# Chicago, Baseball and Paul Erdös

Thursday afternoon, before the 2013 CAE Faculty Conference, Stuart Klugman should invit us to go and watch the Cubs playing, in Chicago. That should be fun. First baseball game, ever. I will be back in Montréal (and on the blog) next week !

That will be an opportunity to discuss with mathematicians and baseball fans. Actually, a colleague told me that there was a nice anecdote about baseball and mathematics. Hank Aaron, “considered to be one of the greatest baseball players of all time” is supposed to have an Erdös number of 1 (see e.g. http://boolesrings.org/mpawliuk/….). Some pretend that it is only because Hank Aaron has signed the same baseball as Paul Erdös (thus, they cosigned something, giving him the Erdös number 1) while both of them were invited in some ceremony to get honorary diplomas… The funny part is that, even if he was not a mathematician (but has an Erdös number of 1), he also has named some numbers, the so-called Ruth–Aaron pairs. The story is nice, actually. On April 1974, Hank Aaron become famous for swatting his 715th home run. The prior record was held by Babe Ruth, with (have a wise guess…) 714 home run. Three mathematicians in Georgia (including Carl Pomerance) notice that 714 × 715 was not a common pair of consecutive number. It consists of two consecutive integers for which the sums of the prime factors of each integer are equal, since

714 = 2 × 3 × 7 × 17

715 = 5 × 11 × 13

and

2 + 3 + 7 + 17 = 5 + 11 + 13    (= 29)

Those are Ruth-Aaron pairs, see e.g. http://mathworld.wolfram.com/… or Pomerance (1999). Note that Carl Pomerance published more than 40 papers with Paul Erdös. End of the loop.

# Chicago, Society of Actuaries

I will be in Chicago for the next few days, at the 2013 CAE Faculty Conference, organized by the Society of Actuaries,

# Visualizing densities of spatial processes

We recently uploaded on http://hal.archives-ouvertes.fr/hal-00725090 a revised version of our work, with Ewen Gallic (a.k.a. @3wen) on Visualizing spatial processes using Ripley’s correction: an application to bodily-injury car accident location

In this paper, we investigate (and extend) Ripley’s circumference method to correct bias of density estimation of edges (or frontiers) of regions. The idea of the method was theoretical and di#cult to implement. We provide a simple technique – based of properties of Gaussian kernels – to compute e#efficiently weights to correct border bias on frontiers of the region of interest, with an automatic selection of an optimal radius for the method. An illustration on location of bodily-injury car accident (and hot spots) in the western part of France is discussed, where a lot of accident occur close to large cities, next to the sea.

Sketches of the R code can be found in the paper, to produce maps, an to describe the impact of our boundary correction. For instance, in Finistère, the distribution of car accident is the following (with a standard kernel on the left, and with correction on the right), with 186 claims (involving bodily injury)

and in Morbihan with 180 claims, observed in a specific year (2008 as far as I remember),

The code is the same as the one mentioned last year, except perhaps plotting functions. First, one needs to defi
ne a color scale and associated breaks

```breaks <- seq( min( result \$ZNA , na.rm = TRUE ) * 0.95 , max ( result\$ZNA , na.rm = TRUE ) * 1.05 , length = 21)
col <- rev( heat . colors (20) )```

to
finally plot the estimation

```image . plot ( result \$X, result \$Y, result \$ZNA , xlim = range (pol[,
1]) , ylim = range (pol[, 2]) , breaks = breaks , col = col ,
xlab = "", ylab = "", xaxt = "n", yaxt = "n", bty = "n",
zlim = range ( breaks ), horizontal = TRUE )```

It is possible to add a contour, the observations, and the border of the polygon

```contour ( result \$X, result \$Y, result \$ZNA , add = TRUE , col = "grey ")
points (X[, 1], X[, 2], pch = 19, cex = 0.2 , col = " dodger blue")
polygon (pol , lwd = 2)```

Now, if one wants to improve the aesthetics of the map, by adding a Google Maps base map, the
first thing to do – after loading ggmap package – is to get the base map

```theMap <- get_map( location =c( left =min (pol [ ,1]) , bottom =min (pol[ ,2]) , right =max (pol [ ,1]) ,
top =max (pol [ ,2])), source =" google ", messaging =F, color ="bw")```

Of course, data need to be put in the right format

```getMelt <- function ( smoothed ){
res <- melt ( smoothed \$ZNA)
res [ ,1] <- smoothed \$X[res [ ,1]]
res [ ,2] <- smoothed \$Y[res [ ,2]]
names (res) <- list ("X","Y","ZNA")
return (res )
}
smCont <- getMelt ( result )```

Breaks and labels should be prepared

```theLabels <- round (breaks ,2)
indLabels <- floor (seq (1, length ( theLabels ),length .out =5))
indLabels [ length ( indLabels )] <- length ( theLabels )
theLabels <- as. character ( theLabels [ indLabels ])
theLabels [ theLabels =="0"] <- " 0.00 "```

Now, the map can be built

```P <- ggmap ( theMap )
P <- P + geom _ point (aes(x=X, y=Y, col=ZNA), alpha =.3 , data =
smCont [!is.na( smCont \$ZNA ) ,], na.rm= TRUE )```

It is possible to add a contour

```P <- P + geom _ contour ( data = smCont [!is.na( smCont \$ZNA) ,] ,aes(x=
X, y=Y, z=ZNA ), alpha =0.5 , colour =" white ")```

and colors need to be updated

```P <- P + scale _ colour _ gradient ( name ="", low=" yellow ", high ="
red", breaks = breaks [ indLabels ], limits = range ( breaks ),
labels = theLabels )```

To remove the axis legends and labels, the theme should be updated

```P <- P + theme ( panel . grid . minor = element _ line ( colour =NA), panel
. grid . minor = element _ line ( colour =NA), panel . background =
element _ rect ( fill =NA , colour =NA), axis . text .x= element _ blank() ,
axis . text .y= element _ blank () , axis . ticks .x= element _ blank() ,
axis . ticks .y= element _ blank () , axis . title = element _ blank() , rect = element _ blank ())```

The
final step, in order to draw the border of the polygon

```polDF <- data . frame (pol)
colnames ( polDF ) <- list ("lon","lat")
(P <- P + geom _ polygon ( data =polDF , mapping =( aes(x=lon , y=lat)), colour =" black ", fill =NA))```

Then, we’ve applied that methodology to estimate the road network density in those two regions, in order to understand if high intensity means that it is a dangerous area, or if it simply because there is a lot of traffic (more traffic, more accident),

We have been using the dataset obtained from the Geofabrik website which provides
Open-StreetMap data. Each observation is a section of a road, and contains a few points identifi
ed by their geographical coordinates that allow to draw lines. We have use those points to estimate a proxy of road intensity, with weight going from 10 (highways) to 1 (service roads).

```splitroad <- function ( listroad , h = 0.0025) {
pts = NULL
weights <- types . weights [ match ( unique ( listroad \$ type ), types .
weights \$ type ), " weight "]
for (i in 1:( length ( listroad ) - 1)) {
d = diag (as. matrix ( dist ( listroad [[i]]))[, 2: nrow ( listroad
[[i ]]) ]))
}}
return (pts )
}```

See Ewen’s blog for more details on the code, http://editerna.free.fr/blog/…. Note that Ewen did publish a poster of the paper (in French), for the http://r2013-lyon.sciencesconf.org/ conference, that will be organized in Lyon on June 27th-28th, see

# Eat a beaver, save a tree

Wednesday, just before leaving the office, I remembered I wanted to buy Andreas Kyprianou’s book, on Lévy processes. A second edition is coming soon, but I just need a simple introduction to Lévy processes, so I thought that this first edition should be complicated enough for me. And when a second edition should appear soon, you can get a discounted version of the (almost) old one. So I went on Springer’s wesite to purchase the book. I did pay for the book, and finished packing, in order to go back home. I did receive my confirmation order, which is standard, and I opened it,

Dear Arthur Charpentier,

Thank you for your email and interest in our products.

This is to inform you that it is irrelevant for us to proceed with your request, because it has already been entered into our database/system.

However, when you have downloaded the PDF copy of the E-book. You can print manually through your printer.

If you want a paper copy of a book, “you can print manually through your printer.” At first, I thought it was some kind of misunderstanding. Or joke, maybe. But no. You cannot cancel a purchase when you order eBooks. And to make sure that I got the book, they did send me the full pdf in my mail box. What I am supposed to do with that file? This is not what I wanted! I wanted a book! a book with paper you can hold in your hands! with paper, made from trees that died so that I can learn stuff!

Anyway, I gave up… I will ask colleagues if I can borrow their copies. Now, I have to fight with Dell since I ordered a laptop (yes, the Ubuntu version), and it did arrive at the office in a wet box. Looks like the computer (at least the box) has been staying in the water for a very long time! I don’t know if people around still believe that researchers actually do research when they have time… trust me, they don’t! They discuss with Customer Services… and it can take a while!

# Bayes, credit scoring and terrorism

Once again, my neighbor Corey did publish a very interesting post on his blog http://bayesianbiologist.com/… on how likely the NSA program will catch a terrorist (a real one). I was working on something similar last weeks, with Stéphane Tufféry, for our chapter, entitled Statistical Learning in Actuarial Science. The idea was to show credit scoring techniques, from logistic regression, classification trees, random forests, etc. Of course, it is more boring, since we talk about loans and not terrorism. In credit scoring, we consider possible loans, and we have to predict if someone is more likely to be a bad guy or a good guy. The idea is the same: based on some covariates, we need to build a score function, that can be related to the probability of being bad. The higher the score, the more likely the person will be a bad guy. Then, of course, we have to discuss errors, namely false positive (good guys that we think are bad) and false negative (bad guys that we think are good). From the company, you do not want to have bad guys in your portfolio, and from everyone else point of view (since everyone believes he is with the good one, this is a classical optimistic bias), we do not want to be confused with those bad guys. Then we can spend hours on classification curves, and criteria to assess if our classifier is good or not, etc. While I was writing the introduction of the chapter, I remember that I found it hard to find proper words (to describe that 0/1 problem). But I did use (like everyone else) the terms good and bad. Like in terrorism. Except that to use this terminology (bad and good), we have to be more specific. In credit scoring, a bad guy is someone who did not pay back, at least once, for instance. But in terrorism, I think it is more difficult to say what a terrorist is.

I mean, in France, we did experiment terrorism too, a few years ago. In December 1996, I was in a RER train, going South, and we reached Cité Universitaire when a bomb did explode in Port Royal. The train following mine I guess. I remember that a couple of days after, I was traveling Paris, in bus, carrying with me a nice plant of… a plant that you’re not supposed to grow. Say I was carrying sandwiches, according to Ted Mosby. So in order to avoid troubles (since I was not suppose to have this kind of plant species), I put it in a large box. I remember that people were starring at me, and some actually asked me what was in the box. So for some reasons, people try to build there own terrorist classifier, based on what they think might be covariates. And dirty trousers, not well shaved, long hair (yes, I used to have long hair) and box in the bus were obviously some of them. Note that I don’t blame them, I do the same! After reading Corey’s post this morning, I took the bus. And I saw someone with a ninja sword.

At first, my terrorist classifier put her (yes, I try to have a gender-free terrorist model) in the bad guy class. Then I understood it was an umbrella. So I put her in the super cool geeky category (that only a few can reach).

When I started to teach non-life insurance in Paris, the last part of the course was dedicated to large risks, natural catastrophes, and a hot topic: terrorism. I was giving this course (probably my best experience, ever) in tandem with François Bucchini, who was working by that time for AXA France. The two of us were giving the course together, interacting: I was the boring guy doing the maths, and François was sharing his experience. And by that time, he was involved in the creation of GAREAT, a market structure, launched in France in 2002, to propose reinsurance against terrorism (for French companies). And one of the first claim was from the CAV (which is a pun for Comité d’Action Viticole) considered as a terrorist group. So, as he told us, be careful of prejudices when you think about terrorism. Cool wine drinkers can be dangerous terrorists…

Actually, I would love to see covariates used by the NSA to predict if you’re a bad guy, or a potentially dangerous terrorist. Let us have a guess… You have asked for a visa for Pakistan? or Afghanistan? or Libya (not Libya, not yet bad guys still have good friends there)? You have a NRA membership? You bought some heavy metal on iTunes? You still have a stop acta sticker on your blog? you have a blog? you wrote a post including the word terrorist in it?

Note: I am supposed to be in Chicago next week. Si if I cannot enter in the U.S., we’ll probably know more about potential covariates.

# How old is the oldest person you know?

Last week, we had a discussion with some colleagues about the fact that – in order to prepare for the SOA exams – we did not have time (so far) to mention results on extreme values in our actuarial program. I did gave an introduction in my nonlife actuarial models class, but it was only an introduction, in three hours, in order to illustrate reinsurance pricing. And I told my students that if they wanted to know more about extreme values, they should start a master program in actuarial science and finance, since I will give a course on extremes (and copulas) next winter.

But actually, extreme values are everywhere ! For instance, there is a Prudential TV commercial where has people place large, round stickers on a number line to represent the age of the oldest person they know. This forms some kind of histogram. The message is to have Prudential prepare you to have adequate money for all these years. And actually, anyone can add his or her own sticker at the Prudential website.

Patrick Honner, on his blog (http://mrhonner.com/…), did mention this interesting representation. But this idea is not new, as mentioned in a post, published three years ago. In 1932, Emil Gumbel gave a talk in France on the “âge limite“. And as he wrote it “on peut donc supposer que la distribution de l’âge limite – c’est à dire la probabilité que cet âge ait une valeur donnée – soit Gaussienne“. In 1932 (not aware of Fisher and Tippett work, he thought that the limiting distribution for a maximum would be Gaussian). But a few years after, he read about Fisher’s work, and observed also that “la distribution d’une valeur extrêmes peut être représentée pour un nombre suffisant d’observations par la formule doublement exponentielle, pourvu que la distribution initiale se comporte asymptotiquement comme une exponentielle. La formule devient rigoureuse si la distribution initiale est exponentielle“, as he wrote in 1935. And in 1937, he wrote a paper on “les centennaires” that can also be related to the work of Bortkiewicz on rare events. One should also mention one of the most important paper in extreme value theory, published in 1974 by Balkema and de Haan, on Residual Life Time at Great Age.

Because in this experiment, the question is “How Old is the Oldest Person You Know?“, so it is the distribution of a maximum. And from Fisher-Tippett theorem, if we assume that the age is bounded (and that there exists some finite upper limit), then the limiting distribution for the maxima (or to be more rigorous, a affine transformation of the maxima) should be Weibull distribution. And this is what it looks like

`> plot(-x,dweibull(x,2.25,4),type="l",lwd=2)`

As an actuary, the only thing I know about demography, is the distribution of the age of death. For instance, consider the following French life table

```> alive <- read.table(
+ "https://perso.univ-rennes1.fr/arthur.charpentier/TV8890.csv",
> nb= -diff(alive)
> ages=0:110
> plot(ages,nb,type="h")```

This is the distribution of the age of the death in a given population. Which is not the same as the distribution mentioned above! What we look for is the following: given that someone is alive, what could be the distribution of his-her age ? Actually, if we assume that the yearly number of birth is constant with time (as well as death probability), then we can compute easily to number of people of age $x$ : we take everyone born (exactly) $x$ years ago, and remove all those who died at at $x$, $x-1$, etc. So the function should be

```> probadeath=nb/sum(nb)
> surv=Vectorize(nbx)(ages)
> distrage=surv/sum(surv)```

which looks like

But this assumption of constant number of birth is not that relevent. And actually, what we need is the distribution of the age within a population… This is a population pyramid, actually. The French one can be downloaded from http://www.insee.fr/fr/ppp/bases-de-donnees/….

```> population <- read.table("popinsee2007.csv",sep=";",header=TRUE)\$POPTOT07
> ages=0:107
> plot(ages,population/sum(population),type="h")```

(the red line being the one obtained previously, using some natality assumptions). Now, let us use this population to generate acquaintances.

```> agemax=function(nsim=1000,size=20){
+ agemax=rep(NA,nsim)
+ for(i in 1:nsim){
+ X=sample(ages,prob=population/sum(population),size=size,replace=TRUE)
+ agemax[i]=max(X)}
+ return(agemax)}```

Here, we assume that everyone knows 20 other people, randomly chosen in the entire population, then we return the age of the oldest. And we do that for 1,000 people. Here is the distribution, we obtain

```> XS=agemax(10000,20)
> plot(table(XS)/length(XS),type="h",xlim=c(0,108))```

where the red line is a Weibull distribution (a transformed one, actually, since in extremely value theory, the distance to the upper bound of the distribution has a Weibull density),

```> library(MASS)
> fit=fitdistr(108-XS,dweibull,list(shape=1,scale=1))
> lines(ages,dweibull(108-ages,fit\$estimate[1],fit\$estimate[2]),col="red")```

Which is quite close to the distribution obtained in the commercial, don’t you think ? But still, it should be possible to be more accurate, since people should think of their parents, or grandparents. So I guess it could be possible to build a more accurate algorithm, to get something closer to the distribution obtained on the Prudential website. But first, let us wait to have more stickers, more observations… and then I’ll be back to play with it !