################
###Exercice 1###
################

#Q1
N <- 500
x <- vector("numeric",N+1)

for (i in 2:(N+1)){
  x[i] <- x[i-1]+ rnorm(1)
}

x <- x[-1]
length(x)

#Q2
plot(1:N,x, type = "l", col = "blue")

#Q3
acf(x,lag.max=100)

#Q4
dx <- diff(x)
plot(dx, type = "l", col = "blue")
acf(dx,lag.max=100)


################
###Exercice 2###
################

#Q1
N <- 500
x <- vector("numeric",N+1)

for (i in 2:N+1){
  x[i] <- 0.25+ 0.75*x[i-1]+ rnorm(1)
}

x <- x[-1]
length(x)
plot(x,t="l")

#Q2
par(mfrow=c(2,1))
acf(x)
title( sub = "Autocorrelation empirique",cex.sub = 1, font.sub = 3, col.sub = "red")
pacf(x)
title( sub = "Autocorrelation partielle empirique",cex.sub = 1, font.sub = 3, col.sub = "red")

#Q3
N <- 500
x <- vector("numeric",N+1)

for (i in 2:N+1){
  x[i] <- 0.25+ 0.5*x[i-1]+0.4*x[i-2]+ rnorm(1)
}

x <- x[-1]
length(x)

par(mfrow=c(2,1))
acf(x)
title( sub = "Autocorrelation empirique",cex.sub = 1, font.sub = 3, col.sub = "red")
pacf(x)
title( sub = "Autocorrelation partielle empirique",cex.sub = 1, font.sub = 3, col.sub = "red")


################
###Exercice 3###
################

N   <- 500
x   <- vector("numeric",N+1)
z   <- rnorm(N+1)

for (i in 2:N+1){
  x[i] <- z[i]+ 0.75*z[i-1]
}

x <- x[-1]
length(x)

#Q2
par(mfrow=c(3,1))
plot(1:N,x, type = "l", col = "blue")
acf(x)
pacf(x)

#Q3
N   <- 500
x   <- vector("numeric",N+2)
z   <- rnorm(N+2)

for (i in 3:N+2){
  x[i] <- 0.5*z[i-2]+ 0.5*z[i-1]+z[i]
}

x <- x[3:(N+2)]

par(mfrow=c(3,1))
plot(1:N,x, type = "l", col = "blue")
acf(x)
pacf(x)