testMLE.m

%Let's simulate some data

t=0:100; % time series

%Paramters of our exponential
a=.1;
b=.8;
c=.1;
w=a+(b-a)*exp(-c*t);
figure(1);
plot(t,w);
title('Likelihood of an event over time');

%simulated measurements
%Randomly draw a number between 0 and 1
% if its smaller than w, than score is 1
%Otherwise score it as zero.
% This is equivalent to getting a 1 with probability w
q=zeros(size(w));
q( find(rand(size(w))<w) )=1;

%%%%%%%%% FIREWALL.. treat t & q as real measurements

figure(2);
plot(t,q,'o');
title('Events');
%now use q, and t to do a Maximum Likelihood Estimate


x0=[(a+.03) , (b-.1) , (c+.4)];  %initial conditions
lbound=[0,0,0];
ubound=[1,1,10];
f=@(x) sum(-logLikelihood(x(1),x(2),x(3),t,q),2);

%Add in constraint that a+b<1
coef=[1 1 0];
limit=1;

x = fmincon(f,x0,coef,limit,[],[],lbound,ubound)


figure(1)
hold on;
plot(t,x(1)+x(2)*exp(-x(3)*t),'r');