs/10.1142/S0218127412501453
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