ex2-1: Logistic Regression

• 0. ex2.m
• 1. plotData.m
• 2. sigmoid.m
• 3. costFuncion.m
• 4. predict.m

0. ex2.m

약식

%% ============ Part 2: Compute Cost and Gradient ============  
% Add intercept term to x and X_test  
X = [ones(m, 1) X];  

% Initialize fitting parameters  
initial_theta = zeros(n + 1, 1);  

% Compute and display initial cost and gradient  
[cost, grad] = costFunction(initial_theta, X, y);  

%% ============= Part 3: Optimizing using fminunc  =============  
%  Set options for fminunc  
options = optimset('GradObj', 'on', 'MaxIter', 400);  

%  Run fminunc to obtain the optimal theta  
%  This function will return theta and the cost   
[theta, cost] = ...  
    fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);  
% Plot Boundary  
plotDecisionBoundary(theta, X, y);  

%% ============== Part 4: Predict and Accuracies ==============  
prob = sigmoid([1 45 85] * theta);  
fprintf(['for a student with scores 45 and 85, we predict an admission ' ...  
         'probability of %f\n'], prob);  
% Compute accuracy on our training set  
p = predict(theta, X);  
fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);

1. plotData.m

function plotData(X, y)  

% Create New Figure  
figure; hold on;  

% ====================== YOUR CODE HERE ======================  

% find indeces of positive and negative examples  
pos = find (y == 1); neg = find(y == 0);  

%plot Examples  
plot(X(pos,1), X(pos, 2), 'k+', 'LineWidth', 2, 'MarkerSize', 7);  
plot(X(neg,1), X(neg, 2), 'ko', 'MarkerFaceColor', 'y', 'MarkerSize', 7);  

% =========================================================================  
hold off;  

end

2. sigmoid.m

function g = sigmoid(z)  
%SIGMOID Compute sigmoid function  
%   g = SIGMOID(z) computes the sigmoid of z.  

% You need to return the following variables correctly   
g = zeros(size(z));  

% ====================== YOUR CODE HERE ======================  
% Instructions: Compute the sigmoid of each value of z (z can be a matrix,  
%               vector or scalar).  

g = 1 ./ (1 + e .^ -z);  

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

end

3. costFuncion.m

function [J, grad] = costFunction(theta, X, y)  

% 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));  

% ====================== YOUR CODE HERE ======================  

nr_itter = 1000;     % 100해도 결과같음  
alpha = 1;         % 때려맞춤  
max_n = size(X,2);  

hx = X * theta;  
gx = sigmoid(hx);  

% Cost function J  
pos = -y .* log(gx);  
neg = (1 - y) .* log(1 - gx);  
J = (1 / m) * sum(pos - neg);  

% Gradient descent  
for i=1:nr_itter  

    for n=1:max_n  
        grad(n,1) = (alpha / m) * sum((gx - y) .* X(:,n));  
    end  

end  

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

end

이건 틀린답이다. fminunc() 를 사용한다면 itteration과 alpha 계산이 전혀 필요없다. 따라서 vectorized된 J()와 J()의 미분계수만 구하면 된다. 답은 아래와 같다.

다음 두개의 vectorized 공식을 사용해야 한다.
$\begin{align*} & h = g(X\theta)\newline & J(\theta) = \frac{1}{m} \cdot \left(-y^{T}\log(h)-(1-y)^{T}\log(1-h)\right) \end{align*}$

-> sum() 사용하지 말고 traspose 해서 곱하면 된다. 둘다 결과물의 크기는 1X1이 됨.

% Cost function J  
pos = -y' * log(gx);  
neg = (1 - y)' * log(1 - gx);  
J = (1 / m) * (pos - neg);  

% Gradient descent  
grad = (1/m) * X' * (gx - y);

4. predict.m

function p = predict(theta, X)  
%   p = PREDICT(theta, X) computes the predictions for X using a   
%   threshold at 0.5 (i.e., if sigmoid(theta`*x) >= 0.5, predict 1)  

m = size(X, 1); % Number of training examples  

% You need to return the following variables correctly  
p = zeros(m, 1);  

% ====================== YOUR CODE HERE ======================  
gx = sigmoid(X * theta);  

for i=1:m  
    if gx(i,1) >= 0.5  
        p(i,1) = 1;  
    else  
        p(i,1) = 0;  
end  

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

end