NaziによるLondonを使ったPoisson分布の実験

昨秋に、コントロールが失われると、合理的な人々が実際には存在しないパターンを見出すようになる話をとりあげた。

==>忘却からの帰還: When seeing IS believing (2008/10/11)

そこで、例のひとつがロンドン空襲：

Even though later statistical analysis clearly demonstrated that the bombs fell randomly across the city, people were certain that parts of the city had been targeted and other parts spared,” he told me. “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.

第2次世界大戦で空襲を受けたときのロンドン市民の反応など歴史事例にも、この傾向は見られる。戦後の統計解析では空襲は都市全体にランダムに行われたことが明確に示されているが、人々は都市の特定の場所が狙われ、別の場所は狙われていないと確信していた。狙われていない場所に住む人々は、ナチのシンパだと疑われ、生活や安全を脅かされた。そして、狙われていると思われる地区から人々は、実際に系統的空襲ではないのに、系統的空襲から逃れようと、そこから出て行った。

[John Tierney: "See a Pattern on Wall Street?"]

V2

で、この「後の統計解析では空襲は都市全体にランダムに行われたことが明確に示した」のが、たった1ページの論文であるClarke(1946)である。これはロンドンを576区画に分割し、合計537回の攻撃について、どの区画に落ちたか数えて、Poisson分布と比較したもの：

AN APPLICATION OF THE POISSON DISTRIBUTION
BY R. D. CLARKE, F.I.A.
of the Prudential Assurance Company, Ltd.
JIA 72, 0481, 1946

READERS of Lidstone’s Notes on the Poisson frequency distribution (J.I.A. Vol. LXXI, p. 284) may be interested in an application of this distribution which I recently had occasion to make in the course of a practical investigation.

During the flying-bomb attack on London, frequent assertions were made that the points of impact of the bombs tended to be grouped in clusters. It was accordingly decided to apply a statistical test to discover whether any support could be found for this allegation.

An area was selected comprising 144 square kilometres of south London over which the basic probability function of the distribution was very nearly constant, i.e. the theoretical mean density was not subject to material variation anywhere within the area examined. The selected area was divided into 576 squares of 1/4 square kilometre each, and a count was made of the numbers of squares containing 0, 1, 2, 3, . . . , etc. flying bombs. Over the period considered the total number of bombs within the area involved was 537. The expected numbers of squares corresponding to the actual numbers yielded by the count were then calculated from the Poisson formula :

Ne^-m(1+m+m²/2!+m³/3!+ ...)

where N=576 and m=537/576.

The result provided a very neat example of conformity to the Poisson law and might afford material to future writers of statistical text-books.

The actual results were as follows:

 -----------------------------------------------------------------
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
 -----------------------------------------------------------------

The occurrence of clustering would have been reflected in the above table by an excess number of squares containing either a high number of flying bombs or none at all, with a deficiency in the intermediate classes. The closeness of fit which in fact appears lends no support to the clustering hypothesis.

Applying the x²test to the comparison of actual with expected figures, we obtain x² = 1.17. There are 4 degrees of freedom, and the probability of obtaining this or a higher value of x² is .88.

きれいにPoisson分布に従っていて、V1/V2の攻撃がランダムだったことが示されている。

この例は有名で、確率統計の講義によく登場する[Franklin, Rodriguez, ...]。