Why fitting a logistic is nearly impossible from early data

Nothing grows exponentially forever. What appears to be an exponential curve often turns out to be some sort of S curve, such as a logistic curve. Suppose you’re collecting data on the left side of the curve. If there’s even a small amount of error in your data, you won’t be able to predict the asymptotic value with any accuracy. But if you have data on both sides of the inflection point, you can make a good prediction of the limiting value. I’ve written about this before, explaining that the problem is hard, but I didn’t say why it’s hard. Here I’d like to give an idea why it’s hard. Suppose you want to fit a logistic equation to three distinct values of t and the corresponding values of y. There is a unique solution, but in general you cannot find a solution in closed form. However, if the values of t are evenly spaced there is a method [1] to solve for the parameters L, k, and t0. For this post we’re only interested in the limiting value L, and it can be found by independent of h.…