Andrew NG 机器学习 练习7-K-means Clustering and Principal Component Analysis

1 K-means Clustering

1.1 Implementing K-means

The K-means algorithm is a method to automatically cluster similar data examples together.

The K-means algorithm is as follows:

% Initialize centroidscentroids = kMeansInitCentroids(X, K);for iter = 1:iterations    % Cluster assignment step: Assign each data point to the    % closest centroid. idx(i) corresponds to cˆ(i), the index    % of the centroid assigned to example i    idx = findClosestCentroids(X, centroids);    % Move centroid step: Compute means based on centroid    % assignments    centroids = computeMeans(X, idx, K);end

1.1.1 Finding closest centoids

%% ================= 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 should complete the code in the findClosestCentroids function. %fprintf('Finding closest centroids.\n\n');% Load an example dataset that we will be usingload('ex7data2.mat');% Select an initial set of centroidsK = 3; % 3 Centroidsinitial_centroids = [3 3; 6 2; 8 5];% Find the closest centroids for the examples using the% initial_centroidsidx = 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;

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 KK = size(centroids, 1);% You need to return the following variables correctly.idx = zeros(size(X,1), 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.%for i=1:size(X,1)    min=100000;    for j=1:K        if sum((X(i,:)-centroids(j,:)).^2)<=min            min=sum((X(i,:)-centroids(j,:)).^2);            idx(i,1)=j;        end    endend% =============================================================end

1.1.2 Computing centroid means

Given assignments of every point to a centroid, the second phase of the algorithm recomputes, for each centroid, the mean of the points that were assigned to it.

重新计算每个类的质心。

属于该类的所有 横坐标 的平均值，即为该类质心的横坐标。所有 纵坐标 的平均值，即为该类质心的纵坐标。

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

computeCentroids.m

function centroids = computeCentroids(X, idx, K)%COMPUTECENTROIDS returns 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);% You need to return the following variables correctly.centroids = zeros(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    list = find(idx==i);    for j=1:size(list,1)        centroids(i,:)=centroids(i,:)+X(list(j),:);    end;    centroids(i,:)=centroids(i,:)./size(list,1);end;% =============================================================end

1.2 K-means on example dataset

%% =================== 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 datasetload('ex7data2.mat');% Settings for running K-MeansK = 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;

runkMeans.m

function [centroids, idx] = runkMeans(X, initial_centroids, ...                                      max_iters, plot_progress)%RUNKMEANS runs the K-Means algorithm on data matrix X, where each row of X%is a single example%   [centroids, idx] = RUNKMEANS(X, initial_centroids, max_iters, ...%   plot_progress) runs the K-Means algorithm on data matrix X, where each %   row of X is a single example. It uses initial_centroids used as the%   initial centroids. max_iters specifies the total number of interactions %   of K-Means to execute. plot_progress is a true/false flag that %   indicates if the function should also plot its progress as the %   learning happens. This is set to false by default. runkMeans returns %   centroids, a Kxn matrix of the computed centroids and idx, a m x 1 %   vector of centroid assignments (i.e. each entry in range [1..K])%% Set default value for plot progressif ~exist('plot_progress', 'var') || isempty(plot_progress)    plot_progress = false;end% Plot the data if we are plotting progressif plot_progress    figure;    hold on;end% Initialize values[m n] = size(X);K = size(initial_centroids, 1);centroids = initial_centroids;previous_centroids = centroids;idx = zeros(m, 1);% Run K-Meansfor i=1:max_iters    % Output progress    fprintf('K-Means iteration %d/%d...\n', i, max_iters);    if exist('OCTAVE_VERSION')        fflush(stdout);    end    % For each example in X, assign it to the closest centroid    idx = findClosestCentroids(X, centroids);    % Optionally, plot progress here    if plot_progress        plotProgresskMeans(X, centroids, previous_centroids, idx, K, i);        previous_centroids = centroids;        fprintf('Press enter to continue.\n');        pause;    end    % Given the memberships, compute new centroids    centroids = computeCentroids(X, idx, K);end% Hold off if we are plotting progressif plot_progress    hold off;endend

1.3 Random initialization

随机初始化聚类中心

kMeansInitCentroids.m

function centroids = kMeansInitCentroids(X, K)%KMEANSINITCENTROIDS This function initializes K centroids that are to be %used in K-Means on the dataset X%   centroids = KMEANSINITCENTROIDS(X, K) returns K initial centroids to be%   used with the K-Means on the dataset X%% You should return this values correctlycentroids = zeros(K, size(X, 2));% ====================== YOUR CODE HERE ======================% Instructions: You should set centroids to randomly chosen examples from%               the dataset X%% Randomly reorder the indices of examplesrandidx = randperm(size(X, 1));% Take the first K examples as centroidscentroids = X(randidx(1:K), :);% =============================================================end

1.4 Image compression with K-means

RGB编码：24-bit 表示每个像素点的颜色，每 8-bit（0-255） 表示（red,green,blue）的编码。

我们的图片有上千种颜色，我们要将其降维到16种颜色。

将图片的每个像素作为 数据样例，使用k-means 算法找到16种颜色最能将像素在3维RGB空间聚类。

每次你计算出聚类中心，你就使用16种颜色替换原始图片的像素点。

1.4.1 K-means on pixels

首先读取图片，将图片重构成 m*3 的像素颜色矩阵（m=128*128=16384）,在这之上运用 k-means.

发现前 K=16 的表示图片的颜色后，将所有像素点归为这16类。将他们的颜色换为其中心点的颜色。

这样的话，减小了需要描述这张图片的空间：
原始，24bits 对于 128*128 个像素点。总共需要：128*128*24=393216 bits.
现在：存储16种颜色需要：16*24bits，每个像素点只需要需要 4bits 存储16种像素的位置来表示使用的是哪一种颜色即可:128*128*4,所以总共需要 16*24+128*128*4=65920 bits.
相当于压缩为了以前的约1/6。

%% ============= 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 onto its 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 birdA = 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 imageimg_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 hereK = 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 proceedinginitial_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 membersidx = 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 its index in idx) to the centroid valueX_recovered = centroids(idx,:);% Reshape the recovered image into proper dimensionsX_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 sidesubplot(1, 2, 2);imagesc(X_recovered)title(sprintf('Compressed, with %d colors.', K));fprintf('Program paused. Press enter to continue.\n');pause;

2 Principal Component Analysis

2.1 Example Dataset

可视化 使用 PCA 将数据从２D 降到１D 这个过程。

2.2 Implementing PCA

PCA 包括两个步骤：
1、计算数据的 协方差（ covariance）矩阵。
2、使用 matlab 的 SVD 方法计算 特征向量（ eigenvectors） U1,U2,...,Un

在使用PCA之前，归一化数据很重要。

%% ================== Part 1: Load Example Dataset  ===================%  We start this exercise by using a small dataset that is easily to%  visualize%fprintf('Visualizing example dataset for PCA.\n\n');%  The following command loads the dataset. You should now have the %  variable X in your environmentload ('ex7data1.mat');%  Visualize the example datasetplot(X(:, 1), X(:, 2), 'bo');axis([0.5 6.5 2 8]); axis square;fprintf('Program paused. Press enter to continue.\n');pause;%% =============== Part 2: Principal Component Analysis ===============%  You should now implement PCA, a dimension reduction technique. You%  should complete the code in pca.m%fprintf('\nRunning PCA on example dataset.\n\n');%  Before running PCA, it is important to first normalize X[X_norm, mu, sigma] = featureNormalize(X);%  Run PCA[U, S] = pca(X_norm);%  Compute mu, the mean of the each feature%  Draw the eigenvectors centered at mean of data. These lines show the%  directions of maximum variations in the dataset.hold on;drawLine(mu, mu + 1.5 * S(1,1) * U(:,1)', '-k', 'LineWidth', 2);drawLine(mu, mu + 1.5 * S(2,2) * U(:,2)', '-k', 'LineWidth', 2);hold off;fprintf('Top eigenvector: \n');fprintf(' U(:,1) = %f %f \n', U(1,1), U(2,1));fprintf('\n(you should expect to see -0.707107 -0.707107)\n');fprintf('Program paused. Press enter to continue.\n');pause;

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);% You need to return the following variables correctly.U = zeros(n);S = zeros(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).%sigma = X' * X / m;     %计算协方差矩阵  [U,S,V] = svd(sigma);   %利用SVD函数计算降维后的特征向量集U和对角矩阵S  % =========================================================================end

2.3 Dimensionality Reduction with PCA

使用 PCA 返回的 特征向量（ eigenvectors），将数据映射到低维空间 x(i)→z(i) (e.g., projecting the data from 2D to 1D)

%% =================== Part 3: Dimension Reduction ===================%  You should now implement the projection step to map the data onto the %  first k eigenvectors. The code will then plot the data in this reduced %  dimensional space.  This will show you what the data looks like when %  using only the corresponding eigenvectors to reconstruct it.%%  You should complete the code in projectData.m%fprintf('\nDimension reduction on example dataset.\n\n');%  Plot the normalized dataset (returned from pca)plot(X_norm(:, 1), X_norm(:, 2), 'bo');axis([-4 3 -4 3]); axis square%  Project the data onto K = 1 dimensionK = 1;Z = projectData(X_norm, U, K);fprintf('Projection of the first example: %f\n', Z(1));fprintf('\n(this value should be about 1.481274)\n\n');X_rec  = recoverData(Z, U, K);fprintf('Approximation of the first example: %f %f\n', X_rec(1, 1), X_rec(1, 2));fprintf('\n(this value should be about  -1.047419 -1.047419)\n\n');%  Draw lines connecting the projected points to the original pointshold on;plot(X_rec(:, 1), X_rec(:, 2), 'ro');for i = 1:size(X_norm, 1)    drawLine(X_norm(i,:), X_rec(i,:), '--k', 'LineWidth', 1);endhold offfprintf('Program paused. Press enter to continue.\n');pause;

2.3.1 Projecting the data onto the principal components

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);% ====================== 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);%Z = X * U(:,1:K);%计算X在新维度下的表示Z  % =============================================================end

2.3.2 Reconstructing an approximation of the data

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));% ====================== 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.%               X_rec = Z * U(:,1:K)'; %重建X,把X从K维度重建为N维度  % =============================================================end

2.4 Face Image Dataset

%% =============== Part 4: Loading and Visualizing Face Data =============%  We start the exercise by first loading and visualizing the dataset.%  The following code will load the dataset into your environment%fprintf('\nLoading face dataset.\n\n');%  Load Face datasetload ('ex7faces.mat')%  Display the first 100 faces in the datasetdisplayData(X(1:100, :));fprintf('Program paused. Press enter to continue.\n');pause;

2.4.1 PCA on Faces

%% =========== Part 5: PCA on Face Data: Eigenfaces  ===================%  Run PCA and visualize the eigenvectors which are in this case eigenfaces%  We display the first 36 eigenfaces.%fprintf(['\nRunning PCA on face dataset.\n' ...         '(this might take a minute or two ...)\n\n']);%  Before running PCA, it is important to first normalize X by subtracting %  the mean value from each feature[X_norm, mu, sigma] = featureNormalize(X);%  Run PCA[U, S] = pca(X_norm);%  Visualize the top 36 eigenvectors founddisplayData(U(:, 1:36)');fprintf('Program paused. Press enter to continue.\n');pause;

2.4.2 Dimensionality Reduction

This allows you to use your learning algorithm with a smaller input size (e.g., 100 dimensions) instead of the original 1024 dimensions. This can help speed up your learning algorithm.

%% ==== Part 7: Visualization of Faces after PCA Dimension Reduction ====%  Project images to the eigen space using the top K eigen vectors and %  visualize only using those K dimensions%  Compare to the original input, which is also displayedfprintf('\nVisualizing the projected (reduced dimension) faces.\n\n');K = 100;X_rec  = recoverData(Z, U, K);% Display normalized datasubplot(1, 2, 1);displayData(X_norm(1:100,:));title('Original faces');axis square;% Display reconstructed data from only k eigenfacessubplot(1, 2, 2);displayData(X_rec(1:100,:));title('Recovered faces');axis square;fprintf('Program paused. Press enter to continue.\n');pause;

2.5 Optional (ungraded) exercise: PCA for visualization

上面我们在3位RGB空间使用了K-means，这里我们使用PCA将3D映射为2D，以便可视化。

%% === Part 8(a): Optional (ungraded) Exercise: PCA for Visualization ===%  One useful application of PCA is to use it to visualize high-dimensional%  data. In the last K-Means exercise you ran K-Means on 3-dimensional %  pixel colors of an image. We first visualize this output in 3D, and then%  apply PCA to obtain a visualization in 2D.close all; close all; clc% Reload the image from the previous exercise and run K-Means on it% For this to work, you need to complete the K-Means assignment firstA = double(imread('bird_small.png'));% If imread does not work for you, you can try instead%   load ('bird_small.mat');A = A / 255;img_size = size(A);X = reshape(A, img_size(1) * img_size(2), 3);K = 16; max_iters = 10;initial_centroids = kMeansInitCentroids(X, K);[centroids, idx] = runkMeans(X, initial_centroids, max_iters);%  Sample 1000 random indexes (since working with all the data is%  too expensive. If you have a fast computer, you may increase this.sel = floor(rand(1000, 1) * size(X, 1)) + 1;%  Setup Color Palettepalette = hsv(K);colors = palette(idx(sel), :);%  Visualize the data and centroid memberships in 3Dfigure;scatter3(X(sel, 1), X(sel, 2), X(sel, 3), 10, colors);title('Pixel dataset plotted in 3D. Color shows centroid memberships');fprintf('Program paused. Press enter to continue.\n');pause;%% === Part 8(b): Optional (ungraded) Exercise: PCA for Visualization ===% Use PCA to project this cloud to 2D for visualization% Subtract the mean to use PCA[X_norm, mu, sigma] = featureNormalize(X);% PCA and project the data to 2D[U, S] = pca(X_norm);Z = projectData(X_norm, U, 2);% Plot in 2Dfigure;plotDataPoints(Z(sel, :), idx(sel), K);title('Pixel dataset plotted in 2D, using PCA for dimensionality reduction');fprintf('Program paused. Press enter to continue.\n');pause;