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Upload Exercise of Week 9
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‎machine-learning-ex8.zip

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‎machine-learning-ex8/ex8.pdf

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‎machine-learning-ex8/ex8/checkCostFunction.m

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function checkCostFunction(lambda)
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%CHECKCOSTFUNCTION Creates a collaborative filering problem
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%to check your cost function and gradients
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% CHECKCOSTFUNCTION(lambda) Creates a collaborative filering problem
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% to check your cost function and gradients, it will output the
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% analytical gradients produced by your code and the numerical gradients
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% (computed using computeNumericalGradient). These two gradient
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% computations should result in very similar values.
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% Set lambda
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if ~exist('lambda', 'var') || isempty(lambda)
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lambda = 0;
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end
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%% Create small problem
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X_t = rand(4, 3);
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Theta_t = rand(5, 3);
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% Zap out most entries
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Y = X_t * Theta_t';
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Y(rand(size(Y)) > 0.5) = 0;
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R = zeros(size(Y));
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R(Y ~= 0) = 1;
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%% Run Gradient Checking
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X = randn(size(X_t));
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Theta = randn(size(Theta_t));
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num_users = size(Y, 2);
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num_movies = size(Y, 1);
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num_features = size(Theta_t, 2);
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numgrad = computeNumericalGradient( ...
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@(t) cofiCostFunc(t, Y, R, num_users, num_movies, ...
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num_features, lambda), [X(:); Theta(:)]);
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[cost, grad] = cofiCostFunc([X(:); Theta(:)], Y, R, num_users, ...
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num_movies, num_features, lambda);
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disp([numgrad grad]);
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fprintf(['The above two columns you get should be very similar.\n' ...
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'(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']);
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diff = norm(numgrad-grad)/norm(numgrad+grad);
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fprintf(['If your cost function implementation is correct, then \n' ...
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'the relative difference will be small (less than 1e-9). \n' ...
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'\nRelative Difference: %g\n'], diff);
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end

‎machine-learning-ex8/ex8/cofiCostFunc.m

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function [J, grad] = cofiCostFunc(params, Y, R, num_users, num_movies, ...
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num_features, lambda)
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%COFICOSTFUNC Collaborative filtering cost function
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% [J, grad] = COFICOSTFUNC(params, Y, R, num_users, num_movies, ...
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% num_features, lambda) returns the cost and gradient for the
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% collaborative filtering problem.
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%
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% Unfold the U and W matrices from params
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X = reshape(params(1:num_movies*num_features), num_movies, num_features);
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Theta = reshape(params(num_movies*num_features+1:end), ...
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num_users, num_features);
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% You need to return the following values correctly
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J = 0;
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X_grad = zeros(size(X));
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Theta_grad = zeros(size(Theta));
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% ====================== YOUR CODE HERE ======================
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% Instructions: Compute the cost function and gradient for collaborative
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% filtering. Concretely, you should first implement the cost
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% function (without regularization) and make sure it is
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% matches our costs. After that, you should implement the
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% gradient and use the checkCostFunction routine to check
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% that the gradient is correct. Finally, you should implement
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% regularization.
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%
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% Notes: X - num_movies x num_features matrix of movie features
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% Theta - num_users x num_features matrix of user features
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% Y - num_movies x num_users matrix of user ratings of movies
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% R - num_movies x num_users matrix, where R(i, j) = 1 if the
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% i-th movie was rated by the j-th user
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%
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% You should set the following variables correctly:
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%
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% X_grad - num_movies x num_features matrix, containing the
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% partial derivatives w.r.t. to each element of X
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% Theta_grad - num_users x num_features matrix, containing the
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% partial derivatives w.r.t. to each element of Theta
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%
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% =============================================================
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grad = [X_grad(:); Theta_grad(:)];
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end

‎machine-learning-ex8/ex8/computeNumericalGradient.m

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function numgrad = computeNumericalGradient(J, theta)
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%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"
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%and gives us a numerical estimate of the gradient.
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% numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical
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% gradient of the function J around theta. Calling y = J(theta) should
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% return the function value at theta.
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% Notes: The following code implements numerical gradient checking, and
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% returns the numerical gradient.It sets numgrad(i) to (a numerical
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% approximation of) the partial derivative of J with respect to the
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% i-th input argument, evaluated at theta. (i.e., numgrad(i) should
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% be the (approximately) the partial derivative of J with respect
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% to theta(i).)
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%
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numgrad = zeros(size(theta));
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perturb = zeros(size(theta));
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e = 1e-4;
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for p = 1:numel(theta)
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% Set perturbation vector
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perturb(p) = e;
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loss1 = J(theta - perturb);
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loss2 = J(theta + perturb);
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% Compute Numerical Gradient
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numgrad(p) = (loss2 - loss1) / (2*e);
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perturb(p) = 0;
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end
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end

‎machine-learning-ex8/ex8/estimateGaussian.m

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function [mu sigma2] = estimateGaussian(X)
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%ESTIMATEGAUSSIAN This function estimates the parameters of a
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%Gaussian distribution using the data in X
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% [mu sigma2] = estimateGaussian(X),
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% The input X is the dataset with each n-dimensional data point in one row
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% The output is an n-dimensional vector mu, the mean of the data set
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% and the variances sigma^2, an n x 1 vector
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%
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% Useful variables
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[m, n] = size(X);
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% You should return these values correctly
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mu = zeros(n, 1);
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sigma2 = zeros(n, 1);
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% ====================== YOUR CODE HERE ======================
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% Instructions: Compute the mean of the data and the variances
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% In particular, mu(i) should contain the mean of
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% the data for the i-th feature and sigma2(i)
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% should contain variance of the i-th feature.
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%
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% =============================================================
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end

‎machine-learning-ex8/ex8/ex8.m

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%% Machine Learning Online Class
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% Exercise 8 | Anomaly Detection and Collaborative Filtering
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%
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% Instructions
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% ------------
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%
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% This file contains code that helps you get started on the
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% exercise. You will need to complete the following functions:
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%
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% estimateGaussian.m
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% selectThreshold.m
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% cofiCostFunc.m
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%
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% For this exercise, you will not need to change any code in this file,
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% or any other files other than those mentioned above.
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%
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%% Initialization
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clear ; close all; clc
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%% ================== Part 1: Load Example Dataset ===================
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% We start this exercise by using a small dataset that is easy to
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% visualize.
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%
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% Our example case consists of 2 network server statistics across
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% several machines: the latency and throughput of each machine.
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% This exercise will help us find possibly faulty (or very fast) machines.
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%
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fprintf('Visualizing example dataset for outlier detection.\n\n');
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% The following command loads the dataset. You should now have the
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% variables X, Xval, yval in your environment
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load('ex8data1.mat');
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% Visualize the example dataset
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plot(X(:, 1), X(:, 2), 'bx');
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axis([0 30 0 30]);
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xlabel('Latency (ms)');
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ylabel('Throughput (mb/s)');
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fprintf('Program paused. Press enter to continue.\n');
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pause
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%% ================== Part 2: Estimate the dataset statistics ===================
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% For this exercise, we assume a Gaussian distribution for the dataset.
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%
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% We first estimate the parameters of our assumed Gaussian distribution,
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% then compute the probabilities for each of the points and then visualize
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% both the overall distribution and where each of the points falls in
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% terms of that distribution.
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%
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fprintf('Visualizing Gaussian fit.\n\n');
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% Estimate my and sigma2
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[mu sigma2] = estimateGaussian(X);
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% Returns the density of the multivariate normal at each data point (row)
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% of X
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p = multivariateGaussian(X, mu, sigma2);
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% Visualize the fit
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visualizeFit(X, mu, sigma2);
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xlabel('Latency (ms)');
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ylabel('Throughput (mb/s)');
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fprintf('Program paused. Press enter to continue.\n');
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pause;
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%% ================== Part 3: Find Outliers ===================
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% Now you will find a good epsilon threshold using a cross-validation set
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% probabilities given the estimated Gaussian distribution
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%
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pval = multivariateGaussian(Xval, mu, sigma2);
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[epsilon F1] = selectThreshold(yval, pval);
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fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
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fprintf('Best F1 on Cross Validation Set: %f\n', F1);
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fprintf(' (you should see a value epsilon of about 8.99e-05)\n');
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fprintf(' (you should see a Best F1 value of 0.875000)\n\n');
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% Find the outliers in the training set and plot the
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outliers = find(p < epsilon);
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% Draw a red circle around those outliers
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hold on
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plot(X(outliers, 1), X(outliers, 2), 'ro', 'LineWidth', 2, 'MarkerSize', 10);
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hold off
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fprintf('Program paused. Press enter to continue.\n');
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pause;
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%% ================== Part 4: Multidimensional Outliers ===================
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% We will now use the code from the previous part and apply it to a
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% harder problem in which more features describe each datapoint and only
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% some features indicate whether a point is an outlier.
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%
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% Loads the second dataset. You should now have the
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% variables X, Xval, yval in your environment
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load('ex8data2.mat');
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% Apply the same steps to the larger dataset
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[mu sigma2] = estimateGaussian(X);
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% Training set
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p = multivariateGaussian(X, mu, sigma2);
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% Cross-validation set
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pval = multivariateGaussian(Xval, mu, sigma2);
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% Find the best threshold
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[epsilon F1] = selectThreshold(yval, pval);
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fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
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fprintf('Best F1 on Cross Validation Set: %f\n', F1);
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fprintf(' (you should see a value epsilon of about 1.38e-18)\n');
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fprintf(' (you should see a Best F1 value of 0.615385)\n');
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fprintf('# Outliers found: %d\n\n', sum(p < epsilon));

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