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 centroids
centroids = 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 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;

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);

% 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
end
end
% =============================================================

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 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;

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 progress
if ~exist('plot_progress', 'var') || isempty(plot_progress)
plot_progress = false;
end

% Plot the data if we are plotting progress
if 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-Means
for 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 progress
if plot_progress
hold off;
end

end

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 correctly
centroids = 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 examples
randidx = randperm(size(X, 1));
% Take the first K examples as centroids
centroids = 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 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 its 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;

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 environment
load ('ex7data1.mat');

%  Visualize the example dataset
plot(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 dimension
K = 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 points
hold 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);
end
hold off

fprintf('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 dataset
load ('ex7faces.mat')

%  Display the first 100 faces in the dataset
displayData(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 found
displayData(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 displayed

fprintf('\nVisualizing the projected (reduced dimension) faces.\n\n');

K = 100;
X_rec  = recoverData(Z, U, K);

% Display normalized data
subplot(1, 2, 1);
displayData(X_norm(1:100,:));
title('Original faces');
axis square;

% Display reconstructed data from only k eigenfaces
subplot(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 first
A = 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 Palette
palette = hsv(K);
colors = palette(idx(sel), :);

%  Visualize the data and centroid memberships in 3D
figure;
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 2D
figure;
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;