機(jī)器學(xué)習(xí)week8 ex7 review

這周學(xué)習(xí)K-means,并將其運(yùn)用于圖片壓縮。

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1 K-means clustering

先從二維的點(diǎn)開(kāi)始，使用K-means進(jìn)行分類。

1.1 Implement K-means

K-means步驟如上，在每次循環(huán)中，先對(duì)所有點(diǎn)更新分類，再更新每一類的中心坐標(biāo)。

1.1.1 Finding closest centroids

對(duì)每個(gè)example，根據(jù)公式：

找到距離它最近的centroid，并標(biāo)記。若有數(shù)個(gè)距離相同且均為最近，任取一個(gè)即可。
代碼如下：

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)dist = pdist([X(i,:);centroids])(:,1:K);[row, col] = find(dist == min(dist));idx(i) = col(1); end;

1.1.2 Compute centroid means

對(duì)每個(gè)centroid，根據(jù)公式：

求出該類所有點(diǎn)的平均值（即中心點(diǎn)）進(jìn)行更新。
代碼如下：

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:Kcentroids(i,:) = mean(X(find(idx == i),:)); end;% =============================================================end

1.2 K-means on example dataset

ex7.m中提供了一個(gè)例子，其中中 K 已經(jīng)被手動(dòng)初始化過(guò)了。

% 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];

如上，我們要把點(diǎn)分成三類，迭代次數(shù)為10次。三類的中心點(diǎn)初始化為.
得到如下圖像。（中間的圖像略去，只展示開(kāi)始和完成時(shí)的圖像）
這是初始圖像：

進(jìn)行10次迭代后的圖像：

可以看到三堆點(diǎn)被很好地分成了三類。圖片上同時(shí)也展示了中心點(diǎn)的移動(dòng)軌跡。

1.3 Random initialization

ex7.m中為了方便檢驗(yàn)結(jié)果正確性，給定了K的初始化。而實(shí)際應(yīng)用中，我們需要隨機(jī)初始化。
完成如下代碼：

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 %% 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), :);% =============================================================end

這樣初始的中心點(diǎn)就是從X中隨機(jī)選擇的K個(gè)點(diǎn)。

1.4 Image compression with K-means

用K-means進(jìn)行圖片壓縮。
用一張的圖片為例，采用RGB，總共需要個(gè)bit。
這里我們對(duì)他進(jìn)行壓縮，把所有顏色分成16類，以其centroid對(duì)應(yīng)的顏色代替整個(gè)一類中的顏色，可以將空間壓縮至 個(gè)bit。
用題目中提供的例子，效果大概如下：

1.5 (Ungraded)Use your own image

隨便找一張本地圖片，先用PS調(diào)整大小，最好在 以下（否則速度會(huì)很慢），運(yùn)行，效果如下：

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2 Principal component analysis

我們使用PCA來(lái)減少向量維數(shù)。

2.1 Example dataset

先對(duì)例子中的二維向量實(shí)現(xiàn)降低到一維。
繪制散點(diǎn)圖如下：

2.2 Implementing PCA

首先需要計(jì)算數(shù)據(jù)的協(xié)方差矩陣（covariance matrix）。
然后使用 Octave/MATLAB中的SVD函數(shù)計(jì)算特征向量（eigenvector）。

可以先對(duì)數(shù)據(jù)進(jìn)行normalization和feature scaling的處理。
協(xié)方差矩陣如下計(jì)算：

然后用SVD函數(shù)求特征向量。
故完成pca.m如下：

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,V] = svd(1/m * X' * X);% =========================================================================end

把求出的特征向量繪制在圖上：

2.3 Dimensionality reduction with PCA

將高維的examples投影到低維上。

2.3.1 Projecting the data onto the principal components

完成projectData.m如下：

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); %Ureduce = U(:,1:K); Z = X * Ureduce;% =============================================================end

將X投影到K維空間上。

2.3.2 Reconstructing an approximation of the data

從投影過(guò)的低維恢復(fù)高維：

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. % Ureduce = U(:, 1:K); X_rec = Z * Ureduce';% =============================================================end

2.3.3 Visualizing the projections

根據(jù)上圖可以看出，恢復(fù)后的圖只保留了其中一個(gè)特征向量上的信息，而垂直方向的信息丟失了。

2.4 Face image dataset

對(duì)人臉圖片進(jìn)行dimension reduction。ex7faces.mat中存有大量人臉的灰度圖（) , 因此每一個(gè)向量的維數(shù)是 。
如下是前一百?gòu)埲四槇D：

2.4.1 PCA on faces

用PCA得到其主成分，將其重新轉(zhuǎn)化為 的矩陣后，對(duì)其可視化，如下：(只展示前36個(gè)）

2.4.2 Dimensionality reduction

取前100個(gè)特征向量進(jìn)行投影，

可以看出，降低維度后，人臉部的大致框架還保留著，但是失去了一些細(xì)節(jié)。這給我們的啟發(fā)是，當(dāng)我們?cè)谟蒙窠?jīng)網(wǎng)絡(luò)訓(xùn)練人臉識(shí)別時(shí)，有時(shí)候可以用這種方式來(lái)提高速度。

2.5 Optional (Ungraded) exercise: PCA for visualization

PCA常用于高維向量的可視化。
如下圖，用K-means對(duì)三維空間上的點(diǎn)進(jìn)行分類。

對(duì)圖片進(jìn)行旋轉(zhuǎn)，可以看出這些點(diǎn)大致在一個(gè)平面上

因此我們使用PCA將其降低到二維，并觀察散點(diǎn)圖：

這樣就更利于觀察分類的情況了。

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轉(zhuǎn)載于:https://www.cnblogs.com/EtoDemerzel/p/7881396.html

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