# ACT6100, analyse non-supervisée On avance tranquillement dans le cours ACT6100 d’analyse des données en actuariat. Les supports de cours sont en ligne sur https://github.com/freakonometrics/ACT6100. Cette session (mais aussi celle d’hiver) étant en distanciel, le cours est asynchrone, et je poste régulièrement des capsules vidéos. Les capsules présentant les principales méthodes d’analyse non-supervisée sont en ligne

1. ACP (1) (PCAvideo pdf (48:42)
2. ACP (2) video pdf (31:10)
3. ACP (3) video pdf (47:51)
4. ACP (4) video pdf (39:59)
5. ACP (5) video pdf (28:53)
6. CA (1) video pdf (38:33)
7. CA (2) video pdf (46:51)
8. MCA (1) video pdf (28:03)
9. Clusters (k-means) video pdf (48:20)
10. Clusters (CAH) video pdf (37:38)
11. NA & Imputations (k-nn) video pdf (17:47)
12. NA & Imputations (ACP) video pdf (15:28)

Si les liens des vidéos ne marchent pas, je renvoie vers l’ensemble des capsules du cours, ici.

# Clusters of Texts

Another popular application of classification techniques is on texmining (see e.g. an old post on French president speaches). Consider the following example,  inspired by Nobert Ryciak’s post, with 12 wikipedia pages, on various topics,

```> library(tm)
> library(stringi)
> library(proxy)
> titles = c("Boosting_(machine_learning)",
+            "Random_forest",
+            "K-nearest_neighbors_algorithm",
+            "Logistic_regression",
+            "Boston_Bruins",
+            "Los_Angeles_Lakers",
+            "Game_of_Thrones",
+            "House_of_Cards_(U.S._TV_series)",
+            "True_Detective_(TV_series)",
+            "Picasso",
+            "Henri_Matisse",
+            "Jackson_Pollock")
> articles = character(length(titles))
> wiki = "http://en.wikipedia.org/wiki/"
> for (i in 1:length(titles)) {
+   articles[i] = stri_flatten(readLines(stri_paste(wiki, titles[i])), col = " ")
+ }```

Here, we store all the contents of the pages in a corpus (from the text mining package).

`> docs = Corpus(VectorSource(articles))`

This is what we have in that corpus

```> a = stri_flatten(readLines(stri_paste(wiki, titles)), col = " ")
> a = Corpus(VectorSource(a))
> a[]

Thoughts on Hypothesis Boosting</i></a>, Unpublished manuscript (Machine Learning class project, December 1988)</span></li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/Michael_Kearns" title="Michael Kearns">Michael Kearns</a>; <a href="/wiki/Leslie_Valiant" title="Leslie Valiant">Leslie Valiant</a> (1989). <a rel="nofollow" class="external text" href="http://dl.acm.org/citation.cfm?id=73049">"Crytographic limitations on learning Boolean formulae and finite automata"</a>. <i>Symposium on T```

This is because we read an html page.

```> a = tm_map(a, function(x) stri_replace_all_fixed(x, "\t", " "))
> a = tm_map(a, PlainTextDocument)
> a = tm_map(a, stripWhitespace)
> a = tm_map(a, removeWords, stopwords("english"))
> a = tm_map(a, removePunctuation)
> a = tm_map(a, tolower)
> a

can  set  weak learners create  single strong learner  a weak learner  defined    classifier    slightly correlated   true classification  can label examples better  random guessing in contrast  strong learner   classifier   arbitrarily wellcorrelated   true classification robert ```

Now we have the text of the wikipedia document. What we did was

• replace all “” elements with a space. We do it because there are not a part of text document but in general a html code.
• replace all “/t” with a space.
• convert previous result (returned type was “string”) to “PlainTextDocument”, so that we can apply the other functions from tm package, which require this type of argument.
• remove extra whitespaces from the documents.
• remove punctuation marks.
• remove from the documents words which we find redundant for text mining (e.g. pronouns, conjunctions). We set this words as stopwords(“english”) which is a built-in list for English language (this argument is passed to the function removeWords.
• transform characters to lower case.

Now we can do it on the entire corpus

```> docs2 = tm_map(docs, function(x) stri_replace_all_regex(x, "<.+?>", " "))
> docs3 = tm_map(docs2, function(x) stri_replace_all_fixed(x, "\t", " "))
> docs4 = tm_map(docs3, PlainTextDocument)
> docs5 = tm_map(docs4, stripWhitespace)
> docs6 = tm_map(docs5, removeWords, stopwords("english"))
> docs7 = tm_map(docs6, removePunctuation)
> docs8 = tm_map(docs7, tolower)```

Now, we simply count words in each page,

```> dtm <- DocumentTermMatrix(docs8)
> dtm2 <- as.matrix(dtm)
> dim(dtm2)
 12 13683
> frequency <- colSums(dtm2)
> frequency <- sort(frequency, decreasing=TRUE)
> mots=frequency[frequency>20]
> s=dtm2[1,which(colnames(dtm2) %in% names(mots))]
> for(i in 2:nrow(dtm2)) s=cbind(s,dtm2[i,which(colnames(dtm2) %in% names(mots))])
> colnames(s)=titles```

Once we have that dataset, we can use a PCA to visualise the ‘variables’ i.e. the pages

```> library(FactoMineR)
> PCA(s)``` We can also use non-supervised classification to group pages. But first, let us normalize the dataset

`> s0=s/apply(s,1,sd)`

Then, we can run a cluster dendrogram, using the Ward distance

```> h <- hclust(dist(t(s0)), method = "ward")
> plot(h, labels = titles, sub = "")``` Groups are consistent with intuition: painters are in the same cluster, as well as TV series, sports teams, and statistical techniques.

# Clusters of (French) Regions

For the data scienec course of tomorrow, I just wanted to post some functions to illustrate cluster analysis. Consider the dataset of the French 2012 elections

```> elections2012=read.table(
> voix=which(substr(names(
+ elections2012),1,11)=="X..Voix.Exp")
> elections2012=elections2012[1:96,]
> X=as.matrix(elections2012[,voix])
> rownames(X)=elections2012[,1]```

The hierarchical cluster analysis is obtained using

```> cah=hclust(dist(X))
> plot(cah,cex=.6)```

To get five groups, we have to prune the tree

```> rect.hclust(cah,k=5)
> groups.5 <- cutree(cah,5)``` We have to zoom-in to visualize the French regions, It is also possible to use

```> library(dendroextras)
> plot(colour_clusters(cah,k=5))``` And again, if we zoom-in, we get The interpretation of the clusters can be obtained using

```> aggregate(X,list(groups.5),mean)
Group.1     JOLY   LE PEN  SARKOZY
1       1 2.185000 18.00042 28.74042
2       2 1.943824 23.22324 25.78029
3       3 2.240667 15.34267 23.45933
4       4 2.620000 21.90600 34.32200
5       5 3.140000  9.05000 33.80000```

It is also possible to visualize those clusters on a map, using

```> library(RColorBrewer)
> CL=brewer.pal(8,"Set3")
> carte_classe <- function(groupes){
+ library(stringr)
+ elections2012\$dep <- elections2012[,2]
+ elections2012\$dep <- tolower(elections2012\$dep)
+ elections2012\$dep <- str_replace_all(elections2012\$dep, pattern = " |-|'|/", replacement = "")
+ library(maps)
+ france<-map(database="france")
+ france\$dep <- france\$names
+ france\$dep <- tolower(france\$dep)
+ france\$dep <- str_replace_all(france\$dep, pattern = " |-|'|/", replacement = "")
+ corresp_noms <- elections2012[, c(1,2, ncol(elections2012))]
+ corresp_noms\$dep[which(corresp_noms\$dep %in% "corsesud")] <- "corsedusud"
+ col2001<-groupes+1
+ names(col2001) <- corresp_noms\$dep[match(names(col2001), corresp_noms[,1])]
+ color <- col2001[match(france\$dep, names(col2001))]
+ map(database="france", fill=TRUE, col=CL[color])
+ }
> carte_classe(cutree(cah,5))``` or, if we simply want 4 clusters

`> carte_classe(cutree(cah,4))` # Visualizing Clusters

Consider the following dataset, with (only) ten points

```x=c(.4,.55,.65,.9,.1,.35,.5,.15,.2,.85)
y=c(.85,.95,.8,.87,.5,.55,.5,.2,.1,.3)
plot(x,y,pch=19,cex=2)``` We want to get – say – two clusters. Or more specifically, two sets of observations, each of them sharing some similarities.

Since the number of observations is rather small, it is actually possible to get an exhaustive list of all partitions, and to minimize some criteria, such as the within variance. Given a vector with clusters, we compute the within variance using

```within_var = function(I){
I0=which(I==0)
I1=which(I==1)
xbar0=mean(x[I0])
xbar1=mean(x[I1])
ybar0=mean(y[I0])
ybar1=mean(y[I1])
w=sum(I0)*sum( (x[I0]-xbar0)^2+(y[I0]-ybar0)^2 )+
sum(I1)*sum( (x[I1]-xbar1)^2+(y[I1]-ybar1)^2 )
return(c(I,w))
}```

Then, to compute all possible partitions, use

```base2=function(z,n=10){
Base.b=rep(0,n)
ndigits=(floor(logb(z, base=2))+1)
for(i in 1:ndigits){
Base.b[ n-i+1]=(z%%2)
z=(z%/%2)}
return(Base.b)}
L=function(x) within_var(base2(x))
S=sapply(1:(2^10),L)```

The cluster indices at the mimimum is here

```I=S[1:n,which.min(S[n+1,])]
```

To visualize those clusters, use

```cluster_viz = function(indices){
library(RColorBrewer)
CL2palette=rev(brewer.pal(n = 9, name = "RdYlBu"))
CL2f=CL2palette[c(1,9)]
plot(x,y,pch=19,xlab="",ylab="",xlim=0:1,ylim=0:1,cex=2,col=CL2f[1+I])
CL2c=CL2palette[c(3,7)]
I0=which(indices==0)
I1=which(indices==1)
xbar0=mean(x[I0])
xbar1=mean(x[I1])
ybar0=mean(y[I0])
ybar1=mean(y[I1])
segments(x[I0],y[I0],xbar0,ybar0,col=CL2c)
segments(x[I1],y[I1],xbar1,ybar1,col=CL2c)
points(xbar0,ybar0,pch=19,cex=1.5,col=CL2c)
points(xbar1,ybar1,pch=19,cex=1.5,col=CL2c)}```

and then, simply

`cluster_viz(I)` But that was possible only because is not to large (since the total number of scenarios – with only 2 clusters – is , or if we changes zeros in ones).

# Analyse des Données et Cartes

Vendredi, on continuera en cours la classification (non supervisée) et en particulier les méthodes hiérarchiques, et les arbres. On va utiliser la base suivante, avec les résultats des élections présidentielles (premier tour) de 2012,

```elections2012 <- read.csv("http://komodo.regardscitoyens.org/public/presidentielles2012/resultats/resultats_departements_final_T1.csv",

La base qu’on garde contient juste les voix exprimées, en pourcentage

```voix <- which(substr(names(elections2012),1,12)=="X..Voix.Exp.")
X <- as.matrix(elections2012[,voix])
colnames(X) <- substr(names(elections2012)[voix],13,nchar(names(elections2012)[voix]))
rownames(X) <- elections2012[,3]```

Parfois, il convient de normaliser les données. Ici, sur les département, ça n’est pas nécessaire. Par contre, sur les candidats, ça le serait (comme pour l’ACP en fait).  Pour visualiser les distances (entre départements), on peut utiliser

`heatmap(X)` On peut faire une CAH, sur notre matrice de distance entre rangs

```cah <- hclust(dist(X))
plot(cah,cex=.6)```

et si on souhaite garder 5 groupes (par exemple) on utilise

```rect.hclust(cah,k=5)
groups.5 <- cutree(cah,5)``` On peut aussi visualiser les classes avec la fonction

```library(dendroextras)
plot(colour_clusters(cah,k=5))```

Pour comprendre qui se trouve dans nos groupes, on peut utiliser

`aggregate(X,list(groups.5),mean)`

qui va nous renvoyer les “votes  moyens” pour chaque candidat, dans chaque groupe. Maintenant, pour en finir avec la visualisation, on peut utiliser le code suivant (un peu long, certes) qui permettra de visualiser sur un carte les différents groupes

```carte_classe <- function(groupes){
library(stringr)
elections2012\$dep <- elections2012\$Libellé.du.département
elections2012\$dep <- tolower(elections2012\$dep)
elections2012\$dep <- str_replace_all(elections2012\$dep, pattern = " |-|'|/", replacement = "")
library(maps)
france<-map(database="france")
france\$dep <- france\$names
france\$dep <- tolower(france\$dep)
france\$dep <- str_replace_all(france\$dep, pattern = " |-|'|/", replacement = "")
corresp_noms <- elections2012[, c("Libellé.du.département", "dep")]
corresp_noms\$dep[which(corresp_noms\$dep %in% "corsesud")] <- "corsedusud"
col2001<-groupes+1
names(col2001) <- corresp_noms\$dep[match(names(col2001), corresp_noms\$Libellé.du.département)]
color <- col2001[match(france\$dep, names(col2001))]
map(database="france", fill=TRUE, col=color)
}

carte_classe(groups.5)``` Avec ces fonctions, on devrait pouvoir tester différentes méthodes pour constituer des groupes.

# Think academic journals look the same ? Well, some do…

We have seen yesterday that finding an optimal strategy to publish is not that simple. And actually, it can be even more difficult in the case the journal rejects the paper (not because it is not correct, but because “it does not fit” with the standards, the quality of the journal, the audience, the editor’s mood, or whatever). The author has basically two choices,

• forget about the article and move to something else (e.g. start a blog where he/she will be the author and the editor)
• pretend that the article is worth publishing and then try to find another journal with similar interests But this last choice is not that easy, since sometimes the author think that this journal was indeed the one that should publish it (e.g. all the articles on the subject have been published in that journal).
So I was wondering if there were clusters of journals, i.e. journals that publish almost the same kind of articles (so that next time one of my paper is rejected by the editor, I just go to for some journal in the same cluster).
So what I did is extremely simple: I looked at articles titles and looked for correlations between words frequency (I could have done that in key words, but I am not a big fan of those key words). I looked at 35 journals (that are somehow related to my areas of interest) and looked at titles of all articles published over the last 20 years. Then I kept the top 1000 of words, and I removed standard short words (“a“, “the“, “is“, etc). Actually, my top words looks like

```"models" "model" "data" "estimation" "analysis" "time"
"processes" "risk" "random" "stochastic" "regression"
"market" "approach" "optimal" "based" "information"
"evidence" "linear" "games" "bayesian" "theory" "effects"
"distribution" "multivariate" "tests" "markets" "markov"
"equilibrium" "dynamic" "process" "distributions"
"application" "stock" "likelihood"```

Then, I ran a principal component analysis on my dataset (containing 960 variables – here words – and 35 observations – here journal names).

```library("FactoMineR")
res.pca = PCA(MATRICE, scale.unit=TRUE, ncp=5,
graph=FALSE)
plot.PCA(res.pca, axes=c(1, 2), choix="ind")```

The projection of the journals on the first two axis looks like that Here, we can clearly observe some clusters : on the up-left Journal of Finance and Journal of Banking and Finance (say financial journals) on the top-right Biometrika, Biometrics, Computational Statistics and Data Analysis and Journal of Econometrics (JASA is not far away, i.e. applied statistics journal). And below, on the right, Stochastic Processes and their Applications, Annals of Applied Probability, Journal of Applied Probability, Annals of Probability, Proceedings of AMS and Topology and Applications (ie more theoretical journal).
Note that the projection is rather robust: if I consider my first 200 words, the graph is the same In order to go further in the interpretation, we can also plot variables, i.e. words from titles, where we cannot distinguish anything. So if I just look at my top 30, here they are, On top left we see market(s), risk or information; on top right analysis, effects, models or tests; while below we see Markov or process(es). And we can observe interesting facts: in finance in statistics, we talk about dynamics while in theoretical (mathematical) journal it is about processes.
But the goal was to find cluster, i.e. classes of journals that publish papers with similar titles.

```DISTANCE = dist(MATRICE)
cah = hclust(DISTANCE)
plot(cah)```

Here we have If some classes a rather natural (Journal of Applied Proba. and Advances in Applied Proba.or Economic Theory, Journal of Economic Theory and Journal of Mathematical Economics) some strong correlation are not simple to understand, (e.g. Insurance: Mathematics and Economics and Management Science or Annals of Statistics and the Journal of Multivariate Analysis).
Again, it might be possible to spend hours on the graphs, but if I want – someday – to submit something to one of those journals, I guess I have to stop here, and move to something else…