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Can a rare heat wave in a big city occur by chance?

The short answer is Yes. What about the long answer?

Roger Pielke Jr wrote a text about rare weather events that I completely agree with:

Why rare events are a certainty
July 2010 was unusually warm in Western Russia - and Moscow. It actually turns out that a bigger portion of Russia was below the normal temperature than the portion of Russia above it. But that can't change the fact that Moscow et al. was really warm.



I picked Mathematica and used the WeatherData function to find out that the average July 2010 temperature in Moscow was 3.5 standard deviations above the mean temperatures for July 2010 - that can be extracted from the record available via Wolfram's software. By a standard deviation, I mean the root mean square of the differences of July temperatures in the past from their overall average.

Now, how likely it is for a quantity to be more than 3.5 sigma separated from the mean? It's useful to memorize the following table of the odds:
  • greater than 1.0 sigma: 1 in 3
  • greater than 1.5 sigma: 1 in 7.5
  • greater than 2.0 sigma: 1 in 22
  • greater than 2.5 sigma: 1 in 81
  • greater than 3.0 sigma: 1 in 370
  • greater than 3.5 sigma: 1 in 2150
  • greater than 4.0 sigma: 1 in 15,800
  • greater than 4.5 sigma: 1 in 147,000
  • greater than 5.0 sigma: 1 in 1.74 million
Great. So the odds that a normally distributed quantity deviates at least by 3.5 sigma from the mean is something like 1 in 2,000. In the article that Pielke refers to, they clearly used a 4-sigma deviation so that the odds are 1 in 15,000. But that doesn't really change the qualitative story.

Now, is it shocking that Moscow has experienced temperatures that are expected once in 2,000 or 15,000 years? Well, if Moscow were the only city that matters, if July were the only month in the year, and if the temperature were the only quantity that can excite us and that can drive the climate alarmism, the answer would be that it would be relatively unusual. It would be as unusual as living in the year when the Son of God was born, among other things. ;-)

That wouldn't be impossible but it would be somewhat exciting.




But Moscow is not the only city, July is not the only month, and temperature is not the only quantity that can be interesting or that can look like a sign from the heavens to some sensitive individuals. How many independent quantities similar to "average July temperature around Moscow" does the world's atmosphere (and ocean) produce every year?

Well, it's many thousands. The population of Moscow (including its largely integrated vicinity) is about 11 million people - the most populous city of Europe - which is still just 1/600 of the world's population. So it's OK to assume that there are "600 places in the world similar to Moscow". And we generously ignore the ocean where almost no one lives but that can also experience extreme weather.

Besides 600 places in the world, there are 12 months in a year and roughly 5 major quantities - temperature, pressure, wind speed, precipitation, and cloudiness - that we could find as interesting as the temperature. All these parameters are pretty much independent. So how many quantities are there that can be potentially as interesting as July temperatures in Moscow? It is
600 x 12 x 5 = 36,000 a year.
So even if you generously say that the Moscow temperature was not 3.5 but 4 sigma above the mean - the odds are 1 in 15,000 - it means that in average, two such "impossibly rare" events should occur somewhere in the world every year! It is actually unlikely ("less likely than yes") that no such event will occur during a given one year.

I think that Roger Pielke Jr should have used the Poisson distribution instead of the binomial one ;-) to calculate the exact odds, but because the probabilities are small and the calculation was just multiplying them by the "number of types", his mistake doesn't really influence the resulting numbers.

The case of the Asian floods is even less spectacular, statistically speaking. During the last 2,000 years, China's Yellow River has flooded 1,000 times. In 1887, 1931, as well as 1938, these floods killed millions of people which is 3 orders of magnitude above the casualties of 2010.

But much like tens of thousands of years ago when people would invent new gods whenever they experienced a somewhat unusual event - solar eclipse, rainbow, hurricane - people remain irrational when it comes to events that only occur "a few times a life" or less frequently. People just can't "instinctively" understand that statistics also holds at time frames that are longer or much longer than our lives.



That's the main reason why it's so easy for certain ideologues to abuse any event that is slightly unusual. In particular, the man-made global warming fears are thriving because of this natural human irrationality.

If you look at the climate phenomena in a cool fashion and if you don't hype the events that are taking place right now, you will see that nothing is detectably changing about the statistical distributions. In 2005, the Atlantic Ocean would experience many hurricanes - and strong ones. Sadly, some of them hit New Orleans which is just bad luck because it's unlikely that elevated CO2 concentrations are able to focus hurricanes more accurately against the big cities. ;-)

So the most obvious irrational behavior is to focus on the dramatic events that are occurring "right now" and extrapolate their rate - as measured during a short time frame (that's the mistake!) - to the long future. By this methodology, we may "predict" that there will be huge and lethal, Katrina-like hurricanes almost every year. And they will surely be getting ever stronger - because 2005 was more violent than 2004. ;-)

But it's usually enough to wait for 5 more years to see that such predictions don't work. We can't use the cherry-picked short-term measurements to make statistical long-term predictions. The hurricane activity in the 5 following seasons was near average and usually well below the average. Nothing is changing. So the fearmongers try to focus the people's attention to the events that were taking place recently; they are shooting at a moving target. There will surely be huge fires in Russia all the time, right?

Isn't the case of the hurricanes enough for the people to start to understand that this reasoning doesn't work? It is completely irrational.

These comments are not meant to say that it is impossible for heat waves and fires to occur in Russia once again. As Pielke Jr has calculated, the probability that a region in Russia will experience a similar heat wave (and fires?) again in the next 10 years - by pure chance, without assuming any trend - is actually not negligible. His result is 1 in 143. It is relatively small but it may surely happen.

Statistics, mathematics, physics, and science continues to work at long timescales. It's the human intuition and rationality that often breaks down.

If you want to properly understand the frequency of events that occur once a century or less, you must abandon your intuition from the everyday life and you must understand that your life experience is just a tiny speck of noise that can't matter in this reasoning. You must focus on the maths and on the actual evidence that tells us what is actually happening in the long run. I assure you that if you do all these things properly, you will conclude that the "rare events" we sometimes see are no signal that "something strange has to be going on". In fact, they inevitably have to occur somewhere.

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reader Harlow said...

Your exact same reasoning can be applied to unusual clusters of diseases.