The point of a model is to be validly predictive of something. Fitting your exponential is validly predictive of local behaviour more often than not. Often, insanely so.[1] You can directly use the numerical model to make precise and relevant predictions.
Your exponential doesn’t tell you when the trend stops, but it’s not trying to, for one because it’s incapable of modelling that. That’s ok, because that’s not its job.
Fitting a sigmoid doesn’t do this. The majority of times, the only additional thing the result of a sigmoid fit tells you is how an arbitrarily chosen dampening model fits to the arbitrary noise in your data. There’s nothing you can do with that, because it’s not predictive of anything of value.
This doesn’t mean you shouldn’t care about limiting behaviour, or dampening factors. It just means this particular tool, fitting a numerical model to numerical data, isn’t the right tool for reasoning about it.
The point of a model is to be validly predictive of something. Fitting your exponential is validly predictive of local behaviour more often than not. Often, insanely so.[1] You can directly use the numerical model to make precise and relevant predictions.
Your exponential doesn’t tell you when the trend stops, but it’s not trying to, for one because it’s incapable of modelling that. That’s ok, because that’s not its job.
Fitting a sigmoid doesn’t do this. The majority of times, the only additional thing the result of a sigmoid fit tells you is how an arbitrarily chosen dampening model fits to the arbitrary noise in your data. There’s nothing you can do with that, because it’s not predictive of anything of value.
This doesn’t mean you shouldn’t care about limiting behaviour, or dampening factors. It just means this particular tool, fitting a numerical model to numerical data, isn’t the right tool for reasoning about it.
“I answered that the Gods Of Straight Lines are more powerful than the Gods Of The Copybook Headings, so if you try to use common sense on this problem you will fail.” — Is Science Slowing Down?, Slate Star Codex, https://slatestarcodex.com/2018/11/26/is-science-slowing-down-2/