一、ex7.m

%% Machine Learning Online Class % Exercise 7 | Principle Component Analysis and K-Means Clustering % % Instructions % ------------ % % This file contains code that helps you get started on the % exercise. You will need to complete the following functions: % % pca.m % projectData.m % recoverData.m % computeCentroids.m % findClosestCentroids.m % kMeansInitCentroids.m % % For this exercise, you will not need to change any code in this file, % or any other files other than those mentioned above. %%% Initialization clear ; close all; clc%% ================= Part 1: Find Closest Centroids ==================== % To help you implement K-Means, we have divided the learning algorithm % into two functions -- findClosestCentroids and computeCentroids. In this % part, you shoudl complete the code in the findClosestCentroids function. % fprintf('Finding closest centroids.\n\n');% Load an example dataset that we will be using load('ex7data2.mat');% Select an initial set of centroids K = 3; % 3 Centroids initial_centroids = [3 3; 6 2; 8 5];% Find the closest centroids for the examples using the % initial_centroids idx = findClosestCentroids(X, initial_centroids);fprintf('Closest centroids for the first 3 examples: \n') fprintf(' %d', idx(1:3)); fprintf('\n(the closest centroids should be 1, 3, 2 respectively)\n');fprintf('Program paused. Press enter to continue.\n'); pause;%% ===================== Part 2: Compute Means ========================= % After implementing the closest centroids function, you should now % complete the computeCentroids function. % fprintf('\nComputing centroids means.\n\n');% Compute means based on the closest centroids found in the previous part. centroids = computeCentroids(X, idx, K);fprintf('Centroids computed after initial finding of closest centroids: \n') fprintf(' %f %f \n' , centroids'); fprintf('\n(the centroids should be\n'); fprintf(' [ 2.428301 3.157924 ]\n'); fprintf(' [ 5.813503 2.633656 ]\n'); fprintf(' [ 7.119387 3.616684 ]\n\n');fprintf('Program paused. Press enter to continue.\n'); pause;%% =================== Part 3: K-Means Clustering ====================== % After you have completed the two functions computeCentroids and % findClosestCentroids, you have all the necessary pieces to run the % kMeans algorithm. In this part, you will run the K-Means algorithm on % the example dataset we have provided. % fprintf('\nRunning K-Means clustering on example dataset.\n\n');% Load an example dataset load('ex7data2.mat');% Settings for running K-Means K = 3; max_iters = 10;% For consistency, here we set centroids to specific values % but in practice you want to generate them automatically, such as by % settings them to be random examples (as can be seen in % kMeansInitCentroids). initial_centroids = [3 3; 6 2; 8 5];% Run K-Means algorithm. The 'true' at the end tells our function to plot % the progress of K-Means [centroids, idx] = runkMeans(X, initial_centroids, max_iters, true); fprintf('\nK-Means Done.\n\n');fprintf('Program paused. Press enter to continue.\n'); pause;%% ============= Part 4: K-Means Clustering on Pixels =============== % In this exercise, you will use K-Means to compress an image. To do this, % you will first run K-Means on the colors of the pixels in the image and % then you will map each pixel on to it's closest centroid. % % You should now complete the code in kMeansInitCentroids.m %fprintf('\nRunning K-Means clustering on pixels from an image.\n\n');% Load an image of a bird A = double(imread('bird_small.png'));% If imread does not work for you, you can try instead % load ('bird_small.mat');A = A / 255; % Divide by 255 so that all values are in the range 0 - 1% Size of the image img_size = size(A);% Reshape the image into an Nx3 matrix where N = number of pixels. % Each row will contain the Red, Green and Blue pixel values % This gives us our dataset matrix X that we will use K-Means on. X = reshape(A, img_size(1) * img_size(2), 3);% Run your K-Means algorithm on this data % You should try different values of K and max_iters here K = 16; max_iters = 10;% When using K-Means, it is important the initialize the centroids % randomly. % You should complete the code in kMeansInitCentroids.m before proceeding initial_centroids = kMeansInitCentroids(X, K);% Run K-Means [centroids, idx] = runkMeans(X, initial_centroids, max_iters);fprintf('Program paused. Press enter to continue.\n'); pause;%% ================= Part 5: Image Compression ====================== % In this part of the exercise, you will use the clusters of K-Means to % compress an image. To do this, we first find the closest clusters for % each example. After that, we fprintf('\nApplying K-Means to compress an image.\n\n');% Find closest cluster members idx = findClosestCentroids(X, centroids);% Essentially, now we have represented the image X as in terms of the % indices in idx. % We can now recover the image from the indices (idx) by mapping each pixel % (specified by it's index in idx) to the centroid value X_recovered = centroids(idx,:);% Reshape the recovered image into proper dimensions X_recovered = reshape(X_recovered, img_size(1), img_size(2), 3);% Display the original image subplot(1, 2, 1); imagesc(A); title('Original');% Display compressed image side by side subplot(1, 2, 2); imagesc(X_recovered) title(sprintf('Compressed, with %d colors.', K));fprintf('Program paused. Press enter to continue.\n'); pause;

二、findClosestCentroids.m

function idx = findClosestCentroids(X, centroids) %FINDCLOSESTCENTROIDS computes the centroid memberships for every example % idx = FINDCLOSESTCENTROIDS (X, centroids) returns the closest centroids % in idx for a dataset X where each row is a single example. idx = m x 1 % vector of centroid assignments (i.e. each entry in range [1..K]) %% Set K K = size(centroids, 1); % centroids*1 i.e. K*1 % K% You need to return the following variables correctly. idx = zeros(size(X,1), 1); % m*1% ====================== YOUR CODE HERE ====================== % Instructions: Go over every example, find its closest centroid, and store % the index inside idx at the appropriate location. % Concretely, idx(i) should contain the index of the centroid % closest to example i. Hence, it should be a value in the % range 1..K % % Note: You can use a for-loop over the examples to compute this. %m = size(X, 1); % m for i = 1:m dist = []; for j = 1:K dist(j) = sum((X(i, :)-centroids(j, :)) .^ 2); end [min_dist, min_idx] = min(dist); idx(i) = min_idx; end% =============================================================end

三、computeCentroids.m

function centroids = computeCentroids(X, idx, K) %COMPUTECENTROIDS returs the new centroids by computing the means of the %data points assigned to each centroid. % centroids = COMPUTECENTROIDS(X, idx, K) returns the new centroids by % computing the means of the data points assigned to each centroid. It is % given a dataset X where each row is a single data point, a vector % idx of centroid assignments (i.e. each entry in range [1..K]) for each % example, and K, the number of centroids. You should return a matrix % centroids, where each row of centroids is the mean of the data points % assigned to it. %% Useful variables [m n] = size(X); % m*n% You need to return the following variables correctly. centroids = zeros(K, n); % k*n% ====================== YOUR CODE HERE ====================== % Instructions: Go over every centroid and compute mean of all points that % belong to it. Concretely, the row vector centroids(i, :) % should contain the mean of the data points assigned to % centroid i. % % Note: You can use a for-loop over the centroids to compute this. %for i = 1:K idx_set = find(i == idx); ck = numel(idx_set); if(0 ~= ck) cen_sum = sum(X(idx_set, :)); centroids(i, :) = cen_sum / ck; end end% =============================================================end

四、pca.m

function [U, S] = pca(X) %PCA Run principal component analysis on the dataset X % [U, S, X] = pca(X) computes eigenvectors of the covariance matrix of X % Returns the eigenvectors U, the eigenvalues (on diagonal) in S %% Useful values [m, n] = size(X); % m*n% You need to return the following variables correctly. U = zeros(n); % n*n S = zeros(n); % n*n% ====================== YOUR CODE HERE ====================== % Instructions: You should first compute the covariance matrix. Then, you % should use the "svd" function to compute the eigenvectors % and eigenvalues of the covariance matrix. % % Note: When computing the covariance matrix, remember to divide by m (the % number of examples). %Omega = X' * X / m; [U S V] = svd(Omega);% =========================================================================end

五、projectData.m

function Z = projectData(X, U, K) %PROJECTDATA Computes the reduced data representation when projecting only %on to the top k eigenvectors % Z = projectData(X, U, K) computes the projection of % the normalized inputs X into the reduced dimensional space spanned by % the first K columns of U. It returns the projected examples in Z. %% You need to return the following variables correctly. Z = zeros(size(X, 1), K); % m*K% ====================== YOUR CODE HERE ====================== % Instructions: Compute the projection of the data using only the top K % eigenvectors in U (first K columns). % For the i-th example X(i,:), the projection on to the k-th % eigenvector is given as follows: % x = X(i, :)'; % projection_k = x' * U(:, k); %Ureduce = U(:, 1:K); x = X'; Z = x' * Ureduce; % i.e. X * Ureduce % =============================================================end

六、recoverData.m

function X_rec = recoverData(Z, U, K) %RECOVERDATA Recovers an approximation of the original data when using the %projected data % X_rec = RECOVERDATA(Z, U, K) recovers an approximation the % original data that has been reduced to K dimensions. It returns the % approximate reconstruction in X_rec. %% You need to return the following variables correctly. X_rec = zeros(size(Z, 1), size(U, 1)); % size(X)% ====================== YOUR CODE HERE ====================== % Instructions: Compute the approximation of the data by projecting back % onto the original space using the top K eigenvectors in U. % % For the i-th example Z(i,:), the (approximate) % recovered data for dimension j is given as follows: % v = Z(i, :)'; % recovered_j = v' * U(j, 1:K)'; % % Notice that U(j, 1:K) is a row vector. % Ureduce = U(:, 1:K); X_rec = Z * Ureduce';% =============================================================end

七、submit results

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