setwd("C:/Users/Daniel Westhoff/Documents/Lehre/Angewandte Stochastik II/Übungsblätter/Blatt3")

#1)

empiricalDistr = function(x, path){
  x=sort(x)
  n=length(x)
  y=c(1:n)
  for(i in 1:length(x)){
    y[i] = length(x[x<=x[i]])/n
  }
  png(filename=paste(path, ".png", sep=""), width=800, height=800, res=180)
  plot(x,y,lwd=2, type="l", main="ECDF" ,xlab="x", ylab="F(x)")
  dev.off()
}

x=rnorm(1000,0,1)
empiricalDistr(x, "test")

#2)
#a)
epanechnikov=function(x){
  if(x>1 | x<(-1)){
    return(0)
  }
  return(0.75*(1-x^2))
}
estDensity=function(t,x,h){
  sum=0
  for(i in 1:length(x)){
    sum=sum+epanechnikov((t-x[i])/h)
  }
  return(1/(length(x)*h)*sum)
}

#b)
x=rnorm(200,0,1)
col=1:8
count=1
values=c(0.1, 0.2,0.5,1,2,4,5)
int=seq(-4,4,0.1)
for(h in values){
  res=c()
  for(i in 1:length(int)){
    res=c(res, estDensity(int[i], x, h))
  }
  if(h==0.1){
    plot(int, res, type="l", col=col[count], lwd=2)
  }else{
    lines(int, res, type="l", col=col[count], lwd=2)
  }
  count=count+1
}
lines(int, dnorm(int, 0,1), col=8, lwd=2)
legend("topleft", legend=c(values, "N(0,1)"), fill=col)

#c)
h_opt=function(x,values){
  n=length(x)
  hopt=0
  prodopt=0
  for(h in values){
    prod=1
    for(i in 1:n){
      y=c(x[1:(i-1)], x[(i+1):n])
      if(i==1){
        y=x[2:n]
      }
      if(i==n){
        y=x[1:(n-1)]
      }
      prod=prod*estDensity(x[i],y,h)
    }
    if(prod>prodopt){
      hopt=h
      prodopt=prod
    }
  }
  return(hopt)
}


#d)
values=seq(0.7,2,0.05)
hopt=h_opt(x,values)
print(hopt)
int=seq(-4,4,0.1)
res=c()
for(i in 1:length(int)){
  res=c(res, estDensity(int[i], x, hopt))
}
plot(int, res, type="l", lwd=2)
lines(int, dnorm(int), col="red", lwd=2)