Machine Learning - Week2 Linear

单变量和多变量从矩阵运算角度讲是一样的，直接看多变量线性回归练习题。

1. featureNormalize.m (原值减去平均值再除以标准差)

function [X_norm, mu, sigma] = featureNormalize(X)

X_norm = X;

mu = zeros(1, size(X, 2));

sigma = zeros(1, size(X, 2));

mu = mean(X, 1);

sigma = std(X, 1);

n = size(X, 2);

for j=1:n,

  X_norm(:, j) = (X(:, j) - mu(j)) / sigma(j);

endfor

end

2. computeCostMulti.m  (两种计算方法)

function J = computeCostMulti(X, y, theta)

m = length(y); % number of training examples

J = 0;

% X size: m * (n+1), theta size: (n+1) * 1, => m * 1;

% Y size: m * 1

J = 1 / (2 * m) * (X * theta - y)' * (X * theta - y)

% p = X * theta;

% J = 1 / (2 * m) * sum((p - y) .^ 2)

end

3. gradientDescentMulti.m  (梯度下降，单变量多变量都一样)

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

m = length(y); % number of training examples

J_history = zeros(num_iters, 1);

for iter = 1:num_iters

    % p size: m * 1, X size: m * (n+1), theta size: (n+1) * 1

    p = X * theta;

    % theta size: (n+1) * 1, (p-y) size: m * 1, X size: m * (n+1)

    theta = theta - alpha / m * ((p - y)' * X)';

    % Save the cost J in every iteration

    J_history(iter) = computeCostMulti(X, y, theta);

end

end

4. normalEqn.m  (正规方程计算，不需迭代梯度下降)

function [theta] = normalEqn(X, y)

theta = zeros(size(X, 2), 1);

theta = pinv(X' * X) * X' * y;

end

5. ex1_multi.m

alpha = 0.1;

num_iters = 400;

% Estimate the price of a 1650 sq-ft, 3 br house 

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

% Recall that the first column of X is all-ones. Thus, it does

% not need to be normalized.

input_x = [1650 3];

norm_x = (input_x - mu) ./ sigma;

norm_x = [1, norm_x];

price = norm_x * theta; % You should change this

% Estimate the price of a 1650 sq-ft, 3 br house

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

input_x = [1, 1650, 3];

price = input_x * theta; % You should change this

梯度下降和正规方程最后预测1650 se-ft 和3 br house 结果y是差不多的：

Theta computed from gradient descent:

340412.659574

109447.795586

-6578.353971

Predicted price of a 1650 sq-ft, 3 br house (using gradient descent):

$293081.464529

Program paused. Press enter to continue.

Solving with normal equations...

Theta computed from the normal equations:

89597.909542

139.210674

-8738.019112

Predicted price of a 1650 sq-ft, 3 br house (using normal equations):

$293081.464335