Machine Learning by Andrew Ng

function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X
%   FEATURENORMALIZE(X) returns a normalized version of X where
%   the mean value of each feature is 0 and the standard deviation
%   is 1. This is often a good preprocessing step to do when
%   working with learning algorithms.

mu = mean(X);
X_norm = bsxfun(@minus, X, mu);

sigma = std(X_norm);
X_norm = bsxfun(@rdivide, X_norm, sigma);


% ============================================================

end

function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear
%regression with multiple variables
%   [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the
%   cost of using theta as the parameter for linear regression to fit the
%   data points in X and y. Returns the cost in J and the gradient in grad

% 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 and gradient of regularized linear
%               regression for a particular choice of theta.
%
%               You should set J to the cost and grad to the gradient.
%

diff = X * theta - y;
J = sum(diff .^ 2) * 0.5 / m;
t = theta;
t(1) = 0;
J += sum(t .^ 2) * lambda * 0.5 / m;
grad = ((X' * diff) + lambda * t) / m;

% =========================================================================

grad = grad(:);

end

function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   nn_params and need to be converted back into the weight matrices.
%
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
m = size(X, 1);

% You need to return the following variables correctly
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m
%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the
%               first time.
%
% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%

X2 = [ones(m, 1)  X];
tx2 = X2 * Theta1';
hx2 = sigmoid(tx2);
X3 = [ones(m, 1) hx2];
tx3 = X3 * Theta2';
hx3 = sigmoid(tx3);
hy = zeros(m, num_labels);
for i = [1:m],
    hy(i, y(i)) = 1;
end;
J = sum(sum(log(hx3) .* (-hy) - log(1 - hx3) .* (1 - hy))) / m;

R = 0;
R += sum(sum(Theta1(:, 2:end) .^ 2));
R += sum(sum(Theta2(:, 2:end) .^ 2));
R *= lambda / m * 0.5;

J += R;

% -------------------------------------------------------------

d3 = hx3 - hy; # M * K
d2 = (d3 * Theta2)(:,2:end) .* sigmoidGradient(tx2); # M * H
Theta2_grad = d3' * X3 / m; # K * M * M * (H+1) = K * (H+1)
Theta1_grad = d2' * X2 / m; # H * M * M * (N+1) = H * (N+1)

t2 = Theta2;
t2(:,1) = 0;
t1 = Theta1;
t1(:,1) = 0;
Theta2_grad += t2 * lambda / m;
Theta1_grad += t1 * lambda / m;

% =========================================================================

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];


end

function [U, S] = pca(X)
%PCA Run principal component analysis on the dataset X
%   [U, S, X] = pca(X) computes eigenvectors of the covariance matrix of X
%   Returns the eigenvectors U, the eigenvalues (on diagonal) in S
%

% Useful values
[m, n] = size(X);

% You need to return the following variables correctly.
U = zeros(n);
S = zeros(n);

% ====================== YOUR CODE HERE ======================
% Instructions: You should first compute the covariance matrix. Then, you
%               should use the "svd" function to compute the eigenvectors
%               and eigenvalues of the covariance matrix.
%
% Note: When computing the covariance matrix, remember to divide by m (the
%       number of examples).
%

sigma = 1.0 / m * X' * X;
[U,S,_ ] = svd(sigma);



% =========================================================================

end

function Z = projectData(X, U, K)
%PROJECTDATA Computes the reduced data representation when projecting only
%on to the top k eigenvectors
%   Z = projectData(X, U, K) computes the projection of
%   the normalized inputs X into the reduced dimensional space spanned by
%   the first K columns of U. It returns the projected examples in Z.
%

% You need to return the following variables correctly.
Z = zeros(size(X, 1), K);

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the projection of the data using only the top K
%               eigenvectors in U (first K columns).
%               For the i-th example X(i,:), the projection on to the k-th
%               eigenvector is given as follows:
%                    x = X(i, :)';
%                    projection_k = x' * U(:, k);
%

U_reduce = U(:, 1:K);

Z = X * U_reduce;



% =============================================================

end

function X_rec = recoverData(Z, U, K)
%RECOVERDATA Recovers an approximation of the original data when using the
%projected data
%   X_rec = RECOVERDATA(Z, U, K) recovers an approximation the
%   original data that has been reduced to K dimensions. It returns the
%   approximate reconstruction in X_rec.
%

% You need to return the following variables correctly.
X_rec = zeros(size(Z, 1), size(U, 1));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the approximation of the data by projecting back
%               onto the original space using the top K eigenvectors in U.
%
%               For the i-th example Z(i,:), the (approximate)
%               recovered data for dimension j is given as follows:
%                    v = Z(i, :)';
%                    recovered_j = v' * U(j, 1:K)';
%
%               Notice that U(j, 1:K) is a row vector.
%

U_reduce = U(:, 1:K);

X_rec = Z * U_reduce';

% =============================================================

end

function [mu sigma2] = estimateGaussian(X)
%ESTIMATEGAUSSIAN This function estimates the parameters of a
%Gaussian distribution using the data in X
%   [mu sigma2] = estimateGaussian(X),
%   The input X is the dataset with each n-dimensional data point in one row
%   The output is an n-dimensional vector mu, the mean of the data set
%   and the variances sigma^2, an n x 1 vector
%

% Useful variables
[m, n] = size(X);

% You should return these values correctly
mu = zeros(n, 1);
sigma2 = zeros(n, 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the mean of the data and the variances
%               In particular, mu(i) should contain the mean of
%               the data for the i-th feature and sigma2(i)
%               should contain variance of the i-th feature.
%

mu = mean(X)';
# xu = X - mu';
# sigma2 = 1.0 / m * sum(xu .^ 2)';
sigma2 = (m-1) / m * var(X)';

% =============================================================


end

function p = multivariateGaussian(X, mu, Sigma2)
%MULTIVARIATEGAUSSIAN Computes the probability density function of the
%multivariate gaussian distribution.
%    p = MULTIVARIATEGAUSSIAN(X, mu, Sigma2) Computes the probability
%    density function of the examples X under the multivariate gaussian
%    distribution with parameters mu and Sigma2. If Sigma2 is a matrix, it is
%    treated as the covariance matrix. If Sigma2 is a vector, it is treated
%    as the \sigma^2 values of the variances in each dimension (a diagonal
%    covariance matrix)
%

k = length(mu);

if (size(Sigma2, 2) == 1) || (size(Sigma2, 1) == 1)
    Sigma2 = diag(Sigma2);
end

X = bsxfun(@minus, X, mu(:)');
p = (2 * pi) ^ (- k / 2) * det(Sigma2) ^ (-0.5) * ...
    exp(-0.5 * sum(bsxfun(@times, X * pinv(Sigma2), X), 2));

end

function [bestEpsilon bestF1] = selectThreshold(yval, pval)
%SELECTTHRESHOLD Find the best threshold (epsilon) to use for selecting
%outliers
%   [bestEpsilon bestF1] = SELECTTHRESHOLD(yval, pval) finds the best
%   threshold to use for selecting outliers based on the results from a
%   validation set (pval) and the ground truth (yval).
%

bestEpsilon = 0;
bestF1 = 0;
F1 = 0;

stepsize = (max(pval) - min(pval)) / 1000;
for epsilon = min(pval):stepsize:max(pval)

    % ====================== YOUR CODE HERE ======================
    % Instructions: Compute the F1 score of choosing epsilon as the
    %               threshold and place the value in F1. The code at the
    %               end of the loop will compare the F1 score for this
    %               choice of epsilon and set it to be the best epsilon if
    %               it is better than the current choice of epsilon.
    %
    % Note: You can use predictions = (pval < epsilon) to get a binary vector
    %       of 0's and 1's of the outlier predictions

    cv_pred = pval < epsilon;
    tp = sum((cv_pred == 1) & (yval == 1));
    fp = sum((cv_pred == 1) & (yval == 0));
    fn = sum((cv_pred == 0) & (yval == 1));
    prec = tp / (tp + fp);
    recall = tp / (tp + fn);
    F1 = 2 * prec * recall / (prec + recall);

    % =============================================================

    if F1 > bestF1
       bestF1 = F1;
       bestEpsilon = epsilon;
    end
end

end

function [J, grad] = cofiCostFunc(params, Y, R, num_users, num_movies, ...
                                  num_features, lambda)
%COFICOSTFUNC Collaborative filtering cost function
%   [J, grad] = COFICOSTFUNC(params, Y, R, num_users, num_movies, ...
%   num_features, lambda) returns the cost and gradient for the
%   collaborative filtering problem.
%

% Unfold the U and W matrices from params
X = reshape(params(1:num_movies*num_features), num_movies, num_features);
Theta = reshape(params(num_movies*num_features+1:end), ...
                num_users, num_features);


% You need to return the following values correctly
J = 0;
X_grad = zeros(size(X));
Theta_grad = zeros(size(Theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost function and gradient for collaborative
%               filtering. Concretely, you should first implement the cost
%               function (without regularization) and make sure it is
%               matches our costs. After that, you should implement the
%               gradient and use the checkCostFunction routine to check
%               that the gradient is correct. Finally, you should implement
%               regularization.
%
% Notes: X - num_movies  x num_features matrix of movie features
%        Theta - num_users  x num_features matrix of user features
%        Y - num_movies x num_users matrix of user ratings of movies
%        R - num_movies x num_users matrix, where R(i, j) = 1 if the
%            i-th movie was rated by the j-th user
%
% You should set the following variables correctly:
%
%        X_grad - num_movies x num_features matrix, containing the
%                 partial derivatives w.r.t. to each element of X
%        Theta_grad - num_users x num_features matrix, containing the
%                     partial derivatives w.r.t. to each element of Theta
%

xt = X * Theta'; % m * n * n * u = m * u
df = (xt- Y) .* R; % m * u
J = 0.5 * sum(sum(df .^ 2));
J += 0.5 * lambda * (sum(sum(Theta .^ 2)) + sum(sum(X .^ 2)));


X_grad = df * Theta; % m * u * u * n = m * n;
X_grad += lambda * X;

Theta_grad = df' * X; % u * m * m * n = u * n;
Theta_grad += lambda * Theta;

% =============================================================

grad = [X_grad(:); Theta_grad(:)];

end