changeset 0:88cb4625c904 default tip

init
author Jordi Gutiérrez Hermoso <jordigh@octave.org>
date Mon, 12 Dec 2011 02:02:57 -0500
parents
children
files checkCostFunction.m cofiCostFunc.m computeNumericalGradient.m estimateGaussian.m ex8.m ex8_cofi.m ex8_movieParams.mat ex8_movies.mat ex8data1.mat ex8data2.mat fmincg.m loadMovieList.m movie_ids.txt multivariateGaussian.m normalizeRatings.m selectThreshold.m submit.m submitWeb.m visualizeFit.m
diffstat 19 files changed, 3234 insertions(+), 0 deletions(-) [+]
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new file mode 100644
--- /dev/null
+++ b/checkCostFunction.m
@@ -0,0 +1,48 @@
+function checkCostFunction(lambda)
+%CHECKCOSTFUNCTION Creates a collaborative filering problem 
+%to check your cost function and gradients
+%   CHECKCOSTFUNCTION(lambda) Creates a collaborative filering problem 
+%   to check your cost function and gradients, it will output the 
+%   analytical gradients produced by your code and the numerical gradients 
+%   (computed using computeNumericalGradient). These two gradient 
+%   computations should result in very similar values.
+
+% Set lambda
+if ~exist('lambda', 'var') || isempty(lambda)
+    lambda = 0;
+end
+
+%% Create small problem
+X_t = rand(4, 3);
+Theta_t = rand(5, 3);
+
+% Zap out most entries
+Y = X_t * Theta_t';
+Y(rand(size(Y)) > 0.5) = 0;
+R = zeros(size(Y));
+R(Y ~= 0) = 1;
+
+%% Run Gradient Checking
+X = randn(size(X_t));
+Theta = randn(size(Theta_t));
+num_users = size(Y, 2);
+num_movies = size(Y, 1);
+num_features = size(Theta_t, 2);
+
+numgrad = computeNumericalGradient( ...
+                @(t) cofiCostFunc(t, Y, R, num_users, num_movies, ...
+                                num_features, lambda), [X(:); Theta(:)]);
+
+[cost, grad] = cofiCostFunc([X(:); Theta(:)],  Y, R, num_users, ...
+                          num_movies, num_features, lambda);
+
+disp([numgrad grad]);
+fprintf(['The above two columns you get should be very similar.\n' ...
+         '(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']);
+
+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
\ No newline at end of file
new file mode 100644
--- /dev/null
+++ b/cofiCostFunc.m
@@ -0,0 +1,62 @@
+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
+%
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+% =============================================================
+
+grad = [X_grad(:); Theta_grad(:)];
+
+end
new file mode 100644
--- /dev/null
+++ b/computeNumericalGradient.m
@@ -0,0 +1,29 @@
+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
new file mode 100644
--- /dev/null
+++ b/estimateGaussian.m
@@ -0,0 +1,36 @@
+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.
+%
+
+
+
+
+
+
+
+
+
+
+% =============================================================
+
+
+end
new file mode 100644
--- /dev/null
+++ b/ex8.m
@@ -0,0 +1,123 @@
+%% Machine Learning Online Class
+%  Exercise 8 | Anomaly Detection and Collaborative Filtering
+%
+%  Instructions
+%  ------------
+%
+%  This file contains code that helps you get started on the
+%  exercise. You will need to complete the following functions:
+%
+%     estimateGaussian.m
+%     selectThreshold.m
+%     cofiCostFunc.m
+%
+%  For this exercise, you will not need to change any code in this file,
+%  or any other files other than those mentioned above.
+%
+
+%% Initialization
+clear ; close all; clc
+
+%% ================== Part 1: Load Example Dataset  ===================
+%  We start this exercise by using a small dataset that is easy to
+%  visualize.
+%
+%  Our example case consists of 2 network server statistics across
+%  several machines: the latency and throughput of each machine.
+%  This exercise will help us find possibly faulty (or very fast) machines.
+%
+
+fprintf('Visualizing example dataset for outlier detection.\n\n');
+
+%  The following command loads the dataset. You should now have the
+%  variables X, Xval, yval in your environment
+load('ex8data1.mat');
+
+%  Visualize the example dataset
+plot(X(:, 1), X(:, 2), 'bx');
+axis([0 30 0 30]);
+xlabel('Latency (ms)');
+ylabel('Throughput (mb/s)');
+
+fprintf('Program paused. Press enter to continue.\n');
+pause
+
+
+%% ================== Part 2: Estimate the dataset statistics ===================
+%  For this exercise, we assume a Gaussian distribution for the dataset.
+%
+%  We first estimate the parameters of our assumed Gaussian distribution, 
+%  then compute the probabilities for each of the points and then visualize 
+%  both the overall distribution and where each of the points falls in 
+%  terms of that distribution.
+%
+fprintf('Visualizing Gaussian fit.\n\n');
+
+%  Estimate my and sigma2
+[mu sigma2] = estimateGaussian(X);
+
+%  Returns the density of the multivariate normal at each data point (row) 
+%  of X
+p = multivariateGaussian(X, mu, sigma2);
+
+%  Visualize the fit
+visualizeFit(X,  mu, sigma2);
+xlabel('Latency (ms)');
+ylabel('Throughput (mb/s)');
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% ================== Part 3: Find Outliers ===================
+%  Now you will find a good epsilon threshold using a cross-validation set
+%  probabilities given the estimated Gaussian distribution
+% 
+
+pval = multivariateGaussian(Xval, mu, sigma2);
+
+[epsilon F1] = selectThreshold(yval, pval);
+fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
+fprintf('Best F1 on Cross Validation Set:  %f\n', F1);
+fprintf('   (you should see a value epsilon of about 8.99e-05)\n\n');
+
+%  Find the outliers in the training set and plot the
+outliers = find(p < epsilon);
+
+%  Draw a red circle around those outliers
+hold on
+plot(X(outliers, 1), X(outliers, 2), 'ro', 'LineWidth', 2, 'MarkerSize', 10);
+hold off
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% ================== Part 4: Multidimensional Outliers ===================
+%  We will now use the code from the previous part and apply it to a 
+%  harder problem in which more features describe each datapoint and only 
+%  some features indicate whether a point is an outlier.
+%
+
+%  Loads the second dataset. You should now have the
+%  variables X, Xval, yval in your environment
+load('ex8data2.mat');
+
+%  Apply the same steps to the larger dataset
+[mu sigma2] = estimateGaussian(X);
+
+%  Training set 
+p = multivariateGaussian(X, mu, sigma2);
+
+%  Cross-validation set
+pval = multivariateGaussian(Xval, mu, sigma2);
+
+%  Find the best threshold
+[epsilon F1] = selectThreshold(yval, pval);
+
+fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
+fprintf('Best F1 on Cross Validation Set:  %f\n', F1);
+fprintf('# Outliers found: %d\n', sum(p < epsilon));
+fprintf('   (you should see a value epsilon of about 1.38e-18)\n\n');
+pause
+
+
+
new file mode 100644
--- /dev/null
+++ b/ex8_cofi.m
@@ -0,0 +1,237 @@
+%% Machine Learning Online Class
+%  Exercise 8 | Anomaly Detection and Collaborative Filtering
+%
+%  Instructions
+%  ------------
+%
+%  This file contains code that helps you get started on the
+%  exercise. You will need to complete the following functions:
+%
+%     estimateGaussian.m
+%     selectThreshold.m
+%     cofiCostFunc.m
+%
+%  For this exercise, you will not need to change any code in this file,
+%  or any other files other than those mentioned above.
+%
+
+%% =============== Part 1: Loading movie ratings dataset ================
+%  You will start by loading the movie ratings dataset to understand the
+%  structure of the data.
+%  
+fprintf('Loading movie ratings dataset.\n\n');
+
+%  Load data
+load ('ex8_movies.mat');
+
+%  Y is a 1682x943 matrix, containing ratings (1-5) of 1682 movies on 
+%  943 users
+%
+%  R is a 1682x943 matrix, where R(i,j) = 1 if and only if user j gave a
+%  rating to movie i
+
+%  From the matrix, we can compute statistics like average rating.
+fprintf('Average rating for movie 1 (Toy Story): %f / 5\n\n', ...
+        mean(Y(1, R(1, :))));
+
+%  We can "visualize" the ratings matrix by plotting it with imagesc
+imagesc(Y);
+ylabel('Movies');
+xlabel('Users');
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+%% ============ Part 2: Collaborative Filtering Cost Function ===========
+%  You will now implement the cost function for collaborative filtering.
+%  To help you debug your cost function, we have included set of weights
+%  that we trained on that. Specifically, you should complete the code in 
+%  cofiCostFunc.m to return J.
+
+%  Load pre-trained weights (X, Theta, num_users, num_movies, num_features)
+load ('ex8_movieParams.mat');
+
+%  Reduce the data set size so that this runs faster
+num_users = 4; num_movies = 5; num_features = 3;
+X = X(1:num_movies, 1:num_features);
+Theta = Theta(1:num_users, 1:num_features);
+Y = Y(1:num_movies, 1:num_users);
+R = R(1:num_movies, 1:num_users);
+
+%  Evaluate cost function
+J = cofiCostFunc([X(:) ; Theta(:)], Y, R, num_users, num_movies, ...
+               num_features, 0);
+           
+fprintf(['Cost at loaded parameters: %f '...
+         '\n(this value should be about 22.22)\n'], J);
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ============== Part 3: Collaborative Filtering Gradient ==============
+%  Once your cost function matches up with ours, you should now implement 
+%  the collaborative filtering gradient function. Specifically, you should 
+%  complete the code in cofiCostFunc.m to return the grad argument.
+%  
+fprintf('\nChecking Gradients (without regularization) ... \n');
+
+%  Check gradients by running checkNNGradients
+checkCostFunction;
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ========= Part 4: Collaborative Filtering Cost Regularization ========
+%  Now, you should implement regularization for the cost function for 
+%  collaborative filtering. You can implement it by adding the cost of
+%  regularization to the original cost computation.
+%  
+
+%  Evaluate cost function
+J = cofiCostFunc([X(:) ; Theta(:)], Y, R, num_users, num_movies, ...
+               num_features, 1.5);
+           
+fprintf(['Cost at loaded parameters (lambda = 1.5): %f '...
+         '\n(this value should be about 31.34)\n'], J);
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ======= Part 5: Collaborative Filtering Gradient Regularization ======
+%  Once your cost matches up with ours, you should proceed to implement 
+%  regularization for the gradient. 
+%
+
+%  
+fprintf('\nChecking Gradients (with regularization) ... \n');
+
+%  Check gradients by running checkNNGradients
+checkCostFunction(1.5);
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ============== Part 6: Entering ratings for a new user ===============
+%  Before we will train the collaborative filtering model, we will first
+%  add ratings that correspond to a new user that we just observed. This
+%  part of the code will also allow you to put in your own ratings for the
+%  movies in our dataset!
+%
+movieList = loadMovieList();
+
+%  Initialize my ratings
+my_ratings = zeros(1682, 1);
+
+% Check the file movie_idx.txt for id of each movie in our dataset
+% For example, Toy Story (1995) has ID 1, so to rate it "4", you can set
+my_ratings(1) = 4;
+
+% Or suppose did not enjoy Silence of the Lambs (1991), you can set
+my_ratings(98) = 2;
+
+% We have selected a few movies we liked / did not like and the ratings we
+% gave are as follows:
+my_ratings(7) = 3;
+my_ratings(12)= 5;
+my_ratings(54) = 4;
+my_ratings(64)= 5;
+my_ratings(66)= 3;
+my_ratings(69) = 5;
+my_ratings(183) = 4;
+my_ratings(226) = 5;
+my_ratings(355)= 5;
+
+fprintf('\n\nNew user ratings:\n');
+for i = 1:length(my_ratings)
+    if my_ratings(i) > 0 
+        fprintf('Rated %d for %s\n', my_ratings(i), ...
+                 movieList{i});
+    end
+end
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ================== Part 7: Learning Movie Ratings ====================
+%  Now, you will train the collaborative filtering model on a movie rating 
+%  dataset of 1682 movies and 943 users
+%
+
+fprintf('\nTraining collaborative filtering...\n');
+
+%  Load data
+load('ex8_movies.mat');
+
+%  Y is a 1682x943 matrix, containing ratings (1-5) of 1682 movies by 
+%  943 users
+%
+%  R is a 1682x943 matrix, where R(i,j) = 1 if and only if user j gave a
+%  rating to movie i
+
+%  Add our own ratings to the data matrix
+Y = [my_ratings Y];
+R = [(my_ratings ~= 0) R];
+
+%  Normalize Ratings
+[Ynorm, Ymean] = normalizeRatings(Y, R);
+
+%  Useful Values
+num_users = size(Y, 2);
+num_movies = size(Y, 1);
+num_features = 10;
+
+% Set Initial Parameters (Theta, X)
+X = randn(num_movies, num_features);
+Theta = randn(num_users, num_features);
+
+initial_parameters = [X(:); Theta(:)];
+
+% Set options for fmincg
+options = optimset('GradObj', 'on', 'MaxIter', 100);
+
+% Set Regularization
+lambda = 10;
+theta = fmincg (@(t)(cofiCostFunc(t, Y, R, num_users, num_movies, ...
+                                num_features, lambda)), ...
+                initial_parameters, options);
+
+% Unfold the returned theta back into U and W
+X = reshape(theta(1:num_movies*num_features), num_movies, num_features);
+Theta = reshape(theta(num_movies*num_features+1:end), ...
+                num_users, num_features);
+
+fprintf('Recommender system learning completed.\n');
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+%% ================== Part 8: Recommendation for you ====================
+%  After training the model, you can now make recommendations by computing
+%  the predictions matrix.
+%
+
+p = X * Theta';
+my_predictions = p(:,1) + Ymean;
+
+movieList = loadMovieList();
+
+[r, ix] = sort(my_predictions, 'descend');
+fprintf('\nTop recommendations for you:\n');
+for i=1:10
+    j = ix(i);
+    fprintf('Predicting rating %.1f for movie %s\n', my_predictions(j), ...
+            movieList{j});
+end
+
+fprintf('\n\nOriginal ratings provided:\n');
+for i = 1:length(my_ratings)
+    if my_ratings(i) > 0 
+        fprintf('Rated %d for %s\n', my_ratings(i), ...
+                 movieList{i});
+    end
+end
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new file mode 100644
--- /dev/null
+++ b/fmincg.m
@@ -0,0 +1,175 @@
+function [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
+% Minimize a continuous differentialble multivariate function. Starting point
+% is given by "X" (D by 1), and the function named in the string "f", must
+% return a function value and a vector of partial derivatives. The Polack-
+% Ribiere flavour of conjugate gradients is used to compute search directions,
+% and a line search using quadratic and cubic polynomial approximations and the
+% Wolfe-Powell stopping criteria is used together with the slope ratio method
+% for guessing initial step sizes. Additionally a bunch of checks are made to
+% make sure that exploration is taking place and that extrapolation will not
+% be unboundedly large. The "length" gives the length of the run: if it is
+% positive, it gives the maximum number of line searches, if negative its
+% absolute gives the maximum allowed number of function evaluations. You can
+% (optionally) give "length" a second component, which will indicate the
+% reduction in function value to be expected in the first line-search (defaults
+% to 1.0). The function returns when either its length is up, or if no further
+% progress can be made (ie, we are at a minimum, or so close that due to
+% numerical problems, we cannot get any closer). If the function terminates
+% within a few iterations, it could be an indication that the function value
+% and derivatives are not consistent (ie, there may be a bug in the
+% implementation of your "f" function). The function returns the found
+% solution "X", a vector of function values "fX" indicating the progress made
+% and "i" the number of iterations (line searches or function evaluations,
+% depending on the sign of "length") used.
+%
+% Usage: [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
+%
+% See also: checkgrad 
+%
+% Copyright (C) 2001 and 2002 by Carl Edward Rasmussen. Date 2002-02-13
+%
+%
+% (C) Copyright 1999, 2000 & 2001, Carl Edward Rasmussen
+% 
+% Permission is granted for anyone to copy, use, or modify these
+% programs and accompanying documents for purposes of research or
+% education, provided this copyright notice is retained, and note is
+% made of any changes that have been made.
+% 
+% These programs and documents are distributed without any warranty,
+% express or implied.  As the programs were written for research
+% purposes only, they have not been tested to the degree that would be
+% advisable in any important application.  All use of these programs is
+% entirely at the user's own risk.
+%
+% [ml-class] Changes Made:
+% 1) Function name and argument specifications
+% 2) Output display
+%
+
+% Read options
+if exist('options', 'var') && ~isempty(options) && isfield(options, 'MaxIter')
+    length = options.MaxIter;
+else
+    length = 100;
+end
+
+
+RHO = 0.01;                            % a bunch of constants for line searches
+SIG = 0.5;       % RHO and SIG are the constants in the Wolfe-Powell conditions
+INT = 0.1;    % don't reevaluate within 0.1 of the limit of the current bracket
+EXT = 3.0;                    % extrapolate maximum 3 times the current bracket
+MAX = 20;                         % max 20 function evaluations per line search
+RATIO = 100;                                      % maximum allowed slope ratio
+
+argstr = ['feval(f, X'];                      % compose string used to call function
+for i = 1:(nargin - 3)
+  argstr = [argstr, ',P', int2str(i)];
+end
+argstr = [argstr, ')'];
+
+if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end
+S=['Iteration '];
+
+i = 0;                                            % zero the run length counter
+ls_failed = 0;                             % no previous line search has failed
+fX = [];
+[f1 df1] = eval(argstr);                      % get function value and gradient
+i = i + (length<0);                                            % count epochs?!
+s = -df1;                                        % search direction is steepest
+d1 = -s'*s;                                                 % this is the slope
+z1 = red/(1-d1);                                  % initial step is red/(|s|+1)
+
+while i < abs(length)                                      % while not finished
+  i = i + (length>0);                                      % count iterations?!
+
+  X0 = X; f0 = f1; df0 = df1;                   % make a copy of current values
+  X = X + z1*s;                                             % begin line search
+  [f2 df2] = eval(argstr);
+  i = i + (length<0);                                          % count epochs?!
+  d2 = df2'*s;
+  f3 = f1; d3 = d1; z3 = -z1;             % initialize point 3 equal to point 1
+  if length>0, M = MAX; else M = min(MAX, -length-i); end
+  success = 0; limit = -1;                     % initialize quanteties
+  while 1
+    while ((f2 > f1+z1*RHO*d1) | (d2 > -SIG*d1)) & (M > 0) 
+      limit = z1;                                         % tighten the bracket
+      if f2 > f1
+        z2 = z3 - (0.5*d3*z3*z3)/(d3*z3+f2-f3);                 % quadratic fit
+      else
+        A = 6*(f2-f3)/z3+3*(d2+d3);                                 % cubic fit
+        B = 3*(f3-f2)-z3*(d3+2*d2);
+        z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A;       % numerical error possible - ok!
+      end
+      if isnan(z2) | isinf(z2)
+        z2 = z3/2;                  % if we had a numerical problem then bisect
+      end
+      z2 = max(min(z2, INT*z3),(1-INT)*z3);  % don't accept too close to limits
+      z1 = z1 + z2;                                           % update the step
+      X = X + z2*s;
+      [f2 df2] = eval(argstr);
+      M = M - 1; i = i + (length<0);                           % count epochs?!
+      d2 = df2'*s;
+      z3 = z3-z2;                    % z3 is now relative to the location of z2
+    end
+    if f2 > f1+z1*RHO*d1 | d2 > -SIG*d1
+      break;                                                % this is a failure
+    elseif d2 > SIG*d1
+      success = 1; break;                                             % success
+    elseif M == 0
+      break;                                                          % failure
+    end
+    A = 6*(f2-f3)/z3+3*(d2+d3);                      % make cubic extrapolation
+    B = 3*(f3-f2)-z3*(d3+2*d2);
+    z2 = -d2*z3*z3/(B+sqrt(B*B-A*d2*z3*z3));        % num. error possible - ok!
+    if ~isreal(z2) | isnan(z2) | isinf(z2) | z2 < 0   % num prob or wrong sign?
+      if limit < -0.5                               % if we have no upper limit
+        z2 = z1 * (EXT-1);                 % the extrapolate the maximum amount
+      else
+        z2 = (limit-z1)/2;                                   % otherwise bisect
+      end
+    elseif (limit > -0.5) & (z2+z1 > limit)          % extraplation beyond max?
+      z2 = (limit-z1)/2;                                               % bisect
+    elseif (limit < -0.5) & (z2+z1 > z1*EXT)       % extrapolation beyond limit
+      z2 = z1*(EXT-1.0);                           % set to extrapolation limit
+    elseif z2 < -z3*INT
+      z2 = -z3*INT;
+    elseif (limit > -0.5) & (z2 < (limit-z1)*(1.0-INT))   % too close to limit?
+      z2 = (limit-z1)*(1.0-INT);
+    end
+    f3 = f2; d3 = d2; z3 = -z2;                  % set point 3 equal to point 2
+    z1 = z1 + z2; X = X + z2*s;                      % update current estimates
+    [f2 df2] = eval(argstr);
+    M = M - 1; i = i + (length<0);                             % count epochs?!
+    d2 = df2'*s;
+  end                                                      % end of line search
+
+  if success                                         % if line search succeeded
+    f1 = f2; fX = [fX' f1]';
+    fprintf('%s %4i | Cost: %4.6e\r', S, i, f1);
+    s = (df2'*df2-df1'*df2)/(df1'*df1)*s - df2;      % Polack-Ribiere direction
+    tmp = df1; df1 = df2; df2 = tmp;                         % swap derivatives
+    d2 = df1'*s;
+    if d2 > 0                                      % new slope must be negative
+      s = -df1;                              % otherwise use steepest direction
+      d2 = -s'*s;    
+    end
+    z1 = z1 * min(RATIO, d1/(d2-realmin));          % slope ratio but max RATIO
+    d1 = d2;
+    ls_failed = 0;                              % this line search did not fail
+  else
+    X = X0; f1 = f0; df1 = df0;  % restore point from before failed line search
+    if ls_failed | i > abs(length)          % line search failed twice in a row
+      break;                             % or we ran out of time, so we give up
+    end
+    tmp = df1; df1 = df2; df2 = tmp;                         % swap derivatives
+    s = -df1;                                                    % try steepest
+    d1 = -s'*s;
+    z1 = 1/(1-d1);                     
+    ls_failed = 1;                                    % this line search failed
+  end
+  if exist('OCTAVE_VERSION')
+    fflush(stdout);
+  end
+end
+fprintf('\n');
new file mode 100644
--- /dev/null
+++ b/loadMovieList.m
@@ -0,0 +1,25 @@
+function movieList = loadMovieList()
+%GETMOVIELIST reads the fixed movie list in movie.txt and returns a
+%cell array of the words
+%   movieList = GETMOVIELIST() reads the fixed movie list in movie.txt 
+%   and returns a cell array of the words in movieList.
+
+
+%% Read the fixed movieulary list
+fid = fopen('movie_ids.txt');
+
+% Store all movies in cell array movie{}
+n = 1682;  % Total number of movies 
+
+movieList = cell(n, 1);
+for i = 1:n
+    % Read line
+    line = fgets(fid);
+    % Word Index (can ignore since it will be = i)
+    [idx, movieName] = strtok(line, ' ');
+    % Actual Word
+    movieList{i} = strtrim(movieName);
+end
+fclose(fid);
+
+end
new file mode 100644
--- /dev/null
+++ b/movie_ids.txt
@@ -0,0 +1,1682 @@
+1 Toy Story (1995)
+2 GoldenEye (1995)
+3 Four Rooms (1995)
+4 Get Shorty (1995)
+5 Copycat (1995)
+6 Shanghai Triad (Yao a yao yao dao waipo qiao) (1995)
+7 Twelve Monkeys (1995)
+8 Babe (1995)
+9 Dead Man Walking (1995)
+10 Richard III (1995)
+11 Seven (Se7en) (1995)
+12 Usual Suspects, The (1995)
+13 Mighty Aphrodite (1995)
+14 Postino, Il (1994)
+15 Mr. Holland's Opus (1995)
+16 French Twist (Gazon maudit) (1995)
+17 From Dusk Till Dawn (1996)
+18 White Balloon, The (1995)
+19 Antonia's Line (1995)
+20 Angels and Insects (1995)
+21 Muppet Treasure Island (1996)
+22 Braveheart (1995)
+23 Taxi Driver (1976)
+24 Rumble in the Bronx (1995)
+25 Birdcage, The (1996)
+26 Brothers McMullen, The (1995)
+27 Bad Boys (1995)
+28 Apollo 13 (1995)
+29 Batman Forever (1995)
+30 Belle de jour (1967)
+31 Crimson Tide (1995)
+32 Crumb (1994)
+33 Desperado (1995)
+34 Doom Generation, The (1995)
+35 Free Willy 2: The Adventure Home (1995)
+36 Mad Love (1995)
+37 Nadja (1994)
+38 Net, The (1995)
+39 Strange Days (1995)
+40 To Wong Foo, Thanks for Everything! Julie Newmar (1995)
+41 Billy Madison (1995)
+42 Clerks (1994)
+43 Disclosure (1994)
+44 Dolores Claiborne (1994)
+45 Eat Drink Man Woman (1994)
+46 Exotica (1994)
+47 Ed Wood (1994)
+48 Hoop Dreams (1994)
+49 I.Q. (1994)
+50 Star Wars (1977)
+51 Legends of the Fall (1994)
+52 Madness of King George, The (1994)
+53 Natural Born Killers (1994)
+54 Outbreak (1995)
+55 Professional, The (1994)
+56 Pulp Fiction (1994)
+57 Priest (1994)
+58 Quiz Show (1994)
+59 Three Colors: Red (1994)
+60 Three Colors: Blue (1993)
+61 Three Colors: White (1994)
+62 Stargate (1994)
+63 Santa Clause, The (1994)
+64 Shawshank Redemption, The (1994)
+65 What's Eating Gilbert Grape (1993)
+66 While You Were Sleeping (1995)
+67 Ace Ventura: Pet Detective (1994)
+68 Crow, The (1994)
+69 Forrest Gump (1994)
+70 Four Weddings and a Funeral (1994)
+71 Lion King, The (1994)
+72 Mask, The (1994)
+73 Maverick (1994)
+74 Faster Pussycat! Kill! Kill! (1965)
+75 Brother Minister: The Assassination of Malcolm X (1994)
+76 Carlito's Way (1993)
+77 Firm, The (1993)
+78 Free Willy (1993)
+79 Fugitive, The (1993)
+80 Hot Shots! Part Deux (1993)
+81 Hudsucker Proxy, The (1994)
+82 Jurassic Park (1993)
+83 Much Ado About Nothing (1993)
+84 Robert A. Heinlein's The Puppet Masters (1994)
+85 Ref, The (1994)
+86 Remains of the Day, The (1993)
+87 Searching for Bobby Fischer (1993)
+88 Sleepless in Seattle (1993)
+89 Blade Runner (1982)
+90 So I Married an Axe Murderer (1993)
+91 Nightmare Before Christmas, The (1993)
+92 True Romance (1993)
+93 Welcome to the Dollhouse (1995)
+94 Home Alone (1990)
+95 Aladdin (1992)
+96 Terminator 2: Judgment Day (1991)
+97 Dances with Wolves (1990)
+98 Silence of the Lambs, The (1991)
+99 Snow White and the Seven Dwarfs (1937)
+100 Fargo (1996)
+101 Heavy Metal (1981)
+102 Aristocats, The (1970)
+103 All Dogs Go to Heaven 2 (1996)
+104 Theodore Rex (1995)
+105 Sgt. Bilko (1996)
+106 Diabolique (1996)
+107 Moll Flanders (1996)
+108 Kids in the Hall: Brain Candy (1996)
+109 Mystery Science Theater 3000: The Movie (1996)
+110 Operation Dumbo Drop (1995)
+111 Truth About Cats & Dogs, The (1996)
+112 Flipper (1996)
+113 Horseman on the Roof, The (Hussard sur le toit, Le) (1995)
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+1474 Nina Takes a Lover (1994)
+1475 Bhaji on the Beach (1993)
+1476 Raw Deal (1948)
+1477 Nightwatch (1997)
+1478 Dead Presidents (1995)
+1479 Reckless (1995)
+1480 Herbie Rides Again (1974)
+1481 S.F.W. (1994)
+1482 Gate of Heavenly Peace, The (1995)
+1483 Man in the Iron Mask, The (1998)
+1484 Jerky Boys, The (1994)
+1485 Colonel Chabert, Le (1994)
+1486 Girl in the Cadillac (1995)
+1487 Even Cowgirls Get the Blues (1993)
+1488 Germinal (1993)
+1489 Chasers (1994)
+1490 Fausto (1993)
+1491 Tough and Deadly (1995)
+1492 Window to Paris (1994)
+1493 Modern Affair, A (1995)
+1494 Mostro, Il (1994)
+1495 Flirt (1995)
+1496 Carpool (1996)
+1497 Line King: Al Hirschfeld, The (1996)
+1498 Farmer & Chase (1995)
+1499 Grosse Fatigue (1994)
+1500 Santa with Muscles (1996)
+1501 Prisoner of the Mountains (Kavkazsky Plennik) (1996)
+1502 Naked in New York (1994)
+1503 Gold Diggers: The Secret of Bear Mountain (1995)
+1504 Bewegte Mann, Der (1994)
+1505 Killer: A Journal of Murder (1995)
+1506 Nelly & Monsieur Arnaud (1995)
+1507 Three Lives and Only One Death (1996)
+1508 Babysitter, The (1995)
+1509 Getting Even with Dad (1994)
+1510 Mad Dog Time (1996)
+1511 Children of the Revolution (1996)
+1512 World of Apu, The (Apur Sansar) (1959)
+1513 Sprung (1997)
+1514 Dream With the Fishes (1997)
+1515 Wings of Courage (1995)
+1516 Wedding Gift, The (1994)
+1517 Race the Sun (1996)
+1518 Losing Isaiah (1995)
+1519 New Jersey Drive (1995)
+1520 Fear, The (1995)
+1521 Mr. Wonderful (1993)
+1522 Trial by Jury (1994)
+1523 Good Man in Africa, A (1994)
+1524 Kaspar Hauser (1993)
+1525 Object of My Affection, The (1998)
+1526 Witness (1985)
+1527 Senseless (1998)
+1528 Nowhere (1997)
+1529 Underground (1995)
+1530 Jefferson in Paris (1995)
+1531 Far From Home: The Adventures of Yellow Dog (1995)
+1532 Foreign Student (1994)
+1533 I Don't Want to Talk About It (De eso no se habla) (1993)
+1534 Twin Town (1997)
+1535 Enfer, L' (1994)
+1536 Aiqing wansui (1994)
+1537 Cosi (1996)
+1538 All Over Me (1997)
+1539 Being Human (1993)
+1540 Amazing Panda Adventure, The (1995)
+1541 Beans of Egypt, Maine, The (1994)
+1542 Scarlet Letter, The (1926)
+1543 Johns (1996)
+1544 It Takes Two (1995)
+1545 Frankie Starlight (1995)
+1546 Shadows (Cienie) (1988)
+1547 Show, The (1995)
+1548 The Courtyard (1995)
+1549 Dream Man (1995)
+1550 Destiny Turns on the Radio (1995)
+1551 Glass Shield, The (1994)
+1552 Hunted, The (1995)
+1553 Underneath, The (1995)
+1554 Safe Passage (1994)
+1555 Secret Adventures of Tom Thumb, The (1993)
+1556 Condition Red (1995)
+1557 Yankee Zulu (1994)
+1558 Aparajito (1956)
+1559 Hostile Intentions (1994)
+1560 Clean Slate (Coup de Torchon) (1981)
+1561 Tigrero: A Film That Was Never Made (1994)
+1562 Eye of Vichy, The (Oeil de Vichy, L') (1993)
+1563 Promise, The (Versprechen, Das) (1994)
+1564 To Cross the Rubicon (1991)
+1565 Daens (1992)
+1566 Man from Down Under, The (1943)
+1567 Careful (1992)
+1568 Vermont Is For Lovers (1992)
+1569 Vie est belle, La (Life is Rosey) (1987)
+1570 Quartier Mozart (1992)
+1571 Touki Bouki (Journey of the Hyena) (1973)
+1572 Wend Kuuni (God's Gift) (1982)
+1573 Spirits of the Dead (Tre passi nel delirio) (1968)
+1574 Pharaoh's Army (1995)
+1575 I, Worst of All (Yo, la peor de todas) (1990)
+1576 Hungarian Fairy Tale, A (1987)
+1577 Death in the Garden (Mort en ce jardin, La) (1956)
+1578 Collectionneuse, La (1967)
+1579 Baton Rouge (1988)
+1580 Liebelei (1933)
+1581 Woman in Question, The (1950)
+1582 T-Men (1947)
+1583 Invitation, The (Zaproszenie) (1986)
+1584 Symphonie pastorale, La (1946)
+1585 American Dream (1990)
+1586 Lashou shentan (1992)
+1587 Terror in a Texas Town (1958)
+1588 Salut cousin! (1996)
+1589 Schizopolis (1996)
+1590 To Have, or Not (1995)
+1591 Duoluo tianshi (1995)
+1592 Magic Hour, The (1998)
+1593 Death in Brunswick (1991)
+1594 Everest (1998)
+1595 Shopping (1994)
+1596 Nemesis 2: Nebula (1995)
+1597 Romper Stomper (1992)
+1598 City of Industry (1997)
+1599 Someone Else's America (1995)
+1600 Guantanamera (1994)
+1601 Office Killer (1997)
+1602 Price Above Rubies, A (1998)
+1603 Angela (1995)
+1604 He Walked by Night (1948)
+1605 Love Serenade (1996)
+1606 Deceiver (1997)
+1607 Hurricane Streets (1998)
+1608 Buddy (1997)
+1609 B*A*P*S (1997)
+1610 Truth or Consequences, N.M. (1997)
+1611 Intimate Relations (1996)
+1612 Leading Man, The (1996)
+1613 Tokyo Fist (1995)
+1614 Reluctant Debutante, The (1958)
+1615 Warriors of Virtue (1997)
+1616 Desert Winds (1995)
+1617 Hugo Pool (1997)
+1618 King of New York (1990)
+1619 All Things Fair (1996)
+1620 Sixth Man, The (1997)
+1621 Butterfly Kiss (1995)
+1622 Paris, France (1993)
+1623 Cérémonie, La (1995)
+1624 Hush (1998)
+1625 Nightwatch (1997)
+1626 Nobody Loves Me (Keiner liebt mich) (1994)
+1627 Wife, The (1995)
+1628 Lamerica (1994)
+1629 Nico Icon (1995)
+1630 Silence of the Palace, The (Saimt el Qusur) (1994)
+1631 Slingshot, The (1993)
+1632 Land and Freedom (Tierra y libertad) (1995)
+1633 Á köldum klaka (Cold Fever) (1994)
+1634 Etz Hadomim Tafus (Under the Domin Tree) (1994)
+1635 Two Friends (1986) 
+1636 Brothers in Trouble (1995)
+1637 Girls Town (1996)
+1638 Normal Life (1996)
+1639 Bitter Sugar (Azucar Amargo) (1996)
+1640 Eighth Day, The (1996)
+1641 Dadetown (1995)
+1642 Some Mother's Son (1996)
+1643 Angel Baby (1995)
+1644 Sudden Manhattan (1996)
+1645 Butcher Boy, The (1998)
+1646 Men With Guns (1997)
+1647 Hana-bi (1997)
+1648 Niagara, Niagara (1997)
+1649 Big One, The (1997)
+1650 Butcher Boy, The (1998)
+1651 Spanish Prisoner, The (1997)
+1652 Temptress Moon (Feng Yue) (1996)
+1653 Entertaining Angels: The Dorothy Day Story (1996)
+1654 Chairman of the Board (1998)
+1655 Favor, The (1994)
+1656 Little City (1998)
+1657 Target (1995)
+1658 Substance of Fire, The (1996)
+1659 Getting Away With Murder (1996)
+1660 Small Faces (1995)
+1661 New Age, The (1994)
+1662 Rough Magic (1995)
+1663 Nothing Personal (1995)
+1664 8 Heads in a Duffel Bag (1997)
+1665 Brother's Kiss, A (1997)
+1666 Ripe (1996)
+1667 Next Step, The (1995)
+1668 Wedding Bell Blues (1996)
+1669 MURDER and murder (1996)
+1670 Tainted (1998)
+1671 Further Gesture, A (1996)
+1672 Kika (1993)
+1673 Mirage (1995)
+1674 Mamma Roma (1962)
+1675 Sunchaser, The (1996)
+1676 War at Home, The (1996)
+1677 Sweet Nothing (1995)
+1678 Mat' i syn (1997)
+1679 B. Monkey (1998)
+1680 Sliding Doors (1998)
+1681 You So Crazy (1994)
+1682 Scream of Stone (Schrei aus Stein) (1991)
new file mode 100644
--- /dev/null
+++ b/multivariateGaussian.m
@@ -0,0 +1,22 @@
+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
\ No newline at end of file
new file mode 100644
--- /dev/null
+++ b/normalizeRatings.m
@@ -0,0 +1,16 @@
+function [Ynorm, Ymean] = normalizeRatings(Y, R)
+%NORMALIZERATINGS Preprocess data by subtracting mean rating for every 
+%movie (every row)
+%   [Ynorm, Ymean] = NORMALIZERATINGS(Y, R) normalized Y so that each movie
+%   has a rating of 0 on average, and returns the mean rating in Ymean.
+%
+
+[m, n] = size(Y);
+Ymean = zeros(m, 1);
+Ynorm = zeros(size(Y));
+for i = 1:m
+    Ymean(i) = mean(Y(i, R(i, :)));
+    Ynorm(i, R(i, :)) = Y(i, R(i, :)) - Ymean(i);
+end
+
+end
\ No newline at end of file
new file mode 100644
--- /dev/null
+++ b/selectThreshold.m
@@ -0,0 +1,46 @@
+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
+
+
+
+
+
+
+
+
+
+
+
+
+
+    % =============================================================
+
+    if F1 > bestF1
+       bestF1 = F1;
+       bestEpsilon = epsilon;
+    end
+end
+
+end
new file mode 100644
--- /dev/null
+++ b/submit.m
@@ -0,0 +1,348 @@
+function submit(partId)
+%SUBMIT Submit your code and output to the ml-class servers
+%   SUBMIT() will connect to the ml-class server and submit your solution
+
+  fprintf('==\n== [ml-class] Submitting Solutions | Programming Exercise %s\n==\n', ...
+          homework_id());
+  if ~exist('partId', 'var') || isempty(partId)
+    partId = promptPart();
+  end
+  
+  % Check valid partId
+  partNames = validParts();
+  if ~isValidPartId(partId)
+    fprintf('!! Invalid homework part selected.\n');
+    fprintf('!! Expected an integer from 1 to %d.\n', numel(partNames) + 1);
+    fprintf('!! Submission Cancelled\n');
+    return
+  end
+
+  [login password] = loginPrompt();
+  if isempty(login)
+    fprintf('!! Submission Cancelled\n');
+    return
+  end
+
+  fprintf('\n== Connecting to ml-class ... '); 
+  if exist('OCTAVE_VERSION') 
+    fflush(stdout);
+  end
+  
+  % Setup submit list
+  if partId == numel(partNames) + 1
+    submitParts = 1:numel(partNames);
+  else
+    submitParts = [partId];
+  end
+
+  for s = 1:numel(submitParts)
+    % Submit this part
+    partId = submitParts(s);
+    
+    % Get Challenge
+    [login, ch, signature] = getChallenge(login);
+    if isempty(login) || isempty(ch) || isempty(signature)
+      % Some error occured, error string in first return element.
+      fprintf('\n!! Error: %s\n\n', login);
+      return
+    end
+  
+    % Attempt Submission with Challenge
+    ch_resp = challengeResponse(login, password, ch);
+    [result, str] = submitSolution(login, ch_resp, partId, output(partId), ...
+                                 source(partId), signature);
+                                 
+    fprintf('\n== [ml-class] Submitted Homework %s - Part %d - %s\n', ...
+            homework_id(), partId, partNames{partId});
+    fprintf('== %s\n', strtrim(str));
+    if exist('OCTAVE_VERSION') 
+      fflush(stdout);
+    end
+  end
+  
+end
+
+% ================== CONFIGURABLES FOR EACH HOMEWORK ==================
+
+function id = homework_id() 
+  id = '8';
+end
+
+function [partNames] = validParts()
+  partNames = { 'Estimate Gaussian Parameters', ...
+                'Select Threshold' ...
+                'Collaborative Filtering Cost', ...
+                'Collaborative Filtering Gradient', ...
+                'Regularized Cost', ...
+                'Regularized Gradient' ...
+                };
+end
+
+function srcs = sources()
+  % Separated by part
+  srcs = { { 'estimateGaussian.m' }, ...
+           { 'selectThreshold.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           };
+end
+
+function out = output(partId)
+  % Random Test Cases
+  n_u = 3; n_m = 4; n = 5;
+  X = reshape(sin(1:n_m*n), n_m, n);
+  Theta = reshape(cos(1:n_u*n), n_u, n);
+  Y = reshape(sin(1:2:2*n_m*n_u), n_m, n_u);
+  R = Y > 0.5;
+  pval = [abs(Y(:)) ; 0.001; 1];
+  yval = [R(:) ; 1; 0];
+  params = [X(:); Theta(:)];
+  if partId == 1
+    [mu sigma2] = estimateGaussian(X);
+    out = sprintf('%0.5f ', [mu(:); sigma2(:)]);
+  elseif partId == 2
+    [bestEpsilon bestF1] = selectThreshold(yval, pval);
+    out = sprintf('%0.5f ', [bestEpsilon(:); bestF1(:)]);
+  elseif partId == 3
+    [J] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                       n, 0);
+    out = sprintf('%0.5f ', J(:));
+  elseif partId == 4
+    [J, grad] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                             n, 0);
+    out = sprintf('%0.5f ', grad(:));
+  elseif partId == 5
+    [J] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                       n, 1.5);
+    out = sprintf('%0.5f ', J(:));
+  elseif partId == 6
+    [J, grad] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                             n, 1.5);
+    out = sprintf('%0.5f ', grad(:));
+  end 
+end
+
+function url = challenge_url()
+  url = 'http://www.ml-class.org/course/homework/challenge';
+end
+
+function url = submit_url()
+  url = 'http://www.ml-class.org/course/homework/submit';
+end
+
+% ========================= CHALLENGE HELPERS =========================
+
+function src = source(partId)
+  src = '';
+  src_files = sources();
+  if partId <= numel(src_files)
+      flist = src_files{partId};
+      for i = 1:numel(flist)
+          fid = fopen(flist{i});
+          while ~feof(fid)
+            line = fgets(fid);
+            src = [src line];
+          end
+          fclose(fid);
+          src = [src '||||||||'];
+      end
+  end
+end
+
+function ret = isValidPartId(partId)
+  partNames = validParts();
+  ret = (~isempty(partId)) && (partId >= 1) && (partId <= numel(partNames) + 1);
+end
+
+function partId = promptPart()
+  fprintf('== Select which part(s) to submit:\n', ...
+          homework_id());
+  partNames = validParts();
+  srcFiles = sources();
+  for i = 1:numel(partNames)
+    fprintf('==   %d) %s [', i, partNames{i});
+    fprintf(' %s ', srcFiles{i}{:});
+    fprintf(']\n');
+  end
+  fprintf('==   %d) All of the above \n==\nEnter your choice [1-%d]: ', ...
+          numel(partNames) + 1, numel(partNames) + 1);
+  selPart = input('', 's');
+  partId = str2num(selPart);
+  if ~isValidPartId(partId)
+    partId = -1;
+  end
+end
+
+function [email,ch,signature] = getChallenge(email)
+  str = urlread(challenge_url(), 'post', {'email_address', email});
+
+  str = strtrim(str);
+  [email, str] = strtok (str, '|');
+  [ch, str] = strtok (str, '|');
+  [signature, str] = strtok (str, '|');
+end
+
+
+function [result, str] = submitSolution(email, ch_resp, part, output, ...
+                                        source, signature)
+
+  params = {'homework', homework_id(), ...
+            'part', num2str(part), ...
+            'email', email, ...
+            'output', output, ...
+            'source', source, ...
+            'challenge_response', ch_resp, ...
+            'signature', signature};
+
+  str = urlread(submit_url(), 'post', params);
+  
+  % Parse str to read for success / failure
+  result = 0;
+
+end
+
+% =========================== LOGIN HELPERS ===========================
+
+function [login password] = loginPrompt()
+  % Prompt for password
+  [login password] = basicPrompt();
+  
+  if isempty(login) || isempty(password)
+    login = []; password = [];
+  end
+end
+
+
+function [login password] = basicPrompt()
+  login = input('Login (Email address): ', 's');
+  password = input('Password: ', 's');
+end
+
+
+function [str] = challengeResponse(email, passwd, challenge)
+  salt = ')~/|]QMB3[!W`?OVt7qC"@+}';
+  str = sha1([challenge sha1([salt email passwd])]);
+  sel = randperm(numel(str));
+  sel = sort(sel(1:16));
+  str = str(sel);
+end
+
+
+% =============================== SHA-1 ================================
+
+function hash = sha1(str)
+  
+  % Initialize variables
+  h0 = uint32(1732584193);
+  h1 = uint32(4023233417);
+  h2 = uint32(2562383102);
+  h3 = uint32(271733878);
+  h4 = uint32(3285377520);
+  
+  % Convert to word array
+  strlen = numel(str);
+
+  % Break string into chars and append the bit 1 to the message
+  mC = [double(str) 128];
+  mC = [mC zeros(1, 4-mod(numel(mC), 4), 'uint8')];
+  
+  numB = strlen * 8;
+  if exist('idivide')
+    numC = idivide(uint32(numB + 65), 512, 'ceil');
+  else
+    numC = ceil(double(numB + 65)/512);
+  end
+  numW = numC * 16;
+  mW = zeros(numW, 1, 'uint32');
+  
+  idx = 1;
+  for i = 1:4:strlen + 1
+    mW(idx) = bitor(bitor(bitor( ...
+                  bitshift(uint32(mC(i)), 24), ...
+                  bitshift(uint32(mC(i+1)), 16)), ...
+                  bitshift(uint32(mC(i+2)), 8)), ...
+                  uint32(mC(i+3)));
+    idx = idx + 1;
+  end
+  
+  % Append length of message
+  mW(numW - 1) = uint32(bitshift(uint64(numB), -32));
+  mW(numW) = uint32(bitshift(bitshift(uint64(numB), 32), -32));
+
+  % Process the message in successive 512-bit chs
+  for cId = 1 : double(numC)
+    cSt = (cId - 1) * 16 + 1;
+    cEnd = cId * 16;
+    ch = mW(cSt : cEnd);
+    
+    % Extend the sixteen 32-bit words into eighty 32-bit words
+    for j = 17 : 80
+      ch(j) = ch(j - 3);
+      ch(j) = bitxor(ch(j), ch(j - 8));
+      ch(j) = bitxor(ch(j), ch(j - 14));
+      ch(j) = bitxor(ch(j), ch(j - 16));
+      ch(j) = bitrotate(ch(j), 1);
+    end
+  
+    % Initialize hash value for this ch
+    a = h0;
+    b = h1;
+    c = h2;
+    d = h3;
+    e = h4;
+    
+    % Main loop
+    for i = 1 : 80
+      if(i >= 1 && i <= 20)
+        f = bitor(bitand(b, c), bitand(bitcmp(b), d));
+        k = uint32(1518500249);
+      elseif(i >= 21 && i <= 40)
+        f = bitxor(bitxor(b, c), d);
+        k = uint32(1859775393);
+      elseif(i >= 41 && i <= 60)
+        f = bitor(bitor(bitand(b, c), bitand(b, d)), bitand(c, d));
+        k = uint32(2400959708);
+      elseif(i >= 61 && i <= 80)
+        f = bitxor(bitxor(b, c), d);
+        k = uint32(3395469782);
+      end
+      
+      t = bitrotate(a, 5);
+      t = bitadd(t, f);
+      t = bitadd(t, e);
+      t = bitadd(t, k);
+      t = bitadd(t, ch(i));
+      e = d;
+      d = c;
+      c = bitrotate(b, 30);
+      b = a;
+      a = t;
+      
+    end
+    h0 = bitadd(h0, a);
+    h1 = bitadd(h1, b);
+    h2 = bitadd(h2, c);
+    h3 = bitadd(h3, d);
+    h4 = bitadd(h4, e);
+
+  end
+
+  hash = reshape(dec2hex(double([h0 h1 h2 h3 h4]), 8)', [1 40]);
+  
+  hash = lower(hash);
+
+end
+
+function ret = bitadd(iA, iB)
+  ret = double(iA) + double(iB);
+  ret = bitset(ret, 33, 0);
+  ret = uint32(ret);
+end
+
+function ret = bitrotate(iA, places)
+  t = bitshift(iA, places - 32);
+  ret = bitshift(iA, places);
+  ret = bitor(ret, t);
+end
new file mode 100644
--- /dev/null
+++ b/submitWeb.m
@@ -0,0 +1,365 @@
+function submitWeb(partId)
+%SUBMITWEB Generates a base64 encoded string for web-based submissions
+%   SUBMITWEB() will generate a base64 encoded string so that you can submit your
+%   solutions via a web form
+
+  fprintf('==\n== [ml-class] Submitting Solutions | Programming Exercise %s\n==\n', ...
+          homework_id());
+  if ~exist('partId', 'var') || isempty(partId)
+    partId = promptPart();
+  end
+  
+  % Check valid partId
+  partNames = validParts();
+  if ~isValidPartId(partId)
+    fprintf('!! Invalid homework part selected.\n');
+    fprintf('!! Expected an integer from 1 to %d.\n', numel(partNames));
+    fprintf('!! Submission Cancelled\n');
+    return
+  end
+
+  [login] = loginPrompt();
+  if isempty(login)
+    fprintf('!! Submission Cancelled\n');
+    return
+  end
+  
+  [result] = submitSolution(login, partId, output(partId), ...
+                            source(partId));
+  result = base64encode(result);
+
+  fprintf('\nSave as submission file [submit_ex%s_part%d.txt]: ', ...
+          homework_id(), partId);
+  saveAsFile = input('', 's');
+  if (isempty(saveAsFile))
+    saveAsFile = sprintf('submit_ex%s_part%d.txt', homework_id(), partId);
+  end
+
+  fid = fopen(saveAsFile, 'w');
+  if (fid)
+    fwrite(fid, result);
+    fclose(fid);
+    fprintf('\nSaved your solutions to %s.\n\n', saveAsFile);
+    fprintf(['You can now submit your solutions through the web \n' ...
+             'form in the programming exercises. Select the corresponding \n' ...
+             'programming exercise to access the form.\n']);
+
+  else
+    fprintf('Unable to save to %s\n\n', saveAsFile);
+    fprintf(['You can create a submission file by saving the \n' ...
+             'following text in a file: (press enter to continue)\n\n']);
+    pause;
+    fprintf(result);
+  end                  
+
+end
+
+% ================== CONFIGURABLES FOR EACH HOMEWORK ==================
+
+
+function id = homework_id() 
+  id = '8';
+end
+
+function [partNames] = validParts()
+  partNames = { 'Estimate Gaussian Parameters', ...
+                'Select Threshold' ...
+                'Collaborative Filtering Cost', ...
+                'Collaborative Filtering Gradient', ...
+                'Regularized Cost', ...
+                'Regularized Gradient' ...
+                };
+end
+
+function srcs = sources()
+  % Separated by part
+  srcs = { { 'estimateGaussian.m' }, ...
+           { 'selectThreshold.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           { 'cofiCostFunc.m' }, ...
+           };
+end
+
+function out = output(partId)
+  % Random Test Cases
+  n_u = 3; n_m = 4; n = 5;
+  X = reshape(sin(1:n_m*n), n_m, n);
+  Theta = reshape(cos(1:n_u*n), n_u, n);
+  Y = reshape(sin(1:2:2*n_m*n_u), n_m, n_u);
+  R = Y > 0.5;
+  pval = [abs(Y(:)) ; 0.001; 1];
+  yval = [R(:) ; 1; 0];
+  params = [X(:); Theta(:)];
+  if partId == 1
+    [mu sigma2] = estimateGaussian(X);
+    out = sprintf('%0.5f ', [mu(:); sigma2(:)]);
+  elseif partId == 2
+    [bestEpsilon bestF1] = selectThreshold(yval, pval);
+    out = sprintf('%0.5f ', [bestEpsilon(:); bestF1(:)]);
+  elseif partId == 3
+    [J] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                       n, 0);
+    out = sprintf('%0.5f ', J(:));
+  elseif partId == 4
+    [J, grad] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                             n, 0);
+    out = sprintf('%0.5f ', grad(:));
+  elseif partId == 5
+    [J] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                       n, 1.5);
+    out = sprintf('%0.5f ', J(:));
+  elseif partId == 6
+    [J, grad] = cofiCostFunc(params, Y, R, n_u, n_m, ...
+                             n, 1.5);
+    out = sprintf('%0.5f ', grad(:));
+  end 
+end
+
+
+% ========================= SUBMIT HELPERS =========================
+
+function src = source(partId)
+  src = '';
+  src_files = sources();
+  if partId <= numel(src_files)
+      flist = src_files{partId};
+      for i = 1:numel(flist)
+          fid = fopen(flist{i});
+          while ~feof(fid)
+            line = fgets(fid);
+            src = [src line];
+          end
+          fclose(fid);
+          src = [src '||||||||'];
+      end
+  end
+end
+
+function ret = isValidPartId(partId)
+  partNames = validParts();
+  ret = (~isempty(partId)) && (partId >= 1) && (partId <= numel(partNames));
+end
+
+function partId = promptPart()
+  fprintf('== Select which part(s) to submit:\n', ...
+          homework_id());
+  partNames = validParts();
+  srcFiles = sources();
+  for i = 1:numel(partNames)
+    fprintf('==   %d) %s [', i, partNames{i});
+    fprintf(' %s ', srcFiles{i}{:});
+    fprintf(']\n');
+  end
+  fprintf('\nEnter your choice [1-%d]: ', ...
+          numel(partNames));
+  selPart = input('', 's');
+  partId = str2num(selPart);
+  if ~isValidPartId(partId)
+    partId = -1;
+  end
+end
+
+
+function [result, str] = submitSolution(email, part, output, source)
+
+  result = ['a:5:{' ...
+            p_s('homework') p_s64(homework_id()) ...
+            p_s('part') p_s64(part) ...
+            p_s('email') p_s64(email) ...
+            p_s('output') p_s64(output) ...
+            p_s('source') p_s64(source) ...
+            '}'];
+
+end
+
+function s = p_s(str)
+   s = ['s:' num2str(numel(str)) ':"' str '";'];
+end
+
+function s = p_s64(str)
+   str = base64encode(str, '');
+   s = ['s:' num2str(numel(str)) ':"' str '";'];
+end
+
+% =========================== LOGIN HELPERS ===========================
+
+function [login] = loginPrompt()
+  % Prompt for password
+  [login] = basicPrompt();
+end
+
+
+function [login] = basicPrompt()
+  login = input('Login (Email address): ', 's');
+end
+
+
+% =========================== Base64 Encoder ============================
+% Thanks to Peter John Acklam
+%
+
+function y = base64encode(x, eol)
+%BASE64ENCODE Perform base64 encoding on a string.
+%
+%   BASE64ENCODE(STR, EOL) encode the given string STR.  EOL is the line ending
+%   sequence to use; it is optional and defaults to '\n' (ASCII decimal 10).
+%   The returned encoded string is broken into lines of no more than 76
+%   characters each, and each line will end with EOL unless it is empty.  Let
+%   EOL be empty if you do not want the encoded string broken into lines.
+%
+%   STR and EOL don't have to be strings (i.e., char arrays).  The only
+%   requirement is that they are vectors containing values in the range 0-255.
+%
+%   This function may be used to encode strings into the Base64 encoding
+%   specified in RFC 2045 - MIME (Multipurpose Internet Mail Extensions).  The
+%   Base64 encoding is designed to represent arbitrary sequences of octets in a
+%   form that need not be humanly readable.  A 65-character subset
+%   ([A-Za-z0-9+/=]) of US-ASCII is used, enabling 6 bits to be represented per
+%   printable character.
+%
+%   Examples
+%   --------
+%
+%   If you want to encode a large file, you should encode it in chunks that are
+%   a multiple of 57 bytes.  This ensures that the base64 lines line up and
+%   that you do not end up with padding in the middle.  57 bytes of data fills
+%   one complete base64 line (76 == 57*4/3):
+%
+%   If ifid and ofid are two file identifiers opened for reading and writing,
+%   respectively, then you can base64 encode the data with
+%
+%      while ~feof(ifid)
+%         fwrite(ofid, base64encode(fread(ifid, 60*57)));
+%      end
+%
+%   or, if you have enough memory,
+%
+%      fwrite(ofid, base64encode(fread(ifid)));
+%
+%   See also BASE64DECODE.
+
+%   Author:      Peter John Acklam
+%   Time-stamp:  2004-02-03 21:36:56 +0100
+%   E-mail:      pjacklam@online.no
+%   URL:         http://home.online.no/~pjacklam
+
+   if isnumeric(x)
+      x = num2str(x);
+   end
+
+   % make sure we have the EOL value
+   if nargin < 2
+      eol = sprintf('\n');
+   else
+      if sum(size(eol) > 1) > 1
+         error('EOL must be a vector.');
+      end
+      if any(eol(:) > 255)
+         error('EOL can not contain values larger than 255.');
+      end
+   end
+
+   if sum(size(x) > 1) > 1
+      error('STR must be a vector.');
+   end
+
+   x   = uint8(x);
+   eol = uint8(eol);
+
+   ndbytes = length(x);                 % number of decoded bytes
+   nchunks = ceil(ndbytes / 3);         % number of chunks/groups
+   nebytes = 4 * nchunks;               % number of encoded bytes
+
+   % add padding if necessary, to make the length of x a multiple of 3
+   if rem(ndbytes, 3)
+      x(end+1 : 3*nchunks) = 0;
+   end
+
+   x = reshape(x, [3, nchunks]);        % reshape the data
+   y = repmat(uint8(0), 4, nchunks);    % for the encoded data
+
+   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+   % Split up every 3 bytes into 4 pieces
+   %
+   %    aaaaaabb bbbbcccc ccdddddd
+   %
+   % to form
+   %
+   %    00aaaaaa 00bbbbbb 00cccccc 00dddddd
+   %
+   y(1,:) = bitshift(x(1,:), -2);                  % 6 highest bits of x(1,:)
+
+   y(2,:) = bitshift(bitand(x(1,:), 3), 4);        % 2 lowest bits of x(1,:)
+   y(2,:) = bitor(y(2,:), bitshift(x(2,:), -4));   % 4 highest bits of x(2,:)
+
+   y(3,:) = bitshift(bitand(x(2,:), 15), 2);       % 4 lowest bits of x(2,:)
+   y(3,:) = bitor(y(3,:), bitshift(x(3,:), -6));   % 2 highest bits of x(3,:)
+
+   y(4,:) = bitand(x(3,:), 63);                    % 6 lowest bits of x(3,:)
+
+   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+   % Now perform the following mapping
+   %
+   %   0  - 25  ->  A-Z
+   %   26 - 51  ->  a-z
+   %   52 - 61  ->  0-9
+   %   62       ->  +
+   %   63       ->  /
+   %
+   % We could use a mapping vector like
+   %
+   %   ['A':'Z', 'a':'z', '0':'9', '+/']
+   %
+   % but that would require an index vector of class double.
+   %
+   z = repmat(uint8(0), size(y));
+   i =           y <= 25;  z(i) = 'A'      + double(y(i));
+   i = 26 <= y & y <= 51;  z(i) = 'a' - 26 + double(y(i));
+   i = 52 <= y & y <= 61;  z(i) = '0' - 52 + double(y(i));
+   i =           y == 62;  z(i) = '+';
+   i =           y == 63;  z(i) = '/';
+   y = z;
+
+   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+   % Add padding if necessary.
+   %
+   npbytes = 3 * nchunks - ndbytes;     % number of padding bytes
+   if npbytes
+      y(end-npbytes+1 : end) = '=';     % '=' is used for padding
+   end
+
+   if isempty(eol)
+
+      % reshape to a row vector
+      y = reshape(y, [1, nebytes]);
+
+   else
+
+      nlines = ceil(nebytes / 76);      % number of lines
+      neolbytes = length(eol);          % number of bytes in eol string
+
+      % pad data so it becomes a multiple of 76 elements
+      y = [y(:) ; zeros(76 * nlines - numel(y), 1)];
+      y(nebytes + 1 : 76 * nlines) = 0;
+      y = reshape(y, 76, nlines);
+
+      % insert eol strings
+      eol = eol(:);
+      y(end + 1 : end + neolbytes, :) = eol(:, ones(1, nlines));
+
+      % remove padding, but keep the last eol string
+      m = nebytes + neolbytes * (nlines - 1);
+      n = (76+neolbytes)*nlines - neolbytes;
+      y(m+1 : n) = '';
+
+      % extract and reshape to row vector
+      y = reshape(y, 1, m+neolbytes);
+
+   end
+
+   % output is a character array
+   y = char(y);
+
+end
new file mode 100644
--- /dev/null
+++ b/visualizeFit.m
@@ -0,0 +1,20 @@
+function visualizeFit(X, mu, sigma2)
+%VISUALIZEFIT Visualize the dataset and its estimated distribution.
+%   VISUALIZEFIT(X, p, mu, sigma2) This visualization shows you the 
+%   probability density function of the Gaussian distribution. Each example
+%   has a location (x1, x2) that depends on its feature values.
+%
+
+[X1,X2] = meshgrid(0:.5:35); 
+Z = multivariateGaussian([X1(:) X2(:)],mu,sigma2);
+Z = reshape(Z,size(X1));
+
+plot(X(:, 1), X(:, 2),'bx');
+hold on;
+% Do not plot if there are infinities
+if (sum(isinf(Z)) == 0)
+    contour(X1, X2, Z, 10.^(-20:3:0)');
+end
+hold off;
+
+end
\ No newline at end of file