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The issue is that x(t) has some dependence on y(t), and I'd like to account for this: if there is a large change in y(t) we expect there to be a corresponding change in x(t), and I'd like the metric to account for this expected change.

The problem is that x(t) takes only discrete values (in fact it is almost always 1 or 2, with a small probability of being any other integer greater than 2) whereas y(t) is a positive real number with an unknown distribution (although of course I can approximate the distribution with a histogram - I have a lot of data available).

What would be an appropriate model for the dependence of x(t) on y(t)?

I've thought about normalizing x(t) to be in the range [0,1] and using a logistic or probit model, but I really have no idea how appropriate this is.

Any ideas?