costFunction.m

function [costJ, gradient] = costFunction(theta, X, y)
%COSTFUNCTION Compute cost and gradient for logistic regression
%   costJ = COSTFUNCTION(theta, X, y) computes the cost of using theta as the
%   parameter for logistic regression and the gradient of the cost
%   w.r.t. to the parameters.

% Initialize some useful values
% y = mx1 column vector
numberOfTrainingExamples = length(y); % = m

% return the following variables correctly  
costJ = 0; % costJ = single number 
gradient = zeros(size(theta)); % gradient = nx1 column vector (same size as theta)

% Compute the costJ of a particular choice of theta

% compute cost costJ
% X = mxn matrix
% theta = nx1 column vector
hypothesis = sigmoid(X*theta); % hypothesis = mx1 column vector
 
% costJ = single number 
costJ = (-1/numberOfTrainingExamples) * sum( y .* log(hypothesis) + (1 - y) .* log(1 - hypothesis) );

% Compute the partial derivatives and set gradiant to the partial
% derivatives of the cost w.r.t. each parameter in theta

% compute the gradient 
for i = 1:numberOfTrainingExamples
	% hypothesis = mx1 column vector
	% y = mx1 column vector
	% X = mxn matrix
	gradient = gradient + ( hypothesis(i) - y(i) ) * X(i, :)';
end

% gradient = nx1 column vector
gradient = (1/numberOfTrainingExamples) * gradient;

end