Maths writer/cowritter needed: how you can't distinguish early exponential from early sigmoid

Another nice example of how this is a known result but not presented in the academic literature:

https://constancecrozier.com/2020/04/16/forecasting-s-curves-is-hard/

The fundamental problem is not even distinguishing exponential from logistic: even if you *know* it is logistic, the parameters that you typically care about (inflexion point location and asymptote) are badly behaved until after the inflection point. As pointed out in the related twitter thread, you gain little information about the latter two in the early phase and only information about the first two in the mid phase: it is the sequential nature of the forecasting that is making this problem.

I find it odd that this does not have a classic paper. There are *lots* of Bass curves used in technology adoption studies, and serious business people are interested in using them to forecast - somebody ought to have told them they will get disappointed. It seems to be a result of the kind that everybody who knows the field would know but rarely mention since it is so obvious.

You may want to look into stiff equations. The logistic equation y'=λy(1-y) is increasingly stiff with increase in λ, meaning, in particular, that the resulting curve depends very sensitively on small relative changes in λ when λ is large. That would give you the exact effect you observe, inability to predict the inflection point from noisy data.

From a numerical methods textbook:

If the value of λ is increased the transition from zero to one occurs much faster, and for large values of λ it looks almost vertical. Such rapid changes in the solution are characteristic of stiff equations, and what this means is that the logistic equation becomes stiffer as λ increases

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I was going to suggest that maybe it could be a known and published result in dynamical systems / population dynamics literature, but I am unable to find anything with Google, and textbooks I have at hand, while plenty mentions of logistic growth models, do not discuss prediction from partial data before inflection point.

On the other hand, it is fundamentally a variation on the themes of difficulty in model selection with partial data and dangers of extrapolation, which are common in many numerical textbooks.

If anyone wishes to flesh it out, I believe this behavior is not limited to trying to distinguish exponentials from logistic curves (or different logistics from each other), but also distinguishing different orders of growth from each other in general. With a judicious choice of data range and constants, it is not difficult to create a set of noisy points which could be either from a particular exponential or a particular quadratic curve. Quick example: https://raw.githubusercontent.com/aa-m-sa/exponential_weirdness/master/exp_vs_x2.png (And if you limit data point range you are looking at to 0 to 2, it is quite impossible to say if a linear model wouldn't also be plausible.)

Isn't this really straight forward? I'm pretty sure ln(e^x) and ln(sigma(x)) only differ by about e^x + O(e^(2x)) when x < 0. You can't tell apart 2 curves that basically make the same predictions.

In the simplest case where the errors are Gaussian, this would probably be covered by standard regression lower bounds? You'd show that exponentials and sigmoids can be made close in L² over a restricted domain, then deduce it requires many samples / low noise to distinguish them.

Or as Aaro says above, maybe better to parametrise the sigmoid, and take the Fisher information of the turning point parameter.

More examples of the difficulty of predicting the future using fitted regression models:

https://shape-of-code.coding-guidelines.com/2020/04/19/predicting-the-future-with-datalogistic-regression/

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