02 Multivariate Linear Regression

1. h(x)
   \[
   \begin{align*}h_\theta(x) =\begin{bmatrix}\theta_0 \hspace{2em} \theta_1 \hspace{2em} ... \hspace{2em} \theta_n\end{bmatrix}\begin{bmatrix}x_0 \newline x_1 \newline \vdots \newline x_n\end{bmatrix}= \theta^T x\end{align*}, x_0^{(i)} = 1
   \]
2. Gradient descent equation
   \[
   \begin{align*}& \text{repeat until convergence:} \; \lbrace \newline \; & \theta_j := \theta_j - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0...n}\newline \rbrace\end{align*}
   \]
3. 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.

4. Learning Rate
   In automatic convergence test, declare convergence if \(J(\theta)\) decreases by less than \(1-^{-3}\) in one iteration.
5. 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
6. Normal Equation 正规方程 不需要特征缩放
   \[
   \theta = (X^TX)^{-1}X^Ty
   \]
7. 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 \(O(n^3)\)

8. 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).

原文地址：https://www.cnblogs.com/QQ-1615160629/p/02-Multivariate-Linear-Regression.html