Coursera Machine Learning Notes

% matlab code for feature scaling and mean normalization
mu = mean(X);
sigma = std(X);
for i = 1:size(X,2)
    X_norm(:,i) = (X(:,i)-mu(i))/sigma(i);
end

% Cost function and gradient for linear regression
J = sum((X * theta - y).^2)/(2*m) + + lambda/(2 * m) * sum(theta(2:end).^2);

grad = X' * (X * theta - y) / m + [0; lambda * theta(2:end) / m];

% matlab code for normal equation
theta = pinv(X' * X + lambda * L) * X' * y;

% matlab code for cost function and gradient of logistic regression
h = sigmoid(X * theta);
n = length(theta);
J = (-y' * log(h)- (1 - y)' * log(1 - h)) / m + lambda/(2 * m) * sum(theta(2:n).^2);
grad = X' * (h - y) / m + [0; lambda * theta(2:n) / m];

function numgrad = computeNumericalGradient(J, theta)
%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"
%and gives us a numerical estimate of the gradient.
%   numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical
%   gradient of the function J around theta. Calling y = J(theta) should
%   return the function value at theta.

% Notes: The following code implements numerical gradient checking, and 
%        returns the numerical gradient.It sets numgrad(i) to (a numerical 
%        approximation of) the partial derivative of J with respect to the 
%        i-th input argument, evaluated at theta. (i.e., numgrad(i) should 
%        be the (approximately) the partial derivative of J with respect 
%        to theta(i).)
%                

numgrad = zeros(size(theta));
perturb = zeros(size(theta));
e = 1e-4;
for p = 1:numel(theta)
    % Set perturbation vector
    perturb(p) = e;
    loss1 = J(theta - perturb);
    loss2 = J(theta + perturb);
    % Compute Numerical Gradient
    numgrad(p) = (loss2 - loss1) / (2*e);
    perturb(p) = 0;
end

end

function checkNNGradients(lambda)
%CHECKNNGRADIENTS Creates a small neural network to check the
%backpropagation gradients
%   CHECKNNGRADIENTS(lambda) Creates a small neural network to check the
%   backpropagation gradients, it will output the analytical gradients
%   produced by your backprop code and the numerical gradients (computed
%   using computeNumericalGradient). These two gradient computations should
%   result in very similar values.
%

if ~exist('lambda', 'var') || isempty(lambda)
    lambda = 0;
end

input_layer_size = 3;
hidden_layer_size = 5;
num_labels = 3;
m = 5;

% We generate some 'random' test data
Theta1 = debugInitializeWeights(hidden_layer_size, input_layer_size);
Theta2 = debugInitializeWeights(num_labels, hidden_layer_size);
% Reusing debugInitializeWeights to generate X
X  = debugInitializeWeights(m, input_layer_size - 1);
y  = 1 + mod(1:m, num_labels)';

% Unroll parameters
nn_params = [Theta1(:) ; Theta2(:)];

% Short hand for cost function
costFunc = @(p) nnCostFunction(p, input_layer_size, hidden_layer_size, ...
                               num_labels, X, y, lambda);

[cost, grad] = costFunc(nn_params);
numgrad = computeNumericalGradient(costFunc, nn_params);

% Visually examine the two gradient computations.  The two columns
% you get should be very similar. 
disp([numgrad grad]);
fprintf(['The above two columns you get should be very similar.\n' ...
         '(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']);

% Evaluate the norm of the difference between two solutions.  
% If you have a correct implementation, and assuming you used EPSILON = 0.0001 
% in computeNumericalGradient.m, then diff below should be less than 1e-9
diff = norm(numgrad-grad)/norm(numgrad+grad);

fprintf(['If your backpropagation implementation is correct, then \n' ...
         'the relative difference will be small (less than 1e-9). \n' ...
         '\nRelative Difference: %g\n'], diff);

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));

% 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.
%

yVec = zeros(m, num_labels);
for i = 1:length(y)
    yVec(i,y(i)) = 1;
end

%% =================== feedforward ===========
% 5000*401
X = [ones(m,1) X];
% 5000*26
A2 = [ones(m,1) sigmoid(X * Theta1')];
% 5000*10
A3 = sigmoid(A2 * Theta2');
J = sum(-dot(yVec', log(A3)') - dot((1 - yVec)', log(1 - A3)')) / m;

% Regularization
th1 = Theta1(:,2:end).^2;
th2 = Theta2(:,2:end).^2;
J = J + lambda / (2 * m) * (sum(th1(:)) +sum(th2(:)));



%% ================== backpropagation ============

for t = 1:m
    % step1 feedforward to calculate a1 z2 a2 z3 a3
    % 401*1
    a1 = X(t,:)';
    % 26*1
    z2 = Theta1 * a1;
    a2 = [1; sigmoid(z2)];
    %10*1
    z3 = Theta2 * a2;
    a3 = sigmoid(z3);
    
    % step2 calculate detal3 and delta2
    % 10*1
    delta3 = a3 - yVec(t,:)';
    % 26*1
    delta2 = Theta2' * delta3 .* [0; sigmoidGradient(z2)];
    
    % step3 accumulate the gradient
    Theta2_grad = Theta2_grad + delta3 * a2';
    Theta1_grad = Theta1_grad + delta2(2:end) * a1';
end

Theta2_grad = Theta2_grad / m + lambda * [zeros(num_labels, 1) Theta2(:, 2:end)] / m;
Theta1_grad = Theta1_grad / m + lambda * [zeros(hidden_layer_size, 1) Theta1(:, 2:end)] / m;


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

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

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


end

function sim = linearKernel(x1, x2)
%LINEARKERNEL returns a linear kernel between x1 and x2
%   sim = linearKernel(x1, x2) returns a linear kernel between x1 and x2
%   and returns the value in sim

% Ensure that x1 and x2 are column vectors
x1 = x1(:); x2 = x2(:);

% Compute the kernel
sim = x1' * x2;  % dot product

end

function sim = gaussianKernel(x1, x2, sigma)
%RBFKERNEL returns a radial basis function kernel between x1 and x2
%   sim = gaussianKernel(x1, x2) returns a gaussian kernel between x1 and x2
%   and returns the value in sim

% Ensure that x1 and x2 are column vectors
x1 = x1(:); x2 = x2(:);

% You need to return the following variables correctly.
sim = 0;

% Instructions: Fill in this function to return the similarity between x1
%               and x2 computed using a Gaussian kernel with bandwidth
%               sigma
%
%
sim = exp(- sum((x1 - x2) .^ 2) /  (sigma ^ 2 * 2));
    
end

function pred = svmPredict(model, X)
%SVMPREDICT returns a vector of predictions using a trained SVM model
%(svmTrain). 
%   pred = SVMPREDICT(model, X) returns a vector of predictions using a 
%   trained SVM model (svmTrain). X is a mxn matrix where there each 
%   example is a row. model is a svm model returned from svmTrain.
%   predictions pred is a m x 1 column of predictions of {0, 1} values.
%

% Check if we are getting a column vector, if so, then assume that we only
% need to do prediction for a single example
if (size(X, 2) == 1)
    % Examples should be in rows
    X = X';
end

% Dataset 
m = size(X, 1);
p = zeros(m, 1);
pred = zeros(m, 1);

if strcmp(func2str(model.kernelFunction), 'linearKernel')
    % We can use the weights and bias directly if working with the 
    % linear kernel
    p = X * model.w + model.b;
elseif strfind(func2str(model.kernelFunction), 'gaussianKernel')
    % Vectorized RBF Kernel
    % This is equivalent to computing the kernel on every pair of examples
    X1 = sum(X.^2, 2);
    X2 = sum(model.X.^2, 2)';
    K = bsxfun(@plus, X1, bsxfun(@plus, X2, - 2 * X * model.X'));
    K = model.kernelFunction(1, 0) .^ K;
    K = bsxfun(@times, model.y', K);
    K = bsxfun(@times, model.alphas', K);
    p = sum(K, 2);
else
    % Other Non-linear kernel
    for i = 1:m
        prediction = 0;
        for j = 1:size(model.X, 1)
            prediction = prediction + ...
                model.alphas(j) * model.y(j) * ...
                model.kernelFunction(X(i,:)', model.X(j,:)');
        end
        p(i) = prediction + model.b;
    end
end

% Convert predictions into 0 / 1
pred(p >= 0) =  1;
pred(p <  0) =  0;

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);
mu = zeros(n, 1);
sigma2 = zeros(n, 1);

% 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)';
sigma2 = (var(X) * (m - 1) / m)';

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