這是習題和答案的下載地址，全網最便宜，只要一積分哦~~~

0.綜述

異常檢測演算法用於檢測異常資料，通常在異常資料的數量遠小於正常資料的數量時使用異常檢測演算法，在兩者數量相差不大的時候，我們通常會選擇邏輯迴歸或神經網路等演算法。

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

這個部分是用來確定各個特徵的高斯分佈。

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

函式[mu sigma2] = estimateGaussian(X)，返回特徵的均值，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
 

函式p = multivariateGaussian(X, mu, sigma2);  用多元高斯分佈計算概率

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轉變為對角矩陣
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

函式 visualizeFit(X, mu, sigma2)；用於畫出概率分佈

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;

函式[epsilon F1] = selectThreshold(yval, pval);   選擇閥值

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;
%   設定步長


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);
    %   計算召回率和精準率

    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