本周课程开始进入无监督学习。
一个重要应用是聚类问题：
随机找K个中心点（红×和蓝×），将样本标记为最近的中心点：
计算每个类别里样本的平均值（mean），作为新的中心点：
循环执行上面两个步骤，直到中心点不再变化，得到聚类结果：
算法伪代码如下：
注意：有可能出现某个类别中没有点的情况，这时通常就删掉这个中心点，就变成了k-1个类别。（如果还是需要k个类别，可以再重新随机出一个中心点）
这里的J也被称为Distortion函数。
目标就是找出一组中心点μ，对样本标记后得到c，使得J(c, μ) = 样本点到相应标记点的总距离平均值最小：
随机从样本点里找K个点，作为初始中心点：
随机选点的不同，K-means的结果也可能不同。右上是全局最优，右下两个都是局部最优：
为了保证效果，需要执行多次K-means，尝试多次随机选点，选出代价函数最低的。一般是50-1000次。
当k较小时，比如k=2-10时，执行多次K-means有明显效果。但当k较大时，比如k=100，多次执行可能效果也不会提高太多：
一般是靠数据可视化，或者看数据特点，人工选择的。
比如下面的样本，选择K=4或K=2都是可以的：
左图，Elbow方法：随着K的变化，找到代价函数下降速率的拐点（像肘一样）。
但实际上，有可能出现右图的形状，很难区分哪个是拐点。
所以Elbow方法仅是值得一试，不要抱太大期望。
能减少空间占用，也能加速机器学习算法。
当你有成百上千个特征时，需要去下重，即缩减高度相关的。
如下cm和inch的特征（由于有四舍五入的误差，二者并没有完美拟合在一条直线上）。
另一个例子，将飞行员的“技巧”和“兴趣”，缩减为“态度”：
三维降成二维的例子：
因为人类只能看到三维以内的图像，所以为了可视化数据，需要将特征降维到2个以内。
PCA（Principal Component Analysis）主成分分析方法降维映射后，距离最短的。
从n维降到k维：找k个向量，使数据映射到k个向量上时，映射距离最小。
使用PCA算法前，要做特征缩放、归一化等处理：
名词解释：
eigenvectors 特征向量
eigenvalues 特征值
- 计算协方差矩阵sigma使用sigma作为输入，调用函数svd(sigma)计算出eigenvectors（特征向量），[U, S, V]得到的U是n * n矩阵，取前k列，得到Ureduce ，n * k 矩阵得到新的特征：z = Ureduce’ * x
Total variation：就是数据离0的距离。
保留足够高的差异度，一般设为99%。
左图，从1开始遍历k，找到第一个符合差异度保留99%以上的k。右图，可以根据[U, S, V] = svd(Sigma)中的S快速计算差异保留度：
将样本的维度从n缩减到k。
注意：只在训练集上运行PCA，找到x->z的映射方法。以后这个映射方法可以再应用到交叉验证集和测试集。
正确运用：1. 数据压缩 2. 可视化
错误运用：缩减特征来避免过拟合
不要上来就用PCA，首先要想不用PCA怎么样？
当内存或硬盘空间不足、算法运行速度过慢时，再考虑加入PCA。
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.
%
m = size(X,1);
for i = 1:m,
minDist = inf;
for k = 1:K,
diff = X(i,:)-centroids(k,:);
dist = diff * diff';
if dist < minDist,
idx(i) = k;
minDist = dist;
end
end
end
% =============================================================
end
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,
centroids(i,:) = mean(X(idx == i, :));
end
% =============================================================
end
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; [U, S, ~] = svd(Sigma); % ========================================================================= endprojectData.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); % ============================================================= endrecoverData.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)'; % ============================================================= end
k-means：
pca映射效果：
