# The odds of a cluster of airplane accidents

Recently, there have been a lot of airplane accidents.

• July, 17th 2014, Hrabove, Ukraine, Malaysia Airlines, Boeing 777, fatalities 298 (/298)
• July, 23rd 2014, Magong, Taiwan, TransAsia Airways, ATR 72-500, fatalities 47 (/58)
• July, 24th 2014, Aguelhok, Mali, Air Algerie, Mc Donnell Douglas MD-83, fatalities 116 (/116)

It is simple to find a lot of datasets about airplane crashes. For instance on http://ntsb.gov/aviationquery. The dataset is nice, with a lot of information,

> planes=read.table(
+ sep="|",header=TRUE)

for instance the exact location of the crashes,

> library(maps)
> map("world", interior = FALSE)
> points(planes$Longitude,planes$Latitude,
+ pch=19,cex=planes$Total.Fatal.Injuries/50, + col="red") But I could not find recent major accidents on that (supposed to be updated) dataset. > X=as.Date(as.character(planes$Event.Date),
+ "%m/%d/%Y")
> Y=planes$Total.Fatal.Injuries > plot(X,Y,type="h") It looks like it is an american dataset, so maybe domestic flights in some countries are not in this dataset? Let us try another one, such as https://opendata.socrata.com/ > planes=read.table( + "Crashes_and_Fatalities_Since_1908-b.csv", + sep=",",header=TRUE) > X=as.Date(as.character(planes$Date),"%m/%d/%Y")
> Y=planes$Fatalities > plot(X,Y,type="h") It is not an updated dataset, but for the purpose of studying the process of airplane accident, it should be just fine. In the recent years, it seems like a stationary process. If we focus on the past 35 years, we have > subplanes=planes[X>as.Date("01/01/1980", + "%m/%d/%Y"),] > X=as.Date(as.character(subplanes$Date),
+ "%m/%d/%Y")
> Y=subplanes\$Fatalities
> plot(X,Y,type="h")

Let us study the process of accident arrival. I will keep only airplane accidents with – say – 10 fatalities, or more (it is purely arbitrary, but we need some threshold here).

> X=X[!is.na(Y)]
> Y=Y[!is.na(Y)]
> T=X[Y>10]

In actuarial science, the standard process we use to model accidents is the Poisson process. Meaning that the time between two (consecutive) accidents is a random variable with an exponential distribution (further, those random variables are independent). If we look at the cumulative distribution function, we get

> W=as.numeric(diff(T))
> mean(W)
[1] 12.66942
> plot(ecdf(W))
> u=0:100
> lines(u,pexp(u,1/mean(W)),col="red",lwd=2)

and the histogram is

> hist(W,breaks=0:100,probability=TRUE)
> lines(u,dexp(u,1/mean(W)),col="red",lwd=2)

The red line is the exponential distribution, with the same mean. We can observe that we have too many zero’s here (i.e. accidents on the same day). If we remove them, it is slightly better,

> Tu=unique(T)
> W=as.numeric(diff(Tu))
> mean(W)
[1] 13.2974
> hist(W,breaks=0:100,probability=TRUE)
> lines(u,dexp(u,1/mean(W)),col="red",lwd=2)

(but to be honest, I have no real motivation to remove those zeros… it is possible to have two airplane crashes on the same day – at different locations, i.e. not necessarily collisions). An alternative is to consider only ‘major’ accidents, with more than 40 casualties.

> T=X[Y>40]
> W=diff(T)

Since we have exponentially distributed inter-arrival duration, and the assumption of independence between duration seems valid

> acf(W)

we will consider here that an homogeneous Poisson process is a valid model. Above 40 fatalities, the average time between two major accidents is a bit more than a month,

> W=as.numeric(diff(T))
> mean(W)
[1] 36.22973

so the probability to have (at least) 3 accidents within 8 days (if we use the proportionality property of the Poisson process, i.e. the number of catastrophes between times $t_1$ and $t_2$ is a Poisson variable $\mathcal{P}(\lambda\cdot[t_2-t_1])$) is

> 1-ppois(2,8/mean(W))
[1] 0.001521955

i.e. a bit more than 1.5 over 1000. Over 8 years, if we consider some blocks of 8 days, what could be the probability that such a cluster of crashed does not occur? The probability not to have 3 crashes, over 365 blocks of 8 days is

> p=1-ppois(2,8/mean(W))
> (1-p)^365
[1] 0.5735347

meaning that over 8 years, there are more than 40% chance to have such a cluster. I think we might call that a coincidence…

## 6 thoughts on “The odds of a cluster of airplane accidents”

1. translator says:

//TRANSLATION:
Hello sir,
In one of your old posts (from the old blog) whose link is as follows:

http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/post/2010/03/02/Actuariat,-ajustement-de-lois-pour-les-co%C3%BBts-individuels

I did not understand the writing of the log-normal at the level of the functions of distribirons plnorm (blue curve) and exp (red curve):

My writing would have given me:
lines (x [x> 1170] 0.25 * (plnorm (x [x> 1170] 7.0679,1.55) -plnorm (1170,7.0679,1.55) / (1-plnorm (1170,7.0679,1.55))

by fitting that fitdistr (X [X> 1204], “lognormal”) and not the translated.

1. TRANSLATOR says:

translation on ulrlch’es comment

2. David Menezes says:

Hi Arthur, it’s not totally surprising to me that this appears coincidental. In my last job within Lloyd’s of London, I built a model to price aviation reinsurance. During the background research, I think I found that perhaps 1 in 7 or 8 million flights has a loss event, i.e. including minor hull damage. The odds of a cluster start to become vanishingly small when viewed in terms of the frequency lens. Even more so when you consider that poor Malaysia Airlines was involved in two of these.

As a complete aside, I helped review Markus G’s reserving section for the Computing in Actuarial Science book, which looks great. Well done and keep up the blog posts, as they’re typically thought provoking.

Best, dM/

3. je ne suis pas certain de comprendre… l’estimation est faite sur

fitdistr(X[X>1204]-1204,”lognormal”)

maintenant ce n’est qu’un modèle, il est possible de faire beaucoup mieux je pense… comparez les distances (sup) entre les deux fonctions de répartition, et utilisez ce qui semble mieux…

1. Ulrich says:

Mon problème n’est pas sur le modèle,
mais sur l’écriture de la F.d.r

Si vous voulez, en restant sur votre modèle, je ne sais pas d’où vient le terme (+0.75)
et je ne comprends pourquoi vous n’avez pas un terme au dénominateur. Un truc de la sorte (F(x)-F(m))/(1-F(m)) avec m=1204.

Je ne sais pas si vous m’avez compris.

4. Ulrich says:

Bonjour Monsieur,
Dans un de votre ancien billet (de l’ancien blog) dont le lien est le suivant:

http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/post/2010/03/02/Actuariat,-ajustement-de-lois-pour-les-co%C3%BBts-individuels

je n’ai pas bien compris l’écriture de la log-normale au niveau des fonctions de répartirons plnorm (courbe bleu) et exp (courbe rouge):

Mon écriture m’aurait donné:
lines(x[x>1170],0.25*(plnorm(x[x>1170],7.0679,1.55)-plnorm(1170,7.0679,1.55)/(1-plnorm(1170,7.0679,1.55))

en ajustant que fitdistr(X[X>1204],”lognormal”) et non la translatée.

Merci d’avance de votre réponse.

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