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Brad Fitzpatrick

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Exponentially more [Feb. 20th, 2005|12:25 pm]
Brad Fitzpatrick
The other day gaal wrote about his pet peeve. Here's mine:

Misuse of "exponentially". As in:

Bob has 4 tacos.
Mary has 7 tacos.
Mary has exponentially more tacos than Bob!

Uh, where the x-axis? There's one frickin' data point.

(and yes, I made up that example, but I see it all the time with other things in place of tacos.)

I need to get back to packing. Down to the last 10% taking 90% of the time, I bet.

[User Picture]From: garrettpalm
2005-02-20 08:38 pm (UTC)
That last 10% takes up exponentially more time than the rest!
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[User Picture]From: adamthebastard
2005-02-20 09:01 pm (UTC)
Problem with that statement is you could be making a prediction. At each increment of the percent finished there could be twice as much time spent making it to the next increment. Which of course would make it exponential.

But we all know you were just trolling... :P

unlike me of course, who would never do such a thing.
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[User Picture]From: garrettpalm
2005-02-21 01:13 am (UTC)
I was trolling?
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[User Picture]From: hughe
2005-02-20 09:03 pm (UTC)
what annoys me more is "Bob has exponentially more tacos than at any other time in history"
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[User Picture]From: confuseme
2005-02-20 09:05 pm (UTC)
There's also a generally annoying tendency for people to use "exponential" and "power law" interchangeably -- they don't actually care whether the curve is x**3 or 3**x, they just mean that it grows real fast. If you put a high enough slope on it, I bet you could get people excited about linear growth.
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[User Picture]From: gaal
2005-02-20 09:20 pm (UTC)
In related moose, the unqualified expression an order of magnitude is likewise meaningless. Like, where's the base? (Belong to us, no doubt.)

"This problem is harder by an order of magnitude".
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[User Picture]From: scosol
2005-02-21 07:31 am (UTC)
unqualified, an "order of magnitude" refers to a power of 10-

IE the problem is now 10x harder than it was
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[User Picture]From: gaal
2005-02-21 12:41 pm (UTC)
Except that for mathematicians, e may be a more natural base; and for compsci folks, 2 perhaps. This depends too much on convention and context; my experience has been that when e.g., a politician uses it, it means "more or less by some factor I won't bother to make explicit".
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[User Picture]From: scosol
2005-02-21 07:56 pm (UTC)

OH OH- we're talking about when *politicians* say things- like nucular!
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[User Picture]From: dakus
2005-02-20 09:54 pm (UTC)

taco science

The x-axis is in the amount of the taco sauce applied??
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[User Picture]From: daberna
2005-02-20 10:32 pm (UTC)
i guess it all really depends on your fitting parameters... i mean, you know how many points you need to fit a line, right? One, that way you can pick the slope.

(Man... i've been waiting years to use that pearl of geek humor)
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[User Picture]From: avva
2005-02-20 10:37 pm (UTC)
The real problem is that "exponentially" is virtually never used correctly. Say you have a bunch of data points on which your function seems to grow exponentially. As long as it's finite amount of data, it could always be a polynomial with a really large degree and very small factors at high degrees. Whatever it is, you can retrofit a polynomial to have that finite set of values.

When we say 'exponential' for finite amount of data, we really mean 'it looks like exponential growth'; but it also looks like an (appropriately retrofitted) polynomial function, too, so why are we so sure to declare it exponential growth and not polynomial? Because deep down we're convinced that there's an underlying reason which causes the exponential growth, which makes it "natural" in this case, whereas really weird polynomials with unusual degrees and factors are not "natural" and there's no reason to suppose that our data points could be caused by anything governed by such an unusual formula. But "natural" and "unusual" are not strict mathematical notions, so we have a methodological problem on our hands here, that we usually are happy to shirk and ignore.
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[User Picture]From: avva
2005-02-21 08:17 am (UTC)
That's kind of silly. If we have 10 data points, we can fit a ninth order polynomial through it, or we can fit an expontential through it, or we can fit an arbitrary number of 10th-order or 11th-order or 12th-order etc. polynomials through it as well. If there's no good reason to choose the exponential there's no good reason to prefer the 9th-order polynomial over the higher-order ones.


If you're fitting curves to data, you're not doing mathematics, you're doing some kind of analysis, e.g. science, and you apply whatever rules apply in that domain. The scientific method covers this handily; produce the simplest possible model (hence "the most natural"),

Can you define "the simplest possible model"?

Look, of course I'm not saying let's stop doing science, wash our hands of the whole deal and go home. My point is that it's not commonly understood or realised that "this is exponential growth" is *not* a purely mathematical statement, that it relies heavily on assumptions brought from the specific domain the data was collected for. You may think it obvious, but many people don't realize that; they think that just because the data points themselves are purely mathematical, you only have numbers, the fitting of the "natural" growth function to them is purely mathematical as well.

My second point is that the notion of "the most natural", or "simplest", is not only domain-specific but problem-specific, and is poorly understood (which doesn't mean it shouldn't be used as part of the scientific method).
By problem-specific I mean that it will seem much more "natural" to you to suppose that the growth of given data points is exponential if you can imagine an underlying explanation, a model that could produce exponential growth of your specific data, be it in physics or demographics or whatever else.
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[User Picture]From: scosol
2005-02-21 07:34 am (UTC)
exponential means "having an exponent" no?

sure, calling a growth pattern "exponential looking" with only 2 datapoints is useless, but with numbers of tacos whats wrong?

bob has 4 tacos?
mary has 16?

mary does in fact have exponentially more tacos than bob hahah
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[User Picture]From: avva
2005-02-21 07:57 am (UTC)
That's meaningless; if Bob has 4 tacos and Mary 5, Mary still has 'exponentially more' tacos in that sense.
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[User Picture]From: scosol
2005-02-21 07:53 pm (UTC)
maybe you misunderstood me- for mary to have "exponentially more" tacos than bob, she would have to have at least 4^2 tacos
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[User Picture]From: nick
2005-02-21 12:11 am (UTC)
ya... well... how about I exponentially kick your ass! What do you think about that biatch?
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