机器学习之线性回归

设线性回归的训练集为

其中 , , m为训练集的样本个数，p为样本特征数。
作线性回归，设其回归模型为

其中均为p+1维向量。
最小化性能指标

梯度为

其中,
Matlab代码实现为

clear ; close all; clc
data = load('ex1data2.txt');
X = data(:, 1:2);
y = data(:, 3);
m = length(y);

%normalization
mu = mean(X);       %  mean value 
sigma = std(X);     %  standard deviation
X_norm  = (X - repmat(mu,size(X,1),1)) ./  repmat(sigma,size(X,1),1);

X = [ones(m, 1) X];     % Add intercept term to X

% Choose some alpha value
alpha = 0.01;
num_iters = 8500;
theta = zeros(3, 1);
J_history = zeros(num_iters, 1);

for iter = 1:num_iters
    theta = theta - alpha / m * X' * (X * theta - y); 
    J_history(iter) = computeCostMulti(X, y, theta);
end

function J = computeCostMulti(X, y, theta)
m = length(y);
J = 0;
J = sum((X * theta - y).^2) / (2*m);
end