# Optimal Portfolios, or sort of…

Last week, we got our first class on portfolio optimization. We’ve seen Markowitz’s theory where expected returns and the covariance matrix are given,

> source("portfolio.r")
> library(zoo)
> library(FRAPO)
> library(IntroCompFinR)
> library(rrcov)
> data( StockIndex )
> pzoo = zoo ( StockIndex , order.by = rownames ( StockIndex ) )
> rzoo = ( pzoo / lag ( pzoo , k = -1) - 1 ) * 100
> Moments <- function ( x , method = c ( "CovClassic" , "CovMcd" , "CovMest" , "CovMMest" , "CovMve" , "CovOgk" , "CovSde" , "CovSest" ) , ... ) {
method <- match.arg ( method )
ans <- do.call ( method , list ( x = x , ... ) ) + return ( getCov ( ans ) )} > covmat=Moments(as.matrix(rzoo),"CovClassic")
> (covmat=round(covmat,1))
SP500 N225 FTSE100 CAC40 GDAX HSI
SP500   17.8 12.7 13.8 17.8 19.5 18.9
N225    12.7 36.6 10.8 15.0 16.2 16.7
FTSE100 13.8 10.8 17.3 18.8 19.4 19.1
CAC40   17.8 15.0 18.8 30.9 29.9 22.8
GDAX    19.5 16.2 19.4 29.9 38.0 26.1
HSI     18.9 16.7 19.1 22.8 26.1 58.1
> er=apply(as.matrix(rzoo),2,mean)
> (er=round(er,1))
SP500 N225 FTSE100 CAC40 GDAX HSI
0.6 -0.2 0.4 0.5 0.8 1.0
> ef <- efficient.frontier(er, covmat, alpha.min=-2.5, alpha.max=2.5, nport=50)

We can now visualize the efficient frontier (and admissible portfolios) below

> u=c(12,ef$sd,12,12) > v=c(5,ef$er,-1,5)
> plot(ef$sd,ef$er,type="l",xlab="Standard Deviation",ylab="Expected Return", xlim=c(3.5,11),ylim=c(0,2.5),col="red",lwd=1.5)
> points(sqrt(diag(covmat)),er,pch=19,col="blue")
> text(sqrt(diag(covmat)),er,names(er),pos=4, col="blue",cex=.6)
> polygon(u,v,border=NA,col=rgb(0,0,1,.3))

That was the starting point of our class. We did also mention that something important was actually hard to visualize on that graph : the correlation between returns. It is not in the points (which are univariate, with expected return and standard deviation), but in the efficient frontier. For instance, here is our correlation matrix

> (cormat=covmat/(sqrt(diag(covmat) %*% t(diag(covmat)))))
SP500 N225 FTSE100 CAC40 GDAX HSI
SP500   1.00 0.50 0.79 0.76 0.75 0.59
N225    0.50 1.00 0.43 0.45 0.43 0.36
FTSE100 0.79 0.43 1.00 0.81 0.76 0.60
CAC40   0.76 0.45 0.81 1.00 0.87 0.54
GDAX    0.75 0.43 0.76 0.87 1.00 0.56
HSI     0.59 0.36 0.60 0.54 0.56 1.00

We can actually change the correlation between FT500 and FTSE100 (which is here .786)

courbe=function(r=.786){
R=cormat
R[1,3]=R[3,1]=r
covmat2=(sqrt(diag(covmat) %*% t(diag(covmat))))*R
ef <- efficient.frontier(er, covmat2, alpha.min=-2.5, alpha.max=2.5, nport=50)
plot(ef$sd,ef$er,type="l",xlab="Standard Deviation",ylab="Expected Return",
xlim=c(3.5,11),ylim=c(0,2.5),col="red",lwd=1.5)
points(sqrt(diag(covmat)),er,pch=19,col=c("blue","red")[c(2,1,2,1,1,1)])
text(sqrt(diag(covmat)),er,names(er),pos=4,col=c("blue","red")[c(2,1,2,1,1,1)],cex=.6)
polygon(u,v,border=NA,col=rgb(0,0,1,.3))
}

for instance, with a correlation of 0.6, we get the following efficient frontier

> courbe(.6)

and with a stronger correlation

> courbe(.9)

So clearly, correlation does matter. A lot. But more important, one should keep in mind that expected returns and covariances are not given, but estimated. Previously, we did use the standard estimator for the variance matrix. But another (more robust) estimator can be considered

covmat=Moments(as.matrix(rzoo),"CovSde")
er=apply(as.matrix(rzoo),2,mean)
ef <- efficient.frontier(er, covmat, alpha.min=-2.5, alpha.max=2.5, nport=50)
plot(ef$sd,ef$er,type="l",xlab="Standard Deviation",ylab="Expected Return",xlim=c(3.5,11),ylim=c(0,2.5),col="red",lwd=1.5)
points(sqrt(diag(covmat)),er,pch=19,col="blue")
text(sqrt(diag(covmat)),er,names(er),pos=4,col="blue",cex=.6)
polygon(u,v,border=NA,col=rgb(0,0,1,.3))

It did influence (horizontal) position of points, since variances are now different, as well as the efficient frontier, with clearly much lower variances that can be reached.

And to illustrate a last point, to illustrate the fact that we do have estimators based on observed returns, what if we had observed different ones? A way to get an idea of what might happened is to use bootstrap, e.g. of daily returns.

> covmat=Moments(as.matrix(rzoo),"CovClassic")
> er=apply(as.matrix(rzoo),2,mean)
> ef <- efficient.frontier(er, covmat, alpha.min=-2.5, alpha.max=2.5, nport=50) > a=sqrt(diag(covmat))
> b=er
> k=1
> plot(ef$sd,ef$er,type="l",xlab="Standard Deviation",ylab="Expected Return", xlim=c(3.5,11),ylim=c(0,2.5),col="white",lwd=1.5)
> polygon(u,v,border=NA,col=rgb(0,0,1,.3))
> for(i in 1:100){
+ id=sample(nrow(rzoo),replace=TRUE)
+ covmat=Moments(as.matrix(rzoo)[id,],"CovClassic")
+ er=apply(as.matrix(rzoo)[id,],2,mean)
+ points(sqrt(diag(covmat))[k],er[k],cex=.5)
+ }

or for another asset

Here is what we got on the (estimated) efficient frontier

> covmat=Moments(as.matrix(rzoo),"CovClassic")
> er=apply(as.matrix(rzoo),2,mean)
> ef <- efficient.frontier(er, covmat, alpha.min=-2.5, alpha.max=2.5, nport=50) > plot(ef$sd,ef$er,type="l",xlab="Standard Deviation",ylab="Expected Return", xlim=c(3.5,11),ylim=c(0,2.5),col="white",lwd=1.5)
> points(sqrt(diag(covmat)),er,pch=19,col="blue")
> text(sqrt(diag(covmat)),er,names(er),pos=4, col="blue",cex=.6)
> polygon(u,v,border=NA,col=rgb(0,0,1,.3))
> for(i in 1:100){
+ id=sample(nrow(rzoo),replace=TRUE)
+ covmat=Moments(as.matrix(rzoo)[id,],"CovClassic")
+ er=apply(as.matrix(rzoo)[id,],2,mean)
+ ef <- efficient.frontier(er, covmat, alpha.min=-2.5, alpha.max=2.5, nport=50)
+ lines(ef$sd,ef$er,col="red")
+ }

Thus, it is somehow rather difficult to assess wheter a portfolio is optimal, or not… At least from a statistical perspective….

# Traffic Flow of Kota Kinabalu (with R)

This morning, we had our first practicals on network flows, using  an example mentioned in some papers published by Noraini Abdullah and Ting Kien Hua, max flow min cut theorem to minimize traffic congestion in Kota Kinabalu and application of the Shortest Path and Maximum Flow with Bottleneck in Traffic Flow of Kota Kinabalu. From the roads mentioned in the articles, I did try my best to locate the nodes on a map,

m=matrix(c(0,5.995910, 116.105520,
1,5.992737, 116.093718,
2,5.992066, 116.109883,
3,5.976947, 116.095760,
4,5.985766, 116.091580,
5,5.988940, 116.080112,
6,5.968318, 116.080764,
7,5.977454, 116.075460,
8,5.974226, 116.073604,
9,5.969651, 116.073753,
10,5.972341, 116.069270,
11,5.978818, 116.072880),3,12)

we can be visualized below

library(OpenStreetMap)
map = openmap(c(lat= 6.000, lon= 116.06),
c(lat= 5.960, lon= 116.12))
map=openproj(map)
plot(map)
points(t(m[3:2,]),col="black", pch=19, cex=3 )
text(t(m[3:2,]),c("s",1:10,"t"),col="white")

If the source is realistic (up north), I do not feel very confortable with the location of the sink (on the west). But let’s pretend it’s find (to do the maths, at least).

To extract information about edge capacity, on that network use the following code that will extract the three tables from the paper

library(devtools)
install_github("ropensci/tabulizer")
library(tabulizer)
location <- 'http://www.jistm.com/PDF/JISTM-2017-04-06-02.pdf'
out <- extract_tables(location)

with Windows, it seems to be necessary to download another package first

library(devtools)
install_github("ropensci/tabulizerjars")
install_github("ropensci/tabulizer")
library(tabulizer)
location <- 'http://www.jistm.com/PDF/JISTM-2017-04-06-02.pdf'
out <- extract_tables(location)

Now we can get out data frame with capacities

B1=as.data.frame(out[[2]])
B2=as.data.frame(out[[3]])
E=data.frame(from=B1[3:20,"V3"],
to=B1[3:20,"V4"])
E=E[-c(6,8),]
capacity=as.character(B2$V3[-1]) capacity[6]="843" capacity[4]="2913" E$capacity=as.numeric(capacity)

We can add those edges on our map (without the arrows to indicate the direction, it would be to heavy to read)

plot(map)
points(t(m[3:2,]),col="black", pch=19, cex=3 )
B=data.frame(i=as.character(c("s",paste("V",1:10,sep=""),"t")),
x=m[3,],y=m[2,])
for(i in 1:nrow(E)){
i1=which(B$i==as.character(E$from[i]))
i2=which(B$i==as.character(E$to[i]))
segments(B[i1,"x"],B[i1,"y"],B[i2,"x"],B[i2,"y"],lwd=3)
}
text(t(m[3:2,]),c("s",1:10,"t"),col="white")

To get the graph with capacities, an alternative is to use

library(igraph)
g=graph_from_data_frame(E)
E(g)$label=E$capacity
plot(g)

but it does not respect geographical locations of nodes. It can actually be done using

plot(g, layout=as.matrix(B[,c("x","y")]))

To get a better understanding of the capacities of the road, use

plot(g, layout=as.matrix(B[,c("x","y")]),
edge.width=E$capacity/200) From that network with capacities, the goal is to determine maximum flow on that network, from the source to the sink. This can be done with R using > (m=max_flow(graph=g, source="s", target="t"))$value
[1] 2571

$flow [1] 1191 1380 1422 1380 231 0 231 0 1149 1422 1149 0 0 1149 1422 [16] 1149 Our maximum flow is here 2571, which is different from was is actually claimed both in the two papers max flow min cut theorem to… and application of the Shortest Path… (“the maximum flow for the capacitated network with 12 nodes and 16 edges of the selected scope in this study was 2598 vehicles per hour“) where there are clearly typos since values in the table and on the graph are different. Here I did use the ones from the tables. E$flux1=m$flow E(g)$label=E$flux1 plot(g, layout=as.matrix(B[,c("x","y")]), edge.width=E$flux1/200)

That is nice, but rather odd. Actually, a much simpler flow can be considered, but the same global value

E$flux2=c(1422,1149,1422,1149,0,0,0,0, 1149,1422,1149,0,0,1149,1422,1149) E(g)$label=E$flux2 plot(g, layout=as.matrix(B[,c("x","y")]), edge.width=E$flux2/200)

Nice, isn’t it. It is actually possible to do exactly the same on another paper they have, on the same city, traffic congestion problem of road networks in Kota Kinabalu.

location <- 'http://www.worldresearchlibrary.org/up_proc/pdf/999-150486366625-30.pdf'
out <- extract_tables(location)
dim(out[[3]])
B1=as.data.frame(out[[3]])
E=data.frame(from=B1[2:61,"V2"],
to=B1[2:61,"V3"],
capacity=B1[2:61,"V4"])
E$capacity=as.numeric( as.character(E$capacity))
library(igraph)
g=graph_from_data_frame(E)
m=max_flow(graph=g,
source="S",
target="T")
E$flux1=m$flow
E(g)$label=E$flux1
plot(g,
edge.width=E$flux1/200, edge.arrow.size=0.15) Here the value of the maximal flow is 4017, just as they found in the original paper # Multinomial Logit as an Iterated Logit Regression For the second section of the course at ENSAE, yesterday, we’ve seen how to run a multinomial logistic regression model. It is simply an extension of the binomial logistic regression. But actually, it is also possible to consider iterative binomial regressions. Consider here a response variable $Y$ with a multinomial distribution (3 factors to have something more general than the binomial), taking values $\{A,B,C\}$, with respective probabilities $\mathbf{p}=(p_A,p_B,p_C)$. Here is a code to generate some multinomial variables msample=function(A,B,C){ Y=rep(NA,B) for(i in 1:B){Y[i]=sample(A,size=1,prob=C[i,])} return(Y) } and here is a code to generate a dataset with $n$ rows, generate3=function(n,x,pb=c(-2,0)){ set.seed(x) X1=runif(n) X2=runif(n) X3=runif(n) s1=pb[1]+X1+X2 s2=pb[2]-X1+X2 P1=exp(s1)/(1+exp(s1)+exp(s2)) P2=exp(s2)/(1+exp(s1)+exp(s2)) Y=msample(0:2,n,cbind(1-P1-P2,P1,P2)) df=data.frame(Y=Y,X1=X1,X2=X2,X3=X3) return(df) } Let us generate a training dataset and a validation one pb=c(.31,.42) DF1=generate3(1000,1,pb=pb) DF2=generate3(500,2,pb=pb) With a multivariate logistic regression $$\mathbb{P}[Y=A|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}$$ $$\mathbb{P}[Y=B|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}$$ $$\mathbb{P}[Y=B|\mathbf{x}]=\frac{1}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}$$ For convenience, consider the most popular factor in our training dataset modalite=names(sort(table(DF1$Y),decreasing = TRUE))

Consider a regression model on the simulated dataset (with several covariates), let us estimate it, and let us get predictions.

library(nnet)
reg=multinom(as.factor(Y) ~ ., data = DF1)
mp1=predict (reg, DF1, "probs")
mp2=predict (reg, DF2, "probs")

An alternative can be the following.
consider a first regression model on the Bernoulli variable $Y_A=\mathbf{1}(Y=A)$. Actually, we will consider the most important factor, but for convenience, assume that it is $A$.
$$\mathbb{P}[Y_A=A|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{a}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{a}]}$$
On our dataset, estimate that model, and get predictions. In the case where $Y\neq A$, define another Bernoulli variable $Y_B=\mathbf{1}(Y=B|Y\neq A)$. We can estimate that model and derive two probabilities, $\mathbb{P}(Y=B|Y\neq A)$ and $\mathbb{P}(Y=C|Y\neq A)$ (the sum of the two being equal to 1). Based on those two models, it is possible to compute the three probabilities we are looking for. $\mathbb{P}[Y=A]$ is obtained from the first model, and we can derive the other two from $\mathbb{P}[Y=B|Y\neq A]\cdot\mathbb{P}[Y\neq A]$ and $\mathbb{P}[Y=C|Y\neq A]\cdot\mathbb{P}[Y\neq A]$.

reg1=glm((Y==modalite[1])~.,data=DF1,family=binomial)
reg2=glm((Y==modalite[2])~.,data=DF1[-which(DF1$Y==modalite[1]),],family=binomial) p11=predict (reg1, newdata=DF1, type="response") p12=predict (reg2, newdata=DF1, type="response") p21=predict (reg1, newdata=DF2, type="response") p22=predict (reg2, newdata=DF2, type="response") mmp1=cbind(p11,(1-p11)*p12,(1-p11)*(1-p12)) mmp2=cbind(p21,(1-p21)*p22,(1-p21)*(1-p22)) colnames(mmp1)=colnames(mmp2)=modalite Let us compare the predicted probabilites, on the same dataset (here the training dataset) > mmp1[1:9,c("0","1","2")] 0 1 2 1 0.19728737 0.4991805 0.3035321 2 0.17244580 0.5648537 0.2627005 3 0.19291753 0.5971058 0.2099767 4 0.09087176 0.7787304 0.1303978 5 0.23400225 0.4083022 0.3576955 6 0.18063647 0.6637352 0.1556283 7 0.13188881 0.7402710 0.1278401 8 0.13776970 0.6524959 0.2097344 9 0.12325864 0.6790336 0.1977078 > mp1[1:9,c("0","1","2")] 0 1 2 1 0.19691036 0.5022692 0.3008205 2 0.17123189 0.5680647 0.2607034 3 0.19293066 0.5984402 0.2086291 4 0.08821851 0.7813318 0.1304497 5 0.23470739 0.4109990 0.3542936 6 0.18249687 0.6602168 0.1572863 7 0.13128711 0.7400898 0.1286231 8 0.13525341 0.6553618 0.2093848 9 0.12090016 0.6815915 0.1975084 The two are very close. So yes, it is possible to see the multinomial regression as some sequential binomial regressions. # Traveling Salesman In the second part of the course on graphs and networks, we will focus on economic applications, and flows. The first series of slides are on the traveling salesman problem. Slides are available online. # Networks with R In order to practice with network data with R, we have been playing with the Padgett (1994) Florentine’s wedding dataset (discussed in the lecture). The dataset is available from > library(network) > data(flo) > nflo=network(flo,directed=FALSE) > plot(nflo, displaylabels = TRUE, + boxed.labels = + FALSE) The next step was to move from the network package to igraph. Since we have the adjacency matrix, we can use it > library(igraph) > iflo=graph_from_adjacency_matrix(flo, + mode = "undirected") > plot(iflo) The good thing is that a lot of functions are available, for instance we can get shortest paths, between two specific nodes. And we can give appropriate colors to the nodes that we’ll cross > AP=all_shortest_paths(iflo, + from="Peruzzi", + to="Ginori") > L=AP$res[[1]]
> V(iflo)$color="yellow" > V(iflo)$color[L[2:4]]="light blue"
> V(iflo)$color[L[c(1,5)]]="blue" > plot(iflo) We can also visualize edges, but I found it slightly more complicated (to extract edges from the output) > liens=c(paste(as.character(L)[1:4], + "--", + as.character(L)[2:5],sep=""), + paste(as.character(L)[2:5], + "--", + as.character(L)[1:4],sep="")) > df=as.data.frame(ends(iflo,E(iflo))) > names(df)=c("src","target") > lstn=sort(unique(c(as.character(df[,1]),as.character(df[,2]),"Pucci"))) > Eliens=paste(as.numeric(factor(df[,1],levels=lstn)),"--", + as.numeric(factor(df[,2],levels=lstn)),sep="") > EU=unlist(lapply(Eliens,function(x) x%in%liens)) > E(iflo)$color=c("grey","black")[1+EU]
> plot(iflo)

But it works. It is also possible to use some D3js visualization

> library( networkD3 )
> simpleNetwork (df)

Then the next question was to add a vertice to the network. The most simple way to do it is probability through the adjacency matrix

> flo2=flo
> flo2["Pucci","Bischeri"]=1
> flo2["Bischeri","Pucci"]=1
> nflo2=network(flo2,directed=FALSE)
> plot(nflo2, displaylabels = TRUE,
+ boxed.labels =
+ FALSE)

Then, we’ve been playing with centrality measures.

> plot(iflo,vertex.size=betweenness(iflo))

The goal was to see how related they were. Here, for all of them, “Medici” is the central node. But what about the others?

> B=betweenness(iflo)
> C=closeness(iflo)
> D=degree(iflo)
> E=eigen_centrality(iflo)$vector > base=data.frame(betw=B,close=C,deg=D,eig=E) > cor(base) betw close deg eig betw 1.0000000 0.5763487 0.8333763 0.6737162 close 0.5763487 1.0000000 0.7572778 0.7989789 deg 0.8333763 0.7572778 1.0000000 0.9404647 eig 0.6737162 0.7989789 0.9404647 1.0000000 Those measures are quite correlated. It is also possible to use a hierarchical graph to visualize how close those centrality measures can be > H=hclust(dist(t(base)), + method="ward") > plot(H) Instead of looking at values of centrality measures, it is possible to looks are ranks > rbase=base > for(i in 1:4) rbase[,i]=rank(base[,i]) > H=hclust(dist(t(rbase)), + method="ward") > plot(H) Here the eigenvector measure is very close to the degree of vertices. Finally, it is possible to seek clusters (in the context of coalition here, in case a war should start between those families) > kc <- fastgreedy.community ( iflo ) Here we have 3 classes (+1 for the node that is disconnected from the other families) > V(iflo)$color=c("yellow","orange",
+ "light blue")[membership ( kc )]
> plot(iflo)

> plot(kc,iflo)

# I Got The Feelin’

Last week, I’ve been going through my CD collection, trying to find records I haven’t been listing for a while. And I got the feeling that music I listen to nowadays is slower than the one I was listening to in my 20’s. I was wondering if that was an age issue, or it was simply the fact that music in the 90s was “faster” than the one released in 2015. So I tried to scrap the BPM database to get a more appropriate answer about this “feeling” I have. I did extract two information: the BPM (beat per minute) and the year (of release).

Here is a function to extract information from the website,

> library(XML)
> extractbpm = function(VBP,P){
+ url=paste("https://www.bpmdatabase.com/music/search/?artist=&title=&bpm=",VBP,"&genre=&page=",P,sep="")
+ return(tables)}

For instance

> extractbpm(115,13)
$track-table Artist Title 1 Eros Ramazzotti y Claudio Guidetti Dimelo A Mi 2 Everclear Volvo Driving Soccer Mom 3 Evils Toy Dear God 4 Expose In Walked Love 5 Fabolous ft. 2 Chainz When I Feel Like It 6 Fabolous ft. 2 Chainz When I Feel Like It 7 Fabolous ft. 2 Chainz When I Feel Like It 8 Fanny Lu Fanfarron 9 Featurecast Ain't My Style 10 Fem 2 Fem Obsession 11 Fernando Villalona Mi Delito 12 Fever Ray Triangle Walks 13 Firstlove Freaky 14 Fito Blanko Pegadito Suavecito 15 Flechazo Del Norte Mariposa Traicionera 16 Fluke Switch/Twitch 17 Flyleaf Something Better 18 FM Static The Next Big Thing 19 Fonseca Eres Mi Sueno 20 Fonseca ft. Maffio & Nayer Eres Mi Sueno 21 Francesca Battistelli Have Yourself A Merry Little Christmas 22 Frankie Ballard Young & Crazy 23 Frankie J. More Than Words 24 Frank Sinatra The Hucklebuck 25 Franz Ferdinand The Dark Of The Matinée Mix BPM Genre Label Year 1 — 115 — Sony 2009 2 — 115 — Capitol Records 2003 3 — 115 — — — 4 — 115 — Arista Records 1994 5 Explicit 115 Urban Def Jam/Island Def Jam 2013 6 — 115 Urban Def Jam/Island Def Jam 2013 7 Radio Edit 115 Urban Def Jam/Island Def Jam 2013 8 — 115 Latin Pop Universal Latino 2011 9 Psychemagik Dub 115 — Jalapeno 2012 10 — 115 — Critique Records 1993 11 — 115 — Mt&vi Records/caminante Records 2001 12 Rex The Dog Remix 115 — Little Idiot/Mute 2012 13 — 115 — Jwp Music 2000 14 — 115 Merengue Mambo Crown Loyalty 2012 15 — 115 — Hacienda 2010 16 Album Version 115 — One Little Indian Records 2004 17 — 115 Alternative A&M/Octone 2013 18 — 115 — Tooth & Nail Records 2007 19 — 115 Merengue Mambo 10 2012 20 Urban Version 115 — 10 2012 21 — 115 — Word/Fervent/Warner Bros 2009 22 — 115 Country Warner Bros 2015 23 Mynt Rocks Radio Edit 115 — Columbia 2005 24 — 115 Jazz Columbia 1950 25 — 115 New Wave — 2004 We have here one of the few old songs, a 1950 tune by Frank Sinatra. If we scrap the website, with a simple loop (where the bpm is from 40 to 200). Start with BASE=NULL > vbp=40 > p=1 and then, a loop based on > while(vbp<=2017){ + F=extractbmp(vbp,p) + if(length(F)==1){ + BASE=rbind(BASE,F[[1]][,c("Artist","Title","BPM","Year")]) + p=p+1} + if(length(F)==0){ + vbp=vbp+1 + p=1}} Then we should clean the dataset BASE=BASE[-duplicated(BASE),] BASE=BASE[-which(BASE$Year=="—"),]
BASE$y=as.numeric(as.character(BASE$Year))
BASE$bpm=as.numeric(as.character(BASE$BPM))
BASE=BASE[BASE$y>=1940,] and we end up with almost 50,000 tunes. boxplot(BASE$bpm~as.factor(BASE$y), col="light blue") Over the past 20 years, it looks like speed of tunes has declined (let us forget tunes of 2017, clearly we have a problem here…) library(mgcv) plot(BASE$y,BASE$bpm) reg=gam(bpm~s(y),data=BASE) B=data.frame(y=1950:2017) p=predict(reg,newdata=B) lines(B$y,p,lwd=3,col="red")

which is confirmed with a (smoothed) regression

p2=predict(reg,newdata=B,se.fit=TRUE)
plot(B$y,p2$fit,lwd=3,col="red",type="l",ylim=c(90,140))
lines(B$y,p2$fit+p2$se.fit) lines(B$y,p2$fit-p2$se.fit)

even when incorporating the confidence band. Bumps are probably related to smoothing parameters, but indeed, it looks like the average speed of music tune has decreased, from 110-115 in the 90’s to less than 100 nowadays. Now to be honest, I would love to have access to personal information from itunes, deezer or spotify, to get a better understanding (eg when in the week, in the day, do we like to listen to faster music for instance). But so far, I could not have access to such data. Too bad…

# Matching, Optimal Transport and Statistical Tests

To explain the “optimal transport” problem, we usually start with Gaspard Monge’s “Mémoire sur la théorie des déblais et des remblais“, where the the problem of transporting a given distribution of matter (a pile of sand for instance) into another (an excavation for instance). This problem is usually formulated using distributions, and we seek the “optimal” transport from one distribution to the other one. The formulation, in the context of distributions has been formulated in the 40’s by Leonid Kantorovich, e.g. from the distribution on the left to the distribution on the right.

Consider now the context of finite sets of points. We want to transport mass from points $\{A_1,\cdots,A_4\}$ to points $\{B_1,\cdots,B_4\}$. It is a complicated combinatorial problem. For 4 points, there are only 24 possible transfer to consider, but it exceeds 20 billions with 15 points (on each side). For instance, the following one is usually seen as inefficient

while the following is usually seen as much better

Of course, it depends on the cost of the transport, which depends on the distance between the origin and the destination. That cost is usually either linear or quadratic.

There are many application of optimal transport in economics, see eg Alfred’s book Optimal Transport Methods in Economics. And there are also applications in statistics, that what I’ve seen while I was discussing with Pierre while I was in Boston, in June. For instance if we want to test whether some sample were drawn from the same distribution,

set.seed(13)
npoints <- 25
mu1 <- c(1,1)
mu2 <- c(0,2)
Sigma1 <- diag(1, 2, 2)
Sigma2 <- diag(1, 2, 2)
Sigma2[2,1] <- Sigma2[1,2] <- -0.5
Sigma1 <- 0.4 * Sigma1
Sigma2 <- 0.4 *Sigma2
library(mnormt)
X1 <- rmnorm(npoints, mean = mu1, Sigma1)
X2 <- rmnorm(npoints, mean = mu2, Sigma2)
plot(X1[,1], X1[,2], ,col="blue")
points(X2[,1], X2[,2], col = "red")

Here we use a parametric model to generate our sample (as always), and we might think of a parametric test (testing whether mean and variance parameters of the two distributions are equal).

or we might prefer a nonparametric test. The idea Pierre mentioned was based on optimal transport. Consider some quadratic loss

ground_p <- 2
p <- 1
w1 <- rep(1/npoints, npoints)
w2 <- rep(1/npoints, npoints)
C <- cost_matrix_Lp(t(X1), t(X2), ground_p)
library(transport)
library(winference)
a <- transport(w1, w2, costm = C^p, method = "shortsimplex")

then it is possible to match points in the two samples

nonzero <- which(a$mass != 0) from_indices <- a$from[nonzero]
to_indices <- a$to[nonzero] for (i in from_indices){ segments(X1[from_indices[i],1], X1[from_indices[i],2], X2[to_indices[i], 1], X2[to_indices[i],2]) } Here we can observe two things. The total cost can be seen as rather large > cost=function(a,X1,X2){ nonzero <- which(a$mass != 0)
naa=a[nonzero,]
d=function(i) (X1[naa$from[i],1]-X2[naa$to[i],1])^2+(X1[naa$from[i],2]-X2[naa$to[i],2])^2
sum(Vectorize(d)(1:npoints))
}
> cost(a,X1,X2)
[1] 9.372472

and the angle of the transport direction is alway in the same direction (more or less)

> angle=function(a,X1,X2){
nonzero <- which(a$mass != 0) naa=a[nonzero,] d=function(i) (X1[naa$from[i],2]-X2[naa$to[i],2])/(X1[naa$from[i],1]-X2[naa$to[i],1]) atan(Vectorize(d)(1:npoints)) } > mean(angle(a,X1,X2)) [1] -0.3266797 > library(plotrix) > ag=(angle(a,X1,X2)/pi)*180 > ag[ag<0]=ag[ag<0]+360 > dag=hist(ag,breaks=seq(0,361,by=1)-.5) > polar.plot(dag$counts,seq(0,360,by=1),main=”Test Polar Plot”,lwd=3,line.col=4)

(actually, the following plot has been obtain by generating a thousand of sample of size 25)

In order to have a decent test, we need to see what happens under the null assumption (when drawing samples from the same distribution), see

Here is the optimal matching

Here is the distribution of the total cost, when drawing a thousand samples,

VC=rep(NA,1000)
VA=rep(NA,1000*npoints)
for(s in 1:1000){
X1a <- rmnorm(npoints, mean = mu1, Sigma1)
X1b <- rmnorm(npoints, mean = mu1, Sigma2)
ground_p <- 2
p <- 1
w1 <- rep(1/npoints, npoints)
w2 <- rep(1/npoints, npoints)
C <- cost_matrix_Lp(t(X1a), t(X1b), ground_p)
ab <- transport(w1, w2, costm = C^p, method = "shortsimplex")
VC[s]=cout(ab,X1a,X1b)
VA[s*npoints-(0:(npoints-1))]=angle(ab,X1a,X1b)
}
plot(density(VC)

So our cost of 9 obtained initially was not that high. Observe that when drawing from the same distribution, there is now no pattern in the optimal transport

ag=(VA/pi)*180
ag[ag<0]=ag[ag<0]+360
dag=hist(ag,breaks=seq(0,361,by=1)-.5)
polar.plot(dag$counts,seq(0,360,by=1),main="Test Polar Plot",lwd=3,line.col=4) Nice isn’t it? I guess I will spend some time next year working on those transport algorithm, since we have great R packages, and hundreds of applications in economics… # R in Insurance, in Paris The 5th conference on R in Insurance will be organized on Thursday 8 June 2017 at ENSAE , Paris. I will attend the conference and the program is really nice (I was in the scientific committee – with Christophe Dutang, Markus Gesmann, Giorgio Alfredo Spedicato and Andreas Tsanakas – and I have to admit that was received many interesting submissions). Furthermore, the gala dinner will take place at the restaurant of Musée d’Orsay. I really can’t miss it… # Proportion of people alive in 1945 that are still alive In demography, we like to use life tables to estimate the probability that someone born in 1945 (say) is still alive nowadays. But another interesting quantity might be the probability that someone alive in 1945 is still alive nowadays. The main difference is that we do not know when that person, alive in 1945, was born. Someone who was old in 1945 is very unlikely still alive in 2017. To compute those probabilities, we can use datasets from http://www.mortality.org/hmd/. More precisely, we need both death and birth data. I assume that datasets (text files) were downloaded (it is necessary to register – for free – to get the data). D=read.table("FRDeaths_1x1.txt",skip=1,header=TRUE) B=read.table("FRBirths.txt",skip=1,header=TRUE) In the death dataset, there is a “110+” for people older than 110 years. For convenience, let us cap our observations at 110 years old, D$Age=as.numeric(as.character(D$Age)) D$Age[is.na(D$Age)]=110 Consider now a first function that will return, for people born in 1930 (say) two informations • the number of people (here, let us consider women only) born in 1930 (from the birth database) • the number of death of people of age 0 in 1930, people of age 1 in 1931, people of age 2 in 1932, etc… The code is simple nb=function(y=1930){ debut=1816 MatDFemale=matrix(D$Female,nrow=111)
colnames(MatDFemale)=debut+0:198
cly=y-debut+1:111
deces=diag(MatDFemale[,cly[cly%in%1:199]])
return(c(B$Female[B$Year==y],deces))}

We have a single number for the number of births, and then a vector for the number of deaths. Consider now another function. Consider the people born in 1930. We want to get two numbers : the number of people still alive in 1945 (say), and the number of people still alive nowadays. The ratio will be the proportion of people born in 1930 that were alive in 1945, that are still alive in 2015.

pop=function(ne=1930,an=1945){
comptage=nb(ne)
s=0
if(an>ne) s=sum(comptage[seq(2,1+an-ne)])
p1=max(comptage[1]-s,0)
p2=max(p1-sum(comptage[seq(2+an-ne,length(comptage))]),0)
c(p1,p2)
}

Then, for a given year (say 1945), to get the proportion of people alive in 1945 that are still alive today, we need to count how many people born in 1944 were still alive in 1945, and in 2015, but also born in 1943, 1942, etc, And we simply consider the ratio of the total number of people alive in 2015 over the total number of people alive in 1945

ptn=function(y=1945){
V=Vectorize(function(x) pop(x,y))(1816:y)
sum(V[2,!is.na(V[2,])])/sum(V[1,!is.na(V[1,])])
}

Hence, 22% of those alive in 1945 are still alive in 2015,

> ptn(1945)
[1] 0.2209435

Actually, instead of looking only at 1945, it is possible to get a plot

P=Vectorize(ptn)(1900:2010)
plot(1900:2010,P,type="l",ylim=0:1)

For instance,

> ptn(1975)
[1] 0.6377413

i.e. 63.7% of those alive in 1975 are stil alive 40 years after. That is a rather interesting function, I was surprised that I couldn’t find it is standard demographical R package…

There are now more than 10,000 R packages available from CRAN, much more if you include those available only on github. So, to be honest, it become difficult to know all of them. But sometimes, you discover a nice function in one of them, and that is really awesome. Consider for instance some (standard) censored lifetime data,

n=10000
idx=sample(1:4,size=n,replace=TRUE)
pd=LETTERS[idx]
lambda=1+(idx-1)/3
t=rexp(n,lambda)
x=rexp(n)
c=t>x
y=pmin(t,x)
df=data.frame(time=y,status=c,product=pd)

(yes, I will generate them here). Consider Kaplan-Meier estimator of the survival function,

library(survival)
km.base = survfit( Surv(time,status) ~ 1  , data = df )
plot(km.base)

This week end, Anat (currently finishing the Data Science for Actuaries program) made me discover a nice R function, to add information to that graph (well, not that graph, since it will be a ggplot version, but the same survival distribution plot)

library(ggplot2)
library(survminer)
ggsurvplot(km.base, main = "", color = "blue" , censor = FALSE, xlim = c(0,3), risk.table = TRUE ,
risk.table.col = "blue" , risk.table.height = 0.2, risk.table.title = "" , legend.labs = "All" , legend.title = "" , break.time.by = 1, xlab = "" , ylab = "")

This is more interesting when we have different lifetimes

km.prod = survfit( Surv(time,status) ~ product  , data = df )
ggsurvplot(km.prod, main = "", censor = FALSE, xlim = c(0,3), risk.table = TRUE , risk.table.col = "strata" , risk.table.height = 0.3, risk.table.title = "" , legend.labs = LETTERS[1:4] , legend.title = "" , break.time.by = 1, xlab = "" , ylab = "")

or, with a different time granularity

ggsurvplot(km.prod, main = "", censor = FALSE, xlim = c(0,3), risk.table = TRUE , risk.table.col = "strata" , risk.table.height = 0.3, risk.table.title = "" , legend.labs = LETTERS[1:4] , legend.title = "" , break.time.by = .5, xlab = "" , ylab = "")

Nice, isn’t it?

# Advanced Econometrics: Quantiles and Expectiles

Next Thursday, I will give the fourth lecture of the PhD course on advanced tools for econometrics, on quantile and expectile regressions. Slides are available online.

# The U.S. Has Been At War 222 Out of 239 Years

This morning, I discovered an interesting statistic, America Has Been At War 93% of the Time – 222 Out of 239 Years – Since 1776,  i.e. the U.S. has only been at peace for less than 20 years total since its birth. I wanted to check, get a better understanding and look at other countries in the world.

As always, we can try to extract information from wikipedia, since there are pages dedicated to that information

url="https://en.wikipedia.org/wiki/List_of_wars_involving_the_United_States"
url="https://en.wikipedia.org/wiki/List_of_wars_involving_France"
url="https://fr.wikipedia.org/wiki/Liste_des_guerres_de_la_France#Premi.C3.A8re_R.C3.A9publique"

If we look at the US page, there are tables, so it should be easy to extract it. For instance,

Even if the war did last 1 day, we will say that the US were at war in 1811. The information we want to confirm can be “there were 21 full years – from Jan 1st till Dec 31st – where the US were not at war, once, during those years“. From the row above, we can claim that the US were at war in 1811. Most of the time, we have

I.e. there is a beginning (here 1775) and an end (1783). So here, the US are said to be at war in 1775, 1776, 1777, 1778, 1779, 1780, 1781, 1782, 1783. To extract the information, we look for regular expressions in the first column, with number, on 4 digits.

Well, sometimes it can be a bit tricky, since we have 3 dates, 1941, 1945 and (in the legend) 1944. But if we consider the minimal and the maximal dates, we have our range of dates.

Now that we we how to extract information, let’s do it. The code will be

library(stringr)
ext_date=function(x){
dates12="[0-9]{4}"
#grep(pattern = dates2, x = col1[1])
L=str_extract_all(as.character(x),dates12)
return_L=list()
if(length(L)>0){
for(j in 1:length(L))
if(length(L[[j]])==1) return_L[[j]]=as.numeric(L[[j]])
if(length(L[[j]])>=2) return_L[[j]]=seq(min(as.numeric(L[[j]])),max((as.numeric(L[[j]]))))
}
return(return_L)}

For the US, we get the following years

library(XML)
list_dates=list()
for(i in 1:length(tables)){
if(!is.null(dim(tables[[i]]))){
if(ncol(tables[[i]])>1){
col1=tables[[i]][,1]
list_dates[[i]]=lapply(col1,ext_date)
}
}}
d=unique(unlist(list_dates))

(red means at war, while green means no-war) and indeed,

> length(d)
[1] 222

there were 222 years with war.  Now, what about another country. Like France. Here I use the French wiki page, since information is not in tables in the English one.

list_dates=list()
for(i in 1:length(tables)){
if(!is.null(dim(tables[[i]]))){
if(ncol(tables[[i]])>1){
col1=tables[[i]][,1]
col2=tables[[i]][,2]
col12=paste(col1,col2)
list_dates[[i]]=lapply(col12,ext_date)
}
}}
d=unique(unlist(list_dates))

On the same period of time (starting in 1775), France was also on war most of the time.

Less than the US, but still: 185 years with war,

> length(d[d>=1775])
[1] 185

And on a longer period of time? Why not start, say, around the Hundred Years’s War,

meaning that since 1337, there were (only) 174 years without a single war where France was involved.

Let’s try another one. Like Canada,

list_dates=list()
for(i in 1:length(tables)){
if(!is.null(dim(tables[[i]]))){
if(ncol(tables[[i]])>1){
col1=tables[[i]][,1]
list_dates[[i]]=lapply(col1,ext_date)
}
}}
d=unique(unlist(list_dates))

Guess what… there’s a lot of green on that graph. Surprised?

It is now very easy to read (automatically) some text that can be found in a pdf file. For instance, consider the program of the conference we had yesterday – and today – in Rennes

> library(pdftools)
> scan_pdf <- pdf_text("http://crem.univ-rennes1.fr/Documents/Docs_sem_divers/2017_03_10-11_JJD/JDD_prog.pdf")
> cat(scan_pdf)
Journées Jeunes Docteurs
Programme du jeudi 9 mars 2017
Faculty of Economics - Rennes - Amphi Henri Krier
9h- 9h30 - Accueil
9h30-10h15 :      Présentation du CREM, de la faculté et des activités de recherche liées du ou laboratoire
10h15-10h50 :     Emmanuel LORENZON (Université de Bordeaux, GREThA)
Collusion with a rent seeking agency in sponsored search auctions
10h50-11h25 :     Julien BERTHOUMIEU (Université de Bordeaux, GREThA)
The Impact of “At-the-Border” and “Behind-the-Border” Policies on Cost-Reducing Research
and Development
Co-écrit avec Antoine Bouët

(etc). As you can see, it is working well, even in French, where we have those weird letters (with accents). Here, it is working well because the pdf is vectorized, i.e. it was generated properly, by open office.

But sometimes, we can have only a scanned version of a letter

or just a picture with some typed text. I will not mention hand-writing because it is much more complex.

The other day, my friend Fleur did show me a picture, and some very simple lines of code,

> library('tesseract')
> pic1="https://f.hypotheses.org/wp-content/blogs.dir/253/files/2017/03/pic1.png"
> text_fr <- ocr(pic1, engine = tesseract("fra"))
> cat(text_fr)
Près de 14.400 décès

Si [épidémie de grippe est un phénomène récurrent. celle de
2016—2017 présente plusieurs spéciﬁcités. outre sa virulence :
une survenue plus précoce que d‘habitude. une activité
modérée en médecine ambulatoire. mais un impact fort en
milieu hospitalier.

It looks like we’ve be able to extract typed text from a picture ! I want to check. I have to admit, first of all that installation on a linux machine is tricky: one has to install first leptonica, and then follow some guidelines to install tesseract (see also Artem‘s advices). It took me some time, but I’ve been able to install the package.

The first important step, it to train the algorithm with some texts in French (because it is in French in my picture)

> library('tesseract')

Then, I did try with the picture that Fleur did send me (the picture was inserted in the core of the message)

> pic2="https://f.hypotheses.org/wp-content/blogs.dir/253/files/2017/03/pic2.png"
> text_fr <- ocr(pic2, engine = tesseract("fra"))
> cat(text_fr)
Près de 14400 decès

s. mm….agw«… ………«…m ……
a……u…u Dhs—ur; ;pmum…. ;: u……
… W»: »… w…q…na… … ……
…… ………u…_ …… mm……
…… nwm/u

… … mm…—mg…»— sa…… su a.…….…
: :mmræwesdæ ; ; m…decnﬂwtülﬂws WWW…
un»… M on m…… . … … .. m...… wma:
.,… … V, …… … …;………yg…gn…
…… pe- le…samemeuuœwpwv m…
mum

Clearly, something went wrong here. When I got that output, I thought that I did not train properly the function. But it was not the answer. As described in that post (in French) it is necessary to have a clean picture, to read it properly

And actually, if we zoom in our picture – the first one, used by Fleur, to show me that package – we have

while for the second one – with a lower resolution – we have

It is necessary to have a scan of a typed text with high resolution… And you have to admit that it is awesome….

The good thing is that I have to work with a judge, in France, to assess quality of experts. And since most of the reports are typed, and then scanned, I am glad to have such a function. I just have to make sure that the resolution is high enough…

On Thursday, March 23rd, I will give the third lecture of the PhD course on advanced tools for econometrics, on model selection and variable selection, where we will focus on ridge and lasso regressions . Slides are available online.

The first part was on on Nonlinearities in Econometric models, and the second one on Simulations.

# Ruin probability and infinite time

A couple of weeks ago, I had a discussion with a practitioner, working in some financial company, about ruin, and infinite time. And it reminded me a weird result. Well, not a weird result, but a result I found disturbing, at first, when I was a student (that I rediscovered with the eyes of someone dealing with computational issues, seeing here a difficult theoretical question). Consider a simple ruin problem. A player has wealth . Then he flips a coin: tails he has a gain of 1, heads he experiences a loss of 1. At time , his wealth is where  is associated to the th coin:  is equal to 1 with probability (tails), and -1 with probability  (heads). It is also possible to write

where  can be interpreted as the net gain of the player. In order to get a good understanding of results that can be obtained. Assume  to be given. Let denote the number of heads and  the number of tails. Then , while . Let  denote the number of paths to go from point A (wealth  at time ) to point B (wealth  at time ). Note that this is a Markovian problem, that can be modeled using Markov chains

But here, we will focus on combinatorial results. Hence,

In order to derive probabilities to reach , let  denote the number of paths going from  to . And let denote the number of paths going from  to  that do reach  at some point between  and . Using a simple reflexion property, then if  and  are positive,

Based on those reflexions, two results can be derived (focusing on probability, instead of counting paths). First, we can obtain that

(given that n and x have the same parity). The second result we can obtain is that

Based on those two expressions, if  denotes the first time  become null, given ,

then

This can be computed easily,

> x=10
> p=.55
> ProbN=function(n){
+ pb=0
+ if(abs(n-x) %% 2 == 0)
+ pb=x/n*choose(n,(n+x)/2)*(1-p)^((n+x)/2)*(p)^((n-x)/2)
+ return(pb)}
> plot(Vectorize(ProbN)(1:1000),type="s")

That looks nice… But if we look closer, we can wonder what

would be ? Since we have the distribution of a probabilty measure, we might expect one. But here

> sum(Vectorize(ProbN)(1:1000))
[1] 0.134385

And this is not due to calculation mistakes that we do not get 1 here. Actually, we should write

which might be interpreted as the probability of ruin, starting from , that we denote  from now on. The term on the left can be approximated using monte-carlo simulations

> p=.55
> x=10
> m=1000
> simul=10000
> S=sample(c(-1,1),size=m*simul,replace=TRUE,prob=c(1-p,p))
> MS=matrix(S,simul,m)
> for(k in 2:m) MS[,k]=MS[,k]+MS[,k-1]
> T0=function(vm) which(vm<=(-x))[1]
> MTmin=apply(MS,1,T0)
> mean(is.na(MTmin)==FALSE)
[1] 0.1328

To check the validity of the relationship above, a simple (theoretical) recursive formula can be derived for the term on the right (ruin probability), namely

with a boundary conditions , and . Then is comes that

Note that it might be tricky to check using monte carlo simulation… since we cannot have an infinite number of runs. And we’re dealing precisely with things that do occur when time is infinite. Actually, we can still check convergence, considering an upper limit  for the number of runs, and then letting  go to infinity. Note that an explicit formula can then be derived (using additional border condition )

Using the following code, it is possible to calculate ruin probability, in order to estimate .

> MSmin=apply(MS,1,min)
> mean(MSmin<=(-x))
[1] 0.1328
> (((1-p)/p)^x-((1-p)/p)^m)/(1-((1-p)/p)^m)
[1] 0.1344306

The following graph shows the evolution of ruin probability as a function of initial wealth (with monte carlo simulation, with a fixed horizon – including a confidence interval – versus the analytical expression)

Hence, with stopping times, one should remember that

and that those two terms can be approximated simply using simulations or standard approximations.