@bjoern, for each fixed V, it seems that your curve is concave and only has positive values. Therefore, my first choice is to assume that Y=A X^r
. The easiest way to estimate this is to apply log in both sides to get the linear regression log Y = log A + r log X
(you probably will find 0<r<1
). Therefore, for each value of V
, I would use the function regress
in matlab applied to the values log Y
and log X
in order to estimate the parameters A
and r
. This function is called Cobb-Douglas and it is very useful in economics: http://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function.
For most curves, it seems that the effect of V
is well behaved, but the behavior of the blue curve, which is very strange. I would say that in general the effect of V
is to translate the points.
If the behavior of V is really linear, maybe you can estimate Y=A V X^r. Therefore, you have to estimate logY = log A + log V + r log X. In this case, your dependent variable is log Y and your independent variables log X and log V.
In both cases, I think that the function regress of matlab does not automatically include the constant of the regression (A for us). So remember to include a vector of ones with the size of your sample as an independent variable as well,
Furthermore, if you really want to test if the behavior of V is linear, just estimate logY = log A + slog V + r log X ehich is equivalent to Y=A V^s X^r
I hope it helps.