Exponential Binning: Plotting Data with f(x) = x^α

[atillaqurd]

Hi,

I need to find out how to plot my data with exponential binning.
To better see the exponent of f(x) = x ^ \alpha, where x and f(x) are given, I am asked to do exponential binning the data.

Would appreciate you help.

Yours
Atilla

 

[mathman]

Use semilog graph paper.

 

[Mute]

It seems the OP hasn't replied, but there are some important issues that need to be addressed here, so I will comment on them for any future posters who stumble across this thread.

If you are generating histograms of something which you expect to follow a power law $y(x) \sim x^\alpha$, you need to use logarithmic binning, not exponential binning.

That is, you want your bins to be equally spaced on a log scale, which means you want the edge of the $k$th bin, B(k), to be given by

$$\log_{10}(B(k)) = m \log_{10} (k) + c,$$
where m is the slope and c is the intercept, which are determined by your bin range and your number of bins. For example, if you want 10 bins between 10^-6 and 10⁰, then $B(0) = 10^{-6}$ and $B(10) = 10^0$, and you can solve for m and c.

Now, this next point is extremely important: when using logarithmic binning, you must divide your y-data by the width of the bin. If you do not do this, the power of $x^\alpha$ that you measure will be wrong.

Furthermore, when estimating power laws from data, if you need anything more than a rough estimate, a linear regression is a terrible way to find the exponent. It is very prone to systematic errors. Maximum likelihood fits are a much better method. See this preprint for a discussion of properly calculating power laws from data (as well as using hypothesis testing to see if you can rule out other behaviors like log-normal distributions).

 

[atillaqurd]

Thanks indeed!