njulia1
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@ -3,47 +3,228 @@ title: "Comparaison of methods"
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output: pdf_document
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---
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# Scan statistique - Méthode de Monte Carlo et calcul de p-value
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## 1. Proposition for simulations under $\mathcal{H}_1$
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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.
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```{r}
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PoissonProcess <- function(lambda,T) {
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return(sort(runif(rpois(1,lambda*T),0,T)))
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}
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pp1=PoissonProcess(lambda0,Ti)
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SimulationH1 <- function(lambda0, lambda1,T,tau){
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ppH0=PoissonProcess(lambda0,T)
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ppH1.segt=PoissonProcess(lambda1,tau)
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dbt=runif(1,0,T-tau)
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ppH0bis=PoissonProcess(lambda0,T)
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ppH1.repo=dbt+ppH1.segt
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ppH0_avant=ppH0bis[which(ppH0bis<ppH1.repo[1])]
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ppH0_apres=ppH0bis[which(ppH0bis>ppH1.repo[length(ppH1.repo)])]
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ppH1=c(ppH0_avant,ppH1.repo,ppH0_apres)
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return (ppH1)
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}
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TimeBetweenEvent <- function(pp){
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n=length(pp)
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tbe=pp[2:n]-pp[1:n1-1]
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tbe=c(0,tbe)
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return (tbe)
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}
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DataFrame <- function(pp,tbe){
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list=data.frame(ProcessusPoisson=pp, TimeBetweenEvent=tbe)
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}
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```
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## 2. Simulation of the sequences under $\mathcal{H}_0$ via a Monte Carlo Method
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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 $\mathcal{H}_0$.
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```{r}
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ScanStat <- function(pp, T, tau){
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n=length(pp)
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stop=n-length(which(pp>(T-tau)))
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ScanStat=0
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for (i in (1:stop)) {
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x=which((pp>=pp[i])&(pp<=(pp[i]+tau)))
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scan=length(x)
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if (scan>ScanStat) {ScanStat=scan}
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}
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return (c(i,ScanStat))
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}
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```
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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.
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```{r}
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EmpDistrib <- function(lambda, n_sample,T,tau){
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pp=PoissonProcess(lambda,T)
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scan=c(ScanStat(pp,T, tau)[2])
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index=c(ScanStat(pp,T, tau)[1])
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for (i in 2:(n_sample)){
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pp=PoissonProcess(lambda,T)
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scan=rbind(scan,ScanStat(pp,T, tau)[2])
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index=rbind(index,ScanStat(pp,T, tau)[1])
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}
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min_scan=min(scan)-1
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max_scan=max(scan)
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table1=table(factor(scan, levels = min_scan:max_scan))
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EmpDis=data.frame(cdf=cumsum(table1)/sum(table1), proba=table1/sum(table1), index_scan=min_scan:max_scan)
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EmpDis<-EmpDis[,-2]
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return(EmpDis)
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}
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```
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```{r}
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library("latex2exp")
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Plot_CDF <- function(lambda,n_sample,T,tau){
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Emp=EmpDistrib(lambda,n_sample,T,tau)
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title=TeX(paste(r'(Cumulative distribution function for $\lambda=$)', lambda))
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plot(Emp$index_scan, Emp$cdf,type="s",xlab="Number of occurrences",ylab="Probability", main=title, col="red")
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return(Emp)
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}
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```
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### 2.1 Test of $\mathcal{H}_0: \lambda=\lambda_0$ against $\mathcal{H}_0: \lambda=\lambda_1$, where $\lambda_1 > \lambda_0$
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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.
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```{r}
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#Empiricial distribution under H0
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n_sample=10**4
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lambda0=3
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T=10
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tau=1
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ppH0=PoissonProcess(lambda0,T)
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CDF=Plot_CDF(lambda0,n_sample,T,tau)
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```
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```{r}
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PValue <- function(Emp,ppH1, T, tau){
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scanH1=ScanStat(ppH1,T,tau)[2]
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index=Emp$index_scan
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n=length(index)
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if (scanH1< min(Emp$index_scan)){
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return (c(scanH1,1))
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} else{
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if(min(Emp$index_scan)<scanH1 && scanH1<=max(Emp$index_scan)){
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return(c(scanH1,1-Emp$cdf[scanH1-min(Emp$index_scan)+1]))
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} else{return (c(scanH1,0))}}
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}
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```
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### 2.2. Simulation under $\mathcal{H}_0$ and computation of p-values
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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.
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```{r}
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NbSeqH0=5
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NbSeqH1=5
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DataH0=vector("list")
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DataH1=vector("list")
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lambda0=3
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lambda1=5
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T=10
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tau=1
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#Creation of a sequence that contains the sequence simulated under the null hypothesis
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for (i in 1:NbSeqH0) {
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ppi=PoissonProcess(lambda0,T)
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DataH0[[i]]=ppi
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}
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#Creation of a sequence that contains the sequence simulated under the alternative hypothesis
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for (i in 1:NbSeqH1) {
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pphi=SimulationH1(lambda0, lambda1,T,tau)
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DataH1[[i]]=pphi
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}
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#Computation of the time between events
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TimeBetweenEventList <- function(list,n_list){
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TBE=vector("list",length=n_list)
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for (i in (1:n_list)) {
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ppi=list[[i]]
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ni=length(ppi)
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tbei=ppi[2:ni]-ppi[1:ni-1]
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TBE[[i]]=tbei
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}
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return (TBE)
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}
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tbe0=TimeBetweenEventList(DataH0,NbSeqH0)
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```
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We compute the p-value associated to all 5 sequences, and stock them in a vector.
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```{r}
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#We start by computing the empirical distribution for lambda0
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Emp=EmpDistrib(lambda0,n_sample,T,tau)
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scan=c()
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pvalue=c()
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#Then, we stock the p-value and the
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for (i in 1:NbSeqH0){
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ppi=DataH0[[i]]
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result=PValue(Emp,DataH0[[i]],T,tau)
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scan=c(scan,result[1])
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pvalue=c(pvalue,result[2])
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cat(paste("\nSimulation for the sequence", i, ", for lambda0=",lambda0, " ,lambda1=", lambda1, " , scan=", result[1] ,"p-value=",result[2]))
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}
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ScS_H0=data.frame(num=1:NbSeqH0, index=scan, pvalue_scan=pvalue, class=(pvalue<0.05))
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ScS_H0
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```
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## 3.Local score
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```{r}
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library("localScore")
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library(Rcpp)
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E = 10
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pvalue=c()
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for (i in 1:NbSeqH0){
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X = floor(E*log(dexp(tbe0[[i]], rate = lambda1)/dexp(tbe0[[i]], rate = lambda0)))
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max_X = max(X)
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min_X = min(X)
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P_X = table(factor(X, levels = min_X:max_X))/length(X)
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LS=localScoreC(X)$localScore[1]
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result = daudin(localScore = LS, score_probabilities = P_X, sequence_length = length(X), sequence_min = min_X, sequence_max =max_X)
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pvalue=c(pvalue,result)
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}
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LS_H0=data.frame(num=1:NbSeqH0, pvalue_scan=pvalue, class=(pvalue<0.05))
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LS_H0
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```
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## A reformater
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# distribtion des scores via MC
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# Nb seq. pp -> Nb seq. tbe -> dist. tbe (vérif) + Nb seq. Scores -> distr scores
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PoissonProcess <- function(lambda,T) {
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return(sort(runif(rpois(1,lambda*T),0,T)))
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}
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lambda0=2
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lambda1=3
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Ti=100000
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pp1=PoissonProcess(lambda1,Ti)
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print(pp1)
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plot(c(0,pp1),0:length(pp1),type="s",xlab="time t",ylab="number of events by time t")
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pp2=PoissonProcess(lambda1,Ti)
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print(pp2)
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plot(c(0,pp2),0:length(pp2),type="s",xlab="time t",ylab="number of events by time t")
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#time between events
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n1=length(pp1)
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tbe1=pp1[2:n1]-pp1[1:n1-1]
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tbe1
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ks.test(tbe1,'exp',lambda)
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x=log(lambda1/lambda0)+(lambda0-lambda1)*tbe1 # ne pas mettre le floor ni le E (certes égale à 1)
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hist(x)
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summary(x)
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n2=length(pp2)
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tbe2=pp2[2:n2]-pp2[1:n2-1]
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tbe2
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# Calcul du maximum des scores
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E=1
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# THEO à faire !!! max.s=log(lambda1/lambda0)
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maxXk = floor(E*(log(lambda1/lambda0)))
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maxXk
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while (maxXk < 3) {
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E = E+1
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maxXk = floor(E*(log(lambda1/lambda0)))
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}
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E
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ks.test(tbe1,pexp,lambda0, alternative="two.sided")
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ks.test(tbe2,pexp,lambda1, alternative="two.sided")
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```
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Local score
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```{r}
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lambda0 = 1
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lambda1 = 2
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library("localScore")
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E = 10
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X = floor(E*log(dexp(tbe1, rate = lambda1)/dexp(tbe1, rate = lambda0)))
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max_X = max(X)
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min_X = min(X)
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P_X = table(factor(X, levels = min_X:max_X))/length(X)
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LS=localScoreC(X)$localScore[1]
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LS
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result = daudin(localScore = LS, score_probabilities = P_X, sequence_length = length(x), sequence_min = min_X, sequence_max = max_X)
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result
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```
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x=floor(E*(log(lambda1/lambda0)+(lambda0-lambda1)*tbe1))
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dist.emp.scores=table(x)/sum(table(x))
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dist.emp.scores
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hist(x)
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range(x)
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x.verif=seq(range(x)[1],range(x)[2],1)
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dist.theo.scores=lambda0*exp(-lambda0*(A*x.verif-B))
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dist.theo.scores
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dist.emp.scores
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