Extraordinary claims. Ordinary investigations.

Seeing Patterns

londresv2dgas

Can you see a pattern in the image above? Click to see it on Google Maps: each star is a point in London where a V2 bomb fell.

The 13-ton Nazi rockets, of which 500+ fell on the British capital, killed more than 9,000 and are represented in an interactive map. Clicking in the satellite view you can see how some impact points are still open craters, six decades later, while most of them have been covered by houses and parking lots. Remember, remember.

V2damagefd

But why have I asked if you could see a pattern? Because how to interpret the places in London where the bombs fell is a classic example of the conflict between our fears and emotions against our science and reason.

In a fascinating study published on Science last year, Jennifer Whitson and Adam Galinsky conducted six experiments showing how lacking control increases illusory pattern perception. From pareidolia to paranoia, amid a stressful environment where we are powerless as to the outcomes, we are prone to see obscure plots and meanings that aren’t really there.

And you can imagine how being at the mercy of flying bombs that fall from the sky without any warning and to which there was no possible defense is a good example of lacking control. The V2 were supersonic weapons, falling from over 100km at 4 times the speed of sound. You would only hear them after they hit their target. Not a single V2 was destroyed after a complete launch.

Predictably, terrorized London citizens started to see some very obscure patterns. As Gallinsky told John Tierney:

People were certain that parts of the city had been targeted and other parts spared. People in those areas of the city seemingly spared came under suspicion as Nazi sympathizers, and their livelihoods and physical safety were threatened. And in those areas seemingly targeted by the bombs, people moved out, attempting to escape systematic bombing that was in fact not systematic.”

As they weren’t systematic. The guiding systems of the V2 were very primitive, and their margin of error was of several miles. The best the Nazis could do was aim at London and hope the city would be hit. The only tactical use of the flying bombs, targeting a specific bridge in Remagen, was a failure: eleven bombs were launched, none hit the bridge. Any belief that the Nazis were targeting specific areas of some dozen meters, or even specific addresses, was all fiction.

We know that today, of course, after the war. But how did the British could have known that during the war, when even the existence of flying bombs from Germany was a very unwelcome surprise?

In an exemplary application of statistics, RD Clarke published a single page that summarized how they did it. Comparing the actual distribution of areas hit by the bombs with a Poisson distribution, he found a nearly perfect match:

--------------------------------------------------------------
No. of flying bombs    Expected no. of squares    Actual no. of
per square                    (Poisson)              squares
--------------------------------------------------------------
       0                       226.74                      229

       1                       211.39                      211

       2                        98.54                       93

       3                        30.62                       35

       4                         7.14                        7

       5 and over                1.57                        1
---------------------------------------------------------------
                               576.00                      576
---------------------------------------------------------------

And that is a statistical distribution that formalizes randomness. The areas hit by the V-2 were distributed at random, not due to some perverse conspiracy of Nazis and traitors.

With such simple test the British were able to safely assume the V-2 lacked precise guidance and acted accordingly. The best they were able to do was to manipulate the news of the areas hit by the bombs, leading the Nazis astray on their aiming.  It kind of worked.

Unfortunately, explaining how such a simple statistical test for randomness proved there was no dark conspiracy didn’t do much for the civilians. Decades later, psychological studies such as the one by Whitson and Gallinsky would help us understand why. Even though a classic Twilight Zone episode already warned us of that.

Curiously, as you may have guessed from the name, the Poisson distribution is named after mathematician Siméon Denis Poisson. A French.

- – -

- AN APPLICATION OF THE POISSON DISTRIBUTION (PDF) BY R. D. CLARKE, F.I.A., JIA 72, 0481, 1946.

- Nazi???London????Poisson?????

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Posted in Science,Skepticism | 7 comments

7 Comments so far

  1. misterkeith January 26th, 2009 2:47 pm

    Thomas Pynchon actually touches on this exact subject in his book, Gravity’s Rainbow. As I remember, the character Roger Mexico is comparing the V rocket bombings to Poisson distributions, to see if there was any true pattern to how they were falling.

    It is an amazing book to read, but being a Pynchon book, it will kick your butt.

  2. Mori January 26th, 2009 3:26 pm

    There’s a hidden Pynchon link buried in the text ;-)

  3. Brett January 27th, 2009 3:53 am

    That’s a very interesting take on the V-2 map (which I recently blogged myself). But I wouldn’t be a pedant worthy of the name if I didn’t point out that when Clarke says “flying bomb”, he’s actually referring to the V-1. I’m sure the analysis would be much the same though …

  4. Enonymous January 27th, 2009 10:27 pm

    There’s clearly a huge cluster roughly parallel to the Thames in South London, which may give some evidence of the flight path while elsewhere the bombs are more evenly spaced. the rich neighborhoods in NW London are the least hit, seemingly an anti-cluster, perhaps due to range limitations. There is a curious cluster of 5 in SW London completely surrounded by blank space, which is not too statistically unlikely, but would sure look like a pattern if you lived there. The Poisson distribution is quite substantially off in the cases of 3 or >=5. Also a unidimensional distribution is not right for this 2-D case – there is clear evidence to my eye that this distribution is elliptical rather than circular.

  5. Joe June 1st, 2009 7:51 am

    Its very true, this is the same reason why I dont buy into conspiracy theories of any kind, as much as I’m interested in this blog. A liberal use of good old Occam’s Razor allays the greatest of fears.

  6. The faith instinct – Telegraph Blogs August 31st, 2012 8:51 am

    [...] There's an ongoing debate (among people who don't believe in God) about whether the religion instinct is an adaptive one, or a byproduct of other instincts. But if it's either of those things, we will be fooled by it into sensing a God where there is none, just as we are fooled by our hyperactive pattern-detector into seeing skill behind the random guesses of stock-pickers or deliberate aiming in the random strikes of V2 bombs. [...]

  7. Alistair September 1st, 2012 6:23 am

    I do this sort of thing for a job.

    The number of V2 across the cells of a specified cell grind will be approximately poisson – in practise it will be an over-distributed poisson because of a central aimpoint (the Germans were aiming for central london).

    The proper way to begin analysing them is CEP around a central aimpoint, effectively using a Rayleigh distribution.

    However, there is clearly different cross-range and down-range errors – most weapons fall in a NW – SE axis. This makes sense – the Germans were firing from the low countries and guidance on V2 was better at stabilsing bearing than attitude. Hence the shots fall long or short but not so much side-to-side.

    Ideally then one should take the principal components of the distribution, then measure the variation on each component, to describe the distribution and central aimpoint. Further cluster analysis could be used to show if there was more than one point of aim.

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