Programming Exercise 8:Anomaly Detection and Recommender Systems 第二部分

        大家好，我是Mac Jiang，今天和大家分享Coursera-Stanford University-Machine Learning-Programming Exercise 8:Anomaly Detection and Recommender Systems的第二部分Recommender Systems的实现过程，第一部分Anomaly Detection的实现过程的网址为： http://blog.csdn.net/a1015553840/article/details/50913824。虽然我的代码通过了系统的测试，但这不一定是最好的，如果博友发现错误或者有更好的实现思路请留言联系，谢谢！希望我的博客能给您的学习带来一些帮助！

       这部分，吴恩达老师主要讲了协同过滤算法（collabrative filtering），先介绍一些符号的基本意思：

j:第j个用户，用来用户计数

i:第i部电影，用于电影计数

r(i,j):第j个用户对第i部电影的评价，r(i,j)=1表示第j个用户对第i部电影有评价，为0反之。

y(i,j):第j个用户对第i部电影的评分

n下标m：用户总数

n下标u：电影总数

theta(j)：用户j的参数

x(i)：电影i的特征

用户j对电影i的评价预测：theta(j)' * x(i)

        协同过滤的算法步骤是利用已有的用户对电影的评价，通过梯度下降算法（B-LFGS/共轭梯度等高级优化算法）求解最优电影特征矩阵X和用户信息矩阵Theta，对r(i,j)=0的对应点进行预测，同时利用这些信息可以得到用户的喜好，为用户推荐电影。注意，在进行学习算法之前需要进行均值归一化！

        这次实验就是基于以上理论，对以上理论的具体实现。

1.实验数据和文件说明

数据集：ex8_movies.mat---用户对电影的评价信息，有两个矩阵，一个为评价矩阵R，另一个为R用来指示哪些位置有评价信息。

              ex8_movieParams.mat---X矩阵信息，Theta矩阵信息，用户人数num_users，电影数num_movies，电影特征数num_features

              movie_ids.txt---用于存储电影名

    文件：ex8_cofi.m---用于控制程序的进行过程

              loadMovieList.m---将movie_ids.txt的电影名导入到电影名数组中

              computeNumericalGradient.m---用求导的方法计算梯度值

              checkCostFuntion.m---确定我们自己用cifiCostFunc.m计算出来的梯度的正确性

              fmincg.m---高级优化函数，通过迭代的方法寻找代价值J的局部最小值或全局最小值，并记录此时的参数Theta和参数X

              normalizeRating.m---在训练之前对举证Y进行均值归一化

              cofiCostFunc.m---计算代价函数J的值和此时对X的梯度值X_grad以及对Theta的梯度值Theta_grad，需要完善代码！

2.ex8_cofi.m的控制过程

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

        Part1：Loading movie ratings dataset---导入用户对电影的评价数据并可视化

        Part2：Colleborative Filtering Cost Function---这里取电影和评价的一个子集，用cofiCostFunc计算代价值J，以及梯度值X_grad，Theta_grad

        Part3：Collaboretive Filtering Gradient---对上面子集用[f(x-epsilon)-f(x+epsilon)]/(2*epsilon)的方法计算梯度，确定我们用举证方法求得梯度的正确性

        Part4：Collaboretive Filtering Regularization---对算法进行正则化，改进cofiCostFunc.m的J加上正则化项

        Part5：Collaboretive Flitering Gradient Rgulazation---对cofiCostFunc.m的求梯度过程进行正则化，对X_grad,Theta_grad加上正则化项

        Part6：Entering ratings for a new user---新建一个用户，增加一些他的评价信息

        Part7：Learning Movie Ratings---利用上面完善的协同过滤算法对数据进行学习

        Part8：Recommendation for you---利用上步得到的学习参数，预测这位新用户可能喜欢的电影

2.cofiCostFunc.m的实现过程

未正则化的代价函数值：J = sum(sum(((X * Theta' - Y) .* R).^2)) / 2;

未正则化的X梯度值：X_grad = (X * Theta' - Y) .* R * Theta;

未正则化的Theta梯度值：Theta_grad = (Theta * X' - Y') .* R' * X;

正则化后的代价函数值：J = J + lambda / 2 * (sum(sum(Theta .^2)) + sum(sum(X .^ 2)))

正则化后的X梯度值：X_grad = X_grad + lambda * X;正则化后的Theta梯度值：Theta_grad = Theta_grad + lambda * Theta;

由于推到过程比较复杂，而且推到过程在这里也不能打出来，所以就不说明了。不过大家可以从矩阵维度的角度寻找一些启发！具体实现代码如下：

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) .* R).^2)) / 2;   %因为到多次调用高级优化函数，所以这里最好采用向量的方法，而不是利用for循环
X_grad = (X * Theta' - Y) .* R * Theta;%利用向量的方法计算对X的梯度
%利用向量的方法计算对Theta的梯度
J = J + lambda / 2 * (sum(sum(Theta .^2)) + sum(sum(X .^ 2)));%对代价值进行正则化
X_grad = X_grad + lambda * X;
Theta_grad = Theta_grad + lambda * Theta;
% =============================================================
grad = [X_grad(:); Theta_grad(:)];
end

From: http://blog.csdn.net/a1015553840/article/details/50916801