There are some methods for prediction of chaotic time series also for long time predictions.
But for long time prediction the cumulative error of each step affects the final result. So, is it useful for us to implement the prediction procedure in phase space of chaotic signals for reducing the cumulative error?
Thank you
Dear Mohammad,
If you have a long enough part of chaotic signal, you can embed it and obtain a strange attractor. You can obtain many of its chaotic features too. However, when you speak about PREDICTION here, you refer to the time domain and in there you encounter the butterfly effect. So I think your answer is No. Instead of your statement, I prefer to say: looking at chaotic time series from topological point of view (phase space, Poincare section, return map, …) has many benefits over dealing with them in the time domain.
Dear Mohammad,
If you have a long enough part of chaotic signal, you can embed it and obtain a strange attractor. You can obtain many of its chaotic features too. However, when you speak about PREDICTION here, you refer to the time domain and in there you encounter the butterfly effect. So I think your answer is No. Instead of your statement, I prefer to say: looking at chaotic time series from topological point of view (phase space, Poincare section, return map, …) has many benefits over dealing with them in the time domain.
Dear Sajad
Thank you for your useful comments. But, there are some papers about this idea. The original time series is transformed into a strange attractor in phase space, and after estimating, the predicted points in state space are transformed into synthesized time series. It is slightly identical to your idea about noise cancelling in phase space.
My question is can these methods use for long time prediction? Did anyone see the comparison of this methods with some classic long time prediction models?
Dear Sajad,
In chaotic time series we have positive Lyapunov exponent and of course we have a serious limit in prediction. On the other hand the evidence of a strange attractor in phase space give to us more information about the dynamical system is studied. New methods such as non-extensive Tsallis statistics, fractal dynamics, fractional mathematics which viewing the singularities of physical space and the multi-fractality of phase space I think it would give better results under the limit of positive Lyapunov exponent.
Dear all,
I think this link can complete this discussion. appreciated if there is any comments.
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CC8QFjAA&url=http%3A%2F%2Fcs.ucf.edu%2F~arslan%2FBasharat_ICCV09.pdf&ei=BxYkUtfWII2hsATnpYDYCg&usg=AFQjCNE2Yv7urelTaM-PEeAtuu-BQ3xZYQ&sig2=xfQptW9w0rhQD4WxRkB5Lw&bvm=bv.51495398,d.d2k
I agree with Sajad and Leonadis, phase space prediction if achievable will not fare better than that of time series for a chaotic time series.
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