I have the following conceptual model:
I want to be able to create an inferential model (e.g., NLM, GAMM, etc.) that can predict the above conceptual model.
To do so, I need my predictive model to account for the curves/structure of the conceptual model.
I don't know how to do that...
I'd seen elsewhere that the assymptotic part of the model could be graphed as $f(x) = b[1 - exp(ax)]$ , where $b$ is the asymptote and $a$ controls the rate of approach to the asymptote.
I also know a 3rd or 4th order polynomial might capture some of the oscillation behavior.
Can anyone help me construct a mathematical model that overlays my theoretical curves?
Theoretically, I want to capture i) the different ascent patterns, ii) the magnitude of the peak, iii) the magnitude of the subsequent dip, and iv) late-term rate of approaching an equilibrium.
You might want to look at Second order LTI systems. Basically the functions of your graph can be modelled by a differential equation of second order, for which the solution looks like one of these three.
For instance the "long lag" one can be modelled by a function that is $y(t) = A\exp(-t/\tau)\sin(\omega_pt+\phi)$
It corresponds to the solution of an equation of the form $\omega_0^2y'' + 2\xi\omega_0y' + y = 0$ where $\Delta = 4\omega_0^2(\xi^2-1) <0$ In this case you can even calculate the overshoot of your signal with respect to the final value. The two other cases correspond to $\Delta = 0$ or $\Delta >0$.
There is a lot of resources on LTI models online so you'll find more precise info there.