Andrew NG 机器学习 练习8-Anomaly Detection and Recommender Systems

1 Anomaly detection

实现一个异常检测算法检测服务器的异常行为
特征是 每个服务器的 吞吐量（throughput）(mb/s) 和 相应延迟（ms）
采集 m=307 台运行中的服务器的特征，{x(1),...,x(m)}
其中大部分是 normal 的服务器特征

你将使用 高斯模型 检测数据集中的异常样例
从 2D 数据集开始，以便可视化算法过程
在那个数据集中你将拟合一个高斯分布，发现低可能性的值，从而找出异常样例
之后，你将在一个大的多维数据集中应用异常检测算法

首先可视化数据，如图：

%% ================== 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 environmentload('ex8data1.mat');%  Visualize the example datasetplot(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

1.1 Gaussian distribution

为了实施异常检测，你需要首先 根据数据分布，拟合一个模型

给一个训练集 {x(1),...,x(m)}，x(i)∈Rn
需要对每一个特征 xi 估算 高斯分布
对于每一个特征，需要计算参数 μi和σ2i

通常如果我们认为变量 x 符合高斯分布 x~N(μ，σ2) 则其概率密度函数为：

其中：μ是均值，σ2是方差

1.2 Estimating parameters for a Gaussian

通过下列公式计算每个特征的 μi和σ2i：

%% ================== 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 Xp = multivariateGaussian(X, mu, sigma2);%  Visualize the fitvisualizeFit(X,  mu, sigma2);xlabel('Latency (ms)');ylabel('Throughput (mb/s)');fprintf('Program paused. Press enter to continue.\n');pause;

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 correctlymu = 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=1/m*sum(X);sigma2=1/m*sum((X-repmat(mu,m,1)).^2);% =============================================================end

1.3 Selecting the threshold, ε

现在我们有了高斯参数，我们就可以调查一下那些样例根据这个分布有高可能性，那些样例有非常低的可能性。有低可能性的样例更有可能是异常的。

决定那些是异常的，一种方法是 根据 交叉验证集 选择一个 阈值。

这部分，实现一个算法选择 ，在交叉验证集中使用 F1 值 来选择 阈值ε。

交叉验证集 {(x(1)cv,y(1)cv),...,(x(mcv)cv,y(mcv)cv)}
标签 y=1 表示是异常样例，y=0 表示是正常样例

对于每一个交叉验证集，计算 p(x(i)cv)

所有的 p(x(1)cv)，…,p(x(mcv)cv) 和 y(1)cv,...,y(mcv)cv 以向量的形式传递到 selectThreshold.m 以计算 阈值 ε，该方法也要返回使用该ε 的 F1值。

• tp 是正确的积极判定（true positives）的数量：标签表明是异常，算法正确分类为异常
• fp 是错误的积极判定（false positives）的数量：标签表明是正常，算法错误的分类为异常
• fn 是错误的消极判定（false negatives）的数量：标签表明是异常，算法错误的分类为正常

%% ================== 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 theoutliers = find(p < epsilon);%  Draw a red circle around those outliershold onplot(X(outliers, 1), X(outliers, 2), 'ro', 'LineWidth', 2, 'MarkerSize', 10);hold offfprintf('Program paused. Press enter to continue.\n');pause;

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;%计算步长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);%概率小于阈值的数量，即预测为异常的数量    fp = sum((predictions == 1) & (yval == 0));%算法错误的分类为异常，标签表明是正常    fn = sum((predictions == 0) & (yval == 1));%算法正确分类为正常,标签表明是异常    tp = sum((predictions == 1) & (yval == 1));%算法正确分类为异常,标签表明是异常    prec = tp / (tp + fp);%准确率    rec = tp / (tp + fn);%召回率    F1 = 2 * prec * rec / (prec + rec);%F1值    % =============================================================    if F1 > bestF1       bestF1 = F1;       bestEpsilon = epsilon;    endendend

1.4 High dimensional dataset

将前面实现的异常检测算法应用在一个更现实、更难的数据集。
一个样例有11个特征，捕捉了服务器更多的属性。

%% ================== 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 environmentload('ex8data2.mat');%  Apply the same steps to the larger dataset[mu sigma2] = estimateGaussian(X);%  Training set p = multivariateGaussian(X, mu, sigma2);%  Cross-validation setpval = 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));

Best epsilon found using cross-validation: 1.377229e-18Best F1 on Cross Validation Set:  0.615385   (you should see a value epsilon of about 1.38e-18)   (you should see a Best F1 value of 0.615385)# Outliers found: 117

2 Recommender Systems

这部分，你将实现协同过滤学习算法，并将其应用在一个电影评分数据集中。

评分范围是1到5。

有 nu=943 个用户；nm=1682 个电影。

在练习的下一部分，你将实现 cofiCostFunc.m 方法，计算协同过滤目标函数和梯度。之后使用 vfmincg.m 学习协同过滤的参数。

2.1 Movie ratings dataset

从 ex8 movies.mat 读取变量 Y 和 R 。

Y矩阵（num_movies × num_users） 保存 评分 y(i,j) 从 1-5。

R矩阵是一个0-1标记矩阵，R(i,j)=1 表示 用户 j 给电影 i 评过分；R(i,j)=0 相反。

协同过滤的目标是预测 没有被评分，（即R(i,j)=0 ）位置的评分。这样就可以推荐预测用户评分最高的电影给这个用户了。

通过这部分练习，你将用 X 和 Theta 这两个矩阵工作：

X矩阵的第 i 行 对应 第 i 个电影的特征向量 x(i)
Theta矩阵的 第 j 行对用 第 j 个用户的参数向量 θ(j)

x(i) 和 θ(j) 都是 n 维向量。

这个练习中 特征数 n=100，相应的 X 是一个 nm*100 维矩阵；Theta是一个 nu*100 维矩阵。

%% =============== 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 dataload ('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, :))));%取Y中第一行中所有有效评分求均值%  We can "visualize" the ratings matrix by plotting it with imagescimagesc(Y);ylabel('Movies');xlabel('Users');fprintf('\nProgram paused. Press enter to continue.\n');pause;

2.2 Collaborative filtering learning algorithm

协同过滤算法在设置电影推荐考虑一套 n维参数向量 x(1),...,x(nm) 和 θ(1),...,θnu

模型预测用户 j 对电影 i 的评分,y(i,j)=(θ(j))Tx(i)。

给一个数据集包含一些用户对一些电影的评分，我们希望学习参数向量 x(1),...,x(nm) 和 θ(1),...,θnu ，产生最佳拟合（最小化平方误差）

cofiCostFunc.m 计算协同过滤的代价函数和梯度。为了使用现成的最小化函数如fmincg，需要将矩阵展开成单个向量参数。在神经网络编程中也使用过这种方式。

2.2.1 Collaborative filtering cost function

协同过滤代价函数（不带正则化）：

应该现在应该定义 cofiCostFunc.m 来返回代价 J。注意你计算用户 j 对电影 i 的代价仅当 R(i,j)=1 时。

2.2.2 Collaborative filtering gradient

代价函数的梯度：

2.2.3 Regularized cost function

协同过滤代价函数（带正则化）：

2.2.4 Regularized gradient

带正则化的代价函数的梯度:

%% ============ 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 fasternum_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 functionJ = 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 checkNNGradientscheckCostFunction;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 functionJ = 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 checkNNGradientscheckCostFunction(1.5);fprintf('\nProgram paused. Press enter to continue.\n');pause;

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 paramsX = 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 correctlyJ = 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 = (1/2).*sum(sum(((X*Theta').*R-Y.*R).^2))+(lambda./2.*sum(sum(Theta.^2)))+(lambda./2.*sum(sum(X.^2)));        % Only predict rating X*Theta' if user has rated (i.e. R=1)X_grad = (((X*Theta').*R*Theta-Y.*R*Theta)+lambda.*X);Theta_grad = ((X'*((X*Theta').*R)-X'*(Y.*R)))'+lambda.*Theta;% Alternative according to exercise handout%[r,c]=size(R);%for i=1:r%        idx = find(R(i,:)==1);%        Theta_temp = Theta(idx,:);%        Y_temp = Y(i,idx);%        X_grad(i,:) = (X(i,:)*Theta_temp'-Y_temp)*Theta_temp;%end% For Theta_grad, similar approach to X_grad% =============================================================grad = [X_grad(:); Theta_grad(:)];end

2.3 Learning movie recommendations

%% ================== 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 dataload('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 matrixY = [my_ratings Y];R = [(my_ratings ~= 0) R];%  Normalize Ratings[Ynorm, Ymean] = normalizeRatings(Y, R);%  Useful Valuesnum_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 fmincgoptions = optimset('GradObj', 'on', 'MaxIter', 100);% Set Regularizationlambda = 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 WX = 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});endfprintf('\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});    endend