Hey tuathal.

Correlation statistics only look at changes in a linear sense: Any non-linear relationship will be represented in a non-linear relationship and the way that non-linear relationships are assessed in general

is through regression modeling.

The benefit with regression modeling is that you can specify any model to fit the data to (what you are doing is essentially "projecting" the data to some high-dimensional space) and you can

look at the fit statistics to evaluate how strong the fit really is.

In the general case, you basically look at the coefficients of the fitted model (for example Y = a + bx + cX^2 + dX^3 will have coefficients a,b,c,d which are themselves are random variables with

distributions) and based on that you can look at how the relationship changes based on the rates of change expressed within these statistics (and their variance measures).

Note that regression modeling can be applied to time series models and those which have correlation structures between observations. Common statistical packages like SAS and R allow you

to specify these constraints if you need to have them. You can also use things like Markov-Chain Monte-Carlo (MCMC) if you have complicated distributions with all sorts of conditional

relationships between the variables themselves. MCMC is based on theorems that dictate that the posterior distribution converges to the stationary distribution and this result basically

allows you to specify all kinds of complicated relationships with a probabilistic guarantee that given enough iterations, you will get the right distribution. If you need to model

complicated conditional distributions, then I would suggest looking at MCMC and something like WinBUGS (which does MCMC) or its frontend programs in R. WinBUGS is free by the way.

In terms of the relationship, after you fit a regression model, then look at the coefficients in terms of the rates of change. Remember that if you are plotting a mean response function

(which is typically what you do in regression: you fit a population mean model), then consider rates of changes in terms of derivatives. Although you can't really do this properly

for random variables (you can but it's a lot more complicated due to them being random variables and having different measures from a measure-theoretic point of view), you can

use simple calculus to get the rates of change if you only focus on the population mean model (which is deterministic) as opposed to the differential for the random variables.

What I mean by the above is that you focus on dP/dx where P(x) is the mean response at a given x instead of focusing on dY/dX where Y and X are random variables. The

stochastic calculus is very complicated, but you do not need it if you are looking for mean response models.