PCA and kmeans MATLAB实现
MATLAB基础知识
l Imread： 读取图片信息；
l axis：轴缩放：axis([xmin xmax ymin ymax zmin zmax cmin cmax]) 设置 x、y 和 z 轴范围以及颜色缩放范围（请参阅 caxis）。v = axis 返回包含 x、y 和 z 轴缩放因子的行矢量。v 具有 4 或 6 个分量，具体分别取决于当前坐标轴是二维还是三维。返回值是当前坐标轴的 XLim、Ylim 和 ZLim 属性。 基于 x、y 和 z 数据的最小值和最大值，axis auto 自动设置 MATLAB® 默认行为以计算当前坐标轴范围。可以将该自动行为限制为特定轴。例如，axis 'auto x' 仅自动计算 x 轴范围；axis'auto yz' 自动计算 y 和 z 轴范围。
l bsxfun
对两个数组应用基于元素的二进制操作（启用单一扩展）
用 bsxfun 减去的矩阵 A 对应列元素的列均值。
A = [1 2 10; 1 4 20;1 6 15] ;
C = bsxfun(@minus, A, mean(A))
C =
0 -2 -5
0 0 5
0 2 0
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@plus |
加 |
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@minus |
减 |
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@times |
数组乘法 |
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@rdivide |
数组右除 |
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@ldivide |
数组左除 |
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@power |
数组幂 |
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@max |
二进制最大值 |
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@min |
二进制最小值 |
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@rem |
除后的余数 |
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@mod |
除后的模数 |
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@atan2 |
四象限反切线；以弧度表示结果 |
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@atan2d |
四象限反切线；以度表示结果 |
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@hypot |
平方和的平方根 |
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@eq |
等于 |
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@ne |
不等于 |
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@lt |
小于 |
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@le |
小于或等于 |
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@gt |
大于 |
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@ge |
大于或等于 |
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@and |
按元素逻辑 AND |
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@or |
按元素逻辑 OR |
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@xor |
逻辑异 OR |
l 奇异值分解：
svd 命令计算矩阵奇异值分解。
s = svd(X) 返回奇异值的矢量。
[U,S,V] = svd(X) 生成维度与 X 相同的对角矩阵 S（包含以降序排列的非负对角线元素）以及单位矩阵 U 和 V，这样 X = U*S*V'。
[U,S,V] = svd(X,0) 生成大小合适的分解。如果 X 是 m×n（其中 m > n），则 svd 仅计算 U 的前 n 列，并且 S 是 n×n。
[U,S,V] = svd(X,'econ') 也生成大小合适的分解。如果 X 是 m×n（其中 m >= n），则它等于 svd(X,0)。对于 m < n，仅计算 V 的前 m 列并且 S 是 m×m。
A. 示例
对于矩阵
X =
1 2
3 4
5 6
7 8
语句
[U,S,V] = svd(X)
生成
U =
-0.1525 -0.8226 -0.3945 -0.3800
-0.3499 -0.4214 0.2428 0.8007
-0.5474 -0.0201 0.6979 -0.4614
-0.7448 0.3812 -0.5462 0.0407
S =
14.2691 0
0 0.6268
0 0
0 0
V =
-0.6414 0.7672
-0.7672 -0.6414
II. K-MEANS
聚类是非监督模型不带标签
KMeans算法的基本思想是初始随机给定K个簇中心，按照最邻近原则把待分类样本点分到各个簇。然后按平均法重新计算各个簇的质心，从而确定新的簇心。一直迭代，直到簇心的移动距离小于某个给定的值。
K-Means聚类算法主要分为三个步骤：
(1)第一步是为待聚类的点寻找聚类中心
(2)第二步是计算每个点到聚类中心的距离，将每个点聚类到离该点最近的聚类中去
(3)第三步是计算每个聚类中所有点的坐标平均值，并将这个平均值作为新的聚类中心
反复执行(2)、(3)，直到聚类中心不再进行大范围移动或者聚类次数达到要求为止
数据kmeans具体过程：
1：设置中心点：初始化，
2：找到每个点的所属最近簇
Function idx = findClosestCentroids(X, initial_centroids);
3:训练数据找到每个簇的数据，将数据进行归属簇i。并且重新求每个簇的中心点
4：将数据迭代计算中心点基本达到中心点不变为；迭代训练找到中心对
测试集和验证集进行验证。
III. PCA算法降维：
人脸识别，
和数据压缩（将100维的数据降到10维压缩率为90%）。
l 假设将数据的维数从 R N 降到 R 3 ，具体的 PCA 分析步骤如下：
数据压缩
l 均值归一化；第一步计算矩阵 X 的样本的协方差矩阵 S :
[X_norm, mu, sigma] = featureNormalize(X);
求平均值；和协方差
mu = mean(X);平均值
X_norm = bsxfun(@minus, X, mu);
sigma = std(X_norm);协方差
X_norm = bsxfun(@rdivide, X_norm, sigma)
l 奇异值分解：
sigma = X'*X/m;
[U, S, V] = svd(sigma);
l 找到平面上的一维平面
Z = X*U(:, 1:K);
将所有数据垂直折射到一维平面上
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
将bird 中的数据进行提取降维
1：读取数据数据归一化：
A = double(imread('bird_small.png'));
A = A / 255;
img_size = size(A);
X = reshape(A, img_size(1) * img_size(2), 3);
2：kmeans训练特征将3d数据投射到2d曲面上
sel = floor(rand(1000, 1) * size(X, 1)) + 1;
palette = hsv(K);
colors = palette(idx(sel), :);
3:画出23d的图像；
人脸识别：
l 基本原理：
1主成分分析（PCA）的原理就是将一个高维向量x,通过一个特殊的特征向量矩阵U，投影到一个低维的向量空间中，表征为一个低维向量y，并且仅仅损失了一些次要信息。也就是说，通过低维表征的向量和特征向量矩阵，可以基本重构出所对应的原始高维向量。
在人脸识别中，特征向量矩阵U称为特征脸（eigenface）空间，因此其中的特征向量ui进行量化后可以看出人脸轮廓，在下面的实验中可以看出。
以人脸识别为例，说明下PCA的应用。
设有N个人脸训练样本，每个样本由其像素灰度值组成一个向量xi，则样本图像的像素点数即为xi的维数，M=width*height ，由向量构成的训练样本集为 。
该样本集的平均向量为：
平均向量又叫平均脸。
样本集的协方差矩阵为：
求出协方差矩阵的特征向量ui和对应的特征值 ，这些特征向量组成的矩阵U就是人脸空间的正交基底，用它们的线性组合可以重构出样本中任意的人脸图像，（如果有朋友不太理解这句话的意思，请看下面的总结2。）并且图像信息集中在特征值大的特征向量中，即使丢弃特征值小的向量也不会影响图像质量。
将协方差矩阵的特征值按大到小排序： 。由大于 的 对应的特征向量构成主成分，主成分构成的变换矩阵为：
这样每一幅人脸图像都可以投影到 构成的特征脸子空间中，U的维数为M×d。有了这样一个降维的子空间，任何一幅人脸图像都可以向其作投影 ，即并获得一组坐标系数，即低维向量y，维数d×1,为称为KL分解系数。这组系数表明了图像在子空间的位置，从而可以作为人脸识别的依据。
有朋友可能不太理解，第一部分讲K-L变换的时候，求的是相关矩阵 的特征向量和特征值，这里怎么求的是协方差矩阵 ?
其实协方差矩阵也是：
，可以看出其实 用代替x就成了相关矩阵R，相当于原始样本向量都减去个平均向量，实质上还是一样的，协方差矩阵也是实对称矩阵。
总结下：
1、在人脸识别过程中，对输入的一个测试样本x，求出它与平均脸的偏差 ，则 在特征脸空间U的投影，可以表示为系数向量y：
U的维数为M×d， 的维数为M×1，y的维数d×1。若M为200*200=40000维，取200个主成分，即200个特征向量，则最后投影的系数向量y维数降维200维。
2、根据1中的式子，可以得出：
这里的x就是根据投影系数向量y重构出的人脸图像，丢失了部分图像信息，但不会影响图像质量。
Matlab 基本函数：
部分函数说明如下：
Mat Mat::reshape(int cn, int rows=0) const
该函数是改变Mat的尺寸，即保持尺寸大小=行数*列数*通道数 不变。其中第一个参数为变换后Mat的通道数，如果为0，代表变换前后通道数不变。第二个参数为变换后Mat的行数，如果为0也是代表变换前后通道数不变。但是该函数本身不复制数据。
void Mat::convertTo(OutputArray m, int rtype, double alpha=1, double beta=0 ) const
该函数其实是对原Mat的每一个值做一个线性变换。参数1为目的矩阵，参数2为目d矩阵的类型，参数3和4变换的系数，看完下面的公式就明白了：
PCA::PCA(InputArray data, InputArray mean, int flags, int maxComponents=0)
该构造函数的第一个参数为要进行PCA变换的输入Mat；参数2为该Mat的均值向量；参数3为输入矩阵数据的存储方式，如果其值为CV_PCA_DATA_AS_ROW则说明输入Mat的每一行代表一个样本，同理当其值为CV_PCA_DATA_AS_COL时，代表输入矩阵的每一列为一个样本；最后一个参数为该PCA计算时保留的最大主成分的个数。如果是缺省值，则表示所有的成分都保留。
Mat PCA::project(InputArray vec) const
该函数的作用是将输入数据vec(该数据是用来提取PCA特征的原始数据)投影到PCA主成分空间中去，返回每一个样本主成分特征组成的矩阵。因为经过PCA处理后，原始数据的维数降低了，因此原始数据集中的每一个样本的维数都变了，由改变后的样本集就组成了本函数的返回值。下面由一个图说明：
Mat PCA::backProject(InputArray vec) const
一般调用backProject（）函数前需调用project()函数，因为backProject()函数的参数vec就是经过PCA投影降维过后的矩阵dst。 因此backProject()函数的作用就是用vec来重构原始数据集（关于该函数的本质就是上面总结2的公式）。由一个图说明如下：
另外PCA类中还有几个成员变量，mean,eigenvectors, eigenvalues等分别对应着原始数据的均值，协方差矩阵的特征值和特征向量。
具体步骤：
1) 加载图片：
load ('ex7faces.mat')
displayData(X(1:100, :))
2) 降维操作
[X_norm, mu, sigma] = featureNormalize(X);
% Run PCA
[U, S] = pca(X_norm);
3) 将降维之后的特征一样的进行聚类然后进行识别为同一个人
重复操作进行分类：
X_rec = Z * U(:, 1:K)';
1:
%% 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 %定义三簇
initial_centroids = [3 3; 6 2; 8 5]; %定义每个簇的中心点
% Find the closest centroids for the examples using the
% initial_centroids
idx = findClosestCentroids(X, initial_centroids);
fprintf('Closest centroids for the firxst 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 找到每个训练集最近的簇节点 训练特征值然后改变节点位置
[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
%将数据归一化
% 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;
2:
%% 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: 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;
%% =================== 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;
%% =============== 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;
%% =========== 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 mght 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;
%% ============= Part 6: Dimension Reduction for Faces =================
% Project images to the eigen space using the top k eigenvectors
% If you are applying a machine learning algorithm
fprintf('\nDimension reduction for face dataset.\n\n');
K = 100;
Z = projectData(X_norm, U, K);
fprintf('The projected data Z has a size of: ')
fprintf('%d ', size(Z));
fprintf('\n\nProgram paused. Press enter to continue.\n');
pause;
%% ==== 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;
%% === 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
% Re-load 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;
3:
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); %接着调用找最近节点的函数指导训练结束
% 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
4:
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);
% 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,
k = find(idx==i);%注意这里不要写成一个等号，第一次就写错了
num = size(k, 1); %分类将属于每个簇的数据放到一起
centroids(i,:) = sum(X(k,:),1)/num; %将每个簇中的数据进行求中心点
end;
% =============================================================
end
5;
function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
% [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
% stored in X in a nice grid. It returns the figure handle h and the
% displayed array if requested.
% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width)
example_width = round(sqrt(size(X, 2)));
end
% Gray Image
colormap(gray);
% Compute rows, cols
[m n] = size(X);
example_height = (n / example_width);
% Compute number of items to display
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);
% Between images padding
pad = 1;
% Setup blank display
display_array = - ones(pad + display_rows * (example_height + pad), ...
pad + display_cols * (example_width + pad));
% Copy each example into a patch on the display array
curr_ex = 1;
for j = 1:display_rows
for i = 1:display_cols
if curr_ex > m,
break;
end
% Copy the patch
% Get the max value of the patch
max_val = max(abs(X(curr_ex, :)));
display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
reshape(X(curr_ex, :), example_height, example_width) / max_val;
curr_ex = curr_ex + 1;
end
if curr_ex > m,
break;
end
end
% Display Image
h = imagesc(display_array, [-1 1]);
% Do not show axis
axis image off
drawnow;
end
6:
function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X
% FEATURENORMALIZE(X) returns a normalized version of X where
% the mean value of each feature is 0 and the standard deviation
% is 1. This is often a good preprocessing step to do when
% working with learning algorithms.
mu = mean(X);
X_norm = bsxfun(@minus, X, mu);
sigma = std(X_norm);
X_norm = bsxfun(@rdivide, X_norm, sigma);
% ============================================================
end
8:
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
%
% =============================================================
end
