lrCostFunction.m
function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with
%regularization
% J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
% theta as the parameter for regularized logistic regression and the
% gradient of the cost w.r.t. to the parameters.
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
grad = zeros(size(theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
% You should set J to the cost.
% Compute the partial derivatives and set grad to the partial
% derivatives of the cost w.r.t. each parameter in theta
%
% Hint: The computation of the cost function and gradients can be
% efficiently vectorized. For example, consider the computation
%
% sigmoid(X * theta)
%
% Each row of the resulting matrix will contain the value of the
% prediction for that example. You can make use of this to vectorize
% the cost function and gradient computations.
%
% Hint: When computing the gradient of the regularized cost function,
% there’re many possible vectorized solutions, but one solution
% looks like:
% grad = (unregularized gradient for logistic regression)
% temp = theta;
% temp(1) = 0; % because we don’t add anything for j = 0
% grad = grad + YOUR_CODE_HERE (using the temp variable)
%
J = 1/m* ( -y’ * log(sigmoid(X * theta)) – (1-y)’ * log(1-sigmoid(X * theta)) ) + lambda/(2*m) * sum((theta(2:end).^2));
grad = 1/m * (( sigmoid(X * theta) – y )’ * X)’;
grad(2:end) = grad(2:end) + lambda/m*theta(2:end);
% =============================================================
grad = grad(:);
end
oneVsAll.m
function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta
%corresponds to the classifier for label i
% [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
% logisitc regression classifiers and returns each of these classifiers
% in a matrix all_theta, where the i-th row of all_theta corresponds
% to the classifier for label i
% Some useful variables
m = size(X, 1);
n = size(X, 2);
% You need to return the following variables correctly
all_theta = zeros(num_labels, n + 1);
% Add ones to the X data matrix
X = [ones(m, 1) X];
% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
% logistic regression classifiers with regularization
% parameter lambda.
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1’s and 0’s that tell use
% whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
% function. It is okay to use a for-loop (for c = 1:num_labels) to
% loop over the different classes.
%
% fmincg works similarly to fminunc, but is more efficient when we
% are dealing with large number of parameters.
%
% Example Code for fmincg:
%
% % Set Initial theta
% initial_theta = zeros(n + 1, 1);
%
% % Set options for fminunc
% options = optimset(‘GradObj’, ‘on’, ‘MaxIter’, 50);
%
% % Run fmincg to obtain the optimal theta
% % This function will return theta and the cost
% [theta] = …
% fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), …
% initial_theta, options);
%
options = optimset(‘GradObj’, ‘on’, ‘MaxIter’, 50);
initial_theta = zeros(n+1,1);
for c=1:num_labels
all_theta(c,:) = fmincg(@(t)(lrCostFunction(t,X,(y == c), lambda)), initial_theta, options);
endfor
% =========================================================================
end
predictOneVsAll.m
function p = predictOneVsAll(all_theta, X)
%PREDICT Predict the label for a trained one-vs-all classifier. The labels
%are in the range 1..K, where K = size(all_theta, 1).
% p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions
% for each example in the matrix X. Note that X contains the examples in
% rows. all_theta is a matrix where the i-th row is a trained logistic
% regression theta vector for the i-th class. You should set p to a vector
% of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2
% for 4 examples)
m = size(X, 1);
num_labels = size(all_theta, 1);
% You need to return the following variables correctly
p = zeros(size(X, 1), 1);
% Add ones to the X data matrix
X = [ones(m, 1) X];
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned logistic regression parameters (one-vs-all).
% You should set p to a vector of predictions (from 1 to
% num_labels).
%
% Hint: This code can be done all vectorized using the max function.
% In particular, the max function can also return the index of the
% max element, for more information see ‘help max’. If your examples
% are in rows, then, you can use max(A, [], 2) to obtain the max
% for each row.
%
results = sigmoid(X * all_theta’);
[value , maxindex] = max(results, [] ,2);
p = maxindex
% ========================================================================
end
predict.m
function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
% p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
% trained weights of a neural network (Theta1, Theta2)
% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);
% You need to return the following variables correctly
p = zeros(size(X, 1), 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned neural network. You should set p to a
% vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. In particular, the max
% function can also return the index of the max element, for more
% information see ‘help max’. If your examples are in rows, then, you
% can use max(A, [], 2) to obtain the max for each row.
%
X = [ones(m,1) X]; % m*401 matrix
a2 = sigmoid(Theta1 * X’); % 25*401 matrix * 401*m matrix = 25*m matrix
a2 = [ones(1, m); a2]; % 26*m matrix
a3 = sigmoid(Theta2 * a2); % 10*26 matrix * 26*m matrix = 10*m matrix
a3 = a3′ ;
[w , index] = max(a3, [], 2);
p = index;
% =========================================================================
end
ML Study Notes 