相关概念

无监督学习

我们前面所学的逻辑回归，线性回归等都需要经过的训练，而无监督学习与我们前面所讲的不同，无监督学习的目的是学习出一个function f ，包括两种：

• 密度估计（density estimation）
• 聚类（clustering）

聚类

聚类顾名思义就是将数据分为多少类，或者给出没一类的概率，不需要提前进行训练，聚类算法就是无监督学习最常见的一种，给定一组数据， 需要聚类算法去发掘数据中的隐藏结构。
通过这张图对聚类有个初步的认识。

聚类

K-means算法

K-means算法的流程

初始数据,

选取k个聚类重心,
样本中心初始化的方法：
从样本中随机选择个使等于它们，然后随机初始化（50~1000次）选择最优的。

样本标号

更新聚类中心

重复第二步第三步直到聚类中心的变化低于阀值。
在聚类结束后，如果一个中心没有得到任何样本，那么需要去除这个中心点，或者重新 初始化。

分类中心个数K的选择

肘部法则：

肘部法则

MatlabCode

随机初始化聚类中心

Code

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
%
randidx = randperm(size(X, 1));
centroids = X(randidx(1:K), :);

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

end

将点分类，求

Code

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)
    idx(i)=1;
    for j =1:K
        if norm(X(i,:)-centroids(idx(i),:)) > norm(X(i,:)-centroids(j,:))
            idx(i)=j;
        end;
    end;
end;     

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

end

更新聚类中心

Code

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
    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

RunK-means

Code

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

运行图示

聚类结果