% Section 7.1.1: Counting problems with Poisson distribution % Boyd & Vandenberghe "Convex Optimization" % JoĆ«lle 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