I am talking from theoretical perspective, by analysing the source and interaction. I feel like most of the things are from non-linear dynamic system even if the empirical evidence is not enough or states otherwise. Because, the accuracy of measurement can always be doubtable. My intention is to apply the ccm technique by Sugihara in order to find causal link between some variables.
Thanks LIVIO for your answer.
I am not an expert in this field and I am only doing Masters by research , so I do not have the chance to go in depth. Also , I just started to look into non-linear dynamics.
What I am saying is by analyzing the source of the variable I feel that they interact with each other in a non-linear dynamic fashion. But I haven't done any empirical tests!
Also , There exists statistically significant , p value =0.0 negative 0.8 correlation.
A -0.8 negative correlation is a high linear interaction and the reduction from -1 might be due to noisy measurements of your variables and not due to nonlinear effects.
A possible way to check is to fit a linear and a nonlinear model to your time series and see how adding the nonlinear interaction terms to the model improves its predictability of test time series. A possible model is Volterra series of order-1 for the linear model, and then order-2 for nonlinear interaction. The linear model should be at least as good as the correlation measure, as long as you do not over-fit.
The problem is , the variables should not be negatively correlated as it is expected to have a positive relation . Because both of them should interact with each other and they should increase each others effect rather then decreasing. There might be a confounding variable causing this negative correlation.
If I know that two variables should interact with each other in a nonlinear way and there outcome is unpredictable although its deterministic then can I say they are non-linear dynamics system although there is no empirical evidence ?
Sometimes its not possible to get a long time series data , as a result empirical analysis becomes unreliable. Can any one please comment on it ?
problem solved , this paper seems to have the ans, thanks everyone!
Patil, D. A. S., Hunt, B. R., & Carton, J. A. (2001). Identifying low-dimensional nonlinear behavior in atmospheric data. Monthly weather review, 129(8), 2116-2125.
Non-linear dynamic systems create rich class of solutions. There might be convergence to multiple equilibria, chaotic behavior, emergence of solitary waves (in the case of nonlinear PDE), etc.
I think that the answer depends very much on the exact model you are doing, both on mathematical formalization and the field of application. If you are doing statistical analysis of some data (for example, in economics) and just rejected linear dependence dependence between variables, this is a simple non-linearity.
Physicists often try to detect power law distributions (instead of Gaussian) as some indicator of presence of non-linear phenomena. But in any case I advice you to go into theory of described phenomenon as well.
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