My solutions to week 8 exercises :

**Part 1 : Find Closest Centroids**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) minDistance = 100000000000; minIndex = -1; for j=1:K vec = ones(size(centroids, 1)) * X(i); thisDistance = sum((X(i,:)-centroids(j, :)).^2); if thisDistance<minDistance minDistance = thisDistance; minIndex = j; end end idx(i) = minIndex; end % ============================================================= end

**Part 2 : Compute Centroid Means**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 add=zeros(1, n); count=0; for j=1:m if idx(j)==i add = add+X(j,:); count = count+1; end end centroids(i, :) = add/count; end % ============================================================= end

**Part 3 : PCA**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). % covarianceMatrixSigma = (X'*X)/m; [U S V] = svd(covarianceMatrixSigma); % ========================================================================= end

**Part 4 : Project Data**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 = X*Ureduced; % ============================================================= end

**Part 5 : Recover Data**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. % U_reducedT = U(:, 1:K)'; X_rec = Z*U_reducedT; % ============================================================= end

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