# Coursera: Machine Learning-Andrew NG (Week 5) [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

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

## sigmoidGradient.m

```
function g = sigmoidGradient(z)
%SIGMOIDGRADIENT returns the gradient of the sigmoid function
%evaluated at z
% g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function
% evaluated at z. This should work regardless if z is a matrix or a
% vector. In particular, if z is a vector or matrix, you should return
% the gradient for each element.
g = zeros(size(z));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the gradient of the sigmoid function evaluated at
% each value of z (z can be a matrix, vector or scalar).
g = sigmoid(z) .* (1 - sigmoid(z));
% =============================================================
end
```

`function g = sigmoidGradient(z) %SIGMOIDGRADIENT returns the gradient of the sigmoid function %evaluated at z % g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function % evaluated at z. This should work regardless if z is a matrix or a % vector. In particular, if z is a vector or matrix, you should return % the gradient for each element. g = zeros(size(z)); % ====================== YOUR CODE HERE ====================== % Instructions: Compute the gradient of the sigmoid function evaluated at % each value of z (z can be a matrix, vector or scalar). g = sigmoid(z) .* (1 - sigmoid(z)); % ============================================================= end`

## randInitializeWeights.m

```
function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
% W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights
% of a layer with L_in incoming connections and L_out outgoing
% connections.
%
% Note that W should be set to a matrix of size(L_out, 1 + L_in) as
% the column row of W handles the "bias" terms
%
% You need to return the following variables correctly
W = zeros(L_out, 1 + L_in);
% ====================== YOUR CODE HERE ======================
% Instructions: Initialize W randomly so that we break the symmetry while
% training the neural network.
%
% Note: The first row of W corresponds to the parameters for the bias units
%
% Randomly initialize the weights to small values
epsilon_init = 0.12;
W = rand(L_out, 1 + L_in) * 2 * epsilon_init - epsilon_init;
% =========================================================================
end
```

```
function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
% W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights
% of a layer with L_in incoming connections and L_out outgoing
% connections.
%
% Note that W should be set to a matrix of size(L_out, 1 + L_in) as
% the column row of W handles the "bias" terms
%
% You need to return the following variables correctly
W = zeros(L_out, 1 + L_in);
% ====================== YOUR CODE HERE ======================
% Instructions: Initialize W randomly so that we break the symmetry while
% training the neural network.
%
% Note: The first row of W corresponds to the parameters for the bias units
%
% Randomly initialize the weights to small values
epsilon_init = 0.12;
W = rand(L_out, 1 + L_in) * 2 * epsilon_init - epsilon_init;
% =========================================================================
end
```

## predictOneVsAll.m

**function [J grad] = nnCostFunction(nn_params, ...
input_layer_size, ...
hidden_layer_size, ...
num_labels, ...
X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
% [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
% X, y, lambda) computes the cost and gradient of the neural network. The
% parameters for the neural network are "unrolled" into the vector
% nn_params and need to be converted back into the weight matrices.
%
% The returned parameter grad should be a "unrolled" vector of the
% partial derivatives of the neural network.
%
% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
hidden_layer_size, (input_layer_size + 1));
Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
num_labels, (hidden_layer_size + 1));
% Setup some useful variables
m = size(X, 1);
% You need to return the following variables correctly
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));
% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
% following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
% variable J. After implementing Part 1, you can verify that your
% cost function computation is correct by verifying the cost
% computed in ex4.m
% Foward propagation
% a1 = X;
X = [ones(m,1) X]; % 5000*401
z2 = Theta1 * X'; % (25*401)*(401*5000)
a2 = sigmoid(z2); % (25*5000)
a2 = [ones(m,1) a2'];
z3 = Theta2 * a2';
h_theta = sigmoid(z3); % h_theta equals a3
% y(k) - the great trick - we need to recode the labels as vectors containing only values 0 or 1 (page 5 of ex4.pdf)
y_new = zeros(num_labels, m); % 10*5000
for i=1:m,
y_new(y(i),i)=1;
end
J = (1/m) * sum ( sum ( (-y_new) .* log(h_theta) - (1-y_new) .* log(1-h_theta) ));
% Note we should not regularize the terms that correspond to the bias.
% For the matrices Theta1 and Theta2, this corresponds to the first column of each matrix.
t1 = Theta1(:,2:size(Theta1,2));
t2 = Theta2(:,2:size(Theta2,2));
% Regularization
Reg = lambda * (sum( sum ( t1.^ 2 )) + sum( sum ( t2.^ 2 ))) / (2*m);
% Regularized cost function
J = J + Reg;
% Part 2: Implement the backpropagation algorithm to compute the gradients
% Theta1_grad and Theta2_grad. You should return the partial derivatives of
% the cost function with respect to Theta1 and Theta2 in Theta1_grad and
% Theta2_grad, respectively. After implementing Part 2, you can check
% that your implementation is correct by running checkNNGradients
%
% Note: The vector y passed into the function is a vector of labels
% containing values from 1..K. You need to map this vector into a
% binary vector of 1's and 0's to be used with the neural network
% cost function.
%
% Hint: We recommend implementing backpropagation using a for-loop
% over the training examples if you are implementing it for the
% first time.
% Back propagation
for t=1:m
% Step 1
a1 = X(t,:); % X already have a bias Line 44 (1*401)
a1 = a1'; % (401*1)
z2 = Theta1 * a1; % (25*401)*(401*1)
a2 = sigmoid(z2); % (25*1)
a2 = [1 ; a2]; % adding a bias (26*1)
z3 = Theta2 * a2; % (10*26)*(26*1)
a3 = sigmoid(z3); % final activation layer a3 == h(theta) (10*1)
% Step 2
delta_3 = a3 - y_new(:,t); % (10*1)
z2=[1; z2]; % bias (26*1)
% Step 3
delta_2 = (Theta2' * delta_3) .* sigmoidGradient(z2); % ((26*10)*(10*1))=(26*1)
% Step 4
delta_2 = delta_2(2:end); % skipping sigma2(0) (25*1)
Theta2_grad = Theta2_grad + delta_3 * a2'; % (10*1)*(1*26)
Theta1_grad = Theta1_grad + delta_2 * a1'; % (25*1)*(1*401)
end;
% Step 5
Theta2_grad = (1/m) * Theta2_grad; % (10*26)
Theta1_grad = (1/m) * Theta1_grad; % (25*401)
% Part 3: Implement regularization with the cost function and gradients.
%
% Hint: You can implement this around the code for
% backpropagation. That is, you can compute the gradients for
% the regularization separately and then add them to Theta1_grad
% and Theta2_grad from Part 2.
% Regularization
% Theta1_grad(:, 1) = Theta1_grad(:, 1) ./ m; % for j = 0
%
Theta1_grad(:, 2:end) = Theta1_grad(:, 2:end) + ((lambda/m) * Theta1(:, 2:end)); % for j >= 1
%
% Theta2_grad(:, 1) = Theta2_grad(:, 1) ./ m; % for j = 0
%
Theta2_grad(:, 2:end) = Theta2_grad(:, 2:end) + ((lambda/m) * Theta2(:, 2:end)); % for j >= 1
% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];
end**

function [J grad] = nnCostFunction(nn_params, ... input_layer_size, ... hidden_layer_size, ... num_labels, ... X, y, lambda) %NNCOSTFUNCTION Implements the neural network cost function for a two layer %neural network which performs classification % [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ... % X, y, lambda) computes the cost and gradient of the neural network. The % parameters for the neural network are "unrolled" into the vector % nn_params and need to be converted back into the weight matrices. % % The returned parameter grad should be a "unrolled" vector of the % partial derivatives of the neural network. % % Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices % for our 2 layer neural network Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ... hidden_layer_size, (input_layer_size + 1)); Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ... num_labels, (hidden_layer_size + 1)); % Setup some useful variables m = size(X, 1); % You need to return the following variables correctly J = 0; Theta1_grad = zeros(size(Theta1)); Theta2_grad = zeros(size(Theta2)); % ====================== YOUR CODE HERE ====================== % Instructions: You should complete the code by working through the % following parts. % % Part 1: Feedforward the neural network and return the cost in the % variable J. After implementing Part 1, you can verify that your % cost function computation is correct by verifying the cost % computed in ex4.m % Foward propagation % a1 = X; X = [ones(m,1) X]; % 5000*401 z2 = Theta1 * X'; % (25*401)*(401*5000) a2 = sigmoid(z2); % (25*5000) a2 = [ones(m,1) a2']; z3 = Theta2 * a2'; h_theta = sigmoid(z3); % h_theta equals a3 % y(k) - the great trick - we need to recode the labels as vectors containing only values 0 or 1 (page 5 of ex4.pdf) y_new = zeros(num_labels, m); % 10*5000 for i=1:m, y_new(y(i),i)=1; end J = (1/m) * sum ( sum ( (-y_new) .* log(h_theta) - (1-y_new) .* log(1-h_theta) )); % Note we should not regularize the terms that correspond to the bias. % For the matrices Theta1 and Theta2, this corresponds to the first column of each matrix. t1 = Theta1(:,2:size(Theta1,2)); t2 = Theta2(:,2:size(Theta2,2)); % Regularization Reg = lambda * (sum( sum ( t1.^ 2 )) + sum( sum ( t2.^ 2 ))) / (2*m); % Regularized cost function J = J + Reg; % Part 2: Implement the backpropagation algorithm to compute the gradients % Theta1_grad and Theta2_grad. You should return the partial derivatives of % the cost function with respect to Theta1 and Theta2 in Theta1_grad and % Theta2_grad, respectively. After implementing Part 2, you can check % that your implementation is correct by running checkNNGradients % % Note: The vector y passed into the function is a vector of labels % containing values from 1..K. You need to map this vector into a % binary vector of 1's and 0's to be used with the neural network % cost function. % % Hint: We recommend implementing backpropagation using a for-loop % over the training examples if you are implementing it for the % first time. % Back propagation for t=1:m % Step 1 a1 = X(t,:); % X already have a bias Line 44 (1*401) a1 = a1'; % (401*1) z2 = Theta1 * a1; % (25*401)*(401*1) a2 = sigmoid(z2); % (25*1) a2 = [1 ; a2]; % adding a bias (26*1) z3 = Theta2 * a2; % (10*26)*(26*1) a3 = sigmoid(z3); % final activation layer a3 == h(theta) (10*1) % Step 2 delta_3 = a3 - y_new(:,t); % (10*1) z2=[1; z2]; % bias (26*1) % Step 3 delta_2 = (Theta2' * delta_3) .* sigmoidGradient(z2); % ((26*10)*(10*1))=(26*1) % Step 4 delta_2 = delta_2(2:end); % skipping sigma2(0) (25*1) Theta2_grad = Theta2_grad + delta_3 * a2'; % (10*1)*(1*26) Theta1_grad = Theta1_grad + delta_2 * a1'; % (25*1)*(1*401) end; % Step 5 Theta2_grad = (1/m) * Theta2_grad; % (10*26) Theta1_grad = (1/m) * Theta1_grad; % (25*401) % Part 3: Implement regularization with the cost function and gradients. % % Hint: You can implement this around the code for % backpropagation. That is, you can compute the gradients for % the regularization separately and then add them to Theta1_grad % and Theta2_grad from Part 2. % Regularization % Theta1_grad(:, 1) = Theta1_grad(:, 1) ./ m; % for j = 0 % Theta1_grad(:, 2:end) = Theta1_grad(:, 2:end) + ((lambda/m) * Theta1(:, 2:end)); % for j >= 1 % % Theta2_grad(:, 1) = Theta2_grad(:, 1) ./ m; % for j = 0 % Theta2_grad(:, 2:end) = Theta2_grad(:, 2:end) + ((lambda/m) * Theta2(:, 2:end)); % for j >= 1 % Unroll gradients grad = [Theta1_grad(:) ; Theta2_grad(:)]; end