% Boyd & Vandenberghe "Convex Optimization"
% Joëlle Skaf - 04/24/08
%
% The random variable y is nonnegative and integer valued with a Poisson
% distribution with mean mu > 0. In a simple statistical model, the mean mu
% is modeled as an affine function of a vector u: mu = a'*u + b.
% We are given a number of observations which consist of pairs (u_i,y_i),
% i = 1,..., m, where y_i is the observed value of y for which the value of
% the explanatory variable is u_i. We find a maximum likelihood estimate of
% the model parameters a and b from these data by solving the problem
%           maximize    sum_{i=1}^m (y_i*log(a'*u_i + b) - (a'*u_i + b))
% where the variables are a and b.

% Input data
rand('state',0);
n = 10;
m = 100;
atrue = rand(n,1);
btrue = rand;

u = rand(n,m);
mu = atrue'*u + btrue;

% Generate random variables y from a Poisson distribution
% (The distribution is actually truncated at 10*max(mu) for simplicity)
L  = exp(-mu);
ns = ceil(max(10*mu));
y  = sum(cumprod(rand(ns,m))>=L(ones(ns,1),:));

% Maximum likelihood estimate of model parameters
cvx_begin
    variables a(n) b(1)
    maximize sum(y.*log(a'*u+b) - (a'*u+b))
cvx_end

 
Successive approximation method to be employed.
   SDPT3 will be called several times to refine the solution.
   Original size: 287 variables, 184 equality constraints
   92 exponentials add 736 variables, 460 equality constraints
-----------------------------------------------------------------
 Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
 92/ 92 | 2.046e+00  2.598e-01  0.000e+00 | Solved
 92/ 92 | 3.402e-01  7.923e-03  0.000e+00 | Solved
 91/ 92 | 2.081e-02  2.914e-05  0.000e+00 | Solved
 81/ 91 | 1.284e-03  1.101e-07  0.000e+00 | Solved
  0/  0 | 0.000e+00  0.000e+00  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +102.57