Coursera-MachineLearning-LinearRegression-MultipleVariables

Coursera機器學習-吳恩達

% 1.對特徵進行特徵縮放：

%首先讀取訓練集數據，有兩列特徵
data = load('ex1data2.txt');
X = data(:, 1:2);
y = data(:, 3);
m = length(y);

%進行特徵縮放
[X mu sigma] = featureNormalize(X);

% 函數featureNormalize：
function [X_norm, mu, sigma] = featureNormalize(X)
X_norm = X;
mu = zeros(1, size(X, 2));
sigma = zeros(1, size(X, 2));

[m n] = size(X);
for i = 1:n
  mu(i) = mean(X_norm(:,i));        % 算每一列的平均值
  sigma(i) = std(X_norm(:,i));      % 算每一列的標準差
  
  % 計算公式：（每個特徵 - 特徵的平均值） / 特徵的標準差
  X_norm(:,i) = (X_norm(:,i) - mu(i)) / sigma(i);	
  
endfor
end

% 2.多元特徵代價函數J：

function J = computeCostMulti(X, y, theta)

m = length(y); 
J = 0;
%計算J，公式如下圖：
J = 1/(2*m) * ( X * theta - y)' * (X * theta - y);

end

% 3.theta使用梯度下降算法更新：

% 選擇學習速率和迭代次數
alpha = 0.01;
num_iters = 400;

% 初始化並計算theta
theta = zeros(3, 1);
[theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);

% 函數gradientDescentMulti：
function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)

m = length(y); 
J_history = zeros(num_iters, 1);

for iter = 1:num_iters
	% 計算J
    theta = theta - alpha/m * X' * (X * theta - y);
   
    % 每次迭代均保存代價函數J：    
    J_history(iter) = computeCostMulti(X, y, theta);

end
end

% 3.用圖表展示代價函數J和迭代次數的關係：

plot(1:numel(J_history), J_history, '-b', 'LineWidth', 2);
xlabel('Number of iterations');
ylabel('Cost J');

% 4.預測一下，如用特徵縮放處理訓練集，則測試集也需要特徵縮放

price = 0; 
% 測試集也需特徵縮放，假設函數h = X'* theta
price  = [1 (1650-mu(1))/sigma(1) (3-mu(1))/sigma(1)] * theta;   

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
         '(using gradient descent):\n $%f\n'], price);

% 5.使用正則化函數處理

data = csvread('ex1data2.txt');
X = data(:, 1:2);
y = data(:, 3);
m = length(y);
X = [ones(m, 1) X];

% 計算theta，用正則化函數
theta = normalEqn(X, y);

% 函數normalEqn：
function [theta] = normalEqn(X, y)

theta = zeros(size(X, 2), 1);
% 計算theta，正則化公式如下：
theta = pinv(X' * X) * X' * y;

end

% 6.預測一波房價

% 正則化後訓練集不用特徵縮放；假設函數h = X' * theta 
price = 0; % You should change this
price = [1 1650 3] * theta;

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
         '(using normal equations):\n $%f\n'], price);