# creation of the sample

alpha <- 0.1; beta <- 0.7; # the following assumes alpha < 0.5 < beta

rexemple <- function(n){
	res <- list();
	res$x <- cbind(runif(n), runif(n));
	res$y <- array(n);
	u <- runif(n);
	T = 2 * res$x[,1] + res$x[,2] < 1.5;
	res$y[T] <- u[T]<alpha;
	res$y[!T] <- u[!T]<beta;
	res;
}

evaluateClassifError <- function(method){
	N <- 10000;
	test <- rexemple(N);
	err <- 0;
	for(i in 1:N){
		err = err + (test$y[i] != method(test$x[i,]));
	}
	err/N;
}

n <- 100;
train <- rexemple(n);
plot(train$x[,1], train$x[,2], col = c("red", "blue")[1+train$y], xlim = c(-0.2, 1.2), ylim = c(-0.2, 1.2), pch = 19)
title("simulated data classification")
lines(c(0, 1), c(1.5, -0.5), col="black")
bayesClassifier <- function(x) {2*x[1] + x[2] > 1.5;}
bayesError <- evaluateClassifError(bayesClassifier)

# method 0: linear regression
linReg <- lm(train$y ~ train$x[,1] + train$x[,2])
cl <- linReg$coeff
lines(c(0, 1), c((0.5-cl[1])/cl[3], (0.5-cl[1]-cl[2])/cl[3]), col="green")
linClassifier <- function(x){cl[1] + x[1]*cl[2] + x[2]*cl[3] > 0.5;}
linError <- evaluateClassifError(linClassifier)

# method 1: logistic regression
logReg <- glm(train$y ~ train$x[,1] + train$x[,2], family = binomial)
cg <- logReg$coeff
lines(c(0, 1), c(-cg[1]/cg[3], (-cg[1]-cg[2])/cg[3]), col="magenta")
logClassifier <- function(x){cg[1] + x[1]*cg[2] + x[2]*cg[3] > 0;}
logError <- evaluateClassifError(logClassifier)


# graph legend and result prints
legend(0.8, 1.2, c("true frontier", "linear classifier", "logistic classifier"), col = c("black", "green", "magenta"),  lty=1);
cat("Bayes classification error:\t", bayesError, "\nLinear classifier error:\t", linError, "\nLogistic classifier error:\t", logError, "\n")