机器学习week9 ex8 review

机器学习 吴恩达

这周学习异常监测, 第一部分完成对一个网络中故障的服务器的监测。第二部分使用协同过滤来实现一个电影推荐系统。

1 Anomaly Detection

监测服务器工作状态的指标：吞吐量（throughput）、延迟（latency）。
我们有 $m=307$ 的无标签数据集，这里认为其中绝大多数都是正常工作的服务器，其中少量是异常状态。
先通过散点图来直观判断。

1.1 Gaussian distribution

对数据的分布情况选择一个模型。
高斯分布的公式如下：

其中 $\mu$ 是平均值，$\sigma$是标准差。

1.2 Estimating parameters for Gaussian distribution

根据如下公式计算高斯分布的参数：


完成estimateGaussian.m如下：

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 = mean(X); 
sigma2 = var(X,1); % choose the way to divide by N rather than N-1
% =============================================================
end

完成之后，脚本文件会执行绘制等高线的操作，即得到如下图像：

1.3 Selecting the threshold $\epsilon$

以 $\epsilon$ 为临界值，$P(x) < \epsilon$ 的情况被认为是异常状况。
通过交叉验证集来选择这样的 $\epsilon$。
交叉验证集中的数据是带标签的。根据之前学到的 $F_1$ 来评价选择的优劣。


其中 $tp,fp,fn$ 分别代表true positive,false positive, false negative。

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
    prediction = (pval < epsilon);
    tp = sum((prediction == 1) & (yval == 1));     % true positive
    fp = sum((prediction == 1) & (yval == 0));     % false positive
    fn = sum((prediction == 0) & (yval == 1));     % false negative
    prec = tp / (tp + fp);                         % precision
    rec = tp / (tp + fn);                          % recall
    F1 = 2 * prec * rec/ (prec + rec);             % F1
    % =============================================================
    if F1 > bestF1
       bestF1 = F1;
       bestEpsilon = epsilon;
    end
end
end

按照选定的 $\epsilon$ ，判断异常情况如下图：

1.4 High dimensional Dataset

对上述函数，换用更高维的数据集。（11 features）
与之前2维的情况并没有什么区别。

2 Recommender system

对关于电影评分的数据集使用协同过滤算法，实现推荐系统。
Datasets来源：MoiveLens 100k Datasets.
对矩阵可视化：

作为对比，四阶单位矩阵可视化情况如下：

2.1 Movie rating dataset

矩阵 $Y$ (大小为num_movies $\times$ num_users);
矩阵 $R$ ($R_{ij} = 1$ 表示电影 $i$ 被用户 $j$ 评分过).

2.2 Collaborating filtering learning algorithm

整个2.2都是对cofiCostFunc.m的处理。
原文件中提供的代码如下：

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

2.2.1 Collaborating filtering cost function

未经过regularization的代价函数如下：

故增加如下代码：

diff = (X * Theta' - Y);
vari = diff.^2;
J = 1/2 * sum(vari(R == 1));

2.2.2 Collaborating filtering gradient

公式如下：

按照文档里的Tips进行向量化，加入如下代码：

for i = 1: num_movies,
  X_grad(i,:) = sum(((diff(i,:).* R(i,:))'.* Theta));
end;
for j = 1: num_users,
  Theta_grad(j,:) = sum(((diff(:,j).* R(:,j)) .* X));
end;

想了一会，发现好像可以更彻底地向量化：

X_grad = diff.* R * Theta;
Theta_grad = (diff.*R)' * X;

2.2.3 Regularized cost function

2.2.4 Regularized gradient

只需要在上述代码中加入regularization的部分即可。
如下：

J = 1/2 * sum(vari(R == 1)) + lambda/2 * (sum((Theta.^2)(:)) + sum((X.^2)(:)));
X_grad = diff.*R*Theta + lambda * X;
Theta_grad = (diff.*R)' * X + lambda * Theta;

2.3 Learning movie recommendations

2.3.1 Recommendations

在脚本文件中填入自己对movie_list.txt中部分电影的评分。
似乎提供的电影都是新世纪以前上映的，因此我没有看过太多。我挑选了如下几部评分：

推荐系统给我推荐了如下电影：

我没有办法判断准不准，因为我一部也没有看过。但随便搜了其中的几部，感觉我可能并不会喜欢。
也许是我提供的样本太小了，也许是这个推荐系统太简陋了吧。