TP2

 # generate data
 
# first generate means for each class
nmix <- 10                    # number of mixture components (nb de paquets de
                              #gaussienne)
x <- rnorm(nmix,0,1)      #10 variables aleat. N(0,1)
y <- rnorm(nmix,1,1)      #10 variables aleat. N(1,1)
means_pos = cbind(x,y)      #colle x et y a côté
x <- rnorm(nmix,1,1)      #10 variables aleat. N(1,1)
y <- rnorm(nmix,0,1)      #10 variables aleat. N(0,1)
means_neg = cbind(x,y)      #colle x et y a côté

# generate n examples
u <- array(1,c(1,nmix))      #tableau à 10 "1"
u <- u/nmix                  #tableau à 10 "0.1"                 
n <- 100

# first generate which mixture components
m_neg = sample(1:nmix, n, replace = TRUE, prob = u) #echantillon de n nombres
                                                    #entre 1 et nmix de proba u
m_pos = sample(1:nmix, n, replace = TRUE, prob = u)
x_neg <- array(0,c(n,2))    #tableau 100 lignes 2 col de "0"
x_pos <- array(0,c(n,2))    #tableau 100 lignes 2 col de "0"

# assigns means
for ( i in 1:n )
  {
  x_neg[i,]=means_neg[m_neg[i],]
  x_pos[i,]=means_pos[m_pos[i],]
  }
x_neg[,1] = x_neg[,1] + rnorm(n,0,1/5)
x_neg[,2] = x_neg[,2] + rnorm(n,0,1/5)
x_pos[,1] = x_pos[,1] + rnorm(n,0,1/5)
x_pos[,2] = x_pos[,2] + rnorm(n,0,1/5)

x <- rbind(x_neg,x_pos);

# plot the data

#plot(x[,1],x[,2],type="n")
#points(x_neg[,1],x_neg[,2],col="blue")
#points(x_pos[,1],x_pos[,2],col="red")


##########################################################################
##########################################################################
# PAS NECESSAIRE POUR CETTE QUESTION
# linear regression
#
x <- rbind(x_neg,x_pos);
x <- cbind(array(1,c(2*n,1)),x)
y <- rbind(array(-1,c(n,1)),array(1,c(n,1)));

beta <- solve(t(x) %*% x, t(x) %*% y)

x1 = array(seq(-2,3, by= 0.1), dim = c(51,1))
y1 = array(seq(-2,3, by= 0.1), dim = c(51,1))

d = dim(x1)[1]

z_reg = array(0,(d*d), dim = c(d,d))

for(i in 1:d)
{
  for(j in 1:d)
  {
    z_reg[i,j] <- t(beta) %*% c(1,x1[i],y1[j])
  }
}


contour(x1,y1,z_reg, level=0, labels=NULL)
points(x_neg[,1],x_neg[,2],col="blue")
points(x_pos[,1],x_pos[,2],col="red")
##########################################################################
##########################################################################

# plot the bayesian ( = optimal ) decision boundary

z_bayes <- array(0,d*d, dim = c(d,d))

for(i in 1:d)
{
  for(j in 1:d)
  {
    for(k in 1:nmix)
    {
      z_bayes[i,j] <- z_bayes[i,j] + ( dnorm(x1[i], means_pos[k,1], 1/5) *
dnorm(y1[j], means_pos[k,2], 1/5)) - ( dnorm(x1[i], means_neg[k,1], 1/5) *
dnorm(y1[j], means_neg[k,2], 1/5))
    }
  }
}

contour(x1,y1,z_bayes, level=0, labels='Bayes')
points(x_neg[,1],x_neg[,2],col="blue")
points(x_pos[,1],x_pos[,2],col="red")






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