Benford’s Law and the power-spectra of natural images

Probability of each of the digits 1 to 9 appearing as the first digit in a number to base 10. Image: Mark R. Diamond

Probability of each of the digits 1 to 9 appearing as the first digit in a number to base 10. Image: Mark R. Diamond

What is the relationship between the power spectra of natural images and Benford’s Law? The preceding question marks this out immediately as a posting for statisticians and vision researchers, but before looking for an answer to the question, one might reasonably ask why anyone would assume that there might be such a connexion. It is on that issue that the rest of this posting focusses.

Benford’s Law [1], which is actually attributable to Simon Newcomb [5], is also known as the “first digit phenomenon”. Specifically, it concerns the fact that in many tables of physical constants and statistical data, but especially in aggregations of such tables, “… the occurrence of numbers is such that the mantissae of their logarithms are equally probable” [5] and that benford

In the five decades following Benford’s publication, dozens of people produced flawed, or inadequate, explanations of the first-digit phenomenon. Nonetheless, beginning in 1995 Theodore Hill [2,3,4] published a series of papers related to Benford’s Law, the culmination of which is a theorem that is likely to become almost as important as the Central Limit Theorem.

Firstly, Hill shows that scale-invariance implies Benford’s Law. Scale invariance corresponds to the notion that, if the first digits of statistical tables follow some universal law, then that law should be independent of the units of measurement chosen. In other words, if data on river areas follow Benford’s Law, then it should not matter whether the areas are measured in square miles, square kilometres, or toe-nail lengths! Secondly, and most importantly, Hill was able to show that if distributions are selected at random in an “unbiased” way, and random samples are then taken from each of these distributions, then the significant digits of the combined sample will converge to the logarithmic distribution, and hence conform to Benford’s Law. Hill then goes on to suggest that rivers, being independently formed by diverse natural processes, might have areas that represent a sample from “random distributions selected in an unbiased way”. Similarly, the way in which natural objects overlap, hide and change one another also (probably) leads to their measurements representing random samples from randomly selected distributions. So much for the first part of the question which is at the heart of this posting.

Now for part two. Many vision researchers [6] have noted that the power spectrum of natural images usually takes the form of a power law in the spatial frequency. Specifically, the power of spatial frequency f is approximately C/(f^(2-eps)), where C is simply a scaling constant that is determined by the image contrast, and eps is a small positive value approximately equal to 0.19. As a consequence of the power-law, natural images also show scale-invariance in their power-spectra. Dan Ruderman [7] showed that artificial images, constructed by creating collages of statistically independent overlapping “objects” whose sizes followed a power-law distribution, also show a power-law distribution in their power spectra. But of course, there is no obvious reason for assuming that two dimensional collages of objects, whose sizes are deliberately chosen to follow one specific distribution, shows anything fundamental about a less carefully constructed, natural, three dimensional world.

Nonetheless, given that Benford’s Law and the power-spectra of natural images both appear to arise as a consequence of object overlap, the two ought to be inherently related. What is missing, however, is a mathematical proof that the images of agglomerations of objects whose sizes are selected at random from distributions which themselves are chosen at random in an unbiased way, will show the same power-spectra as natural images. If that can be done, then Benford’s Law and image power-spectra will be seen to arise from the fundamental mathematical structure of the universe.

I do not have the requisite statistical knowledge even to commence an appropriate analysis, but I have created a variety of computer simulations that strongly suggest that an analysis (or proof?) is worth attempting.

References

[1] Benford, F. (1938). The law of anomalous numbers. Proceedings of the American Philosophical Society, 78, 551-572.

[2] Hill, T. (1995). The significant-digit phenomenon. American Mathematical Monthly, 102, 322-327.

[3] Hill, T. (1995). Base-invariance implies Benford’s Law. Proceedings of the American Mathematical Society, 123(3), 887-895.

[4] Hill, T. (1996). A statistical derivation of the significant-digit law. Statistical Science, 10, 354-363.

[5] Newcomb, S. (1881). Note on the frequency of use of the different digits in natural numbers. American Journal of Mathematics, 4, 39-40.

[6] Ruderman, D. L., & Bialek, W. (1994). Statistics of natural images: scaling in the woods. Physical Review Letters, 73(6), 814-817.

[7] Ruderman, D. L. (1997). Origins of scaling in natural images. Vision Research, 37(23), 3385-3398.

Contributors: Mark R. Diamond

More unblocking

Unlike writer’s block, unblocking a drain can be done easily. Photo: Mark Diamond

Unlike writer’s block, unblocking a drain can be done easily. Photo: Mark Diamond

Last time, I commented that there are a multitude of ways to overcome writer’s block, and having describe one method in the previous blog, I propose to focus on a few more over the coming weeks.

Today, I have decided to focus on what might be termed “investment free” writing. Far too many people, and especially, though not exclusively, those who are early in their writing career, confuse evaluations of their writing with evaluations of themselves. For example, you struggle hard to produce a piece of writing with which you are not entirely satisfied, and then reach conclusions like “I’m hopeless”, “I knew I was no good”, and so on. Or your supervisor criticizes your punctuation and you think, “I’m not smart enough to get a PhD”. Not only are conclusions like this self-defeating, in that they lead you to feel miserable and demotivated (instead of more energized as you would want), but they are focussed on you as a person, rather than on your writing.

One way that some people have found of disengaging themselves from their writing product, is to practice writing something that is deliberately “wrong”. It might be wrong in any number of ways, and the particular way is entirely up to you, but you might consider the following —

Write as if English (if that is the language of your thesis) is not your first language. Of course, it might actually be true that English isn’t your first language, but that is not the point. The purpose is to write in a way that makes it show in an exaggerated way that writing in English doesn’t come easily to you. That much will be true, otherwise you wouldn’t have writer’s block. Here is example. I copy from Tom Lehrer song called “Lobachevsky”. If you not hear this song before, you must listen. Tom Lehrer he is very good. I am remember the time I first hear him. He make big impression to me. I am thinking, “If this man, he can make money with voice like that, then I can be millionaire.” But first, I am giving supervisor new song draft!

Another approach is to take a viewpoint with which you disagree entirely, and write a spoof or a parody. Put in jokes. Sure, whatever you write might not end up in the final version of your thesis, but (a) it might just succeed in unblocking the writing flow, (b) I’d be surprised if writing a spoof did not help you clarify what you really do think, and (c) there might be parts that you can use just as you have written them.

A third, related way, is just to write badly. Deliberately badly, like, you know, with no punctuation and ummm no kind of structure thing (Gee this sounds really bad doesn’t it) where you kinda dont even check your speling of stick in your own stuff about what it all sounds like. You might even start … You could try starting your sentence … You can just experiments with different beginnings and false starts and finally discover that in doing so you have managed to get on a roll and express, at least to some degree, the concepts that you are trying to convey. Get it? Sentences that go nowhere. False starts to a paragraph written one after the other, no back editing. Oh yes, I almost forgot. Lots of critical comments in the text like, “This is really crap”, “I really haven’t a clue about this stuff”, “I guess it must be pretty obvious that I can’t even write two sentences together.”

The critical thing about all these suggestions is that they are aimed at separating your evaluation of yourself from your evaluation of your writing. If you are trying to write like a non-native speaker, then when your writing looks as if you can’t write English, that’s good! And if you accidentally make it look as if you can write English, then you will be able to use a part of it in your thesis. If you write a viewpoint with which you disagree, sounds like you haven’t a clue about your thesis topic, or looks as if you are poking fun at the material, again, that’s good.

The main thing is to produce words. When you feel as if you can’t write, then writing anything at all is great. Later on, you can focus on producing some writing that says what you want to say in the way that you want to say it. But first off, just write.

Contributors: Mark R. Diamond