Exercise 2: Linear Regression

话说LaTex用起来好爽

Matlab代码

迭代并且画出拟合曲线

Linear regression 公式如下

（i是代表x的个数）

batch gradient descent update rule

x = load('L:\\MachineLearning2016\\ex2x.dat');

y = load('L:\\MachineLearning2016\\ex2y.dat');

m = length(y);

x = [ones(m, 1), x];

theta=[0,0];%row vector

figure % open a new figure window

plot(x(:,2), y, 'o');

ylabel('Height in meters')

xlabel('Age in years')

hold on % Plot new data without clearing old plot

plot(x(:,2), x*transpose(theta), '-')

legend('Training data', 'Linear regression')

%迭代方式1

 newTheta1 =theta(1,1) - transpose(x(:,1)) * (x*transpose(theta) -y) *0.07 * 0.02;

 newTheta2 =theta(1,2) - transpose(x(:,2)) * (x*transpose(theta) -y) *0.07 * 0.02;

 theta=[newTheta1,newTheta2];

 %迭代方式2

 for ii = 1:1500

     theta  = theta -  transpose( x*transpose(theta) -y  ) * x * 0.07 * 0.02;

    if rem(ii,100) == 0

    hold on % Plot new data without clearing old plot

    plot(x(:,2), x*transpose(theta), '-')

    end

 end

  hold on % 打印最后一条拟合曲线

  plot(x(:,2), x*transpose(theta), '+')

画出J(θ)的图像

Understanding J(θ)

J_vals = zeros(100, 100);   % initialize Jvals to 100x100 matrix of 0's

theta0_vals = linspace(-3, 3, 100);

theta1_vals = linspace(-1, 1, 100);

for i = 1:length(theta0_vals)

      for j = 1:length(theta1_vals)

      t = [theta0_vals(i); theta1_vals(j)];%column vector

      J_vals(i,j) = sum( (x*t' -y).^2  ) * 0.01;

    end

end

% Plot the surface plot

% Because of the way meshgrids work in the surf command, we need to

% transpose J_vals before calling surf, or else the axes will be flipped

J_vals = J_vals';

figure;

surf(theta0_vals, theta1_vals, J_vals)

xlabel('\theta_0'); ylabel('\theta_1')