Finding a good parametric form for a model 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?
 A: (This is supposed to be a comment.)
I would say that without knowing the physical process(es) that generated the $y$'s for each corresponding $x$'s, any number of functions would be admissible. Barring that, one usually graphs the data first before even thinking about models...
A: First, draw a distinction between what you expect and what you measure.
Since the expectations can't sculpt a functional form, investigate a generic method like polynomial approximation. Then you can look at the result and see how well it matches your expectations.
Start with a set of $m$ measurements $\left\{ x_{k}, y_{k} \right\}_{k=1}^{m}$ and $d$th order approximation.
Model
$$
 y(x) = a_{0} + \sum_{\mu=1}^{d} a_{\mu} x^{\mu}
$$
Linear system
$$
\begin{align}
  \mathbf{A} a &= y \\
%
\left[ \begin{array}{cccc}
  1 & x_{1} & x_{1}^{2} & \dots & x_{1}^{d} \\
  1 & x_{2} & x_{2}^{2} & \dots & x_{2}^{d} \\
  \vdots & \vdots & \vdots & & \vdots \\
  1 & x_{m} & x_{m}^{2} & \dots & x_{m}^{d} \\
\end{array} \right]
%
\left[ \begin{array}{c}
  a_{0} \\ a_{1} \\ \vdots \\ a_{d}
\end{array} \right]
%
&=
%
\left[ \begin{array}{c}
  y_{1} \\ y_{2} \\ \vdots \\ y_{m}
\end{array} \right]
%
\end{align}
$$
Least squares solution
The least squares solution is
$$
 a_{LS} = \mathbf{A}^{+}b +
\left(
 \mathbf{I}_{d+1} - \mathbf{A}^{+} \mathbf{A}
\right) z, \quad z \in \mathbb{C}^{d+1}
$$
