# Coursera: Machine Learning-Andrew NG (Week 4) [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

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

## lrCostFunction.m

```
function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with
%regularization
% J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
% theta as the parameter for regularized logistic regression and the
% gradient of the cost w.r.t. to the parameters.
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
grad = zeros(size(theta));
h_theta = sigmoid(X * theta);
% Cost
J = (1/m) * (((-1 * y') * log(h_theta)) - ((1-y') * log(1-h_theta))) + (lambda/(2*m)) * (sum(theta(2:end) .^ 2));
temp = theta;
temp(1) = 0;
grad = ((1 / m) * X' * (h_theta - y)) + (lambda / m) * temp;
grad = grad(:);
end
```

`function [J, grad] = lrCostFunction(theta, X, y, lambda) %LRCOSTFUNCTION Compute cost and gradient for logistic regression with %regularization % J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using % theta as the parameter for regularized logistic regression and the % gradient of the cost w.r.t. to the parameters. % Initialize some useful values m = length(y); % number of training examples % You need to return the following variables correctly J = 0; grad = zeros(size(theta)); h_theta = sigmoid(X * theta); % Cost J = (1/m) * (((-1 * y') * log(h_theta)) - ((1-y') * log(1-h_theta))) + (lambda/(2*m)) * (sum(theta(2:end) .^ 2)); temp = theta; temp(1) = 0; grad = ((1 / m) * X' * (h_theta - y)) + (lambda / m) * temp; grad = grad(:); end`

## oneVsAll.m

**function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta
%corresponds to the classifier for label i
% [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
% logisitc regression classifiers and returns each of these classifiers
% in a matrix all_theta, where the i-th row of all_theta corresponds
% to the classifier for label i
% Some useful variables
m = size(X, 1);
n = size(X, 2);
% You need to return the following variables correctly
all_theta = zeros(num_labels, n + 1); % 10 * 401
% Add ones to the X data matrix
X = [ones(m, 1) X];
% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
% logistic regression classifiers with regularization
% parameter lambda.
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell use
% whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
% function. It is okay to use a for-loop (for c = 1:num_labels) to
% loop over the different classes.
%
% fmincg works similarly to fminunc, but is more efficient when we
% are dealing with large number of parameters.
%
% Example Code for fmincg:
%
% Set Initial theta
initial_theta = zeros(n + 1, 1);
% Set options for fminunc
options = optimset('GradObj', 'on', 'MaxIter', 50);
% Run fmincg to obtain the optimal theta
% This function will return theta and the cost
% Variable 'X' contains data in dimension (5000 * 400).
% 5000 = Total no. of training examples, 400 = 400 pixels / training sample (digit image)
% Total no. Features = 400
for c = 1:num_labels
all_theta(c,:) = fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), initial_theta, options);
% remember y (5000*1) is an array of labels i.e. it contains actual
% digit names (y==c) will return a vector with values 0 or 1. 1 at places where y==c
% 't' is passed as dummy parameter which is initialized with 'initial_theta' first
% then subsequent values are choosen by fmincg [Note: Its not a builtin function like fminunc
% fmincg will consider all training 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
end**

**function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta
%corresponds to the classifier for label i
% [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
% logisitc regression classifiers and returns each of these classifiers
% in a matrix all_theta, where the i-th row of all_theta corresponds
% to the classifier for label i
% Some useful variables
m = size(X, 1);
n = size(X, 2);
% You need to return the following variables correctly
all_theta = zeros(num_labels, n + 1); % 10 * 401
% Add ones to the X data matrix
X = [ones(m, 1) X];
% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
% logistic regression classifiers with regularization
% parameter lambda.
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell use
% whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
% function. It is okay to use a for-loop (for c = 1:num_labels) to
% loop over the different classes.
%
% fmincg works similarly to fminunc, but is more efficient when we
% are dealing with large number of parameters.
%
% Example Code for fmincg:
%
% Set Initial theta
initial_theta = zeros(n + 1, 1);
% Set options for fminunc
options = optimset('GradObj', 'on', 'MaxIter', 50);
% Run fmincg to obtain the optimal theta
% This function will return theta and the cost
% Variable 'X' contains data in dimension (5000 * 400).
% 5000 = Total no. of training examples, 400 = 400 pixels / training sample (digit image)
% Total no. Features = 400
for c = 1:num_labels
all_theta(c,:) = fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), initial_theta, options);
% remember y (5000*1) is an array of labels i.e. it contains actual
% digit names (y==c) will return a vector with values 0 or 1. 1 at places where y==c
% 't' is passed as dummy parameter which is initialized with 'initial_theta' first
% then subsequent values are choosen by fmincg [Note: Its not a builtin function like fminunc
% fmincg will consider all training 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
end**

## predictOneVsAll.m

**function p = predictOneVsAll(all_theta, X)
%PREDICT Predict the label for a trained one-vs-all classifier. The labels
%are in the range 1..K, where K = size(all_theta, 1).
% p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions
% for each example in the matrix X. Note that X contains the examples in
% rows. all_theta is a matrix where the i-th row is a trained logistic
% regression theta vector for the i-th class. You should set p to a vector
% of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2
% for 4 examples)
m = size(X, 1);
num_labels = size(all_theta, 1); % 10
% You need to return the following variables correctly
p = zeros(size(X, 1), 1); % 5000 * 1
% Add ones to the X data matrix
X = [ones(m, 1) X]; % 5000 * 401
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned logistic regression parameters (one-vs-all).
% You should set p to a vector of predictions (from 1 to
% num_labels).
%
% Hint: This code can be done all vectorized using the max function.
% In particular, the max function can also return the index of the
% max element, for more information see 'help max'. If your examples
% are in rows, then, you can use max(A, [], 2) to obtain the max
% for each row.
%
predict = sigmoid(X*all_theta'); % 5000 * 401 by 401*10
[~, p] = max(predict, [], 2);
% M = max(A,[],dim) returns the largest elements along dimension dim.
% For example, if A is a matrix, then max(A,[],2) is a column vector
% containing the maximum value of each row.
end**

function p = predictOneVsAll(all_theta, X) %PREDICT Predict the label for a trained one-vs-all classifier. The labels %are in the range 1..K, where K = size(all_theta, 1). % p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions % for each example in the matrix X. Note that X contains the examples in % rows. all_theta is a matrix where the i-th row is a trained logistic % regression theta vector for the i-th class. You should set p to a vector % of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2 % for 4 examples) m = size(X, 1); num_labels = size(all_theta, 1); % 10 % You need to return the following variables correctly p = zeros(size(X, 1), 1); % 5000 * 1 % Add ones to the X data matrix X = [ones(m, 1) X]; % 5000 * 401 % ====================== YOUR CODE HERE ====================== % Instructions: Complete the following code to make predictions using % your learned logistic regression parameters (one-vs-all). % You should set p to a vector of predictions (from 1 to % num_labels). % % Hint: This code can be done all vectorized using the max function. % In particular, the max function can also return the index of the % max element, for more information see 'help max'. If your examples % are in rows, then, you can use max(A, [], 2) to obtain the max % for each row. % predict = sigmoid(X*all_theta'); % 5000 * 401 by 401*10 [~, p] = max(predict, [], 2); % M = max(A,[],dim) returns the largest elements along dimension dim. % For example, if A is a matrix, then max(A,[],2) is a column vector % containing the maximum value of each row. end

## predict.m

```
function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
% p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
% trained weights of a neural network (Theta1, Theta2)
% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);
% You need to return the following variables correctly
p = zeros(size(X, 1), 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned neural network. You should set p to a
% vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. In particular, the max
% function can also return the index of the max element, for more
% information see 'help max'. If your examples are in rows, then, you
% can use max(A, [], 2) to obtain the max for each row.
%
X = [ones(m, 1) X];
t1 = sigmoid(X * Theta1');
t1 = [ones(m, 1) t1];
t2 = sigmoid( t1 * Theta2');
[~, p] = max(t2, [], 2);
end
```

`function p = predict(Theta1, Theta2, X) %PREDICT Predict the label of an input given a trained neural network % p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the % trained weights of a neural network (Theta1, Theta2) % Useful values m = size(X, 1); num_labels = size(Theta2, 1); % You need to return the following variables correctly p = zeros(size(X, 1), 1); % ====================== YOUR CODE HERE ====================== % Instructions: Complete the following code to make predictions using % your learned neural network. You should set p to a % vector containing labels between 1 to num_labels. % % Hint: The max function might come in useful. In particular, the max % function can also return the index of the max element, for more % information see 'help max'. If your examples are in rows, then, you % can use max(A, [], 2) to obtain the max for each row. % X = [ones(m, 1) X]; t1 = sigmoid(X * Theta1'); t1 = [ones(m, 1) t1]; t2 = sigmoid( t1 * Theta2'); [~, p] = max(t2, [], 2); end`