编程作业(七)
K均值算法与主成分分析算法
K均值分析算法
在本部分练习中,你将实现K均值算法并将该算法用于图像压缩。最初,你通过使用2D数据集帮助你理解K均值算法。在此之后,你将使用K均值算法以减少颜色数量的方式压缩图像。
任务一 实现K均值算法
K均值算法是一种自动将相似数据聚类在一起的方法,其基本代码如下:
% 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
上述代码中,循环分为了两步:1)将样本数据x(i)聚类至最近的聚类中心;2)计算每一聚类的均值再将样本数据重新分配。
Step 1:寻找最近的聚类中心
根据上述公式,我们在findClosestCentroids.m文件中添加如下代码:
for i = 1 : length(idx)
T = [];
for j = 1 : K
T = [T; X(i, :)];
end
[t, idx(i)] = min(sum((T - centroids) .^ 2, 2)); % sum(A, 2)表示对矩阵A按列求和
end
Step 2:计算每一聚类的均值
根据上述公式,我们在computeCentroids.m文件中添加如下代码:
for k = 1 : K
T = find(idx == k);
centroids(k, :) = sum(X(T, :)) / size(T, 1); % size(A, 1)表示求矩阵A的行数
end
最后,我们运行该任务部分代码,其中ex7.m文件中的代码如下:
%% ================= 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 = [];
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;
运行结果为:
任务二 随机初始化(该部分任务不需提交)
在实际应用中,我们更为推荐对训练集进行随机初始化操作。我们在kMeansInitCentroids.m文件中添加如下代码:
% Initialize the centroids to be random examples
% Randomly reorder the indices of examples
randidx = randperm(size(X, 1));
% Take the first K examples as centroids
centroids = X(randidx(1:K), :);
上述代码通过使用randperm()函数随机排列了样本数据的索引,然后其基于索引的随机排列选择最初的K个样本数据。这种方式避免了出现两次随机选择相同样本数据的问题。
任务三 使用K均值算法对图像压缩(该部分任务不需提交)
128*128的原始图片在24位彩色图像中,每个像素表示为红色、绿色和蓝色强度值的三个8位无符号整数(其取值范围为0~255)。我们的图像包括数千种颜色,在这部分练习中,你需将颜色的数量减少至16种。通过这种方式,可以有效地压缩图像。在该方式下,你只需存储16个所选颜色地RGB值,并且对于图像中地每个像素,你只需存储该位置地颜色索引即可。
在本练习中,你将使用K均值算法来选择用于表示压缩图像的16中颜色。具体来说,你应将原始图像中的每个像素视为样本数据,通过使用K均值算法找出最佳的16种颜色。
在Octave或 Matlab中,我们使用如下代码来导入图像:
% Load 128x128 color image (bird small.png)
A = imread('bird small.png');
% You will need to have installed the image package to used
% imread. If you do not have the image package installed, you
% should instead change the following line to
%
% load('bird small.mat'); % Loads the image into the variable A
这创建了一个三维矩阵A,其前两个索引标识像素位置,最后一个索引表示红色、绿色或蓝色。例如:A(50, 33, 3)表示行50和列33处像素的蓝色强度。
该部分代码如下:
%% ============= 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;
运行结果为:
主成分分析算法
在部分练习中,你将使用主成分分析算法实现降维。最初,你将在2D数据集上了解PCA算法,然后在一个大数据集上使用PCA算法。
任务一 可视化数据集
该部分代码如下:
%% ================== 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;
运行结果为:
任务二 实现PCA算法
PCA算法由两部分组成:首先计算数据的协方差矩阵,然后使用Octave或Matlab的svd()函数来计算特征向量U1, ···, Un。
不过在使用PCA算法之前,我们应当对训练集进行均值归一化操作,其具体实现代码可在featureNormalize.m文件中查看。
求取协方差矩阵的数学公式为:
svd()函数为:[U, S, V] = svd(Sigma)
我们在pca.m文件中添加如下代码:
Sigma = X' * X / m;
[U, S, V] = svd(Sigma);
该部分在ex7_pac.m文件的代码如下:
%% =============== 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;
运行结果为:
任务三 使用PCA算法降维
这部分你需要将数据集X中的每个数据投影到主成分矩阵U中的前K个分量上。projectData.m文件中的代码如下:
U_reduce = U(:, 1 : K);
Z = X * U_reduce;
任务四 重建数据的近似值
recoverData.m添加的代码如下:
U_reduce = U(:, 1 : K);
X_rec = Z * U_reduce';
任务三、四部分的运行结果如下:
本次编程作业需要提交的部分到此结束。文档中的后续任务,此处不再叙述。请自行查阅和动手实践,谢谢!