# Hell and High Histogramming – Mastering an Interesting Heat Wave Puzzle

Guest Post by Willis Eschenbach

Anthony Watts, Lucia Liljegren , and Michael Tobis have all done a good job blogging about Jeff Masters’ egregious math error. His error was that he claimed that a run of high US temperatures had only a chance of 1 in 1.6 million of being a natural occurrence. Here’s his claim:

U.S. heat over the past 13 months: a one in 1.6 million event

Each of the 13 months from June 2011 through June 2012 ranked among the warmest third of their historical distribution for the first time in the 1895 – present record. According to NCDC, the odds of this occurring randomly during any particular month are 1 in 1,594,323. Thus, we should only see one more 13-month period so warm between now and 124,652 AD–assuming the climate is staying the same as it did during the past 118 years. These are ridiculously long odds, and it is highly unlikely that the extremity of the heat during the past 13 months could have occurred without a warming climate.

All of the other commenters pointed out reasons why he was wrong … but they didn’t get to what is right.

Let me propose a different way of analyzing the situation … the old-fashioned way, by actually looking at the observations themselves. There are a couple of oddities to be found there. To analyze this, I calculated, for each year of the record, how many of the months from June to June inclusive were in the top third of the historical record. Figure 1 shows the histogram of that data, that is to say, it shows how many June-to-June periods had one month in the top third, two months in the top third, and so on.

Figure 1. Histogram of the number of June-to-June months with temperatures in the top third (tercile) of the historical record, for each of the past 116 years. Red line shows the expected number if they have a Poisson distribution with lambda = 5.206, and N (number of 13-month intervals) = 116. The value of lambda has been fit to give the best results. Photo Source.

The first thing I noticed when I plotted the histogram is that it looked like a Poisson distribution. This is a very common distribution for data which represents discrete occurrences, as in this case. Poisson distributions cover things like how many people you’ll find in line in a bank at any given instant, for example. So I overlaid the data with a Poisson distribution, and I got a good match

Now, looking at that histogram, the finding of one period in which all thirteen were in the warmest third doesn’t seem so unusual. In fact, with the number of years that we are investigating, the Poisson distribution gives an expected value of 0.2 occurrences. In this case, we find one occurrence where all thirteen were in the warmest third, so that’s not unusual at all.

Once I did that analysis, though, I thought “Wait a minute. Why June to June? Why not August to August, or April to April?” I realized I wasn’t looking at the full universe from which we were selecting the 13-month periods. I needed to look at all of the 13 month periods, from January-to-January to December-to-December.

So I took a second look, and this time I looked at all of the possible contiguous 13-month periods in the historical data. Figure 2 shows a histogram of all of the results, along with the corresponding Poisson distribution.

Figure 2. Histogram of the number of months with temperatures in the top third (tercile) of the historical record for all possible contiguous 13-month periods. Red line shows the expected number if they have a Poisson distribution with lambda = 5.213, and N (number of 13-month intervals) = 1374. Once again, the value of lambda has been fit to give the best results. Photo Source

Note that the total number of periods is much larger (1374 instead of 116) because we are looking, not just at June-to-June, but at all possible 13-month periods. Note also that the fit to the theoretical Poisson distribution is better, with Figure 2 showing only about 2/3 of the RMS error of the first dataset.

The most interesting thing to me is that in both cases, I used an iterative fit (Excel solver) to calculate the value for lambda. And despite there being 12 times as much data in the second analysis, the values of the two lambdas agreed to two decimal places. I see this as strong confirmation that indeed we are looking at a Poisson distribution.

Finally, the sting in the end of the tale. With 1374 contiguous 13-month periods and a Poisson distribution, the number of periods with 13 winners that we would expect to find is 2.6 … so in fact, far from Jeff Masters claim that finding 13 in the top third is a one in a million chance, my results show finding only one case with all thirteen in the top third is actually below the number that we would expect given the size and the nature of the dataset …

w.

Data Source, NOAA US Temperatures, thanks to Lucia for the link.

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I’m impressed at how nicely your 13 month samples fit the Poisson distribution. Nice review.

jorgekafkazar

Lucia has an update based on low serial autocorrelation data for the 48 states. It’s apparently a lot closer to white noise than she thought previously.

As always, astounding reasoning, Willis. I find your conclusion flawless. I find that it also supports something that I have long suspected — few people are actually qualified to work with statistics or make statistical pronouncements. From what I recall, Jeff was only quoting some ass at NOAA, so perhaps it isn’t his fault. However, you really should communicate your reasoning to him. I think that there is absolutely no question that you have demonstrated that it is a Poisson process with significant autocorrelation — indeed, from the histogram (exactly as one would expect) and that as you say if anything it suggests that there have (probably) been other thirteen month stretches. It is also interesting to note that the distribution peaks at 5 months. That is, the most likely number of months in a year to be in the top 1/3 is between 1/3 and 1/2 of them!
Yet according the reasoning of the unknown statistician at NOAA, the odds of having any interval of 5 months in the top third are $(1/3)^5 \approx 4/1000$. They seem to think that every month is an independent trial or something.
Sigh.
rgb

1 in 2.6 is close enough to 1 in 1.6 million for the average climate alarmist; what’s your beef? Nothing that a little data adjustment won’t fix.

captainfish

Willis, Can I please borrow your brain for a few days.. I could make a gazillion dollars with all that extra smarts and speed of thought. I can’t even comprehend the amount of work and effort that it even took to come up with the line of analysis, let alone sit down with the data. But then, I still don’t have your brain. But then, unfortunately, not many scientists do either.
Thank you, Sir.

John F. Hultquist

Say you investigate a sample of teen age boys. You will find that each has a height. Do the measurements. Run the numbers.
Now investigate a sample of time intervals with lightening strikes. Not every time interval has a strike. There’s the rub!
Your example (people lined up at a bank teller’s window) is the same idea. Somewhere (long ago), I believe hearing or reading that this is exactly why the Poisson distribution was invented. Not that many folks make it that far in their education.

Robert

Willis, I think you are one of the most clever people I have ever read……very well done and keep up the good work,
Robert

Steve R

This whole 1 in 1.6 million issue has been great entertainment. It’s also been an eye opener, to see so many climate scientists struggling with a fairly basic statistical concept.

dp

Nice work, Willis – I can’t see a flaw in your work but it’s been a long day so I’ll look again for the sake of due diligence in the morning. The data has Poisson written all over as I head off to bed, and that doesn’t leave a lot of wiggle room.

Bart

I’m not getting the controversy. We’re not dealing with a stationary process here. The guy said: “These are ridiculously long odds, and it is highly unlikely that the extremity of the heat during the past 13 months could have occurred without a warming climate.” No kidding. And, we are currently at a plateau in temperature which is the result of combining the steady warming since the LIA with the peak of the ~60 year temperature cycle. So, what’s the gripe? The globe has been warming. Everyone knows it’s been warming. The disagreement is over the cause.

It would appear that inadvertently Mr. Masters, or whoever provided him with his numbers, has arrived at a ratio that is quite correct, the only problem being the ratio is applied to the wrong query. If you ask “what are the odds of a story about a human caused plague of horrendous heatwaves, which appears in any Lamestream Media source, NOT being complete BS?” the ratio of 1 in 1.6 million appears, to my eye at least, to be just about spot on.

Venter

Brilliant work, Willis.

Steve R

Bart: The point is that the claimed 1 in 2.6 million chance that this June to June “event” is BS. and regardless of whether there has been warming or not, this “event” is indistinguishable from random.

as in the song:

S’il fait du soleil à Paris il en fait partout…

You may have it warm in the US. Here in Europe we have a rather cold and wet early summer.
Is there also a 1 in 2.6 probability to experience a series of coldest 13 continuous months within a 116 years period? of wettest? of windiest? of cloudiest? of …?
Anthropogenic warming seems to be more frequent in Northern America than elsewhere (or there are more blogs telling this there than elsewhere).

Weather is not climate. What climate would you like? let’s change it!

Willis Eschenbach

Robert Brown says:
July 10, 2012 at 10:24 pm

As always, astounding reasoning, Willis. I find your conclusion flawless. I find that it also supports something that I have long suspected — few people are actually qualified to work with statistics or make statistical pronouncements. From what I recall, Jeff was only quoting some ass at NOAA, so perhaps it isn’t his fault. However, you really should communicate your reasoning to him.

Thanks as always for your comments, Robert. I actually feel sorry for Jeff Masters, because I’ve made foolish public errors myself. It’s not easy, it’s painful, and it is the risk we all take when we blog. So I’ll pass on contacting him, it would look like kicking a man when he’s down.
I suspect he will hear about my analysis in any case. Most people who are truly interested in climate science read WUWT, regardless of their position on the AGW supporter/skeptic continuum, if only to find out what fools these mortals be today.
However, I doubt that in general I am “actually qualified to work with statistics”, as my knowledge of statistics (as with many things) is quite wide but is only deep in some places. However, I am well served by my habit of starting (and perhaps even finishing) by using my Mark One Eyeballs. I get as much data as I can, stretching back as far as I can, and then I put it up on the silver screen and I just think about it. I give it the smell test. I re-plot it from some other angle, I’m a graphical thinker. I give it the laugh test. I use color to give it another dimension. If necessary, I look at and consider each and every case or station record or proxy of the data individually. The work is often very boring, but also has flashes of fire and insight.
I look at the data first, before theorizing about the data, or calculating the statistics of the data, or analyzing the situation using pseudo data or red noise, or doing a monte-carlo analysis. I just look at the data in as many graphical representations as I can dream up. Then I look at it again.
I look because at the core, I’m trying to understand the data, not to measure its waistline. Oh, I may get around to that, but before I pull out the cloth tape and start taking the measurements, I want to know the habits and the relationships and the linkages and the interactions and ultimately the very form and meaning of the data.
Because at the end of the day, statistics are a model of reality. As a result, picking the appropriate model for the situation is the central, crucial, indispensable, and often overlooked first step of any statistical analysis. If you don’t start out with the correct understanding of what’s going on, all the statistics in the world won’t help you. And for me, the only way to get that understanding is to look at the longest dataset I can find, from as many angles as I can, and to think about the data in as many ways as I can.
All the best to you,
w.

P. Solar

Damn, a poisson distribution. I always new these climatologists were hiding something fishy !
Very good analysis Willis. It’s a shame that some of these cargo cult scientists are not capable of applying appropriate maths.

P. Solar

Doesn’t this analysis basically point out that the warming trend far smaller than the magnitude of variations? The fact that it peaks between 4 and 5 months tells us that the dominant variation is of that timescale. This is what we commonly call seasons.
None of this is surprising but it still does a very good job of pointing out how rediculous and inappropriate Masters’ comment was.
Maybe he should have thought about it before using it. The fact he apparently got it from someone at NOAA does not excuse his need to think whether it makes sense before using it himself.

SasjaL

AWG: a chance of 1 in 1.6 million of being close to a correct analyze …

David

Six months of record recent heat wave was a once in 800,000 year event????
There have been 372,989 correctly recorded daily high temperature records in the US since 1895. 84% of them were set when CO2 was below 350ppm.
http://stevengoddard.wordpress.com/2012/07/08/heatwaves-were-much-worse-through-most-of-us-history/
http://stevengoddard.wordpress.com/2012/07/08/ushcn-thermometer-data-shows-no-warming-since-1900/
Lots of things are very rare….
ttp://stevengoddard.wordpress.com/2012/07/11/1970s-global-cooling-was-a-one-in-525000-event/
sheesh!

Poisson is by far the most important and least known mathematician for all aspects of real (non-relativistic) life.

Willis Eschenbach

Bart says:
July 10, 2012 at 11:32 pm

I’m not getting the controversy. We’re not dealing with a stationary process here. The guy said: “These are ridiculously long odds, and it is highly unlikely that the extremity of the heat during the past 13 months could have occurred without a warming climate.”

The controversy is that the “ridiculously long odds” he refers to are wildly incorrect …
w.

tonyb

Robert Brown
I think you make a very good point when you say that few people are qualified to talk about statistics. That is so even on quite simple statistics, but at the sort of level of tree ring analysis and many other facets of climate science I think the maths is quite beyond most scientists as it is a special separate field that they are unlikely to have learnt in detail.
It would be interesting to know who is actually qualified to interpret statistics arising from their work or whether they get in genuine experts to check it through. I suspect the number that do is very small indeed.
tonyb

cd_uk

Willis
I have to be a pedant here but what you plotted is a bar chart not a histogram 😉
I think the main problem with pushing out the trite case of probability with a time series is that you have no base. For example where, and at what length of time, would you determine a norm. He assumes that time series are not second order stationary (why?) and hence you can get a change in the distribution with time (dirft) which is implicitly suggested in his (paraphraising) “…you can only get this if you’re in a warming world…”. This assumption is only the result of incomplete information as it only appears as such when choosing a small portion of the time series. So absolute nonsense, there is no need for any more statistics in my view as it is completely flawed because of incomplete data although you make a more appropriate case here.

Brian H

Since, as you say, “one case with all thirteen in the top third is actually below the number that we would expect given the size and the nature of the dataset …” it follows that warming climate reduces the likelihood of severe weather. Which is what the “heat gradient flattening” POV (mine) predicts.

Mike

What Willis missed is that Jeff sold his website to the weather channel for some beaucoup buxes and he needs to deliver this sort of bs in a technical fashion so they can feel they are getting a good deal. Model that in your Poisson distribution. big guy!

Mike

tonyb

Willis
Nice work. As Phil Jones is unable to use a spreadsheet I doubt if his high profile work is statistically sound. Don’t know about others like Mann as there is so much sound and light surrounding his work. I see him as the lynch pin so his expertise in statistics and analysis is obviously highly relevant
tonyb

Bruce of Newcastle

Mr Masters is an intelligent weatherman, so he must already know that the US heatwave is primarily due to blocking, as was the Moscow heatwave a couple years ago.
Therefore the real question is “does CO2 cause an increase in blocking events?”.
Mr Masters may be able to tell me otherwise, but I’ve seen no hint that this is the case from the literature. But I’ve seen many times that low solar activity is linked with increased jet stream blocking.
I would therefore put the onus on Mr Masters to show that the null hypothesis is false: ie this event is related to solar activity, given the Ap index recently hit its lowest value for over 150 years.

JR

It surely looks like a Poisson process and the 1 in 1.6 million figure is absolutely bollocks, but isn’t the interesting question how / if the lambda has changed over the years? Let’s say calculated from the data of a 30-year-or-so sliding period?

Dr Burns

Unless you are certain of the underlying distribution, curve fitting and reading off the tails may lead to large errors. For example, you might try fitting a Burr distribution. It would be interesting to see if you get a similar result.

Uno2Three4

2.6 times in 1374 trials is a 1 in 528 chance.
Curiously, I happen to turn 44 years old this year which is also 528 months. (12 * 44)

Willis Eschenbach

cd_uk says:
July 11, 2012 at 1:45 am

Willis
I have to be a pedant here but what you plotted is a bar chart not a histogram 😉

Naw, you don’t have to be a pedant, cd_uk, it’s a choice, and one I’d advise you to give a miss.
But if you are going to be a pendant, you should at least be a good one. Mathworld says a histogram is:

The grouping of data into bins (spaced apart by the so-called class interval) plotting the number of members in each bin versus the bin number.

Since that is exactly what I’ve done, it is indeed a histogram. Take a look at the Mathworld example, looks like mine.
w.
PS—By the way, what I believe you are talking about is generally called a “column chart”. In a “bar chart” the bars run horizontally, while in a column chart they run vertically.

Ian_UK

The original mis-use of statistics is the sort that landed an innocent woman in prison for child abuse. Though eventually exonerated and released, she died by suicide. That’s how dangerous these people are!

Kasuha

It looks like a very nice analysis, thanks for it.
I’d just like to see a somewhat more solid proof that poisson distribution is the correct one to use in this case than “The first thing I noticed when I plotted the histogram is that it looked like a Poisson distribution.”. Both number of extreme records and number of their streaks is going to decline over time on normal data, but I suspect streaks are going to decline way faster because we’re working with fixed interval rather than portion of the record. Is the poisson distribution invariant to that?

Kerry McCauley

Reading Willis’ response to rgb at 12:02 above puts me in mind of Richard Feynman talking about what his wife said with regard to Feynman joining the team to find out what happened to bring Challenger down….something along the lines of “You better do it…you won’t let go….you’ll keep circling around, looking at it from a different perspective than others…and it needs doing.” Ah, Willis, what a treasure you are….and in such good company.

mb

I do agree with the general feeling that the “ridiculously long odds” are on very shaky ground, But I don’t agree with Willis’ statistical model. It may be that for n distinctly smaller than n the distribution of “n months in top third out of 13” is roughly approximated by a Poisson distribution, but it’s a leap of faith that the approximation is valid for n equal to 13. There is bound to be edge phenomenons.
For instance, the model predicts that “14 months out of 13 would be in the top third” happens about once in 1374 tries. On the other hand, we can be absolutely sure that this won’t happen until we get two Mondays in a week. This is an edge effect, The model breaks down for completely trivial reasons as soon as n is greater than 14, so we should not trust it too much for n equal to 13.

GraemeG

Great analysis. What really saddens me is that so many after years of study appear to have not done so well at basic statistics which is part of most courses where some analysis is likely to be required. May be it is just that many climatologists just can’t do statistics. I don’t know just looking for some rational explanation for the outrageous claims of so many over the last few years.

David C

Willis – Beautiful work simply and elegantly explained. Entirely as we’ve come to expect from you.

Gary

Such calculations as p^N are based on independent random events which this is not.
With 20 and 60 year oscillations high event will occur roughly 60 years apart and the change of one being higher than the next is 1 / [the number of such events].
Over the last “118 years” there have been 3 (maybe even only 2) such events. Placing the odds of a 13 months of such significant highs occurring (without warming) at close to – 1 in 3 – sometime over this short time period at the height of the harmonic cycle.
With a small natural warming such has been going on since the LIA the chance is very close to 100%.
So if there is a claim that 13 months of co-joined, not independent, warm whether at the peak of this 60 year cycle is proof of a small natural warming trend (since 1895 or even the LIA) – the chances of that are approximately 66% likely.
To be fair though such limited datasets are usually rated against [N-1] events making even that an overstatement of the chance: more like 50% likely.

Statistics is for those who understand its intricacies, and I do not. It appears to me that extreme events may happen (but not necessarily) if certain principal conditions are satisfied. From my (non-climate related) experience this is most likely happen with cyclical events when one of two extreme plateaux (plural ?) is reached. If there is a 65 year cycle in the climate events (AMO at peak etc) , than it looks as the conditions are right for such events to be more frequent than usual.
Just a speculation of an idle mind.

Dermot O'Logical

So just to clarify – the actual odds of a running 13 month “month in top third” based on historical observational data is 2.6 out of 1374, or 1:528 (ish) ?
Maybe I should wait on Jeff opening a betting shop.

Nigel Harris

Willis, you say that picking the appropriate model for the situation is the central, crucial, indispensable, and often overlooked first step of any statistical analysis.
But the Poisson distribution is unbounded at the upper end. So the distribution that you fitted to your histogram also suggests that we should expect to find one instance (0.939) of a 13-month period in which 14 of the temperatures are in the top third. And it wouldn’t be that surprising to find a 13-month period with 15 (expected frequency 0.326) or 16 (expected frequency 0.106) of the individual months in the top third.
Does this really sound like the appropriate model?
Also, lambda in the Poisson distribution is the expected value of the mean of the data. So if you fit a Poisson distribution, you are determining that the mean number of months falling in their top third in a 13-month period is 5.213. (Note: the fact that you arrive at 5.213 seems odd to me, as I’d expect only 4.333 months out of every 13 on average to be in the top third. Am I missing something here?). However, it seems to me that your discovery that the mean of your distribution (5.213) remains the same when you oversample the same dataset is unsurprising. And it doesn’t really endorse the choice of Poisson as a distribution.

John Brookes

Nice work Willis. I like your approach.
So in over 1300 13 month periods, there has been just one where every month is in the top 3rd.

KevinM

Yes, this was the right way to do the math (critic of earlier posts).
Have to look at the details more, like the reasoning for using 13 months, and using The warmest third instead of, say, warmest 10 percent, but at least the tools are good.

A C Osborn

Willis, is your and his data Raw or after it has been mangled by Quality Control algorithms?

This sort of statistical mistake is reminiscent of certain other problems that do not necessarily follow simple intuition about stochastic events. The famous ‘birthday’ problem comes to mind– in a group of 50 people, there is about a 95% probability of two having the same birthday. In the case of the birthday problem, it is the difference between the probability of two particular people having the same birthday compared with any two people in a larger group. It raises the interesting question: If you included all possible combinations of six months out of a sequence of thirteen (rather than sequential months), would you arrive at an even higher probability? I suspect so, despite the fact that you would be reducing the correlations by spreading out the sample.

Bob Layson

Major cities might almost have been designed to set higher max and higher min temperatures. Paint them black and carbon-dioxide could be awarded anothers recordbreaker’s medal – by dumb judges.

John West

Willis Eschenbach says:
“I look because at the core, I’m trying to understand the data”
And that makes all the difference! It seems to me that the climate change action advocates are not trying to understand the data but are trying to understand how to use the data to promote the cause.
Nice work debunking this “wolf cry”.
Also, even if he were right about the stats, evidence for a warming climate is not proof it’s man made nor quantify how dangerous it might be.

Steve in SC

Good job Willis.
The same weather pattern has been in place in the south for at least 30 years.
I recall my wife getting her citizenship on the steps of Tom Jefferson’s house in 1977 in 105 degree heat. Happens every year just a little more intense this year. Shade is strategic terrain.

cd_uk

Willis
I can see you’re conversational skills are as about as an inept as your stats.
Histograms deal in bins not categories: i.e. ranges 0-1, 1-2, 2-3 hence the x-axis labelling should be at each tick not between ticks you are plotting categories (0, 1, 2, 3, 4) where’s the bin range. As for Mathsworld it should know better. If you don’t believe me you can look at even elementary statistical packages such Excel: histogram vs bar chart functionality – note they are not the same.