Hell and High Histogramming – Mastering an Interesting Heat Wave Puzzle

u.k. (us) says:
July 11, 2012 at 10:01 pm
If you’ve come up with a system which allows you to say a given horse is 81%-90% unlikely to win, that could be considerably valuable. Pick races with two strong contenders, then eliminate one of them with your algorithm, and Bob’s your uncle.

Sorry KR…

It’s sad to see that what’s happening on these pages exactly matches what Al Gore was complaining about in his famous “They pay pseudo-scientists…” monologue. No, I don’t think Mr. Eschenbach is paid by anyone for the “science” he’s presenting here but over time I’ve come to conclusion that what’s he presenting here is not science. It looks like science and it gives results likely everybody here wants to see but so far every single such result was mathematically unfounded and questionable at best.
I understand and share Mr. Eschenbach’s approach of ‘looking at and understanding the data’. But there is one more step he’s refusing to do – ask myself ‘okay, now after I got this result, let’s find out why is it wrong’, find all possible loopholes in the approach and prove that they don’t affect the result. Instead, he lets others to find out and when they do he steps up to defend his approach regardless how much evidence is against him.
I’ve been doing some analyses in climate data myself and I know it’s HARD. There are many ways how to process the data and by careful selection you can always find a way to get the conclusion you want to see. The real science is not in the conclusion – it is the art of using the right approach. And sure enough, the position “I’m right because you have not convinced me that I’m wrong” is not scientific at all.

pjie2
As I said to KR, in the purist sense he should’ve normal score transformed the distribution into Gaussian space, interrogate the data there and then back transfrom the outputs. But phew, that would make for an incredibly boring post. It’s just a blog, lighten up. Your point is probably correct, but then he isn’t running a Poisson experiment just waht looks like an attempt to parameterise his distribution. I think for “here’s something that might be of interest” stuff, it isn’t too bad – you’ll see a lot worse in peer reviewed literature.

Well, the limiting case is to truncate the data to 13 months for a probability of 1. Also 1 for all 13 years in the middle and lower thirds. So the nature of the distribution changes over time even if the trend is flat. My head started to hurt so I dropped back to simple N of M analysis.
No, 13 months means a probability of zero that all of them are in the top third of 13 months. Also, it is by no means 1 for years in the upper third, because we’re looking at months, not years. Even a warm year can (and “usually” does) have a cold, or at least a normal, month). Under most circumstances, but in particular under the circumstance of non-trended data with gaussian fluctuations of some assumed width around some assumed mean, the probability of encountering one is zero until one hits month 39 (so one CAN have 13 in a row) and then monotonically increases with the size of the sample (but is very small initially and grows slowly).
If the data is trended everything is very different. Suppose the trend is perfectly linear with slope 0.01, which is a decent enough approximation to the 100 year actual data. Suppose that the noise on the data is — ah, we discover a key parameter — is the noise gaussian? Is it skewed? What’s its kurtosis? And above all, what is its width? Let’s say the noise is pure gaussian, width 0.1. In that case the probability of finding a hit in the first half of the data is essentially zero. In year 33 for example, the data mean for each month is exactly at the lower third boundary. You then draw 13 marbles from the gaussian hat. Each marble has to add 0.33 to barely make it to the top third. That is 3 sigma, so you are basically looking at rolling a 0.001 uniform deviate 13 times in a row in a Bernoulli trial. Your odds of winning the lotter or the Earth being struck by a civilization-ending asteroid are higher. Now suppose sigma is 0.33. Now it is a 1 sigma jump to the top third. This is good — your chances of making it are up to a whopping , which is the mean lifetime of a snowball in hell. Now make , so that the jump to the top third is a once sided . Now our Bernoulli trial probability is around 1/3 (finally) and we get approximately and we are as good as the Masters etc prediction — for this point at the boundary of the bottom third.
Note that as we make larger, we increase this (for this point) until we reach the limit for of — now a true coin flip Bernoulli trial and still a sucker bet (for this point).
Now consider the top point. It is there sitting at the top of the top third. Now small guarantees that it will be a thirteen month run. In fact, you have to make quite large to have a good chance of making the thirteen trial run back to the second third. However, it is indeed a lot easier to fall back (given a large than it was to move forward, because to move forward you had to win thirteen flips in a row, to lose and fall back you only have to lose one time in thirteen tries.
Now consider the latest 13 month run as a datum. Suppose you reach into the hat and pull out a thirteen month rabbit. Forget all modeling, it’s a real rabbit, sitting there looking at you. What does it teach you?
Well, one thing it does is it tells you something important about sigma and/or autocorrelation, or it tells you something about the data itself. What it does not do is tell you anything about the trend itself — it only tells you something about the trend compared to sigma!.
Either it is a random rabbit, p happens, bad/good luck pip pip, or else — and I’m just throwing this out there — it tells you something about the underlying temperature trend.
For example, suppose that the real temperature trend were just 0.066 per year, but I was nefarious and adjusted data (or failed to correctly account for instrumentation) so that my reported trend were much higher, and strongly biased at the end so that it was highest at the end. In that case the trend might well overrun sigma so that it becomes a lot more likely that the rabbit is pulled! Observing the rabbit is thus indicative of a problem with the data.
There is a lovely example in The Black Swan, where Taleb describes two people who are asked almost exactly this question. Moe (or whatever, book not handy) the taxi driver is asked what the chances are of flipping heads on a coin, given the information that the last 100 flips here heads. Dr. Smartaass (again, wrong name, but you get the idea) who is a Real Scientist is asked exactly the same question. Dr. S replies “Fifty percent, because the coin has no memory”. Moe says “It’s a mugs game. The coin has two heads, because there is no friggin’ way you can flip 100 heads in a row on a two sided coin.”
Bayes, Jaynes, Shannon, Cox all agree with Moe, not Dr. S! It’s a mug’s game. What one should conclude from the observation of 13 months in a row given precisely the presented analysis is that the temperature series used to compute it is seriously biased!
We will now return to your regular presentation.
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Willis Eschenbach – The question Masters was investigating was how likely the 13 months in a row of top 1/3 temperatures was absent a trend? And to do that he (and Tamino, and Lucia) looked at the variance and behavior of the monthly data and estimated how likely the observations are give that behavior.
[Incidentally, insofar as the Shapiro-Wilk test goes, monthly anomalies standardized by their SD (which is reasonable considering that the top 1/3 check is on a monthly basis) do follow the normal distribution. See http://tamino.wordpress.com/2012/07/11/thirteen/%5D
The question you asked (and answered) is how much do the observations look like the observations? You fit a Poisson distribution – you might as well have fit a skewed Gaussian, a spline curve, or a Nth order polynomial; each would be in that case descriptions of the observations. And, oddly enough, the observations match that description at 1:1, +/- your smoothing of those observations. You have no expectations in your evaluation, and hence nothing to compare the observations or their probability to.
You’ve put the observations in a mirror – and they look just like that reflection. You haven’t compared them to any expectations, or you would notice the antenna and extra limbs, and perhaps find them a bit unlikely…
—
As said by multiple posters here and elsewhere – the 13 month period of high temperatures is extremely unlikely without a climate trend. With the warming trend, it goes from a 5-6 sigma event to a 2-3 sigma. And that is the point that Masters was making.

Not at all Nigel
As I said I would’ve done things differently, but it is a blog; it isn’t being used to inform decision makers, and he hasn’t stated that he’s a statistician. I don’t think he’s right but then as I’ve stated many times above doing any type of statistical analysis such as this on a time series without determining whether its first and second order stationary is a waste of time (in both pro and anti camps). But it is just a blog post where at least he’s making an attempt to approach the issue “casually” without dismissing things out-of-hand.
I think everyone seems to be getting a bit hot under the collar. Make your criticism, suggest a better way and move on.

Coming at it from a slightly different angle. It is summer, close to the summer solstice (June 20th) given the observed change in the Jet Streams over the past couple of years, blocking highs with consecutive high temperatures are to be expected.

….During the autumn and winter months the waves of the jet stream are directed towards the UK, bringing bands of wet and windy weather quite typical of these seasons. However, during the summer, the jet stream usually shifts northwards and steers the depressions away from the UK. This northward shift also allows an area of high pressure, known as the Azores high, to nudge northwards, bringing more summery dry, sunny and warm weather to the UK.
Both in 2009 and last 2008 the jet stream stayed to the south (see above) which continued to steer Atlantic depression after Atlantic depression towards the UK. This meant that the summer was plagued by spells of unseasonably wet and windy conditions which were more apt for the autumn and winter months……
http://www.geogonline.org.uk/g3a_ki4.1.htm

This article also states:

…Rossby Waves are like rivers of air in the upper troposphere and they gradually meander. The meander loops get bigger and bigger until their wavelength from trough to trough could be as much as 8000 kms. When the Waves are well developed and cover a wide range of latitude they are said to have a low zonal index – which leads to the formation of ridges of blocking, high pressure systems and dry stable conditions. When they are almost straight and cover a narrow zone of latitude they are said to have a high zonal index – which leads to a succession of low pressure systems and unsettled weather. The waves evolve then they straighten up and then meanders form again in an endless cycle. The wave evolution cycle lasts about 6 weeks….
….cold Polar air is dragged southwards and surrounded by warmer Tropical air. Similarly loops of warmer Tropical air are moving north and being cut-off by cold Polar air. In this way heat transference is occurring – cold air moving south and warming; warmer moving north and cooling. When the loops become very pronounced, they detach the masses of cold, or warm, air that become cyclones (depressions/low pressure) and anticyclones (high pressure) areas that are responsible for day-to-day weather patterns at mid-latitudes….. http://www.geogonline.org.uk/g3a_ki4.1.htm

In other words given a low zonal index, that is a loopy jet stream, the chances of having a blocking high with several days of hot dry weather (here in NC humidity was often below 50%) is to be expected. Also to be expected is the cool rainy weather in the UK.
Of interest is this statement from Astronomy Online: The Tropopause can shift position due to seasonal changes, and marks the location of the Jet Streams – rivers of high winds energized by UV radiation.
And is echoed here:

A quantitative understanding of stratosphere-troposphere coupling is important for three main reasons:
On intraseasonal time scales, weather, storm tracks, and phase of the Northern Annular Mode (NAM) are affected by variability of the stratosphere (e.g., Baldwin and Dunkerton, 2001; Thompson et al., 2005). The slowly varying stratosphere provides an element of improved predictability on a time scale of at least two months, with a magnitude approaching that of ENSO (Thompson et al., 2005).
As the climate of the stratosphere has changed, the positions of tropospheric jets have shifted and the width of the tropics has expanded (e.g., Thompson and Solomon, 2002; Gillett and Thompson, 2003; Son et al., 2008; Lu et al., 2009). Depletion of the ozone layer over Antarctica has caused a poleward shift of wind and precipitation patterns (Perlwitz, 2011). The depletion of Antarctic ozone occurs primarily during late winter/early spring, causing a cooling of the polar stratosphere owing to reduced absorption of ultraviolet radiation. This cooling leads to a delayed summertime response in the lower atmosphere, characterized by a poleward shift of the jet stream. Kang et al. (2011) concluded that roughly one third of the recent Australian drought can be attributed to stratospheric ozone loss.
http://www.sparc-climate.org/about/themes/stratosphere-troposphere-dynamical-coupling/

Sounds a little like Stephen Wilde doesn’t it?
And finally we have NASA on the sun:

Solar Wind Loses Power, Hits 50-year Low
…The change in pressure comes mainly from reductions in temperature and density. The solar wind is 13% cooler and 20% less dense.
“What we’re seeing is a long term trend, a steady decrease in pressure that began sometime in the mid-1990s,” explains Arik Posner, NASA’s Ulysses Program Scientist in Washington DC.
How unusual is this event?
“It’s hard to say. We’ve only been monitoring solar wind since the early years of the Space Age—from the early 60s to the present,” says Posner. “Over that period of time, it’s unique….
“The solar wind isn’t inflating the heliosphere as much as it used to,” says McComas. “That means less shielding against cosmic rays.”
In addition to weakened solar wind, “Ulysses also finds that the sun’s underlying magnetic field has weakened by more than 30% since the mid-1990s,” says Posner. “This reduces natural shielding even more.”…

As the solar wind and the sun’s magnetic field weaken the amount of cosmic rays striking the earth increase. However the current theory is cosmic rays DESTROY the ozone not create it, but not all agree.
New theory predicts the largest ozone hole over Antarctica will occur this month – cosmic rays at fault

Cosmic rays and stratospheric ozone
Energetic particle events including cosmic rays penetrate the terrestrial atmosphere and perturb its chemical stability. Specially, the balance of nitrogen (NOx) and hydrogen (HOx) components are changed and the O3 destruction begins via catalytic process [71,72]…..
depletion starts within few hours of the arrival of charged particles, (iv) the solar particle induced effects in the atmosphere could last days or weeks, but no relevant long-lived effects were claimed.
Kozin et al. [73] showed that during Forbush decreases the total ozone content, registered by 29 stations situated in the latitude range 350 – 600, decreased practically synchronously with the galactic CR intensity. Contrary to it, Shumilov et al. [74] have shown an increase of the total ozone up to 10% at high latitudes and an insignificant effect at mid latitude during Forbush decreases ….
htp://arxiv.org/pdf/0908.4156

Whether or not it is cosmic rays causing an increase in ozone or the “less puffed up” atmosphere, a negative PDO or something else, the pattern of the Jet Stream has changed from a high zonal index to a low zonal index. Even a lowly farmer would notice the winds are no longer steady out of the west in NC but from all points of the compass including from the east and this has been going on for a couple of years now. Therefore the possibility of blocking highs with record breaking consecutive temperatures is not 1 in a million but an expected occurrence.

– Looks like some good work and analysis here. A few hot months are to be expected. I’d only add that a Kolmogorov-Smirnov test could be useful to see how well the Poisson distribution is fitting the data. (e.g. http://mathworld.wolfram.com/Kolmogorov-SmirnovTest.html)

Nigel Harris says:
July 12, 2012 at 1:53 am
“As several commenters have pointed out (with greater or lesser degrees of condescension), your analysis is tautologous. “
That’s not quite right either, though. IF these data fit the requirements for the particular distribution, it would be quite possible to estimate a non-trivial probability for an event which had not been observed, and the mean frequency of such events in any case.
JJ says:
July 12, 2012 at 6:42 am
IME, your posts are always perspicuous and perspicacious.

As said by multiple posters here and elsewhere – the 13 month period of high temperatures is extremely unlikely without a climate trend. With the warming trend, it goes from a 5-6 sigma event to a 2-3 sigma. And that is the point that Masters was making.
And I agree (although what the sigma is depends, as noted, on parameters of the model estimation process and their best interpretation is that the data fails the null hypothesis of unbiased data, BTW, precisely because it is a 2-3 sigma event, infinitely more so as a 5-6 sigma event).
A secondary, but absolutely fascinating possibility is that the natural variance of the weather was strongly suppressed in the US for that period by an equally natural process. This reduces the probability of the event to “completely irrelevant” as it is a consequence of complicated chaotic non-Markovian dynamics with no predictive value whatsoever (it is a “black swan”).
As noted above, as a demonstration of warming trend it is a big “so what” point. One can visit:
http://commons.wikimedia.org/wiki/File:Holocene_Temperature_Variations.png
on up (any of the figures linked therein) and the mere thermometric data shows the roughly 0.01 C/year trend over the last 100-150 years. And anybody too stupid or paranoid to believe the mere thermometric data isn’t going to understand, or believe, Masters’ argument.
Don’t get me wrong — I’m a long time WU subscriber, and generally I like Masters’ blog, especially when he waxes on about tropical storms (something of his specialty and the main reason I subscribed originally, as I’m sitting here looking out my window in the direction of Cuba across the Atlantic and tropical storms sometimes roll right up to the back door of the house I’m living in). But comparing apples to oranges to prove that bananas attract flies? Not good. Overtly bad if it is done to fool those too ignorant to be able to understand the inanity of the argument that bananas attract flies. Where the bananas bit isn’t to establish that there is a warming trend, it is to establish that there is an anthropogenic warming trend that will lead to catastrophe if we fail to spend lots of money in certain specific ways.
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