Hello I am trying to model the relationship between two variables, say x and y. I have a number of subjects - for each subjectm I have a number of x and corresponding y, both of which are always positive. This data tends to be very sparse. There are some problem specific constraints: 1) y(0) = 0 (or very close to it) 2) y is increasing as a function of x 3) y' is decreasing as a function of x
This is rather nebulous, but I have a feeling that the most important difference between subjects is in the height of the curve, not in the slope. Because of the sparsity, I think I can get away with forcing each subject to have the same "slope" (perhaps at a specified x), but allowing the height to vary. I have been playing around with various sorts of logistic functions, but the asymptote isn't really justifiable. I have also been looking at things like a*log(x+b), but this doesn't really conform to the intuition delineated above. Does anyone have any suggestions?