## Lognormality Revisited

I was looking up the reference for an old paper of mine on ADS yesterday and was surprised to find that it is continuing to attract citations. Thinking about the paper reminds me off the fun time I had in Copenhagen while it was written. I was invited there in 1990 by Bernard Jones, who used to work at the Niels Bohr Institute. I stayed there several weeks over the May/June period which is the best time of year for Denmark; it’s sufficiently far North (about the same latitude as Aberdeen) that the summer days are very long, and when it’s light until almost midnight it’s very tempting to spend a lot of time out late at night..

As well as being great fun, that little visit also produced what has turned out to be my most-cited paper. In fact the whole project was conceived, work done, written up and submitted in the space of a couple of months. I’ve never been very good at grabbing citations – I’m more likely to fall off bandwagons rather than jump onto them – but this little paper seems to keep getting citations. It hasn’t got that many by the standards of some papers, but it’s carried on being referred to for almost twenty years, which I’m quite proud of; you can see the citations-per-year statistics even seen to be have increased recently. The model we proposed turned out to be extremely useful in a range of situations, which I suppose accounts for the citation longevity:

I don’t think this is my best paper, but it’s definitely the one I had most fun working on. I remember we had the idea of doing something with lognormal distributions over coffee one day, and just a few weeks later the paper was finished. In some ways it’s the most simple-minded paper I’ve ever written – and that’s up against some pretty stiff competition – but there you go.

The lognormal seemed an interesting idea to explore because it applies to non-linear processes in much the same way as the normal distribution does to linear ones. What I mean is that if you have a quantity *Y* which is the sum of n independent effects, *Y=X _{1}+X_{2}+…+X*

_{n}, then the distribution of

*Y*tends to be normal by virtue of the

*Central Limit Theorem*regardless of what the distribution of the X

_{i}is If, however, the process is multiplicative so

*Y=X*then since

_{1}×X_{2}×…×X_{n}*log Y = log X*then the Central Limit Theorem tends to make

_{1}+ log X_{2}+ …+log X_{n}*log Y*normal, which is what the lognormal distribution means.

The lognormal is a good distribution for things produced by multiplicative processes, such as hierarchical fragmentation or coagulation processes: the distribution of sizes of the pebbles on Brighton beach is quite a good example. It also crops up quite often in the theory of turbulence.

I’ll mention one other thing about this distribution, just because it’s fun. The lognormal distribution is an example of a distribution that’s not completely determined by knowledge of its moments. Most people assume that if you know all the moments of a distribution then that has to specify the distribution uniquely, but it ain’t necessarily so.

If you’re wondering why I mentioned citations, it’s because it looks like they’re going to play a big part in the Research Excellence Framework, yet another new bureaucratical exercise to attempt to measure the quality of research done in UK universities. Unfortunately, using citations isn’t straightforward. Different disciplines have hugely different citation rates, for one thing. Should one count self-citations?. Also how do you aportion citations to multi-author papers? Suppose a paper with a thousand citations has 25 authors. Does each of them get the thousand citations, or should each get 1000/25? Or, put it another way, how does a single-author paper with 100 citations compare to a 50 author paper with 101?

Or perhaps the REF panels should use the logarithm of the number of citations instead?

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January 14, 2015 at 1:46 pm

Lognormal is a great distribution!

January 14, 2015 at 4:30 pm

It’s my favourite distribution too, also a topic of one of my highest-cited papers: http://adsabs.harvard.edu/abs/2005MNRAS.359..345U

In fact, I’m even tempted to buy the plushie…

https://www.etsy.com/listing/62334561/log-normal-distribution-plushie?ref=related-4

January 16, 2015 at 6:49 pm

The distribution also has a lot of excellent social science applications, but is rarely used there since few social scientists are familiar with it.

January 18, 2015 at 5:54 pm

Peter, nowadays papers which gets lots of citations are those which claim evidence for dark matter annihilation (or indirect evidence for dark matter).

shantanu

January 19, 2015 at 9:35 am

I’ll get on to it.