nonlinear dynamics

[Sina Mehdizadeh]

Hi everyone

I have a question related to the analysis of the time series using nonlinear tools. I wish to know if we can use nonlinear analysis tools such as Lyapunov Exponent, etc. for analyzing multi-directional running or sprinting data of COM of an athlete which seems to be nonstationary? Can anyone help me please? I really need help.

Thanks,

Sina Mehdizadeh
Amirkabir University of Technology,
Tehran
Iran

[Ross Miller]

Re: nonlinear dynamics

Hi Sina,

You can calculate a Lyapunov Exponent (LE) for basically any signal, although there are some issues if the signal is too short. I don't think stationarity is a requirement, although it may affect the interpretation. Some popular algorithms are those by Wolf and Rosenstein, you can google them for more info. They are used quite often.

What do you want to use the LE for (i.e. what is your study about)?

[Jonathan B. Dingwell]

Re: nonlinear dynamics

Hello Sina:

The question of non-stationarity in these analyses is a critical one. In general, all algorithms for computing Lyapunov exponents *assume* / *require* stationary processes. More specifically, a Lyapunov exponent is a direct measure of the local divergence of neighboring trajectories in some reasonably-defined state space.... Nothing more. The algorithm itself cannot tell you *why* these trajectories are diverging! In the case of non-stationary processes (e.g., exactly like the acceleration (increase in speed) that occurs across multiple consecutive steps in sprinting), it is the non-stationarity of the signal itself, and *not* the underlying dynamics, that will create local divergence and lead to large positive exponents. Any interpretation of Lyapunov exponents in such cases must therefore be *completely* different than their interpretation for stationary processes!

Ross Miller's comment that you can calculate Lyapunov exponents for "for basically any signal" is somewhat correct, but potentially misleading also. As with *any* algorithm, you can nearly always feed data in one end and get "an answer" out the back end... Yes, the Rosenstein and/or Wolf algorithms will give you *AN* answer... But again, they cannot tell you *why* you got that answer... And unless you have some other additional information about the underlying dynamics that is causing the local divergence in your signal, you will be left with simply "garbage in = garbage out." The critical part in all of these analyses is not getting the algorithm to give you an answer... It is finding the appropriate and correct interpretation of the answer(s) you get!

One of the best texts I know of on this is "Nonlinear Time Series Analysis" by Holger Kantz & Thomas Schreiber (Cambridge University Press, 2003). Their Chapter 2 starts off with two sections on (2.1) "Stationarity and Sampling" and (2.2) "Testing for Stationarity" precisely to discuss these issues in detail. I strongly recommend you get your hands on a copy of that book and read those first 2 chapters.

Let me know if you have additional questions.

Regards,
Jon Dingwell

[Ross Miller]

Re: nonlinear dynamics

I am not suggesting Sina blindly use an algorithm. Obviously that is foolish, which is why I recommended looking up more info on the algorithms. I'm not sure what is "misleading" about suggesting to look up more information on a technique that one wishes to use.

Ross

[Jingyi Zhou]

Re: nonlinear dynamics

Wavelet Transform can be used to analyze nonstationary signals.

[Sina Mehdizadeh]

Re: nonlinear dynamics

Hello Sina:

The question of non-stationarity in these analyses is a critical one. In general, all algorithms for computing Lyapunov exponents *assume* / *require* stationary processes. More specifically, a Lyapunov exponent is a direct measure of the local divergence of neighboring trajectories in some reasonably-defined state space.... Nothing more. The algorithm itself cannot tell you *why* these trajectories are diverging! In the case of non-stationary processes (e.g., exactly like the acceleration (increase in speed) that occurs across multiple consecutive steps in sprinting), it is the non-stationarity of the signal itself, and *not* the underlying dynamics, that will create local divergence and lead to large positive exponents. Any interpretation of Lyapunov exponents in such cases must therefore be *completely* different than their interpretation for stationary processes!

Ross Miller's comment that you can calculate Lyapunov exponents for "for basically any signal" is somewhat correct, but potentially misleading also. As with *any* algorithm, you can nearly always feed data in one end and get "an answer" out the back end... Yes, the Rosenstein and/or Wolf algorithms will give you *AN* answer... But again, they cannot tell you *why* you got that answer... And unless you have some other additional information about the underlying dynamics that is causing the local divergence in your signal, you will be left with simply "garbage in = garbage out." The critical part in all of these analyses is not getting the algorithm to give you an answer... It is finding the appropriate and correct interpretation of the answer(s) you get!

One of the best texts I know of on this is "Nonlinear Time Series Analysis" by Holger Kantz & Thomas Schreiber (Cambridge University Press, 2003). Their Chapter 2 starts off with two sections on (2.1) "Stationarity and Sampling" and (2.2) "Testing for Stationarity" precisely to discuss these issues in detail. I strongly recommend you get your hands on a copy of that book and read those first 2 chapters.

Let me know if you have additional questions.

Regards,
Jon Dingwell

Dear Prof. Dingwell

It's my pleasure to see here. I read your comment many times and also I reviewed the sections you mentioned from the professor Kantz's book again and again. Generally speaking, the time series stationarity means reproducibility. That is, the dynamics of the system remains unchanged (with some acceptable errors) during the task.

As I realized from your comment, if there is nonstationary due to the time series itself (e.g. acceleration data), it can be treated using nonlinear tools with some modification and the remaining is its interpretation.

A great deal of studies in the field of nonlinear dynamics is focused on the rhythmic goal-directed hand movement between two targets. In these studies, the "system" simply consists of two coupled links and the kinematics of the system is assumed to be stationary which seems feasible. For now, consider the COM of a subject who is running between to targets, repetitively. In this task, the subjects should consecutively accelerate and decelerate. In this case, the "system" is different from the previously mentioned one. The question which arises here is that are kinematics of the latter one stationary too? That is, are these two systems the same regarding the stationarity?

Thanks for your kind attention.

Regards,
Sina
Amirkabir University of Technology,
Tehran
Iran