吴恩达机器学习第八次作业： 异常检测Anomaly Detection

这是习题和答案的下载地址，全网最便宜，只要一积分哦~~~

https://download.youkuaiyun.com/download/wukongakk/10602657

0.综述

异常检测算法用于检测异常数据，通常在异常数据的数量远小于正常数据的数量时使用异常检测算法，在两者数量相差不大的时候，我们通常会选择逻辑回归或神经网络等算法。

1.Load Example Dataset

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

2.Estimate the dataset statistics

这个部分是用来确定各个特征的高斯分布。

%% ================== 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;

函数[mu sigma2] = estimateGaussian(X)，返回特征的均值，方差的平方

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 = 1 / m * sum ( bsxfun(@minus, X, mu) .^2 );


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


end

函数p = multivariateGaussian(X, mu, sigma2);  用多元高斯分布计算概率

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);

%    如果Sigma2是向量，就把Sigma2转变为对角矩阵
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

函数 visualizeFit(X, mu, sigma2)；用于画出概率分布

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

3.Find Outliers

%% ================== 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\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;

函数[epsilon F1] = selectThreshold(yval, pval);   选择阀值

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;
%   设置步长


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

    prec = tp / (tp + fp);
    rec = tp / (tp + fn);
    %   计算召回率和精准率

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

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

end

4. Multidimensional Outliers

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