[HN Gopher] Is growth linear, not exponential?
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Is growth linear, not exponential?
Author : jasoncrawford
Score : 31 points
Date : 2022-04-20 19:18 UTC (3 hours ago)
(HTM) web link (rootsofprogress.org)
(TXT) w3m dump (rootsofprogress.org)
| robocat wrote:
| And perhaps there is a new linear section starting with the
| computer revolution?
| carapace wrote:
| If you like this do yourself a favor and grab a copy of Vaclav
| Smil's "Growth - From Microorganisms to Megacities"
| https://mitpress.mit.edu/books/growth
| georgewsinger wrote:
| This idea was originally discussed by Alexey Guzey back in 2021:
| https://guzey.com/economics/bloom/#bloom-et-al-appear-to-not...
|
| Perhaps there's something I'm missing, but it's weird that he
| isn't getting any credit for it.
| apolloartemis wrote:
| Can't you model almost anything as a piecewise linear function? I
| don't know if this claim is saying much of substance.
| melling wrote:
| I've heard this too. How is modeling as a piecewise linear
| function best done in practice?
| jasoncrawford wrote:
| 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
| ajuc wrote:
| For any sampled data you'll get guaranteed 100% fit by making
| it pieceways constant with N fragments where N=number of data
| points. It says nothing about the function you sampled, it's
| just a way to cheat by overfitting.
| Fomite wrote:
| 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.
| LeegleechN wrote:
| Every apparent exponential in the real universe must
| actually be a logistic or some other bounded curve.
| apolloartemis wrote:
| True, fair point.
|
| Yeah like the article mentions, they are basically making an
| analogy to the idea of "punctuated equilibrium" from
| evolutionary biology. Here's a good exploration of how
| punctuated equilibrium works, vs the alternative which is
| called gradualism.
|
| https://gvpress.com/journals/IJBSBT/vol3_no4/3.pdf
| Enginerrrd wrote:
| I'm not sure that's true in general, nor even frequently. In
| fact, I'd say it's provably false in general.
|
| 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.
| dredmorbius wrote:
| NB: "Note that both of these charts are on a log scale."
|
| Appears to apply to the two preceding _linear-scale_ charts.
| adamsmith143 wrote:
| If you zoom in far enough every curve looks linear.
| mathieutd wrote:
| Not quite!
| https://en.wikipedia.org/wiki/Weierstrass_function
| apolloartemis wrote:
| Wow that's super cool, didn't know about this
| foobarian wrote:
| Heh. Reminds me of how gamers manage to find exploits to
| cheese speed runs while developers react in dismay.
| AlchemistCamp wrote:
| Thank you :)
| aidenn0 wrote:
| 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.
| mrtnmcc wrote:
| You can get an exponential growth if the time until the next
| technology decreases linearly with each technology.
|
| 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.
| dash2 wrote:
| 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://marg
| inalrevolution.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 of capital and L the amount of labour,
| A is just the unexplained "scaling factor" and it gets called
| TFP. But what TFP actually is... is a bit of a whatever-you-
| like.
| pishpash wrote:
| Why is TFP assumed exponential by default? What's the theory
| behind that? A time lag in the transfer between exponential
| input growth (e.g. population) and exponential output growth?
| heavenlyblue wrote:
| How is aircraft limited to the speed of sound still?
| leetcrew wrote:
| they're not? it just turns out that the most efficient
| cruising airspeed for passenger jets is in the upper end of
| the subsonic range. most people can't afford supersonic
| travel, and the ones that can overwhelming prefer to spend
| the extra money on a more comfortable subsonic flight.
|
| interestingly, extremely high speeds turn out not to be that
| useful for military aircraft either. the fastest production
| fighter jet of all time, the mig-25, was introduced in 1970.
| the f-22 has a considerably lower top speed, and it's not
| because they lacked the funding to make it go faster.
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