Help with generalized linear model

[Cexy]

I have a data set {X(t) = (x(t), y(t))}_t=1,...,N and I'm interested in modelling the changes from t to t+1, using some metric d(X(t),X(t+1))

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?

 

[bpet]

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

It's difficult to model a general stochastic process by data exploration alone. If anything is known about the underlying physical system, so that appropriate assumptions can be made, it will help in selecting a range of candidate models (possibly including but not limited to GLM).