I have a series of concentration measurements, that are normalized such that they attain their maximum concentration $y_{max} = 1$ at the location $x_{ \text{max}} = 1$. For $x > 1$ the concentration decays to a known asymptote concentration $y_{ \text{stat}}$. I want to fit a function to the decay part of those measurements, so that I can eventually predict the concentration decay by predicting the parameters of the fitted function.
The exponential decay function $$f_{ \text{exp}} = (y_{ \text{max}} - y_{ \text{stat}}) \cdot exp\left(\frac{(-x+x_{ \text{max}})}{p_1}\right)+y_{ \text{stat}}$$ has one fitting parameter $p_1$ and yields too inaccurate fits. See green dotted line and blue solid line in Fig1.
Let's superimpose $f_{ \text{exp}}$ with a Gamma distribution $$f_{ \text{gam}} = p_{ \text{amp}}\cdot\left(p_{ \text{scale}}^{-p_{a}}\cdot \Gamma(p_{a})^{-1} \cdot (x - x_{ \text{max}})^{p_a - 1}\cdot exp\left(\frac{-x - x_{ \text{max}}}{p_{ \text{scale}}}\right)\right)$$ and call it $f_{ \text{xpgam}}$. Here $\Gamma()$ is the gamma function and $f_{ \text{xpgam}}$ has the parameters $p_1$, $p_{ \text{amp}}$, $p_{ \text{scale}}$ and $p_a$. This 4-parameter fitting function yields very good fits for almost all experiments (see black and blue solid lines in Fig 1; red dashed lines represent the individual components). However, those 4 parameters turn out to be too many for the subsequent predictive model. I imagine this is, because multiple parameter combinations would yield a similar curve.
Fig 1: example of one measurement with according fits $f_{ \text{exp}}$ and $f_{ \text{xpgam}}$
The weibull distribution $$f_{ \text{weib}} = (p_{w,\,amp} - y_{ \text{stat}}) \cdot \frac{p_k}{p_{\lambda}} \cdot (\frac{x}{p_{\lambda}})^{p_k -1} \cdot exp\left(-(\frac{x}{p_{\lambda}})^{p_k} \right) + y_{ \text{stat}}$$ has three parameters, $p_{w, \, amp}$, $p_k$ and $p_{\lambda}$. The fits are suitably good, however, the problem with that function is, that I can't prescribe the fit to go through the point ($x_{ \text{max}}$, $y_{ \text{max}}$) as you see in Fig 2.
Fig 2: Another exemplary measurement with weibull fit $f_{ \text{weib}}$
So my question: Is there a 2-3 parameter function, that can be constructed to go through the point ($x_{ \text{max}}$, $y_{ \text{max}}$) while still yielding adequate fits to the decay?
