I use non-linear regression to model decay in a population, but I have not been able to find a good functional form form. Assume:
- The decay is not exponential (not a constant percentage each time unit; Usually steeper initially than an exponential function, then flatter)
- The function should have a limit that is above zero. Infinite time does not imply complete decay. At some point the decay stops and the line is flat.
- At t=0, the function should be 1
I tried a function of the following form, used in hyperbolic discounting:
y=b/(1 + c t)
where t is time, and b and c are parameters that are estimated based on the observed points, but this function converges to zero and adding a constant parameter means that it is no longer one at t=0.
Any suggestions for functions that could fit would be greatly appreciated.
Best Answer
Here's a possibility: consider $$ y = \begin{cases} b + (1-b)\exp(a/T)\exp\left(\frac{a}{t - T}\right) & t < T\\ b & t\geq T \end{cases} $$ Here, $b$ is the desired minimum value, $T$ is the time at which $y$ reaches its minimum, and $a$ determines the rate of decrease (larger $a$ means a faster decrease). Per my discussion in the second part of my answer, I recommend taking $a > 2T$.
The function is designed so that the transition from $t < T$ to $t \geq T$ is smooth. Here's a graph of the resulting curve for $b = 0.1, T=1,$ and several values of $a$.
As an aside, you can see from the $a=1$ graph that for values of $a$ that are too small, the function noticeably fails to be convex, which may be undesirable. Here's a calculation to determine for which values of $a$ that occurs.
We find that the second derivative of $y = \exp\left(\frac{a}{t - T}\right)$ is given by $$ \frac{d^2y}{dt^2} = \frac{a e^{\frac{a}{t - T}}}{\left(t - T\right)^{4}} \left(a + 2 (t - T)\right), $$ which means that an inflection point occurs where $$ a + 2(t - T) = 0 \implies t = T - \frac a2. $$ If we take $a > 2T$, then we guarantee that this inflection point does not occur within the range $0 \leq t \leq T$.