%X包含1，theta包含x0
ldm =  lambda/m;
dm = ldm/lambda;
m1 = length(theta);
sig = sigmoid(X*theta);
grad = dm.*X'*(sig - y) + ldm.*theta;
 %第一项求导没有尾巴，需要把尾巴减掉
grad(1,:) = grad(1,:) -  ldm*theta(1,:);
 %注意cost的最后规则化求和里面不包括第一项，所以theta0求导没有最后一个尾巴
J = -dm.*(log(sig)'*y+log(1-sig)'*(1-y)) + ldm*0.5* sum(theta(2:m1,:).^2);
%return J,grad

%训练一组预测num_label个数字的参数
function [all_theta] = oneVsAll(X, y, num_labels, lambda)
    m = size(X, 1);n = size(X, 2);
    all_theta = zeros(num_labels, n + 1);
    X = [ones(m, 1) X];
    initial_theta = zeros(n + 1, 1);
    options = optimset('GradObj', 'on', 'MaxIter', 50);
    for i=1:num_labels
         %y==i为逻辑数组
         [theta] = fmincg (@(t)(lrCostFunction(t, X, (y == i),  lambda)),initial_theta, options);
         %返回的是列向量
         all_theta(i,:) = theta';
    end
end

%预测
function p = predictOneVsAll(all_theta, X)
    m = size(X, 1);
    num_labels = size(all_theta, 1);
    p = zeros(size(X, 1), 1);
    X = [ones(m, 1) X];
    Mat = all_theta*X';
    for i=1:m
        [~,pos] = max(Mat(:,i));
        p(i,:) = pos;
    end
end

%Training Set Accuracy: 94.960000

function [jVal, gradient] = costFunction(theta)
  jVal = [...code to compute J(theta)...];
  gradient = [...code to compute derivative of J(theta)...];
end

options = optimset('GradObj', 'on', 'MaxIter', 100);
initialTheta = zeros(2,1);
   [optTheta, functionVal, exitFlag] = fminunc(@costFunction, initialTheta, options);

function p = predict(Theta1, Theta2, X)
    m = size(X, 1);
    num_labels = size(Theta2, 1);
    p = zeros(size(X, 1), 1);
    l = ones(m,1);
    X = [l,X];
    A = Theta1*X';
    A = sigmoid(A)';
    A = [l,A];
    A = Theta2*A';
    A = sigmoid(A);
    [~,p] = max(A',[],2);
    %就是做矩阵乘法然后在结果中挑一个值最大的，位置即为值，矩阵转置过来转置过去可能性能很低。
end

function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));
m = size(X, 1);
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));
l = ones(m,1);
%加一列bais
Xn = [l,X];
Z1 = Xn';
A1 = Xn';
%已经有了bais的参数
A = Theta1*Xn';
Z2 = A;
A = sigmoid(A);
A = [l,A'];
A2 = A';
A = Theta2*A';
Z3 = A;
A3 = A;
A = sigmoid(A);
%1列一组数据

ldm =  lambda/m;
dm = ldm/lambda;
%y为1x10
%m lines
%装换y格式
p = randperm(num_labels);
p = sort(p);
yn = p(ones(m,1),:);
for idx=1:m
    yn(idx,:) = yn(idx,:)==y(idx);
end
a = diag(yn*log(A));
b = diag((1.-yn)*log(1.-A));
J= -sum(a+b)/m;

[n1,n] = size(Theta1);
k1 = 0;
for i=1:n1
    for j=2:n
        k1= k1+Theta1(i,j)*Theta1(i,j);
    end
end

[n1,n] = size(Theta2);
k2 = 0;
for i=1:n1
    for j=2:n
        k2=k2+Theta2(i,j)*Theta2(i,j);
    end
end
J=J+(k1+k2)*ldm*0.5;

%Backpropagation
for i=1:m
    delta3 = A(:,i) - yn(i,:)';
    delta2 = Theta2'*delta3.*sigmoidGradient([1;Z2(:,i)]);
    Theta1_grad = Theta1_grad + (A1(:,i)*delta2(2:end)')';
    temp = A2(:,i)*delta3';
    Theta2_grad = Theta2_grad + temp';
end

Theta1_grad = Theta1_grad./m;
Theta2_grad = Theta2_grad./m;

[~,n2]=size(Theta1);
Theta1_grad(:,2:n2) = Theta1_grad(:,2:n2) + ldm*Theta1(:,2:n2);
[~,n2]=size(Theta2);
Theta2_grad(:,2:n2) = Theta2_grad(:,2:n2) + ldm*Theta2(:,2:n2);
% Unroll
grad = [Theta1_grad(:) ; Theta2_grad(:)];
end

function [error_train, error_val] = ...
    learningCurve(X, y, Xval, yval, lambda)
    m = size(X, 1);
    error_train = zeros(m, 1);
    error_val   = zeros(m, 1);
    for i=1:m
        %learning theta
        t = trainLinearReg(X(1:i,:),y(1:i,:),lambda);
        %train
        [j,~]=linearRegCostFunction(X(1:i,:), y(1:i,:), t, 0);
        error_train(i)=j;
        %cross validation
        [j,~]=linearRegCostFunction(Xval, yval, t, 0);
        error_val(i)=j;
    end
end

function centroids = computeCentroids(X, idx, K)
%COMPUTECENTROIDS returns the new centroids by computing the means of the 
%data points assigned to each centroid.
%   centroids = COMPUTECENTROIDS(X, idx, K) returns the new centroids by 
%   computing the means of the data points assigned to each centroid. It is
%   given a dataset X where each row is a single data point, a vector
%   idx of centroid assignments (i.e. each entry in range [1..K]) for each
%   example, and K, the number of centroids. You should return a matrix
%   centroids, where each row of centroids is the mean of the data points
%   assigned to it.
%

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

% You need to return the following variables correctly.
centroids = zeros(K, n);


% ====================== YOUR CODE HERE ======================
% Instructions: Go over every centroid and compute mean of all points that
%               belong to it. Concretely, the row vector centroids(i, :)
%               should contain the mean of the data points assigned to
%               centroid i.
%
% Note: You can use a for-loop over the centroids to compute this.
%

for k=1:K
    s = [0,0];
    n = 0;
    for i=1:m
        index = idx(i,:);
        if index==k
            centroids(k,:) = centroids(k,:) + X(i,:);
            n = n + 1;
        end
    end
    centroids(k,:) = centroids(k,:)/(n);
end



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


end


function centroids = kMeansInitCentroids(X, K)
%KMEANSINITCENTROIDS This function initializes K centroids that are to be 
%used in K-Means on the dataset X
%   centroids = KMEANSINITCENTROIDS(X, K) returns K initial centroids to be
%   used with the K-Means on the dataset X
%

% You should return this values correctly
centroids = zeros(K, size(X, 2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should set centroids to randomly chosen examples from
%               the dataset X
%


randidx = randperm(size(X, 1));

centroids = X(randidx(1:K), :);




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

end


function idx = findClosestCentroids(X, centroids)
%FINDCLOSESTCENTROIDS computes the centroid memberships for every example
%   idx = FINDCLOSESTCENTROIDS (X, centroids) returns the closest centroids
%   in idx for a dataset X where each row is a single example. idx = m x 1 
%   vector of centroid assignments (i.e. each entry in range [1..K])
%

% Set K
K = size(centroids, 1);

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

% ====================== YOUR CODE HERE ======================
% Instructions: Go over every example, find its closest centroid, and store
%               the index inside idx at the appropriate location.
%               Concretely, idx(i) should contain the index of the centroid
%               closest to example i. Hence, it should be a value in the 
%               range 1..K
%
% Note: You can use a for-loop over the examples to compute this.
%


m = size(X,1);
for i = 1:m
    min_v = 99999;
    min_index = 0;
    for k=1:K
        A = X(i,:)-centroids(k,:);
        t = A*A';
        if t<min_v
            min_v = t;
            min_index = k;
        end
    end
    idx(i)=min_index;
end


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

end



function [centroids, idx] = runkMeans(X, initial_centroids, ...
                                      max_iters, plot_progress)
%RUNKMEANS runs the K-Means algorithm on data matrix X, where each row of X
%is a single example
%   [centroids, idx] = RUNKMEANS(X, initial_centroids, max_iters, ...
%   plot_progress) runs the K-Means algorithm on data matrix X, where each 
%   row of X is a single example. It uses initial_centroids used as the
%   initial centroids. max_iters specifies the total number of interactions 
%   of K-Means to execute. plot_progress is a true/false flag that 
%   indicates if the function should also plot its progress as the 
%   learning happens. This is set to false by default. runkMeans returns 
%   centroids, a Kxn matrix of the computed centroids and idx, a m x 1 
%   vector of centroid assignments (i.e. each entry in range [1..K])
%

% Set default value for plot progress
if ~exist('plot_progress', 'var') || isempty(plot_progress)
    plot_progress = false;
end

% Plot the data if we are plotting progress
if plot_progress
    figure;
    hold on;
end

% Initialize values
[m n] = size(X);
K = size(initial_centroids, 1);
centroids = initial_centroids;
previous_centroids = centroids;
idx = zeros(m, 1);

% Run K-Means
for i=1:max_iters
    
    % Output progress
    fprintf('K-Means iteration %d/%d...\n', i, max_iters);
    if exist('OCTAVE_VERSION')
        fflush(stdout);
    end
    
    % For each example in X, assign it to the closest centroid
    idx = findClosestCentroids(X, centroids);
    
    % Optionally, plot progress here
    if plot_progress
        plotProgresskMeans(X, centroids, previous_centroids, idx, K, i);
        previous_centroids = centroids;
        fprintf('Press enter to continue.\n');
        pause;
    end
    
    % Given the memberships, compute new centroids
    centroids = computeCentroids(X, idx, K);
end

% Hold off if we are plotting progress
if plot_progress
    hold off;
end

end

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

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

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

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



Sigma = X'*X./m;
[U,S,~] = svd(Sigma);


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

end

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

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

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

%n*n
U_reduce = U(:,1:K);
%n*k
Z = X*U_reduce;


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

end

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

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

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

X_rec = Z*U(:,1:K)';

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

end

%% 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 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

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


predictions = (pval < epsilon);



tp = sum((predictions==1)&(yval==1));
fp = sum((predictions==1)&(yval==0));
fn = sum((predictions==0)&(yval==1));


prec = tp/(tp+fp);
rec = tp/(tp+fn);

F1 = 2*prec*rec/(prec+rec);





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

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

end

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

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

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

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


M = X*Theta'- Y;
T = M.*M;
J = sum(T(R==1))/2;

%regularized
l1 = 0;
for i=1:num_users
    for j=1:num_features
       l1=l1+Theta(i,j)*Theta(i,j);
    end
end
l1 = l1*0.5*lambda;

l2 = 0;
for i=1:num_movies
    for j=1:num_features
       l2=l2+X(i,j)*X(i,j);
    end
end
l2 = l2*0.5*lambda;

J = J + l1 + l2;
        

for i=1:num_movies
    idx = find(R(i,:)==1);
    tt = Theta(idx,:);
    yt = Y(i,idx);
    X_grad(i,:) = (X(i,:)*tt'-yt)*tt + lambda*X(i,:);
end


for i=1:num_users
    idx = find(R(:,i)==1);
    xt = X(idx,:);
    yt = Y(idx,i);
    %theta 1*f
    %theta_g u*f
    %xt idx*f
    %yt = idx*1
    Theta_grad(i,:) = (xt*Theta(i,:)'-yt)'*xt + lambda*Theta(i,:);
end



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

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

end

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

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

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

for i = 1:n
    mu(i) = sum(X(:,i))/m;
    K = X(:,i)-mu(i);
    sigma2(i) = K'*K/m;
end
% =============================================================
end