The supposed explanation here is not very good. It amounts to this: "To get those first digits, you took the actual numbers and scaled them all by powers of 10 to get values between 1 and 10. That's kinda logarithmic, and Benford's law is kinda logarithmic, so it's no surprise that the results end up obeying Benford's law. All you really need, kinda, is for the probability distribution to span several powers of 10." This is hand-wavy and, not to put too fine a point on it, wrong. For instance, suppose we generate random numbers uniformly distributed between 1 and 1000000; they will not obey Benford's law.
The author also claims that looking at the Fourier transform of the probability distribution is key to understanding what's going on. But the full extent of his Fourier-based analysis is this: Consider the probability distribution function for log_10(data). Then Benford's law holds if this is constant (editorial note: it cannot in fact be constant) and holds roughly if it's roughly constant. That happens, kinda, when the probability distribution is very broad (editorial note: no, not really; see the example above). What, you didn't see anything about Fourier transforms there? Well, that's because the Fourier stuff is really almost all window-dressing.
For a brief account of Benford's law and related matters written by someone with a better grasp of what's going on, you could turn to http://terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-... whose author is one of the best mathematicians currently living and also a very good expositor.
Comments
The supposed explanation here is not very good. It amounts to this: "To get those first digits, you took the actual numbers and scaled them all by powers of 10 to get values between 1 and 10. That's kinda logarithmic, and Benford's law is kinda logarithmic, so it's no surprise that the results end up obeying Benford's law. All you really need, kinda, is for the probability distribution to span several powers of 10." This is hand-wavy and, not to put too fine a point on it, wrong. For instance, suppose we generate random numbers uniformly distributed between 1 and 1000000; they will not obey Benford's law.
The author also claims that looking at the Fourier transform of the probability distribution is key to understanding what's going on. But the full extent of his Fourier-based analysis is this: Consider the probability distribution function for log_10(data). Then Benford's law holds if this is constant (editorial note: it cannot in fact be constant) and holds roughly if it's roughly constant. That happens, kinda, when the probability distribution is very broad (editorial note: no, not really; see the example above). What, you didn't see anything about Fourier transforms there? Well, that's because the Fourier stuff is really almost all window-dressing.
For a brief account of Benford's law and related matters written by someone with a better grasp of what's going on, you could turn to http://terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-... whose author is one of the best mathematicians currently living and also a very good expositor.
Thanks much for the link.