---
title: "Comparaison of methods"
output: pdf_document
---

# Scan statistique - Méthode de Monte Carlo et calcul de p-value

## 1. Proposition for simulations under $\mathcal{H}_1$

In this part, we propose a method that simulates a Poisson process under the hypothesis $\mathcal{H}_1$. The idea is to simulate a sample under $\mathcal{H}_0$, and add randomly a subsequence under the alternative hypothesis in this sequence. 
```{r}
PoissonProcess <- function(lambda,T) {
  return(sort(runif(rpois(1,lambda*T),0,T)))
}

SimulationH1 <- function(lambda0, lambda1,T,tau){
    ppH0=PoissonProcess(lambda0,T)
    ppH1.segt=PoissonProcess(lambda1,tau)
    dbt=runif(1,0,T-tau)
    ppH0bis=PoissonProcess(lambda0,T)
    ppH1.repo=dbt+ppH1.segt
    ppH0_avant=ppH0bis[which(ppH0bis<ppH1.repo[1])]
    ppH0_apres=ppH0bis[which(ppH0bis>ppH1.repo[length(ppH1.repo)])]
    ppH1=c(ppH0_avant,ppH1.repo,ppH0_apres)
    return (ppH1)
}

TimeBetweenEvent <- function(pp){
    n=length(pp)
    tbe=pp[2:n]-pp[1:n1-1]
    tbe=c(0,tbe)
    return (tbe)
}

DataFrame <- function(pp,tbe){
    list=data.frame(ProcessusPoisson=pp, TimeBetweenEvent=tbe)
}
```

## 2. Simulation of the sequences under $\mathcal{H}_0$ via a Monte Carlo Method
In this part, we will try to simulate, using a Monte Carlo method, a set of $10^5$ independant samples, under the assumption that $\lambda=\lambda_0$, hence, that we are under the null hypothesis $\mathcal{H}_0$.  
```{r}
ScanStat <- function(pp, T, tau){
    n=length(pp)
    stop=n-length(which(pp>(T-tau)))
    ScanStat=0
    for (i in (1:stop)) {
        x=which((pp>=pp[i])&(pp<=(pp[i]+tau)))
        scan=length(x)
        if (scan>ScanStat) {ScanStat=scan}
  }   
    return (c(i,ScanStat))
}
```

We test the scan statistic method for different values of $\lambda_0$. The method of scan statistic we implemented will allow us to have access to the scan test statistic and where it happens in the sequence. 
```{r}
EmpDistrib <- function(lambda, n_sample,T,tau){
    pp=PoissonProcess(lambda,T)
    scan=c(ScanStat(pp,T, tau)[2])
    index=c(ScanStat(pp,T, tau)[1])
    for (i in 2:(n_sample)){
        pp=PoissonProcess(lambda,T)
        scan=rbind(scan,ScanStat(pp,T, tau)[2])
        index=rbind(index,ScanStat(pp,T, tau)[1])
    }
    min_scan=min(scan)-1
    max_scan=max(scan)
    table1=table(factor(scan, levels = min_scan:max_scan))
    EmpDis=data.frame(cdf=cumsum(table1)/sum(table1), proba=table1/sum(table1), index_scan=min_scan:max_scan)
    EmpDis<-EmpDis[,-2]
    return(EmpDis)
    }
```
```{r}
library("latex2exp")
Plot_CDF <- function(lambda,n_sample,T,tau){
    Emp=EmpDistrib(lambda,n_sample,T,tau)
    title=TeX(paste(r'(Cumulative distribution function for $\lambda=$)', lambda))
    plot(Emp$index_scan, Emp$cdf,type="s",xlab="Number of occurrences",ylab="Probability", main=title, col="red")
    return(Emp)
}
```
### 2.1 Test of $\mathcal{H}_0: \lambda=\lambda_0$ against $\mathcal{H}_0: \lambda=\lambda_1$, where $\lambda_1 > \lambda_0$ 
In this part, we will test different values for $\lambda_0$ and $\lambda_1$, and compute the probability of occurrence of a certain scan statistic.

```{r}
#Empiricial distribution under H0
n_sample=10**4
lambda0=3
T=10
tau=1
ppH0=PoissonProcess(lambda0,T)
CDF=Plot_CDF(lambda0,n_sample,T,tau)
```

```{r}
PValue <- function(Emp,ppH1, T, tau){
    scanH1=ScanStat(ppH1,T,tau)[2]
    index=Emp$index_scan
    n=length(index)
    if (scanH1< min(Emp$index_scan)){
        return (c(scanH1,1))
        } else{
            if(min(Emp$index_scan)<scanH1 && scanH1<=max(Emp$index_scan)){
                return(c(scanH1,1-Emp$cdf[scanH1-min(Emp$index_scan)+1]))
            } else{return (c(scanH1,0))}}
}
```

### 2.2. Simulation under $\mathcal{H}_0$ and computation of p-values
On simule des séquences sous $\mathcal{H}_0$, que l'on stocke. On calcule la valeur de la scan stat et de la p-value, que l'on stocke aussi. On a une séquence de p-valeur des scans et une séquence de score local.
```{r}
NbSeqH0=5
NbSeqH1=5
DataH0=vector("list")
DataH1=vector("list")
lambda0=3
lambda1=5
T=10
tau=1

#Creation of a sequence that contains the sequence simulated under the null hypothesis
for (i in 1:NbSeqH0) {
    ppi=PoissonProcess(lambda0,T)
    DataH0[[i]]=ppi
}

#Creation of a sequence that contains the sequence simulated under the alternative hypothesis
for (i in 1:NbSeqH1) {
    pphi=SimulationH1(lambda0, lambda1,T,tau)
    DataH1[[i]]=pphi
}

#Computation of the time between events
TimeBetweenEventList <- function(list,n_list){
    TBE=vector("list",length=n_list)
    for (i in (1:n_list)) {
        ppi=list[[i]]
        ni=length(ppi)
        tbei=ppi[2:ni]-ppi[1:ni-1]
        TBE[[i]]=tbei
    }
    return (TBE)
}
tbe0=TimeBetweenEventList(DataH0,NbSeqH0)
```
We compute the p-value associated to all 5 sequences, and stock them in a vector. 

```{r}
#We start by computing the empirical distribution for lambda0
Emp=EmpDistrib(lambda0,n_sample,T,tau)
scan=c()
pvalue=c()

#Then, we stock the p-value and the 
for (i in 1:NbSeqH0){
    ppi=DataH0[[i]]
    result=PValue(Emp,DataH0[[i]],T,tau)
    scan=c(scan,result[1])
    pvalue=c(pvalue,result[2])
    cat(paste("\nSimulation for the sequence", i, ", for lambda0=",lambda0, " ,lambda1=", lambda1, " , scan=", result[1] ,"p-value=",result[2]))
}
ScS_H0=data.frame(num=1:NbSeqH0, index=scan, pvalue_scan=pvalue, class=(pvalue<0.05))
ScS_H0
```
## 3.Local score
```{r}
library("localScore")
library(Rcpp)
E = 10
pvalue=c()

for (i in 1:NbSeqH0){
    X = floor(E*log(dexp(tbe0[[i]], rate = lambda1)/dexp(tbe0[[i]], rate = lambda0)))
    
    max_X = max(X)
    min_X = min(X)
    P_X = table(factor(X, levels = min_X:max_X))/length(X)
    
    LS=localScoreC(X)$localScore[1]
    
    result = daudin(localScore = LS, score_probabilities = P_X, sequence_length = length(X), sequence_min = min_X, sequence_max =max_X)
    
    pvalue=c(pvalue,result)
}
LS_H0=data.frame(num=1:NbSeqH0, pvalue_scan=pvalue, class=(pvalue<0.05))
LS_H0
```

## A reformater
```{r}
# distribtion des scores via MC
# Nb seq. pp -> Nb seq. tbe -> dist. tbe (vérif) + Nb seq. Scores -> distr scores

A = 1/(lambda0-lambda1)
B = A*log(lambda1/lambda0)

ppH1 = PoissonProcess(lambda1,T)
n1 = length(ppH1)
tbe1 = ppH1[2:n1]-ppH1[1:n1-1]
print(tbe1)
print(ks.test(tbe1,'exp'))
x = log(lambda1/lambda0)+(lambda0-lambda1)*tbe1 # ne pas mettre le floor ni le E (certes égale à 1)
hist(x)
print(summary(x))

# Calcul du maximum des scores
E = 1
# THEO à faire !!! max.s = log(lambda1/lambda0)
maxXk = floor(E*(log(lambda1/lambda0)))
maxXk
while (maxXk < 3) {
    E = E+1
    maxXk = floor(E*(log(lambda1/lambda0)))
}
print(E)

x = floor(E*(log(lambda1/lambda0)+(lambda0-lambda1)*tbe1))
dist.emp.scores = table(x)/sum(table(x))
dist.emp.scores
hist(x)
print(range(x))
x.verif = seq(range(x)[1],range(x)[2],1)
dist.theo.scores = lambda0*exp(-lambda0*(A*x.verif-B))
print(dist.theo.scores)
print(dist.emp.scores)
```