11–20 of 79 posts
Can’t you model almost anything as a piecewise linear function? I don’t know if this claim is saying much of substance.
Yes, but if your data is actually exponential, the linear segments are not going to be better approximations than an exponential curve. That's what's going on here
Appears to apply to the two preceding linear-scale charts.
Can’t you model almost anything as a piecewise linear function? I don’t know if this claim is saying much of substance.
Yes, but if your data is actually exponential, the linear segments are not going to be better approximations than an exponential curve. That's what's going on here
Is this new? I remember reading an article a long time ago (maybe 15 years?) discussing how growth in aircraft speed was overall exponential, but linear for each new technology that was introduced.
For example the time until the next technology is 1/n, then the technologies per unit time increase exponentially.
This is true within a constant which is https://wikipedia.org/wiki/Euler%27s_constant
A justification for maybe why this is the right model is that the previous n technologies all help make the next technology possible in time ~1/n because they (n existing technologies) all contribute to innovating the next technology.
Can’t you model almost anything as a piecewise linear function? I don’t know if this claim is saying much of substance.
Is this new? I remember reading an article a long time ago (maybe 15 years?) discussing how growth in aircraft speed was overall exponential, but linear for each new technology that was introduced.
Earlier quoted context omitted.
Yes, but if your data is actually exponential, the linear segments are not going to be better approximations than an exponential curve. That's what's going on here
The dangerous bit is that an exponential curve will also be a fairly good fit for a logistic function that's not yet fully observed.
Is this new? I remember reading an article a long time ago (maybe 15 years?) discussing how growth in aircraft speed was overall exponential, but linear for each new technology that was introduced.
It would be a change to macro models of economic growth, if accepted. The default is that TFP grows exponentially. There's also a useful comment by Marginal Revolution here ( https://marginalrevolution.com/marginalrevolution/2022/04/ad... ) where he points out that TFP itself is really just a residual, and perhaps not a very well-defined concept. That is, if you regress Y = AK^beta L^(1-beta) with K being the amount…
Can’t you model almost anything as a piecewise linear function? I don’t know if this claim is saying much of substance.
Yes, but if your data is actually exponential, the linear segments are not going to be better approximations than an exponential curve. That's what's going on here
The big issue is that you get MANY more curve-fitting parameters to play with if you use a piece-wise linear model vs. an exponential model. (You get to choose HOW MANY breaks to make, what the slope is for each section, and WHERE to make the breaks.)
So... Let's say you created some synthetic data using an underlying exponential plus a normally distributed random number. Obviously, the BEST predictive model is an exponential one. However, for any arbitrary number of observations, I guarantee you there's trivially at least one piece-wise linear model that will have less error than the exponential one. Consider the one that is simply a straight line between EVERY point. Obviously that has zero error compared to the exponential model. Yet, it has very little predictive power compared to the exponential model.
Now, that's not what was done here... but there's actually quite a few parameters in the form of where to make the breaks and how many to make. Doesn't seem like a fair comparison.