# Coursera: Machine Learning- Andrew NG (Week 8) [Assignment Solution]

These solutions are for reference only.try to solve on your ownbut if you get stuck in between than you can refer these solutions

## --------------------------------------------------------------------

## 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 = 1.0/m .* X' * X;
[U, S, V] = svd(Sigma);
% =========================================================================
end
```

```
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 = 1.0/m .* X' * X;
[U, S, V] = svd(Sigma);
% =========================================================================
end
```

## projectData.m

```
function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the claning data having label c (1-10 note
% 0 is mapped to 10) and find the optimal theta vector for it (Classifying white pixels with gray pixels). same
% process is repeated for other classes
end
endfunction 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);
%
for i=1:size(X, 1),
for j=1:K,
x = X(i, :)';
projection_k = x' * U(:, j);
Z(i, j) = projection_k;
end
end
% =============================================================
end
```

```
function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the claning data having label c (1-10 note
% 0 is mapped to 10) and find the optimal theta vector for it (Classifying white pixels with gray pixels). same
% process is repeated for other classes
end
endfunction 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);
%
for i=1:size(X, 1),
for j=1:K,
x = X(i, :)';
projection_k = x' * U(:, j);
Z(i, j) = projection_k;
end
end
% =============================================================
end
```

## recoverData.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.
%
for i=1:size(Z, 1),
for j=1:size(U,1),
v = Z(i, :)';
recovered_j = v' * U(j, 1:K)';
X_rec(i, j) = recovered_j;
end
end
% =============================================================
end**

**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.
%
for i=1:size(Z, 1),
for j=1:size(U,1),
v = Z(i, :)';
recovered_j = v' * U(j, 1:K)';
X_rec(i, j) = recovered_j;
end
end
% =============================================================
end**

## findClosestCentroids.m

```
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
distance_array = zeros(1,K);
for j = 1:K
distance_array(1,j) = sqrt(sum(power((X(i,:)-centroids(j,:)),2)));
end
[~, d_idx] = min(distance_array);
idx(i,1) = d_idx;
end
% =============================================================
end
```

```
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
distance_array = zeros(1,K);
for j = 1:K
distance_array(1,j) = sqrt(sum(power((X(i,:)-centroids(j,:)),2)));
end
[~, d_idx] = min(distance_array);
idx(i,1) = d_idx;
end
% =============================================================
end
```

## computeCentroids.m

```
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 k=1:K % for-loop over the centroids
centroids(k, :) = mean(X(idx==k, :));
end
% =============================================================
end
```

```
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 k=1:K % for-loop over the centroids
centroids(k, :) = mean(X(idx==k, :));
end
% =============================================================
end
```

## kMeansInitCentroids.m

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

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