02 Multivariate Linear Regression

Posted 2021-12-17 qq-1615160629

1. h(x)
   \[
   \beginalign*h_\theta(x) =\beginbmatrix\theta_0 \hspace2em \theta_1 \hspace2em ... \hspace2em \theta_n\endbmatrix\beginbmatrixx_0 \newline x_1 \newline \vdots \newline x_n\endbmatrix= \theta^T x\endalign*, x_0^(i) = 1
   \]

2. Gradient descent equation
   \[
   \beginalign*& \textrepeat until convergence: \; \lbrace \newline \; & \theta_j := \theta_j - \alpha \frac1m \sum\limits_i=1^m (h_\theta(x^(i)) - y^(i)) \cdot x_j^(i) \; & \textfor j := 0...n\newline \rbrace\endalign*
   \]

3. 当不同特征的值差距过大\((>10^5)\)时，需要特征缩放(Feature Scaling)

   \[
   x_i := \fracx_i - \mu_is_i
   \]
   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
   \[
   \theta = (X^TX)^-1X^Ty
   \]

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).