Exploring Chaos Theory & Statistics: How Do They Intersect?

[adwodon]

Good afternoon,

I've come here to hopefully resolve an issue that cropped up in a debate with a friend.
We were discussing weather patterns and chaos theory was brought up.

My understanding of chaos theory is that it is a way of explaining behavior of certain deterministic systems aka deterministic chaos. A bit of further reading seemed to show that chaos theory was now being applied to stochastic systems as well although I'm not really sure I understand this at all, if someone could give me something insight into this it would be appreciated.

My question really boils down to something quite simple though, what does chaos theory have to do with statistics? I feel that could well be a stupid question but it was raised and now I've managed to confused myself so I decided to turn to this forum.

Do we use statistical methods when applying chaos theory? I know its applied in economics, biology etc, how? Are there any good resources about these applications? I can find a few papers dotted about but they tend to require more background understanding than I feel I possesses on the matter.

I'm asking in the statistics section as really that is the main problem I'm having, my friend insists they are linked but failed to provide any evidence, dismissing me as 'not understanding chaos theory' when I said I didn't see how statistics had anything to do with it.

I suppose as a supplemental question, what statistical methods are used when it comes to macroscopic weather patterns?

If anyone could shine some light on this I would be very appreciative.

 

[MarneMath]

I don't think the question "what does chaos theory do with statistics" but more" how is statistics used in chaos theory." The most common and well-known example is as a test to filter out 'noise' in some time series to help eliminate what is a deterministic system and what isn't. This is non-trivial, and extremely difficult in a non-linear system. The single most important aspect of statistics currently in chaos theory is to try to determine if a system is deterministic but chaotic or if it is random.

Other issues deals with a more fundamental aspect of probability. The common example of flipping a fair coin with .5 probability of heads or tails becomes a bit muddy. While most elementary students of the subject may stumble on this if they rely on a frequencies approach to statistics, a Bayesian approach can usually handle this problem. Nevertheless, one can argue about the notation of what randomness is and say that randomness is a reflection of the uncertainty in the initial condition.

With regards to weather, I'm not an expert, but I'm sure there many. One that pops into my mind is times series analysis and thus naturally regression analysis.

 

[chiro]

Hey adwoden and welcome to the forums.

One specific issue you should consider is whether you are looking at micro or macro level behaviours.

In statistics, the macro level behavior can be found by setting up a statistical model and looking at expectations and/or confidence intervals to get an idea of the macro-behavior of an evolving system.

The different approaches tell us different things and by using the appropriate random variable models (stochastic processes), we can look at the mean behavior to get that understanding of macro-level characteristics.