% script Logistic_regression
% performs a logistic regression for the 
% categorial values in y.
% the output parameters are mu and s such that
% cfit(1) = mu, cfit(2) = s
% for the logistic function in standard form
x = 0.5:0.25:5.5; x(20)=[];
y = [0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1];
plot(x,y,'.r','markersize',12)
% the logistic (probability distribution) function
p = @(c,v) 1./(1+exp(-(v-c(1))/c(2)));
% the loss function to be minimized, 
% i.e. the cross entropy of the predicted distribution
loss = @(c) sum(-y.*log(p(c,x))-(1-y).*log(1-p(c,x)));
cfit = fminsearch(loss,[2 1]);
% display the fitted parameters of the logistic function
cfit
hold on
title('Logistic regression')
xx = linspace(x(1),x(end),200);
yy = p(cfit,xx);
% plot the best (fitted)logistic (probability distribution) function
plot(xx,yy,'b','linewidth',2)
line([cfit(1) cfit(1) 0],[0 0.5 0.5],'Color','green')
hold off