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I have a bunch of data, and I'd like to find the closest exponential aproximation I can to fit the points. I'm guessing there's a (relatively) straightforward way to do this.

For example, if I have the data points (5, 9, 17, 33), the equation I'd be looking for would be f(n) = 2^x + 1, since that would fit the data points exactly. Of course, if I had a similar set of points (5, 9, 16, 33), the equation wouldn't work quite right: f(n) = 2^x + 1 would still be "close", but presumably some other numbers than 2 and 1 could get me ever so slightly "closer".

Is there a way to figure this out? (I'm guessing "yes"). Ideally, I'd like to have it be in the form described above (something^x + possible_constant), though methods for figuring this out in a more general way are welcome, so long as I can tie down some of the parameters (so that any other factors would be forced to be 0).

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Take the logarithm and do a linear regression. – Qiaochu Yuan Feb 28 '11 at 20:50
@Qiaochu: This works if you multiply rather than add the constant. – Yuval Filmus Feb 28 '11 at 20:52
up vote 2 down vote accepted

Here is an online exponential regression calculator.

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This uses Least Squares Fit! I thought that might be the tool I'd need, but it was so long ago that I used it (~20 years) that I was afraid of embarrassing myself by mentioning it. – Beska Feb 28 '11 at 21:04
This solved my immediate problem. And I can re-look up how to do LSF if I need it. Thanks. – Beska Feb 28 '11 at 21:18

You can solve for $f(x) = Ca^x$ using linear regression, since $\log f(x) = (\log a)x + \log C$. Here $x$ should be $1,2,\ldots$, and $y$ should be your data points.

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One of these days I'll find out why my browser insists on displaying equations as garbage...after I decode this answer, I'll let you know. – Beska Feb 28 '11 at 20:58
@Beska: try downloading MathJax to see the equations – Ross Millikan Feb 28 '11 at 21:12
@Ross: Domo arigato gozaimasu. – Beska Feb 28 '11 at 21:21

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