I am looking for a nice decreasing function $f(t)$ for modelling, which starts at a maximum value of $1$ initially i.e. $f(0) = 1$, then decreases slowly initially until a value of $m$ ($0 < m < 1$) in time $t_{1}$ i.e $f(t_{1} = m)$. After $t_{1}$, the function quickly decreases and its value at time $t_{2}$ is near $0$ ($f(t_{2}) \approx 0$).
I have few points so fitting spline or other polynomials overshoot the maximum value of 1. I can threshold the value to 1, but the straight line looks ugly. The second constraint of $f(t_{2} \approx 0$) is not strict, provided the value falls rapidly after $t_{1}$.
Should I try to generate more points and try curve fitting or is there any other solution?
Best Answer
Hint:
A simple family of curve is $$ y=1-\left(\frac{x}{a}\right)^n $$
where you can use the parameter $a$ to fit the point $t_2$ in your graph and the value of $n\ge2$ determine how ''flat'' is the curve at the top and can be used to fit the point $(t_1,m)$.