What methods are there for measuring the strength of arbitrary, highly non-linear relationships between two paired variables? By highly non-linear, I mean relationships that can't sensibly or reliably be modelled by regression to a known model. I'm particularly interested in time-series, but I imagine any thing that works for bi-variate data would work here (if we treat the two time-series as a set of pair data points)
Two that I am aware of are Mean Square Difference (ie. mean square error, treating one time-series as the "expected" value, and one as the observed), as and Distance Covariance. What others are there?
Clarification: I'm basically asking about dependence between series, where linear correlation or simple non-linear correlation (after log, exp, trig, other simple analytic transformations) doesn't really mean much.
Best Answer
Plain old linear regression has a nice non-parametric interpretation as the average linear trend across all pairs of observations; see Berman 1988, "A theorem of Jacobi and its generalization". So, the data doesn't have to look linear in order to use it; any (broadly) monotonic trend could be summarized this way.
You could also use the Spearman rank correlation... and probably much else besides.