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new file mode 100644
index 0000000..4e78526
--- /dev/null
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new file mode 100644
index 0000000..bf514eb
--- /dev/null
+++ b/assures.csv
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diff --git a/census_division.csv b/census_division.csv
new file mode 100644
index 0000000..5c69e1e
--- /dev/null
+++ b/census_division.csv
@@ -0,0 +1,20 @@
+Division,Population,Superficie
+1,78694,20526
+2,156536,17662
+3,38566,13866
+4,10078,21467
+5,53263,16775
+6,1311022,12646
+7,40232,1921
+8,189243,9909
+9,2129,18922
+10,93039,2045
+11,1203115,15768
+12,63427,30047
+13,68919,24374
+14,28584,26965
+15,35983,284
+16,67516,97255
+17,61504,192116
+18,14534,33205
+19,109712,2052
diff --git a/cooperators.Rnw b/cooperators.Rnw
new file mode 100644
index 0000000..15803d0
--- /dev/null
+++ b/cooperators.Rnw
@@ -0,0 +1,366 @@
+% 
+\documentclass[letter]{article}
+\usepackage[margin=0.75in]{geometry}
+\usepackage{Sweave}
+\usepackage{tikz} \usepackage{multirow}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\numberwithin{equation}{section}
+\usepackage{amsfonts} \usepackage{amsthm} \usepackage{thmtools}
+\usepackage{hyperref}
+\usepackage{cleveref}
+\usepackage[hypcap]{caption}
+\usepackage[off]{auto-pst-pdf}
+
+\title{Risque de concentration en Alberta}
+\author{François Pelletier}
+
+\begin{document}
+
+\maketitle
+
+<<librairie,echo=FALSE>>=
+set.seed(908144032)
+library("actuar")
+library("multicore")
+source("mcsapply.R")
+nb.conc <- 50
+@
+
+\section{Risque associé aux tempêtes de grêle}
+
+Une tempête de grêle est un évènement de météo extrême qui peut créer beaucoup
+de dommages dans une zone concentrée en peu de temps. Plusieurs facteurs
+influencent la sévérité des dommages causés par ces tempêtes, dont le principal
+est le diamètre moyen des grêlons. Le facteur influençant le nombre d'assurés
+qui encourent un dommage est principalement la taille de l'orage de grêle et la
+direction des vents.
+
+J'ai choisi de modéliser les orages de grêle selon une force de 1 à 4, qui
+influencera à la fois la taille moyenne de l'orage et la  taille des grêlons.
+Afin de pouvoir obtenir facilement une formule analytique pour déterminer si un
+assuré est touché par l'orage, j'ai choisi de lui donner une forme elliptique.
+Ainsi, l'orage de force 4 sera formée de 4 ellipses concentriques, chacune ayant
+un niveau de dommages allant de 1 à 4.
+
+Je n'ai pas modélisé le lieu de formation des orages. Ainsi, un orage a la même
+possibilité de se former à n'importe quel point à l'intérieur d'un rectangle
+regroupant les assurés.
+
+Les coordonnées en longitude et en latitude des assurés seront les seules
+données utilisées ici.
+La position des agents n'est pas nécessaire.
+L'ensemble des montants et des paramètres des distributions statistiques ont été
+attribués arbitraitement afin de donner des montants raisonnables à mon avis.
+
+Enfin, afin de représenter le risque d'un assuré moins concentré, j'ai
+simplement décidé de former un sous-groupe d'assurés formé d'un assuré sur
+\Sexpr{nb.conc} dans l'ordre fourni, ce qui a pour effet de diminuer la concentration 
+du risque tout en conservant l'effet de la densité de population, qui est déjà
+concentrée sur le territoire.
+
+\subsection{Définition des ellipses}
+
+L'ellipse est une section conique qui peut être représentée selon une formule
+paramétrique \ref{eq:ellipse} qui sert à définir les
+coordonnées en fonction des paramêtres. L'ellipse est alors définie par son
+centre $X_c,Y_c$ et la mesure de son plus grand rayon $a$ et de son plus petit
+rayon $b$, qui sont perpendiculaires. L'inclinaison du plus grand rayon par
+rapport à l'abscisse est donnée par l'angle $\varphi$ en radian.
+
+\begin{eqnarray}
+  \label{eq:ellipse}
+  X(t)&=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi\\
+  Y(t)&=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi
+\end{eqnarray}
+
+La formule canonique \ref{eq:canonique} est une seconde forme qui sert à
+déterminer si un point $(x,y)$ fait partie de l'ellipse.
+
+\begin{eqnarray}
+  \label{eq:canonique}
+  \frac{\left((x-X_c)\cos \varphi-(y-Y_c)\sin \varphi\right)^2}{a^2} + 
+  \frac{\left((x-X_c)\sin \varphi+(y-Y_c)\cos \varphi\right)^2}{b^2} = 1
+\end{eqnarray}
+
+Ces deux dernières équations nous permettrons de simuler des ellipses,
+représentant les orages, et de déterminer si des assurés sont touchés par les
+orages. Les functions suivantes seront utilisées à cette fin:
+
+<<ellipse1>>=
+ellipse_param <- function(t,X,Y,a,b,phi)
+{
+	cbind(X + a * cos(t)*cos(phi) - b*sin(t)*sin(phi),Y + a * 
+					cos(t)*sin(phi) + b*sin(t)*cos(phi))
+}
+
+dans_ellipse_param <- function(coord,param)
+{
+	x <- coord[,1]
+	y <- coord[,2]
+	xc <- param[1]
+	yc <- param[2]
+	a <- param[3]
+	b <- param[4]
+	phi <- param[5]
+	(((x-xc)*cos(phi)-(y-yc)*sin(phi))^2/a^2 + 
+				((x-xc)*sin(phi)+(y-yc)*cos(phi))^2/b^2) <= 1
+}
+@
+
+\subsection{Importation des données}
+
+On importe des données dans une structure
+<<>>=
+agents <- read.csv("agents.csv",sep=":")
+assures <- read.csv("assures.csv",sep=":")
+n.assures <- nrow(assures)
+@
+
+On visualise les données sur une carte. Les agents sont représentés par un
+triangle rouge et les assurés par une croix noire.
+
+<<fig=TRUE>>=
+plot(assures$Longitude_Assure,assures$Latitude_Assure,pch=3,
+		main="Carte des assurés et des agents",xlab="Longitude",
+		ylab="Latitude")
+points(agents$Longitude_Agent,agents$Latitude_Agent,pch=25,
+		col="red")
+@
+
+On définit la zone où se situent les assurés par un rectangle:
+<<>>=
+coord_assures <- cbind(assures$Longitude_Assure,
+		assures$Latitude_Assure)
+range_longitude <- c(min(coord_assures[,1]),
+		max(coord_assures[,1]))
+range_latitude <- c(min(coord_assures[,2]),
+		max(coord_assures[,2]))
+@
+
+On définit le groupe restreint pount l'évaluation avec moins de concentration
+<<>>=
+coord_assures2 <- coord_assures[seq(1,32669,by=nb.conc),]
+@
+
+La longitude est située entre \Sexpr{range_longitude[1]} et \Sexpr{range_longitude[2]}.
+La latitude est située entre \Sexpr{range_latitude[1]} et \Sexpr{range_latitude[2]}.
+
+\subsection{Hypothèses}
+
+On pose plusieurs hypothèses de modélisation:
+
+<<>>=
+franchise <- 500
+n.annees <- 100
+n.moyen.orages <- 10
+longueur.orage <- 1
+largeur.orage <- .2
+angle.dominant <- 1/2
+ecart.angle <- 1
+prob_forces <- c(.8, .15, .04, .01)
+facteur_forces <- c(1,1.5,2,2.5)
+param_dommages <- c(50,1000,20000,400000)
+param_forme <- 2
+@
+
+\begin{enumerate}
+  \item La franchise est de \Sexpr{franchise}\$.
+  \item Le nombre d'années simulées est de \Sexpr{n.annees} années.
+\item Le nombre d'orages par année suit une loi Poisson de moyenne
+\Sexpr{n.moyen.orages}.
+\item La longitude et la latitude du centre de l'orage sont distribués
+uniformément sur le rectangle défini précédemment.
+\item La longueur et la largeur suivent une loi exponentielle de paramètres
+\Sexpr{1/longueur.orage} et \Sexpr{1/largeur.orage}. L'unité est en degrés de
+latitude.
+\item Les vents dominants sont de direction \Sexpr{angle.dominant} radians.
+\item L'angle varie dans un invervalle de \Sexpr{ecart.angle} radian de part et
+d'autre selon la distribution $\beta\left(2,2\right)$.
+\item La force entre 1 et 4 suit une distribution discrète où les probabilités
+sont respectivement \Sexpr{prob_forces[1]},
+\Sexpr{prob_forces[2]}, \Sexpr{prob_forces[3]} et \Sexpr{prob_forces[4]}.
+\item La force multiplie les dimensions de l'orage respectivement par
+\Sexpr{facteur_forces[1]}, \Sexpr{facteur_forces[2]}, \Sexpr{facteur_forces[3]}
+et \Sexpr{facteur_forces[4]}.
+\item Les dommages suivent une distribution Pareto de paramètre de forme
+\Sexpr{param_forme} et de paramètre d'échelle dépendant de la force, prenant les
+valeurs
+\Sexpr{param_dommages[1]}, \Sexpr{param_dommages[2]}, \Sexpr{param_dommages[3]}
+et \Sexpr{param_dommages[4]}.
+\end{enumerate}
+
+\subsection{Simulation des orages}
+
+En utilisant les hypothèse définies précédemment, on simule des orages
+
+<<>>=
+orages_annees <- rpois(n.annees,n.moyen.orages)
+n.orages <- sum(orages_annees)
+sim_xc <- runif(n.orages,range_longitude[1],range_longitude[2])
+sim_yc <- runif(n.orages,range_latitude[1],range_latitude[2])
+sim_a <- rexp(n.orages,1/longueur.orage)
+sim_b <- rexp(n.orages,1/largeur.orage)
+sim_angle <- angle.dominant - ecart.angle + 2*ecart.angle*
+		rbeta(n.orages,2,2)
+sim_force <- sample(x=1:4,size=n.orages,replace=TRUE,
+		prob=prob_forces)
+sim_param_orages_1 <- cbind(sim_xc,sim_yc,
+		facteur_forces[1]*sim_a,
+		facteur_forces[1]*sim_b,sim_angle)
+sim_param_orages_2 <- cbind(sim_xc,sim_yc,
+		facteur_forces[2]*sim_a,
+		facteur_forces[2]*sim_b,sim_angle)
+sim_param_orages_3 <- cbind(sim_xc,sim_yc,
+		facteur_forces[3]*sim_a,
+		facteur_forces[3]*sim_b,sim_angle)
+sim_param_orages_4 <- cbind(sim_xc,sim_yc,
+		facteur_forces[4]*sim_a,
+		facteur_forces[4]*sim_b,sim_angle)
+@
+
+Pour chaque orage, on vérifie le nombre d'assurés touchés. On utilise une boucle
+ici pour éviter de surcharger la mémoire vive. C'est plus rapide ainsi.
+
+<<>>=
+touches1 <- numeric(n.orages)
+touches2 <- numeric(n.orages)
+touches3 <- numeric(n.orages)
+touches4 <- numeric(n.orages)
+for(i in 1:n.orages)
+{
+	touches1[i] <- sum((sim_force[i] >= 1) * 
+					dans_ellipse_param(coord_assures,
+					sim_param_orages_1[i,]))
+	touches2[i] <- sum((sim_force[i] >= 2) * 
+					dans_ellipse_param(coord_assures,
+					sim_param_orages_2[i,]))
+	touches3[i] <- sum((sim_force[i] >= 3) * 
+					dans_ellipse_param(coord_assures,
+					sim_param_orages_3[i,]))
+	touches4[i] <- sum((sim_force[i] >= 4) * 
+					dans_ellipse_param(coord_assures,
+					sim_param_orages_4[i,]))
+}
+
+rpareto_tronque <- function(n,shape,scale,deductible)
+{
+	pmax(rpareto(n,shape,scale)-deductible,0)
+}
+
+dommages <- mcsapply(mcsapply(as.list(touches1),
+						rpareto_tronque,param_forme,
+						param_dommages[1],franchise),sum)+
+		mcsapply(mcsapply(as.list(touches2),
+						rpareto_tronque,param_forme,
+						param_dommages[2],franchise),sum)+
+		mcsapply(mcsapply(as.list(touches3),
+						rpareto_tronque,param_forme,
+						param_dommages[3],franchise),sum)+
+		mcsapply(mcsapply(as.list(touches4),
+						rpareto_tronque,param_forme,
+						param_dommages[4],franchise),sum)
+dommages_cum <- cumsum(dommages)
+orages_annees_cum <- cumsum(orages_annees)
+dommages_annuels <- diff(dommages_cum[orages_annees_cum])
+@
+
+On obtient une moyenne de \Sexpr{mean(dommages_annuels)}, un écart-type de
+\Sexpr{sd(dommages_annuels)} et les quantiles suivants:
+
+<<>>=
+quantile(dommages_annuels,c(.5,.75,.9,.95,.99))
+@
+
+\subsection{Comparaison avec moins de concentration}
+
+On sélectionne un assuré sur \Sexpr{nb.conc} afin de réduire l'exposition et on
+peut comparar les résultats avec les mêmes orages.
+
+<<>>=
+touches21 <- numeric(n.orages)
+touches22 <- numeric(n.orages)
+touches23 <- numeric(n.orages)
+touches24 <- numeric(n.orages)
+for(i in 1:n.orages)
+{
+	touches21[i] <- sum((sim_force[i] >= 1) * 
+					dans_ellipse_param(coord_assures2,
+					sim_param_orages_1[i,]))
+	touches22[i] <- sum((sim_force[i] >= 2) * 
+					dans_ellipse_param(coord_assures2,
+					sim_param_orages_2[i,]))
+	touches23[i] <- sum((sim_force[i] >= 3) * 
+					dans_ellipse_param(coord_assures2,
+					sim_param_orages_3[i,]))
+	touches24[i] <- sum((sim_force[i] >= 4) * 
+					dans_ellipse_param(coord_assures2,
+					sim_param_orages_4[i,]))
+}
+
+dommages2 <- mcsapply(mcsapply(as.list(touches21),
+						rpareto_tronque,param_forme,
+						param_dommages[1],franchise),sum)+
+		mcsapply(mcsapply(as.list(touches22),
+						rpareto_tronque,param_forme,
+						param_dommages[2],franchise),sum)+
+		mcsapply(mcsapply(as.list(touches23),
+						rpareto_tronque,param_forme,
+						param_dommages[3],franchise),sum)+
+		mcsapply(mcsapply(as.list(touches24),
+						rpareto_tronque,param_forme,
+						param_dommages[4],franchise),sum)
+dommages2_cum <- cumsum(dommages2)
+dommages2_annuels <- diff(dommages2_cum[orages_annees_cum])
+@
+
+On obtient une moyenne de \Sexpr{mean(dommages2_annuels)}, un écart-type de
+\Sexpr{sd(dommages2_annuels)} et les quantiles suivants:
+
+<<>>=
+quantile(dommages2_annuels,c(.5,.75,.9,.95,.99))
+@
+
+\section{Risque associé au vol}
+
+<<>>=
+	taux_vol_petits <- 1600/100000
+	taux_vol_grands <- 60/100000
+@
+
+\subsection{Hypothèses}
+Le risque associé au vol peut être modélisé à l'aide d'une approche
+fréquence-sévérité régulière. On pose donc comme hypothèse une distrubution
+poisson pour la fréquence et une distribution gamma pour la sévérité.
+On divise les vols en petits vols et en grands vols.
+<<>>=
+vols_petits <- taux_vol_petits * n.assures
+freq_vols_petits <- rpois(n.annees,vols_petits)
+vols_petits_totaux <- sum(freq_vols_petits)
+sev_vols_petits <- rgamma(vols_petits_totaux,2,.005)
+sev_vols_petits_cum <- cumsum(sev_vols_petits)
+freq_petits_vols_cum <- cumsum(freq_vols_petits)
+sev_vols_petits_annuels <- diff(sev_vols_petits_cum[freq_petits_vols_cum])
+
+vols_grands <- taux_vol_grands * n.assures
+freq_vols_grands <- rpois(n.annees,vols_grands)
+vols_grands_totaux <- sum(freq_vols_grands)
+sev_vols_grands <- rgamma(vols_grands_totaux,2,.0001)
+sev_vols_grands_cum <- cumsum(sev_vols_grands)
+freq_grands_vols_cum <- cumsum(freq_vols_grands)
+sev_vols_grands_annuels <- diff(sev_vols_grands_cum[freq_grands_vols_cum])
+
+vols_totaux <- rowSums(cbind(sev_vols_petits_annuels,sev_vols_grands_annuels))
+@
+
+On obtient une moyenne de vols annuels de \Sexpr{mean(vols_totaux)} avec un
+écart-type de \Sexpr{sd(vols_totaux)}. Les quantiles de la distirubtion sont les
+suivants:
+
+<<>>=
+quantile(vols_totaux,c(.5,.75,.9,.95,.99))
+@
+
+
+
+\end{document}
diff --git a/ellipse.R b/ellipse.R
new file mode 100644
index 0000000..67f6e1a
--- /dev/null
+++ b/ellipse.R
@@ -0,0 +1,6 @@
+# TODO: Add comment
+# 
+# Author: francois
+###############################################################################
+
+
diff --git a/mcsapply.R b/mcsapply.R
new file mode 100644
index 0000000..e726315
--- /dev/null
+++ b/mcsapply.R
@@ -0,0 +1,12 @@
+require("multicore")
+
+mcsapply <- function (X, FUN, ..., simplify = TRUE, USE.NAMES = TRUE) 
+{
+	FUN <- match.fun(FUN)
+	answer <- mclapply(X = X, FUN = FUN, ...)
+	if (USE.NAMES && is.character(X) && is.null(names(answer))) 
+		names(answer) <- X
+	if (!identical(simplify, FALSE) && length(answer)) 
+		simplify2array(answer, higher = (simplify == "array"))
+	else answer
+}
diff --git a/projet.R b/projet.R
new file mode 100644
index 0000000..10f82fd
--- /dev/null
+++ b/projet.R
@@ -0,0 +1,90 @@
+# TODO: Add comment
+# 
+# Author: francois
+###############################################################################
+
+## Importer les données dans un data.frame
+agents <- read.csv("agents.csv",sep=":")
+assures <- read.csv("assures.csv",sep=":")
+
+## Assurés plus uniformes: en prendre le 1/8, multiplier la fréquence par 8
+assures8 <- assures[seq(from=1,to=32669,by=8),]
+
+## Visualiser les agents
+plot(agents$Longitude_Agent,agents$Latitude_Agent,pch=6,col="red")
+## Visualiser les assurés
+points(assures$Longitude_Assure,assures$Latitude_Assure,pch=20)
+
+#fonction orage
+sim_dommages_orage <- function(assures)
+{
+	franchise <- 0
+	## Définir un orage
+	sim_centre <- cbind(runif(1,range_longitude[1],range_longitude[2]),runif(1,range_latitude[1],range_latitude[2]))
+	sim_taille <- cbind(rexp(1,.25),rexp(1,4))
+	sim_angle <- runif(1,0,2*pi)
+	sim_force <- sample(x=1:4,size=1,replace=TRUE,prob=c(.95,.03,.016,.004))
+	
+	if(sim_force==1)
+	{
+		ellipse1 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],sim_taille[1],sim_taille[2],sim_angle)
+	}
+	if(sim_force==2)
+	{
+		ellipse2 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],sim_taille[1],sim_taille[2],sim_angle)
+		ellipse1 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],2*sim_taille[1],2*sim_taille[2],sim_angle)
+	}
+	if(sim_force==3)
+	{
+		ellipse3 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],sim_taille[1],sim_taille[2],sim_angle)
+		ellipse2 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],2*sim_taille[1],2*sim_taille[2],sim_angle)
+		ellipse1 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],3*sim_taille[1],3*sim_taille[2],sim_angle)
+	}
+	if(sim_force==4)
+	{
+		ellipse4 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],sim_taille[1],sim_taille[2],sim_angle)
+		ellipse3 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],2*sim_taille[1],2*sim_taille[2],sim_angle)
+		ellipse2 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],3*sim_taille[1],3*sim_taille[2],sim_angle)
+		ellipse1 <- ellipse_param(seq(0,2*pi,by=0.1),sim_centre[1],sim_centre[2],4*sim_taille[1],4*sim_taille[2],sim_angle)
+	}
+	
+	## Définir le nombre d'assurés touchés
+	touches1 <- FALSE
+	touches2 <- FALSE
+	touches3 <- FALSE
+	touches4 <- FALSE
+	
+	touches1 <- dans_ellipse_param(assures$Longitude_Assure,assures$Latitude_Assure,ellipse1)
+	if(sim_force>1)
+	{
+		touches2 <- dans_ellipse_param(assures$Longitude_Assure,assures$Latitude_Assure,ellipse2)
+		if(sim_force>2)
+		{
+			touches3 <- dans_ellipse_param(assures$Longitude_Assure,assures$Latitude_Assure,ellipse3)
+			if(sim_force>3)
+			{
+				touches4 <- dans_ellipse_param(assures$Longitude_Assure,assures$Latitude_Assure,ellipse4)
+			}
+		}
+	}
+	
+	sum(max(rpareto(sum(touches1*!touches2),2,50)-franchise,0)) + 
+			sum(max(rpareto(sum(touches2*!touches3),2,1000)-franchise,0)) + 
+			sum(max(rpareto(sum(touches3*!touches4),2,20000)-franchise,0)) + 
+			sum(max(rpareto(sum(touches4),2,400000)-franchise,0))
+}
+
+set.seed(123)
+dommages <- replicate(1000,sim_dommages_orage(assures),simplify=TRUE)
+set.seed(123)
+dommages8 <- replicate(1000,sim_dommages_orage(assures8),simplify=TRUE)
+
+
+mean(dommages)
+var(dommages)
+quantile(dommages,c(.25,.5,.75,.9,.95,.99,.995))
+
+
+mean(dommages8)
+var(dommages8)
+quantile(dommages8,c(.25,.5,.75,.9,.95,.99,.995))
\ No newline at end of file