## Log-Normality in Gotland

I’ve recently shown some histograms for site ring widths and opined that they loooked somewhat gamma-ish. Louis Hissink said that they looked log-normal to him. Louis is right for Gotland anyway.

Esper stated that the variance in Gotland was "heteroskedastic" and they did exercised to eliminate the supposed heteroskedasticity by a power-transformation by a power of 0.32. I’ll discuss heteroskedasticity in a minute, but first here are qqnorm- plots for:

Left – ring widths rw ; middle – with a power transformation of rw^0.32 and right – log(rw+1,2). Pretty convincing log-normality.

If this data set is representative, then there is pretty wide scale treatment of log-normally distributed ring widths as though they were normally distributed. This stuff is endless.

I’m not sure that Esper has correctly distinguished between heteroskedasticity and non-normality. The ring width distribution is obviously non-normal; the power transformation mitigates the non-normality, but nearly as well as a log-transformation. "Heteroskedasticity" is different than non-normality: it’s a situation where the variance is not uniform and, in this context, the variance is related to the mean: high average widths are associated in any given year with higher variance. I haven’t fully worked through this, but intuitively, it seems to me that either of these transformations will mitigate non-normality but not necessarily heteroskedasticity.

Mentioning logarithms and Gotland reminds me of an ancient joke, back from the time when we still used slide rules – yes, when I was at university, we still used slide rules. It was something about God telling the snakes to multiply, but they didn’t know how; so he gave them log tables and then the adders could multiply.

1. TCO

1. Lots of stuff in the real world is log normal. Distribution of particle sizes for ceramics for instance.

2. Gotta be a bit careful with log or semilog plots as “it’s easy to get a straight line”. Not sure I can quantify that, but something I learned in school. Particularly if the actual range of the variables is small. Dangerous to extract power laws that way for instance.

2. Steve McIntyre

TCO, if you look at the left-hand plot, we can be fairly sure that the distribution is not normal – whether it’s log-normal is a different issue entirely. It certainly looks more log-normal than normal as an approximation.

3. Douglas Hoyt

How you checked to see if tree ring widths follow a Poisson distribution? As I recall, precipitation can be well approximated with a Poisson distribution, but you better check on that. If true, it would suggest most tree ring variations are primarily controlled by precipitation rather than temperature. Only if you have a series of years with nearly unvarying precipitation would you then see a temperature signal.

4. Steve McIntyre

Doug, I’ve pondered quite a bit about how to construct a stochastic process that would replicate tree ring processes and it’s beyond my present skills. There are a lot of inter-related issues: autocorrelation, mixed effects, aging before you even get to climate, where you have temperature, precipitation, cloudiness, and other things like spruce budworm. It’s something that I’m working on. It would be a great project for a smart young stochastic process guy, which would probably yield some interesting results in stochastic process terms as well. I’ve got the problem laid out and the data organized if you know anyone who’d be interested.

5. fFreddy

I’ve got the problem laid out and the data organized if you know anyone who’d be interested.

Post them ! See what happens …

6. Douglas Hoyt

Steve,
Visually the tree ring width distribution looks a lot like a precipitation amount distribution and both look like Poisson distributions, so I think further investigation would be worthwhile. My inclination would be to eliminate data showing a Poisson-like distribution from a temperature reconstruction since temperature distributions are normal. Perhaps a way to check these ideas would be to look at the bristlecone pine tree widths before 1900 and see if they are following a normal distribution. That would indicate that at one time they were responding predominantly to temperature.

7. Dave Eaton

There’s an interesting site with some software that demonstrates log normal distributions and their origins in many fields:
http://www.inf.ethz.ch/personal/gut/lognormal/brochure.html

I also want to mention in passing mathworld.wolfram.com , which is where I run to when someone says something like ‘heteroskedastic’. Sometimes the I have to follow links a while, when it defines something I don’t know about in terms of something else I don’t know about, but I find it a very valuable resource when I get out of my depth mathematically.

8. Douglas Hoyt

Well, it seems like the lognormal and Poisson distributions are much alike except that the lognormal applies to a continuous distribution whereas Poisson applies to a discrete distribution. That being said, I would say that the Poisson distribution would apply to a discrete distribution of ring widths. Normal and lognormal distributions apply to continuous functions and the corresponding discrete distributions are Binomial and Poisson.

More importantly, I don’t see how one can go from a Poisson (ring widths or RW) distribution to a Binomial (temperature or T) distribution using a linear transformation. A linear transformation would imply that the resulting T would have a Poisson distribution when it actually has a Binomial distribution. It would require some complicated nonlinear transformation to convert an incoming Poisson distribution to an output Binomial distribution. Since this is not done, the transformation of RWs to Ts cannot be correct. It seems to me that people need to recognize that proxies that have Poisson or lognormal distributions are not proxies for temperature.

9. Steve McIntyre