斯坦福大学-回归规则化处理_Exercise Code

Regularization 规则化（过拟合处理方法：一是减少特征，二是规则化）

http://openclassroom.stanford.edu/MainFolder/DocumentPage.php?course=MachineLearning&doc=exercises/ex5/ex5.html

clear all; close all; clc

x = load('ex5Logx.dat');
y = load('ex5Logy.dat');

% Plot the training data
% Use different markers for positives and negatives
figure
pos = find(y); neg = find(y == 0);
plot(x(pos, 1), x(pos, 2), 'k+','LineWidth', 2, 'MarkerSize', 7)
hold on
plot(x(neg, 1), x(neg, 2), 'ko', 'MarkerFaceColor', 'y', 'MarkerSize', 7)

% Add polynomial features to x by
% calling the feature mapping function
% provided in separate m-file
x = map_feature(x(:,1), x(:,2));

[m, n] = size(x);

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

% Define the sigmoid function
g = inline('1.0 ./ (1.0 + exp(-z))');

% setup for Newton's method
MAX_ITR = 15;
J = zeros(MAX_ITR, 1);

% Lambda is the regularization parameter
lambda = 0;

% Newton's Method
for i = 1:MAX_ITR
% Calculate the hypothesis function
z = x * theta;
h = g(z);

% Calculate J (for testing convergence)
J(i) =(1/m)*sum(-y.*log(h) - (1-y).*log(1-h))+ ...
(lambda/(2*m))*norm(theta([2:end]))^2;

% Calculate gradient and hessian.
G = (lambda/m).*theta; G(1) = 0; % extra term for gradient
L = (lambda/m).*eye(n); L(1) = 0;% extra term for Hessian
grad = ((1/m).*x' * (h-y)) + G;
H = ((1/m).*x' * diag(h) * diag(1-h) * x) + L;

% Here is the actual update
theta = theta - H\grad;

end
% Show J to determine if algorithm has converged
J
% display the norm of our parameters
norm_theta = norm(theta)

% Plot the results
% We will evaluate theta*x over a
% grid of features and plot the contour
% where theta*x equals zero

% Here is the grid range
u = linspace(-1, 1.5, 200);
v = linspace(-1, 1.5, 200);

z = zeros(length(u), length(v));
% Evaluate z = theta*x over the grid
for i = 1:length(u)
for j = 1:length(v)
z(i,j) = map_feature(u(i), v(j))*theta;
end
end
z = z'; % important to transpose z before calling contour

% Plot z = 0
% Notice you need to specify the range [0, 0]
contour(u, v, z, [0, 0], 'LineWidth', 2)
legend('y = 1', 'y = 0', 'Decision boundary')
title(sprintf('\\lambda = %g', lambda), 'FontSize', 14)

hold off

% Uncomment to plot J
% figure
% plot(0:MAX_ITR-1, J, 'o--', 'MarkerFaceColor', 'r', 'MarkerSize', 8)
% xlabel('Iteration'); ylabel('J')