h(x)
\[ \begin{align*}h_\theta(x) =\begin{bmatrix}\theta_0 \hspace{2em} \theta_1 \hspace{2em} ... \hspace{2em} \theta_n\end{bmatrix}\begin{bmatrix}x_0 \newline x_1 \newline \vdots \newline x_n\end{bmatrix}= \theta^T x\end{align*}, x_0^{(i)} = 1 \]Gradient descent equation
\[ \begin{align*}& \text{repeat until convergence:} \; \lbrace \newline \; & \theta_j := \theta_j - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0...n}\newline \rbrace\end{align*} \]当不同特征的值差距过大\((>10^5)\)时，需要特征缩放(Feature Scaling)
\[
x_i := \frac{x_i - \mu_i}{s_i} \] Where \(\mu_i\) is the average of all the values for feature(i) and \(s_i\) is the range of values(max - min), or \(s_i\) is the standard deviation.Learning Rate
In automatic convergence test, declare convergence if \(J(\theta)\) decreases by less than \(1-^{-3}\) in one iteration.Features and Polynomial Regression
可以将不同的特征值组合来更好的拟合数据，同时因为数据的组合，更加需要特征缩放来加快几何提高精度Normal Equation 正规方程 不需要特征缩放
\[ \theta = (X^TX)^{-1}X^Ty \]Comparation
Gradient Descent Normal Equation need to choose \(\alpha\) No need to choose \(\alpha\) Needs many iterations Don’t need to iterate Works well even when n is large (\(>10^4\)) Need to compute \((X^TX)^{-1}\) \(O(kn^2)\) Slow if n is very large \(O(n^3)\) If \(X^TX\) is noninvertible, the common causes might be having :
- Redundant features, where two features are very closely related (i.e. they are linearly dependent)
- Too many features (e.g. m ≤ n). In this case, delete some features or use "regularization" (to be explained in a later lesson).