Programming Exercise 7: K-means Clustering and Principal Component Analysis
K-Means聚類
findClosestCentroids
給出若干組數據點X，矩陣X每一行表明一組數據，以及K個聚類中心centroids，尋找距離每一個點最近的聚類中心點，換言之：算法
$$\arg\ \min_{idx(1),\cdots,idx(m)}J(idx(1),\cdots,idx(m),\mu_1,\cdots,\mu_K)$$ $$=\frac 1 m \sum_{i=1}^m |x^{(i)}-\mu_{idx(i)}|^2$$app
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.
%
mindis=zeros(size(X,1),1);
mindis(:)=1e9;
idx(:)=0;
for i=1:size(X,1)
for j=1:K
nowdis=(X(i,:)-centroids(j,:))*(X(i,:)-centroids(j,:))';
if(nowdis<mindis(i))
mindis(i)=nowdis;
idx(i)=j;
end
end
end
% =============================================================
end
computeCentroids
給出若干組數據點X，矩陣X每一行表明一組數據，以及K個聚類中心centroids，更新K個聚類中心點，使得代價函數最小，換言之：dom
$$\arg\ \min_{\mu_1,\cdots,\mu_K}J(idx(1),\cdots,idx(m),\mu_1,\cdots,\mu_K)$$ $$=\frac 1 m \sum_{i=1}^m |x^{(i)}-\mu_{idx(i)}|^2$$ide
$$\mu_t:=\frac {\sum_{idx(i)=t}x^{(i)}}{\sum_{idx(i)=t}1}$$函數
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.
%
cluster_num=zeros(K,1); %cluster_num(i)=the point number of the ith cluster
for i=1:size(X,1)
centroids(idx(i),:)=centroids(idx(i),:)+X(i,:);
cluster_num(idx(i))=cluster_num(idx(i))+1;
end
for i=1:K
centroids(i,:)=centroids(i,:)/cluster_num(i);
end
% =============================================================
end
kMeansInitCentroids
隨機從全部數據點中選K個點做爲初始聚類中心點，具體看代碼oop
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
%
idx=randperm(size(X,1));
centroids=X(idx(1:K),:);
% =============================================================
end
最終測試結果
Fig 1. K-Means聚類10次迭代過程當中，3個聚類中心的變化路徑 
測試
Fig 2.保留16色、32色後壓縮獲得的圖片 this
主成分分析
pca
函數要求返回全部數據構成的協方差矩陣$\Sigma$的特徵向量U，每一個特徵向量對應的特徵值構成的對角陣Sspa
$$\Sigma=\frac 1 m X^T X$$3d
對$\Sigma$奇異值分解便可獲得U和S
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).
%
[U,S,~]=svd((X'*X)/m);
% =========================================================================
end
projectData
用前K個主成分(即U的前K列向量)對數據矩陣X降維，X的每一行表明一組數據
以前求得的U裏每一列都是單位向量，對於某組數據(列向量$x^{(i)}$)，對其降維就是將其投影到$C(u^{(1)},\cdots,u^{(K)})$子空間，即降維後的向量爲：
$$z^{(i)}=\begin{pmatrix}u^{(1)T}\\vdots\u^{(K)T}\end{pmatrix}x^{(i)}$$
令$U_{reduced}=(u^{(1)},\cdots,u^{(K)})$,則
$$Z^T=(z^{(1)},\cdots,z^{(m)})=\begin{pmatrix}u^{(1)T}\\vdots\u^{(K)T}\end{pmatrix}(x^{(1)},\cdots,x^{(m)})=U_{reduced}^TX^T$$
$$Z=XU_{reduced}$$
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);
%
Ureduced=U(:,1:K);
Z=(Ureduced'*X')';
% =============================================================
end
recoverData
給出降維到K維後的若干組數據構成的矩陣Z，Z中每一行表明一組降維後的數據，以及協方差矩陣的特徵向量(列向量)構成的矩陣U，恢復出原始數據X_rec
對於每組降維後的數據$z^{(i)}$，只需將前K個特徵向量按$z^{(i)}$線性組合便可恢復數據
$$x_{rec}^{(i)}=(u^{(1)},\cdots,u^{(K)})z^{(i)}$$
$$X_{rec}^T=(x_{rec}^{(1)},\cdots,x_{rec}^{(m)})=(u^{(1)},\cdots,u^{(K)})(z^{(1)},\cdots,z^{(m)})=U_{reduced}Z^T$$
$$X_{rec}=(U_{reduced}Z^T)^T$$
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.
%
Ureduced=U(:,1:K);
X_rec=(Ureduced*(Z'))';
% =============================================================
end
最終測試結果
Fig 1.PCA降維獲得的兩個主成分
Fig 2.原始數據點(藍色)與PCA降維後重構出的數據點(紅色)
Fig 3.PCA降維獲得的主成分臉
Fig 4.原始臉部圖像與PCA降維重構後的臉部圖像
Fig 5.原始數據點(3種特徵)
Fig 6.經PCA降維可視化後的數據(兩種特徵)
Programming Exercise 8: Anomaly Detection and Recommender Systems
單變量高斯分佈實現異常檢測
estimateGaussian
對於m組、n種特徵的數據，假設其全部特徵都是相互獨立的，$P(x|\mu;\sigma^2)$是數據x正常的機率，那麼
$$P(x|\mu;\sigma^2)=P(x_1|\mu_1;\sigma_1^2)\cdots P(x_n|\mu_n;\sigma_n^2)$$
其中
$$\mu_t=\frac 1 m \sum_{i=1}^m x_t^{(i)}$$
$$\sigma_t^2=\frac 1 m \sum_{i=1}^m (x_t^{(i)}-\mu_t)^2$$
function [mu sigma2] = estimateGaussian(X)
%ESTIMATEGAUSSIAN This function estimates the parameters of a
%Gaussian distribution using the data in X
% [mu sigma2] = estimateGaussian(X),
% The input X is the dataset with each n-dimensional data point in one row
% The output is an n-dimensional vector mu, the mean of the data set
% and the variances sigma^2, an n x 1 vector
%
% Useful variables
[m, n] = size(X);
% You should return these values correctly
mu = zeros(n, 1);
sigma2 = zeros(n, 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the mean of the data and the variances
% In particular, mu(i) should contain the mean of
% the data for the i-th feature and sigma2(i)
% should contain variance of the i-th feature.
%
mu=(mean(X))';
for i=1:n
sigma2(i)=sum((X(:,i)-mu(i)).*(X(:,i)-mu(i)))/m;
end
% =============================================================
end
selectThreshold
用預測機率pval和Ground Truth yval選取最合適的$\epsilon$，使得F1 Score最大
令Positive爲異常(出現頻率極小)，Negtive爲正常(出現頻率大)，則
TP=True Positive,預測爲Positive而且預測正確的個數
TN=True Negative,預測爲Negative而且預測正確的個數
FP=False Positive,預測爲Positive而且預測錯誤的個數
FN=False Negative,預測爲Negative而且預測錯誤的個數
Precision爲準確率(查準率)，Recall爲召回率(查全率)
$$Precision=\frac {TP}{TP+FP}$$
$$Recall=\frac {TP}{TP+FN}$$
$$F_1=2\frac {Precision·Recall}{Precision+Recall}$$
function [bestEpsilon bestF1] = selectThreshold(yval, pval)
%SELECTTHRESHOLD Find the best threshold (epsilon) to use for selecting
%outliers
% [bestEpsilon bestF1] = SELECTTHRESHOLD(yval, pval) finds the best
% threshold to use for selecting outliers based on the results from a
% validation set (pval) and the ground truth (yval).
%
bestEpsilon = 0;
bestF1 = 0;
F1 = 0;
stepsize = (max(pval) - min(pval)) / 1000;
for epsilon = min(pval):stepsize:max(pval)
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the F1 score of choosing epsilon as the
% threshold and place the value in F1. The code at the
% end of the loop will compare the F1 score for this
% choice of epsilon and set it to be the best epsilon if
% it is better than the current choice of epsilon.
%
% Note: You can use predictions = (pval < epsilon) to get a binary vector
% of 0's and 1's of the outlier predictions
predictions=(pval<epsilon);
truePositive=sum((predictions==yval)&(predictions));
falsePositive=sum((predictions~=yval)&(predictions));
trueNegative=sum((predictions==yval)&(~predictions));
falseNegative=sum((predictions~=yval)&(~predictions));
precision=truePositive/(truePositive+falsePositive);
recall=truePositive/(truePositive+falseNegative);
F1=2*(precision*recall)/(precision+recall);
% =============================================================
if F1 > bestF1
bestF1 = F1;
bestEpsilon = epsilon;
end
end
end
最終測試結果
Fig 1.用訓練樣本擬合出的高斯分佈等高線圖
Fig 2.用自動選取的閾值篩選出的異常點
基於內容的推薦系統、協同過濾算法
cofiCostFunc
代價函數
$$J(\theta^{(1)},\cdots,\theta^{(u_n)},x^{(1)},\cdots,x^{(u_m)})=$$ $$\frac 1 2 \sum_{i,j:R(i,j)=1}(\theta^{(j)T}x^{(i)}-y^{(i,j)})^2+\frac \lambda 2 \sum_{j=1}^{n_u}\sum_{k=1}^{n_f}\theta_k^{(j)2}+\frac \lambda 2 \sum_{i=1}^{n_m}\sum_{k=1}^{n_f}x_k^{(i)2}$$
梯度降低：
$$\frac {\partial J}{\partial x_k^{(i)}}=\sum_{j:R(i,j)=1}(\theta^{(j)T}x^{(i)}-y^{(i,j)})\theta_k^{(j)}+\lambda x_k^{(i)}$$
$$\frac {\partial J}{\partial \theta_k^{(j)}}=\sum_{i:R(i,j)=1}(\theta^{(j)T}x^{(i)}-y^{(i,j)})x_k^{(i)}+\lambda\theta_k^{(j)}$$
先嚐試用一堆for循環實現梯度降低的公式，發現matlab效率過低，訓練速度太慢
function [J, grad] = cofiCostFunc(params, Y, R, num_users, num_movies, ...
num_features, lambda)
%COFICOSTFUNC Collaborative filtering cost function
% [J, grad] = COFICOSTFUNC(params, Y, R, num_users, num_movies, ...
% num_features, lambda) returns the cost and gradient for the
% collaborative filtering problem.
%
% Unfold the U and W matrices from params
X = reshape(params(1:num_movies*num_features), num_movies, num_features);
Theta = reshape(params(num_movies*num_features+1:end), ...
num_users, num_features);
% You need to return the following values correctly
J = 0;
X_grad = zeros(size(X));
Theta_grad = zeros(size(Theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost function and gradient for collaborative
% filtering. Concretely, you should first implement the cost
% function (without regularization) and make sure it is
% matches our costs. After that, you should implement the
% gradient and use the checkCostFunction routine to check
% that the gradient is correct. Finally, you should implement
% regularization.
%
% Notes: X - num_movies x num_features matrix of movie features
% Theta - num_users x num_features matrix of user features
% Y - num_movies x num_users matrix of user ratings of movies
% R - num_movies x num_users matrix, where R(i, j) = 1 if the
% i-th movie was rated by the j-th user
%
% You should set the following variables correctly:
%
% X_grad - num_movies x num_features matrix, containing the
% partial derivatives w.r.t. to each element of X
% Theta_grad - num_users x num_features matrix, containing the
% partial derivatives w.r.t. to each element of Theta
%
nm=size(Y,1);
nf=size(X,2);
nu=size(Y,2);
J=sum(sum(((Theta*X'-Y').*R').^2))/2+...
(lambda*sum(sum(Theta.^2))/2)+(lambda*sum(sum(X.^2))/2);
for i=1:nm
for k=1:nf
for j=1:nu
if(R(i,j)==1)
X_grad(i,k)=X_grad(i,k)+(Theta(j,:)*(X(i,:))'-Y(i,j))*Theta(j,k);
end
end
X_grad(i,k)=X_grad(i,k)+lambda*X(i,k);
end
end
for j=1:nu
for k=1:nf
for i=1:nm
if(R(i,j)==1)
Theta_grad(j,k)=Theta_grad(j,k)+(Theta(j,:)*(X(i,:))'-Y(i,j))*X(i,k);
end
end
Theta_grad(j,k)=Theta_grad(j,k)+lambda*Theta(j,k);
end
end
% =============================================================
grad = [X_grad(:); Theta_grad(:)];
end
以後只保留一層for循環，內部其他循環所有用矩陣運算代替： 一、 $$y_{temp}=(y^{(i,j_1)},\cdots,y^{(i,j_t)}),R(i,j_1),\cdots,R(i,j_t)=1$$
$$\theta^T_{temp}=(\theta^{(j_1)},\cdots,\theta^{(j_t)}),R(i,j_1),\cdots,R(i,j_t)=1$$
則有：
$$\frac {\partial J}{\partial x_k^{(i)}}=\sum_{j:R(i,j)=1}(\theta^{(j)T}x^{(i)}-y^{(i,j)})\theta_k^{(j)}+\lambda x_k^{(i)}=\sum_{j:R(i,j)=1}(x^{(i)T}\theta^{(j)}-y^{(i,j)})\theta_k^{(j)}+\lambda x_k^{(i)}$$
$$(\frac {\partial J}{\partial x_1^{(i)}},\cdots,\frac {\partial J}{\partial x_{n_f}^{(i)}})=(x^{(i)}\theta^T_{temp}-y_{temp})\theta_{temp}+\lambda x^{(i)T}$$
二、
$$y_{temp}=(y^{(i_1,j)},\cdots,y^{(i_t,j)}),R(i_1,j),\cdots,R(i_t,j)=1$$
$$X^T_{temp}=(x^{(i_1)},\cdots,x^{(i_t)}),R(i_1,j),\cdots,R(i_t,j)=1$$
$$\frac {\partial J}{\partial \theta_k^{(j)}}=\sum_{i:R(i,j)=1}(\theta^{(j)T}x^{(i)}-y^{(i,j)})x_k^{(i)}+\lambda\theta_k^{(j)}$$
$$(\frac {\partial J}{\partial \theta_1^{(j)}},\cdots,\frac {\partial J}{\partial \theta_{n_f}^{(j)}})=(\theta^{(j)T}X^T_{temp}-y_{temp})X_{temp}+\lambda \theta^{(j)T}$$
function [J, grad] = cofiCostFunc(params, Y, R, num_users, num_movies, ...
num_features, lambda)
%COFICOSTFUNC Collaborative filtering cost function
% [J, grad] = COFICOSTFUNC(params, Y, R, num_users, num_movies, ...
% num_features, lambda) returns the cost and gradient for the
% collaborative filtering problem.
%
% Unfold the U and W matrices from params
X = reshape(params(1:num_movies*num_features), num_movies, num_features);
Theta = reshape(params(num_movies*num_features+1:end), ...
num_users, num_features);
% You need to return the following values correctly
J = 0;
X_grad = zeros(size(X));
Theta_grad = zeros(size(Theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost function and gradient for collaborative
% filtering. Concretely, you should first implement the cost
% function (without regularization) and make sure it is
% matches our costs. After that, you should implement the
% gradient and use the checkCostFunction routine to check
% that the gradient is correct. Finally, you should implement
% regularization.
%
% Notes: X - num_movies x num_features matrix of movie features
% Theta - num_users x num_features matrix of user features
% Y - num_movies x num_users matrix of user ratings of movies
% R - num_movies x num_users matrix, where R(i, j) = 1 if the
% i-th movie was rated by the j-th user
%
% You should set the following variables correctly:
%
% X_grad - num_movies x num_features matrix, containing the
% partial derivatives w.r.t. to each element of X
% Theta_grad - num_users x num_features matrix, containing the
% partial derivatives w.r.t. to each element of Theta
%
nm=size(Y,1);
nf=size(X,2);
nu=size(Y,2);
J=sum(sum(((Theta*X'-Y').*R').^2))/2+...
(lambda*sum(sum(Theta.^2))/2)+(lambda*sum(sum(X.^2))/2);
for i=1:nm
idx=find(R(i,:)==1);
Theta_tmp=Theta(idx,:);
Y_tmp=Y(i,idx);
X_grad(i,:)=(X(i,:)*Theta_tmp'-Y_tmp)*Theta_tmp+lambda*X(i,:);
end
for j=1:nu
idx=find(R(:,j)==1);
X_tmp=X(idx,:);
Y_tmp=Y(idx,j)';
Theta_grad(j,:)=(((Theta(j,:))*X_tmp')-Y_tmp)*X_tmp+lambda*Theta(j,:);
end
% =============================================================
grad = [X_grad(:); Theta_grad(:)];
end
最終測試結果