线性回归假设函数（hypothesis for linear regression）： \(h_{\theta}(x) = \sum_{j=0}^n{\theta}_{j}x_{j} \\=\theta^Tx\)
非向量化的实现
prediction = 0.0
for j = 1:n+1 % matlab中向量下标从1开始
prediction = predictrion +
theta(j) * x(j)
end;
向量化的实现
设 \(\Theta = \begin{bmatrix} {\theta_0} \\ {\theta_1} \\ {\theta_2} \end{bmatrix}\) , \(X = \begin{bmatrix} {x_0}\\ {x_1}\\ {x_2} \end{bmatrix}\) (matlab中下标从1开始)
prediction = theta' * x;
一个线性回归问题的梯度下降的更新方法(the update rule for a gradient descent of a linear regression)
\[\theta_j := \theta_j - \alpha \frac{1}{m} \sum_{i=1}^m(h_\theta(x^ {(i)}) - y^{(i)})x_j^{(i)} \quad (for \ all \ j)\]注意 \(\theta_0\),\(\theta_1\),\(\theta_2\) … 是同时更新的
向量化的实现： \(\Theta := \Theta - \alpha \delta \\ \Theta = {[\theta_0,\, \theta_1, \, \dots\, \theta_n]}^T \\ \begin{align} \delta &= \frac{1}{m} \sum_{i=1}^m(h_{\theta}(x^{(i)})-y^{(i)})x^{(i)} \\ &= \begin{bmatrix} \delta_0 \\ \delta_1 \\ \delta_2 \\ \end{bmatrix} \end{align}\\ \delta_0 = \frac{1}{m}\sum_{i=1}^m( h_{\theta}(x^{(i)})-y^{(i)}) \, x_0^{(i)}\) 注意 \(x^{(i)}\) 是向量，\(h_{\theta}(x^{(i)})\) 和 \(y^{(i)}\) 是标量