added for verification

Anomaly Detection and Recommender Systems/checkCostFunction.m

+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 cost function 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

Anomaly Detection and Recommender Systems/cofiCostFunc.m

+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
+%
+
+
+J=sum(sum(((X*(Theta')-Y).^2).*R))/2+lambda*(sum(sum(Theta.^2)))/2 +lambda*(sum(sum(X.^2)))/2;
+for i=1:num_movies
+idx = find(R(i, :)==1);
+Thetatemp = Theta(idx, :);
+Ytemp = Y(i, idx);
+
+X_grad(i, :) = (X(i, :) * (Thetatemp') - Ytemp) * Thetatemp;
+end
+for i=1:num_users
+idx = find(R(:,i)==1);
+Xtemp = X(idx, :);
+Ytemp = Y(idx, i);
+
+Theta_grad(i, :) = (Theta(i, :) * (Xtemp') - Ytemp') * Xtemp;
+end
+
+
+X_grad=X_grad+lambda.*X;
+Theta_grad=Theta_grad+lambda.*Theta;
+
+
+
+
+
+
+% =============================================================
+
+grad = [X_grad(:); Theta_grad(:)];
+
+end

Anomaly Detection and Recommender Systems/computeNumericalGradient.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

Anomaly Detection and Recommender Systems/estimateGaussian.m

+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=(sum(X))'/m;
+for i=1:n
+    for j=1:m
+sigma2(i)=sigma2(i)+ (X(j,i)-mu(i)).^2;
+    end
+end
+sigma2=sigma2/m;
+
+
+
+
+
+
+
+
+% =============================================================
+
+
+end

Anomaly Detection and Recommender Systems/ex8.m

+%% 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');
+fprintf('   (you should see a Best F1 value of  0.875000)\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('   (you should see a value epsilon of about 1.38e-18)\n');
+fprintf('   (you should see a Best F1 value of 0.615385)\n');
+fprintf('# Outliers found: %d\n\n', sum(p < epsilon));

Anomaly Detection and Recommender Systems/ex8_cofi.m

+%% 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, Ynorm, 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

Anomaly Detection and Recommender Systems/fmincg.m

+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');

Anomaly Detection and Recommender Systems/lib/jsonlab/AUTHORS.txt

+The author of "jsonlab" toolbox is Qianqian Fang. Qianqian
+is currently an Assistant Professor at Massachusetts General Hospital, 
+Harvard Medical School.
+
+Address: Martinos Center for Biomedical Imaging, 
+         Massachusetts General Hospital, 
+         Harvard Medical School
+         Bldg 149, 13th St, Charlestown, MA 02129, USA
+URL: http://nmr.mgh.harvard.edu/~fangq/
+Email: <fangq at nmr.mgh.harvard.edu> or <fangqq at gmail.com>
+
+
+The script loadjson.m was built upon previous works by
+
+- Nedialko Krouchev: http://www.mathworks.com/matlabcentral/fileexchange/25713
+       date: 2009/11/02
+- François Glineur: http://www.mathworks.com/matlabcentral/fileexchange/23393
+       date: 2009/03/22
+- Joel Feenstra: http://www.mathworks.com/matlabcentral/fileexchange/20565
+       date: 2008/07/03
+
+
+This toolbox contains patches submitted by the following contributors:
+
+- Blake Johnson <bjohnso at bbn.com>
+  part of revision 341
+
+- Niclas Borlin <Niclas.Borlin at cs.umu.se>
+  various fixes in revision 394, including
+  - loadjson crashes for all-zero sparse matrix.
+  - loadjson crashes for empty sparse matrix.
+  - Non-zero size of 0-by-N and N-by-0 empty matrices is lost after savejson/loadjson.
+  - loadjson crashes for sparse real column vector.
+  - loadjson crashes for sparse complex column vector.
+  - Data is corrupted by savejson for sparse real row vector.
+  - savejson crashes for sparse complex row vector. 
+
+- Yul Kang <yul.kang.on at gmail.com>
+  patches for svn revision 415.
+  - savejson saves an empty cell array as [] instead of null
+  - loadjson differentiates an empty struct from an empty array

Anomaly Detection and Recommender Systems/lib/jsonlab/ChangeLog.txt

+============================================================================
+
+   JSONlab - a toolbox to encode/decode JSON/UBJSON files in MATLAB/Octave
+
+----------------------------------------------------------------------------
+
+JSONlab ChangeLog (key features marked by *):
+
+== JSONlab 1.0 (codename: Optimus - Final), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2015/01/02 polish help info for all major functions, update examples, finalize 1.0
+ 2014/12/19 fix a bug to strictly respect NoRowBracket in savejson
+
+== JSONlab 1.0.0-RC2 (codename: Optimus - RC2), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/11/22 show progress bar in loadjson ('ShowProgress') 
+ 2014/11/17 add Compact option in savejson to output compact JSON format ('Compact')
+ 2014/11/17 add FastArrayParser in loadjson to specify fast parser applicable levels
+ 2014/09/18 start official github mirror: https://github.com/fangq/jsonlab
+
+== JSONlab 1.0.0-RC1 (codename: Optimus - RC1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/09/17  fix several compatibility issues when running on octave versions 3.2-3.8
+ 2014/09/17  support 2D cell and struct arrays in both savejson and saveubjson
+ 2014/08/04  escape special characters in a JSON string
+ 2014/02/16  fix a bug when saving ubjson files
+
+== JSONlab 0.9.9 (codename: Optimus - beta), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/01/22  use binary read and write in saveubjson and loadubjson
+
+== JSONlab 0.9.8-1 (codename: Optimus - alpha update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2013/10/07 better round-trip conservation for empty arrays and structs (patch submitted by Yul Kang)
+
+== JSONlab 0.9.8 (codename: Optimus - alpha), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+ 2013/08/23 *universal Binary JSON (UBJSON) support, including both saveubjson and loadubjson
+
+== JSONlab 0.9.1 (codename: Rodimus, update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+ 2012/12/18 *handling of various empty and sparse matrices (fixes submitted by Niclas Borlin)
+
+== JSONlab 0.9.0 (codename: Rodimus), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/06/17 *new format for an invalid leading char, unpacking hex code in savejson
+ 2012/06/01  support JSONP in savejson
+ 2012/05/25  fix the empty cell bug (reported by Cyril Davin)
+ 2012/04/05  savejson can save to a file (suggested by Patrick Rapin)
+
+== JSONlab 0.8.1 (codename: Sentiel, Update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/02/28  loadjson quotation mark escape bug, see http://bit.ly/yyk1nS
+ 2012/01/25  patch to handle root-less objects, contributed by Blake Johnson
+
+== JSONlab 0.8.0 (codename: Sentiel), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/01/13 *speed up loadjson by 20 fold when parsing large data arrays in matlab
+ 2012/01/11  remove row bracket if an array has 1 element, suggested by Mykel Kochenderfer
+ 2011/12/22 *accept sequence of 'param',value input in savejson and loadjson
+ 2011/11/18  fix struct array bug reported by Mykel Kochenderfer
+
+== JSONlab 0.5.1 (codename: Nexus Update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2011/10/21  fix a bug in loadjson, previous code does not use any of the acceleration
+ 2011/10/20  loadjson supports JSON collections - concatenated JSON objects
+
+== JSONlab 0.5.0 (codename: Nexus), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2011/10/16  package and release jsonlab 0.5.0
+ 2011/10/15 *add json demo and regression test, support cpx numbers, fix double quote bug
+ 2011/10/11 *speed up readjson dramatically, interpret _Array* tags, show data in root level
+ 2011/10/10  create jsonlab project, start jsonlab website, add online documentation
+ 2011/10/07 *speed up savejson by 25x using sprintf instead of mat2str, add options support
+ 2011/10/06 *savejson works for structs, cells and arrays
+ 2011/09/09  derive loadjson from JSON parser from MATLAB Central, draft savejson.m

Anomaly Detection and Recommender Systems/lib/jsonlab/LICENSE_BSD.txt

+Copyright 2011-2015 Qianqian Fang <fangq at nmr.mgh.harvard.edu>. All rights reserved.
+
+Redistribution and use in source and binary forms, with or without modification, are
+permitted provided that the following conditions are met:
+
+   1. Redistributions of source code must retain the above copyright notice, this list of
+      conditions and the following disclaimer.
+
+   2. Redistributions in binary form must reproduce the above copyright notice, this list
+      of conditions and the following disclaimer in the documentation and/or other materials
+      provided with the distribution.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS ''AS IS'' AND ANY EXPRESS OR IMPLIED
+WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND
+FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS 
+OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
+SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
+ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
+ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+The views and conclusions contained in the software and documentation are those of the
+authors and should not be interpreted as representing official policies, either expressed
+or implied, of the copyright holders.

Anomaly Detection and Recommender Systems/lib/jsonlab/jsonopt.m

+function val=jsonopt(key,default,varargin)
+%
+% val=jsonopt(key,default,optstruct)
+%
+% setting options based on a struct. The struct can be produced
+% by varargin2struct from a list of 'param','value' pairs
+%
+% authors:Qianqian Fang (fangq<at> nmr.mgh.harvard.edu)
+%
+% $Id: loadjson.m 371 2012-06-20 12:43:06Z fangq $
+%
+% input:
+%      key: a string with which one look up a value from a struct
+%      default: if the key does not exist, return default
+%      optstruct: a struct where each sub-field is a key 
+%
+% output:
+%      val: if key exists, val=optstruct.key; otherwise val=default
+%
+% license:
+%     BSD, see LICENSE_BSD.txt files for details
+%
+% -- this function is part of jsonlab toolbox (http://iso2mesh.sf.net/cgi-bin/index.cgi?jsonlab)
+% 
+
+val=default;
+if(nargin<=2) return; end
+opt=varargin{1};
+if(isstruct(opt) && isfield(opt,key))
+    val=getfield(opt,key);
+end
+

Anomaly Detection and Recommender Systems/lib/jsonlab/mergestruct.m

+function s=mergestruct(s1,s2)
+%
+% s=mergestruct(s1,s2)
+%
+% merge two struct objects into one
+%
+% authors:Qianqian Fang (fangq<at> nmr.mgh.harvard.edu)
+% date: 2012/12/22
+%
+% input:
+%      s1,s2: a struct object, s1 and s2 can not be arrays
+%
+% output:
+%      s: the merged struct object. fields in s1 and s2 will be combined in s.
+%
+% license:
+%     BSD, see LICENSE_BSD.txt files for details 
+%
+% -- this function is part of jsonlab toolbox (http://iso2mesh.sf.net/cgi-bin/index.cgi?jsonlab)
+%
+
+if(~isstruct(s1) || ~isstruct(s2))
+    error('input parameters contain non-struct');
+end
+if(length(s1)>1 || length(s2)>1)
+    error('can not merge struct arrays');
+end
+fn=fieldnames(s2);
+s=s1;
+for i=1:length(fn)              
+    s=setfield(s,fn{i},getfield(s2,fn{i}));
+end
+

Anomaly Detection and Recommender Systems/lib/jsonlab/varargin2struct.m

+function opt=varargin2struct(varargin)
+%
+% opt=varargin2struct('param1',value1,'param2',value2,...)
+%   or
+% opt=varargin2struct(...,optstruct,...)
+%
+% convert a series of input parameters into a structure
+%
+% authors:Qianqian Fang (fangq<at> nmr.mgh.harvard.edu)
+% date: 2012/12/22
+%
+% input:
+%      'param', value: the input parameters should be pairs of a string and a value
+%       optstruct: if a parameter is a struct, the fields will be merged to the output struct
+%
+% output:
+%      opt: a struct where opt.param1=value1, opt.param2=value2 ...
+%
+% license:
+%     BSD, see LICENSE_BSD.txt files for details 
+%
+% -- this function is part of jsonlab toolbox (http://iso2mesh.sf.net/cgi-bin/index.cgi?jsonlab)
+%
+
+len=length(varargin);
+opt=struct;
+if(len==0) return; end
+i=1;
+while(i<=len)
+    if(isstruct(varargin{i}))
+        opt=mergestruct(opt,varargin{i});
+    elseif(ischar(varargin{i}) && i<len)
+        opt=setfield(opt,varargin{i},varargin{i+1});
+        i=i+1;
+    else
+        error('input must be in the form of ...,''name'',value,... pairs or structs');
+    end
+    i=i+1;
+end
+

Anomaly Detection and Recommender Systems/lib/makeValidFieldName.m

+function str = makeValidFieldName(str)
+% From MATLAB doc: field names must begin with a letter, which may be
+% followed by any combination of letters, digits, and underscores.
+% Invalid characters will be converted to underscores, and the prefix
+% "x0x[Hex code]_" will be added if the first character is not a letter.
+    isoct=exist('OCTAVE_VERSION','builtin');
+    pos=regexp(str,'^[^A-Za-z]','once');
+    if(~isempty(pos))
+        if(~isoct)
+            str=regexprep(str,'^([^A-Za-z])','x0x${sprintf(''%X'',unicode2native($1))}_','once');
+        else
+            str=sprintf('x0x%X_%s',char(str(1)),str(2:end));
+        end
+    end
+    if(isempty(regexp(str,'[^0-9A-Za-z_]', 'once' ))) return;  end
+    if(~isoct)
+        str=regexprep(str,'([^0-9A-Za-z_])','_0x${sprintf(''%X'',unicode2native($1))}_');
+    else
+        pos=regexp(str,'[^0-9A-Za-z_]');
+        if(isempty(pos)) return; end
+        str0=str;
+        pos0=[0 pos(:)' length(str)];
+        str='';
+        for i=1:length(pos)
+            str=[str str0(pos0(i)+1:pos(i)-1) sprintf('_0x%X_',str0(pos(i)))];
+        end
+        if(pos(end)~=length(str))
+            str=[str str0(pos0(end-1)+1:pos0(end))];
+        end
+    end

Anomaly Detection and Recommender Systems/lib/submitWithConfiguration.m

+function submitWithConfiguration(conf)
+  addpath('./lib/jsonlab');
+
+  parts = parts(conf);
+
+  fprintf('== Submitting solutions | %s...\n', conf.itemName);
+
+  tokenFile = 'token.mat';
+  if exist(tokenFile, 'file')
+    load(tokenFile);
+    [email token] = promptToken(email, token, tokenFile);
+  else
+    [email token] = promptToken('', '', tokenFile);
+  end
+
+  if isempty(token)
+    fprintf('!! Submission Cancelled\n');
+    return
+  end
+
+  try
+    response = submitParts(conf, email, token, parts);
+  catch
+    e = lasterror();
+    fprintf('\n!! Submission failed: %s\n', e.message);
+    fprintf('\n\nFunction: %s\nFileName: %s\nLineNumber: %d\n', ...
+      e.stack(1,1).name, e.stack(1,1).file, e.stack(1,1).line);
+    fprintf('\nPlease correct your code and resubmit.\n');
+    return
+  end
+
+  if isfield(response, 'errorMessage')
+    fprintf('!! Submission failed: %s\n', response.errorMessage);
+  elseif isfield(response, 'errorCode')
+    fprintf('!! Submission failed: %s\n', response.message);
+  else
+    showFeedback(parts, response);
+    save(tokenFile, 'email', 'token');
+  end
+end
+
+function [email token] = promptToken(email, existingToken, tokenFile)
+  if (~isempty(email) && ~isempty(existingToken))
+    prompt = sprintf( ...
+      'Use token from last successful submission (%s)? (Y/n): ', ...
+      email);
+    reenter = input(prompt, 's');
+
+    if (isempty(reenter) || reenter(1) == 'Y' || reenter(1) == 'y')
+      token = existingToken;
+      return;
+    else
+      delete(tokenFile);
+    end
+  end
+  email = input('Login (email address): ', 's');
+  token = input('Token: ', 's');
+end
+
+function isValid = isValidPartOptionIndex(partOptions, i)
+  isValid = (~isempty(i)) && (1 <= i) && (i <= numel(partOptions));
+end
+
+function response = submitParts(conf, email, token, parts)
+  body = makePostBody(conf, email, token, parts);
+  submissionUrl = submissionUrl();
+
+  responseBody = getResponse(submissionUrl, body);
+  jsonResponse = validateResponse(responseBody);
+  response = loadjson(jsonResponse);
+end
+
+function body = makePostBody(conf, email, token, parts)
+  bodyStruct.assignmentSlug = conf.assignmentSlug;
+  bodyStruct.submitterEmail = email;
+  bodyStruct.secret = token;
+  bodyStruct.parts = makePartsStruct(conf, parts);
+
+  opt.Compact = 1;
+  body = savejson('', bodyStruct, opt);
+end
+
+function partsStruct = makePartsStruct(conf, parts)
+  for part = parts
+    partId = part{:}.id;
+    fieldName = makeValidFieldName(partId);
+    outputStruct.output = conf.output(partId);
+    partsStruct.(fieldName) = outputStruct;
+  end
+end
+
+function [parts] = parts(conf)
+  parts = {};
+  for partArray = conf.partArrays
+    part.id = partArray{:}{1};
+    part.sourceFiles = partArray{:}{2};
+    part.name = partArray{:}{3};
+    parts{end + 1} = part;
+  end
+end
+
+function showFeedback(parts, response)
+  fprintf('== \n');
+  fprintf('== %43s | %9s | %-s\n', 'Part Name', 'Score', 'Feedback');
+  fprintf('== %43s | %9s | %-s\n', '---------', '-----', '--------');
+  for part = parts
+    score = '';
+    partFeedback = '';
+    partFeedback = response.partFeedbacks.(makeValidFieldName(part{:}.id));
+    partEvaluation = response.partEvaluations.(makeValidFieldName(part{:}.id));
+    score = sprintf('%d / %3d', partEvaluation.score, partEvaluation.maxScore);
+    fprintf('== %43s | %9s | %-s\n', part{:}.name, score, partFeedback);
+  end
+  evaluation = response.evaluation;
+  totalScore = sprintf('%d / %d', evaluation.score, evaluation.maxScore);
+  fprintf('==                                   --------------------------------\n');
+  fprintf('== %43s | %9s | %-s\n', '', totalScore, '');
+  fprintf('== \n');
+end
+
+% use urlread or curl to send submit results to the grader and get a response
+function response = getResponse(url, body)
+% try using urlread() and a secure connection
+  params = {'jsonBody', body};
+  [response, success] = urlread(url, 'post', params);
+
+  if (success == 0)
+    % urlread didn't work, try curl & the peer certificate patch
+    if ispc
+      % testing note: use 'jsonBody =' for a test case
+      json_command = sprintf('echo jsonBody=%s | curl -k -X POST -d @- %s', body, url);
+    else
+      % it's linux/OS X, so use the other form
+      json_command = sprintf('echo ''jsonBody=%s'' | curl -k -X POST -d @- %s', body, url);
+    end
+    % get the response body for the peer certificate patch method
+    [code, response] = system(json_command);
+    % test the success code
+    if (code ~= 0)
+      fprintf('[error] submission with curl() was not successful\n');
+    end
+  end
+end
+
+% validate the grader's response
+function response = validateResponse(resp)
+  % test if the response is json or an HTML page
+  isJson = length(resp) > 0 && resp(1) == '{';
+  isHtml = findstr(lower(resp), '<html');
+
+  if (isJson)
+    response = resp;
+  elseif (isHtml)
+    % the response is html, so it's probably an error message
+    printHTMLContents(resp);
+    error('Grader response is an HTML message');
+  else
+    error('Grader sent no response');
+  end
+end
+
+% parse a HTML response and print it's contents
+function printHTMLContents(response)
+  strippedResponse = regexprep(response, '<[^>]+>', ' ');
+  strippedResponse = regexprep(strippedResponse, '[\t ]+', ' ');
+  fprintf(strippedResponse);
+end
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% Service configuration
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+function submissionUrl = submissionUrl()
+  submissionUrl = 'https://www-origin.coursera.org/api/onDemandProgrammingImmediateFormSubmissions.v1';
+end

Anomaly Detection and Recommender Systems/loadMovieList.m

+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

Anomaly Detection and Recommender Systems/multivariateGaussian.m

+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

Anomaly Detection and Recommender Systems/normalizeRatings.m

+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
+    idx = find(R(i, :) == 1);
+    Ymean(i) = mean(Y(i, idx));
+    Ynorm(i, idx) = Y(i, idx) - Ymean(i);
+end
+
+end

Anomaly Detection and Recommender Systems/selectThreshold.m

+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
+
+
+
+tp=sum(floor(((pval<epsilon)+yval)/2));
+
+fp=sum(ceil(((pval<epsilon)-yval)/2));
+
+fn=-sum(floor(((pval<epsilon)-yval)/2));
+
+
+prec=tp/(tp+fp);
+rec=tp/(tp+fn);
+F1=2*prec*rec/(prec+rec);
+
+
+
+    % =============================================================
+
+    if F1 > bestF1
+       bestF1 = F1;
+       bestEpsilon = epsilon;
+    end
+end
+
+end

Anomaly Detection and Recommender Systems/submit.m

+function submit()
+  addpath('./lib');
+
+  conf.assignmentSlug = 'anomaly-detection-and-recommender-systems';
+  conf.itemName = 'Anomaly Detection and Recommender Systems';
+  conf.partArrays = { ...
+    { ...
+      '1', ...
+      { 'estimateGaussian.m' }, ...
+      'Estimate Gaussian Parameters', ...
+    }, ...
+    { ...
+      '2', ...
+      { 'selectThreshold.m' }, ...
+      'Select Threshold', ...
+    }, ...
+    { ...
+      '3', ...
+      { 'cofiCostFunc.m' }, ...
+      'Collaborative Filtering Cost', ...
+    }, ...
+    { ...
+      '4', ...
+      { 'cofiCostFunc.m' }, ...
+      'Collaborative Filtering Gradient', ...
+    }, ...
+    { ...
+      '5', ...
+      { 'cofiCostFunc.m' }, ...
+      'Regularized Cost', ...
+    }, ...
+    { ...
+      '6', ...
+      { 'cofiCostFunc.m' }, ...
+      'Regularized Gradient', ...
+    }, ...
+  };
+  conf.output = @output;
+
+  submitWithConfiguration(conf);
+end
+
+function out = output(partId, auxstring)
+  % 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];
+  Y = (Y .* double(R));  % set 'Y' values to 0 for movies not reviewed
+  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

Anomaly Detection and Recommender Systems/visualizeFit.m

+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