Differences

This shows you the differences between two versions of the page.

--- data_mining:regression [2014/07/13 03:11] – [Learning rate $\alpha$] phreazer
+++ data_mining:regression [2019/02/10 17:12] – [Cost function] phreazer
@@ Line 17: / Line 17: @@
 ==== Cost function ====
+$\displaystyle\min_{\theta_0,\theta_1} \sum_{i=1}^m (h_\theta(x^{(i)})-y^{(i)})^2$
-$\text{minimize}_{\theta_0,\theta_1} \sum_{i=1}^m (h_\theta(x^{(i)})-y^{(i)})^2$
 Vereinfachtes Problem:
-$\text{minimize}_{\theta_0,\theta_1} \frac{1}{2*m} \sum_{i=1}^m (h_\theta(x^{(i)})-y^{(i)})^2$
+$\displaystyle\min_{\theta_0,\theta_1} \frac{1}{2*m} \sum_{i=1}^m (h_\theta(x^{(i)})-y^{(i)})^2$
 $h_\theta(x^{(i)}) = \theta_0 +\theta_1x^{(i)}$
-Cost function (Squared error cost function):
+Cost function (Squared error cost function) $J$:
 $J(\theta_0,\theta_1) = \frac{1}{2*m} \sum_{i=1}^m (h_\theta(x^{(i)})-y^{(i)})^2$
-Goal: $\text{minimize}_{\theta_0,\theta_1} J(\theta_0,\theta_1)$
+Goal: $\displaystyle\min_{\theta_0,\theta_1} J(\theta_0,\theta_1)$
 === Functions (example with only $\theta_1$): ===
@@ Line 104: / Line 103: @@
 $\theta_j := \theta_j - alpha \frac{\partial}{\partial\theta_j} J(\theta)$
+==== Normalengleichungen ====
+  * Feature-/Designmatrix X (Dim: m x (n+1))
+  * Vector y (Dim: m)
+$\theta = (X^TX)^{-1}X^Ty$
+  * Feature scaling nicht notwendig.
+Was wenn $X^TX$ singulär (nicht invertierbar)?
+(pinv in Octave)
+**Gründe für Singularität:**
+  * Redundante Features (lineare Abhängigkeit)
+  * Zu viele Features (z.B. $m <= n$)
+    * Lösung: Features weglassen oder regularisieren
+**Wann was benutzten?**
+  * m training tupel, n features
+  * GD funktioniert bei großem n (> 1000) gut, Normalengleichung muss (n x n) Matrix invertieren, liegt ungefähr in $O(n^3)$.
 ===== Gradient Descent Improvements =====