Thread by @anderssandberg

#FridayPhysicsFun – Stretching the definition of physics a fair bit, sigmoid growth curves are useful… except for predicting the future.

Sigmoids get the name from being S-shaped curves, taking the name from Greek letter sigma. They are also called logistic curves, ogives, s-curves, Gompertz curves, Bass curves, Verhulst growth...
en.wikipedia.org/wiki/Logistic_…

First, an initially accelerating growth period, leading up to a turning point. Then the growth slows and the curve tends towards an asymptote (or maximum/ saturation level). There are many formulas that give such curves.

Most sigmoids show up as growth of something but they are all over chemistry, biology & pharmacology because they can also describe the binding of ligands to enzymes or receptors as a function of their concentration. Or titration.
en.wikipedia.org/wiki/Hill_equa…
en.wikipedia.org/wiki/Titration…

The classic application is population models in ecology: something reproduces, but the reproduction rate declines as the carrying capacity of the environment is approached. First rapid spread, then stabilization. nnhsbiology.pbworks.com/w/page/1238878…

This works for technology diffusion too: at first a few neophiles get the device. Others copy them when it looks useful. More and more get it, until few people are left who have not bought it. en.wikipedia.org/wiki/Bass_diff… researchgate.net/publication/22…

Same thing for epidemics: at first infected people come into contact with mostly uninfected and disease spreads exponentially, but after a while there are few uninfected left and most are immune, slowing down the spread. arxiv.org/ftp/arxiv/pape…

This leads to the paper I wrote with some friends: if we have such nice models that fit past data well, has few parameters, and model growth dynamics sensibly, shouldn’t they be great for predicting where things are going? NOOOO!!! arxiv.org/abs/2109.08065

2020 produced far too many papers saying “here is Covid cases until now; fit the sigmoid; ah, look, it is peaking about now and the total will just be about twice the present cases. The pandemic is almost over!” They were all badly wrong, underestimating the length and size.

What went wrong? Backcasting, fitting a curve to a historical dataset, can be very accurate and principled, but forecasting means fitting a curve to a *partial* dataset where a very non-random part (the future) is missing. Sigmoids are extra bad for this.

In particular, fits often suggest (1) the midpoint is about now, (2) the curve will reach about twice its current height. Unless we have already passed the midpoint and it is super-obvious. aleph.se/papers/A%20Sig…

This is known "folklore" or obvious to good data scientists, but unfortunately nobody told the epidemologists, peak oilers, tech forecasters and others. growth-dynamics.com/articles/stren… constancecrozier.com/2020/04/16/for…

We explain it by dividing the equation into an "acceleration" part and a "damping" part. Early data gives good info about the acceleration but hardly any about damping.

In proper statistics lingo, "sigmoid models are not identifiable". royalsocietypublishing.org/doi/10.1098/rs…

Knowing things about reality can sometimes help constrain the model, but it is usually floppy in just the wrong way to make it good for forecasting.

Many models in physics or biology are badly behaved. Or mixed, like "sloppy models" where a few parameters determine behavior but the rest can be set to nearly any values and things still work. lassp.cornell.edu/sethna/Sloppy/

There are other ways things can go wrong. Noise can be heavy-tailed, adding lots of outliers. There can be sudden transitions where entirely new dynamics shows up. Or everything is super-sensitive to some parameter. flic.kr/p/eh3cqJ

Actually knowing about the strengths, weaknesses and quirks of a kind of model is what you pay the expert for. Anybody can fit a model, but it takes a bit of expertise to know if it is a good idea. flic.kr/p/4r62Ye

There are other things called sigmoids in physics, like the twisted magnetic field structures that are precursors to coronal mass ejections on the Sun. phys.org/news/2009-04-s… doi.org/10.1016/S1364-…

...but they have nothing to do with the curves above. Just because a shape fits two things doesn't mean it has the same meaning or cause. But sigmoid shapes at least mean there is a zero crossing of the second derivative. In one dimension.