## Lognormality Revisited (Again)

Today provided me with a (sadly rare) opportunity to join in our weekly Cosmology Journal Club at the University of Sussex. I don’t often get to go because of meetings and other commitments. Anyway, one of the papers we looked at (by Clerkin et al.) was entitled *Testing the Lognormality of the Galaxy Distribution and weak lensing convergence distributions from Dark Energy Survey maps*. This provides yet more examples of the unreasonable effectiveness of the lognormal distribution in cosmology. Here’s one of the diagrams, just to illustrate the point:

The points here are from MICE simulations. Not simulations of mice, of course, but simulations of MICE (Marenostrum Institut de Ciencies de l’Espai). Note how well the curves from a simple lognormal model fit the calculations that need a supercomputer to perform them!

The lognormal model used in the paper is basically the same as the one I developed in 1990 with Bernard Jones in 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 during a lovely visit to the fine city of Copenhagen. 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:

Citations die away for most papers, but this one is actually attracting more interest as time goes on! 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 they’re playing an increasing role in attempts to measure the quality of research done in UK universities. Citations definitely contain some information, but interpreting them isn’t at all straightforward. Different disciplines have hugely different citation rates, for one thing. Should one count self-citations?. Also how do you apportion 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 a better metric would be the logarithm of the number of citations?

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May 10, 2016 at 6:52 pm

I just realised this paper was written 25 years ago….

…where did all that time go?

May 10, 2016 at 7:08 pm

Actually, my first scientific conference was in Cargese, Corsica in the summer of 1986. Almost 30 years ago…

May 11, 2016 at 9:04 am

Citation statistics is a very rough measure of quality. I don’t think number of authors has much to do with quality of the research. Perhaps number of _significant_ authors (who really contributed) has some effect, but given the uncertainty in what citations actually measure, it is not the dominant uncertainty. Number of citations does depend on activity in the field: it is difficult to get more citations than there are papers published in the field – some normalization for this might be useful.

Name recognition is crucial: better known people attract citations by that route. Perhaps we could define a ‘handicap’ depending on seniority?

A paper with zero citations after a few years is probably not that great. Above that, a better way to judge a paper, at least better than counting authors or using bibliometrics, is by reading it.

May 11, 2016 at 2:13 pm

The fit looks good… but was it compared with the fit of other distributions having a similar number of adjustable parameters in order to see just how much better?

May 11, 2016 at 2:31 pm

The only other fit shown is a Gaussian and both it and the lognormal have the same number of parameters

May 11, 2016 at 5:33 pm

The main argument for doing this is to reduce the contribution of citations from papers that people hardly did any work on, but normalising by number of authors is quite a crude approach and would be very unfair on people doing the key work but who happen to be in a large collaboration. In fields where the main contributors are higher up the list it would be much better to apply a weighting based on position in the author list, perhaps some kind of logarithmic dependence since I doubt that it is linear at all. Many of those 1000 authors in a big collaboration may deserve some payoff for what may have been years or work on the project, with no other public recognition. I think similar ideas have been suggested before, but it is probably impossible to come up with a one-size-fits-all approach due to the very different sociology of different fields.