## Friday, June 12, 2015

### Neural Network Learning : Machine Learning

My solutions to Week 5 assignment questions.

```function g = sigmoidGradient(z)
%evaluated at z
%   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).

sigmoidZ = zeros(size(z, 1), size(z, 2));
for i=1:size(sigmoidZ, 1)
for j=1:size(sigmoidZ, 2)
sigmoidZ(i,j) = sigmoid(z(i,j));
end
end

oneMinus = 1-sigmoidZ;

for i=1:size(g,1)
for j=1:size(g,2)
g(i,j) = sigmoidZ(i,j)*oneMinus(i, j);
end
end

% =============================================================

end```

nnCostFunction(#1. #2. #4. #5):

```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;

% ====================== 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
%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%
%         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.
%
% 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
%               and Theta2_grad from Part 2.
%

% Ex1
X = [ones(m, 1) X];
Y = zeros(size(y, 1), num_labels);

for i=1:size(Y, 1) Y(i, y(i)) = 1; end

a2 = zeros(hidden_layer_size, m+1);
a3 = zeros(num_labels, hidden_layer_size+1);

a2 = (Theta1*X');
z2 = a2;
a2 = sigmoid(a2);

a2 = a2';
a2WithoutOnes = a2;
a2 = [ones(size(a2, 1), 1) a2];
a3 = a2*Theta2';
a3 = sigmoid(a3);

predictions = a3;
logPredictions = log(predictions);

tempLeftProd = zeros(size(a3, 1), 1);
tempRightProd = zeros(size(a3, 1), 1);

oneMinusY = 1-Y;
oneMinusPredictions = 1-predictions;

for i=1:size(a3, 1)
tempLeftProd(i) = logPredictions(i,:)*Y(i,:)';
tempRightProd(i) = log(oneMinusPredictions(i,:))*(oneMinusY(i,:)');
end

brackets = tempLeftProd+tempRightProd;
sumAllExamples = sum(brackets);
J = (-1/m)*sumAllExamples;

% Ex2

for i=1:hidden_layer_size
for j=1:size(Theta1, 2)-1
end
end

for i=1:num_labels
for j=1:size(Theta2, 2)-1
end
end

% Ex4
Delta1=0;
Delta2=0;

Theta1WithoutBias = Theta1(:, 2:end);
Theta2WithoutBias = Theta2(:, 2:end);

for t=1:m
a1 = X(t, :)';
z2 = Theta1*a1;
a2 = [1; sigmoid(z2)];
z3 = Theta2*a2;
a3 = [sigmoid(z3)];

d3 = a3-Y(t, :)';

%d2 = d2(2:end); % No need to do that. Theta2WithoutBias
% and z2(we add bias to a2, not z2) have
% already taken care of that

Delta2 = Delta2 + d3*a2';
Delta1 = Delta1 + d2*a1';
end

% Ex5 - regularization
% -------------------------------------------------------------

% =========================================================================