K-Means和 PCA

1 K-means Clustering

首先从2D数据入手来得到一个直观的感受。

1.1 计算均值

训练集: {x(1),...,x(m)} (where x(i) ∈ Rn)
已经给出的均值算法

kMeansInitCentroids.m

% 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

我们要做的是分步实现具体的算法

1.1.1 找到最近的中心点

对于每个训练集，用如下算法
c⁽ⁱ⁾ := j that minimizes ||x⁽ⁱ⁾ − μ_j||^{2</suP>.
c(i)是距离第i个样本x(i)最近的中心点，μ_j第j个中心点的值(位置)。注意，c(i)对应代码中的idx(i)。}

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)
    A = X(i,:) - centroids;
    [v,index] = min(diag(A*A'));
    idx(i)=index;
end




% =============================================================

end

1.1.2 计算中心点均值

算法的第二阶段就该计算均值了。对于每个中心店有：
μ_k:= （1/C_k） Σ x(i) i∈C_k
具体的来说，若果x(1)和x(3)都是离k=2(第二k点)的那个点最近，那么就需要更新μ₂ = 1 /2 x``(x(3) + x(5)) (μ的下标即是第k个中心点)

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.
%
counter = zeros(K,n+1);
for i = 1:m
     counter(idx(i),:) += [X(i,:) 1];   
end


centroids =  counter(:,1:n)./counter(:,n+1);



%   =============================================================


end

1.3 随机初始化

不会两次选取同一个k数

kMeansInitCentroids.m

% 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), :); 

1.4 均值压缩图片

1.4.1 像素均值

% 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

2. 主成分分析法

2.2 PCA算法实现

在PCA之前，最好是先均值和正规化数据。

计算公式：
Σ= 1／m (X‘X)

使用[U, S, V] = svd(Sigma) 函数计算。
其中U是主成分，S是对角矩阵。

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 = (1/m).*(X'*X);
[U,S,V] = svd(sigma);




%   =========================================================================

end

2.3.1 将数据投影到主成分

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(:, K);


% =============================================================

end

2.3.2 复原数据

recoverData

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)';


% =============================================================

end

2.4 面部图像数据

2.4.1 面部主成分分析

在实例数据中，对面部图像运行PCA

图像与数学分析一致，显示出了特征