latext in text

Author: Mallikarjunarao Kosuri

extra/latext_functions.txt

@@ -1,6 +1,91 @@
-Linear Regression
-Hypothesis
 
-Cost Fucntion
+# Supervise Learning
 
+## Linear Regression
+
+### Hypothesis
+
+### Cost Fucntion
+```
+J(\theta_0,\theta_1)=\frac{1}{2m}\sum_{i=1}^{m}(\hat{y}_i-y_i)^2=\frac{1}{2m}\sum_{i=1}^{m}(h_\theta(x_i)-y_i)^2
+```
+
+
+## Linear Regression with multiple variables
+
+### Hypothesis
+
+### Cost Fucntion
+```
 J(\theta_0,\theta_1)=\frac{1}{2m}\sum_{i=1}^{m}(\hat{y}_i-y_i)^2=\frac{1}{2m}\sum_{i=1}^{m}(h_\theta(x_i)-y_i)^2
+```
+
+### Gradient Descent
+```
+repeat \hspace*{1mm} untill \hspace*{1mm} convergence: \{\\
+\hspace*{20mm} \theta_j:=\theta_j-\alpha\frac{1}{m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)}).x_j^{(i)} \hspace*{8mm} for \hspace*{1mm} j:=0..n
+\\\hspace*{6mm}\}
+```
+
+## Logistic Regression
+
+### Hypothesis
+```
+h_\theta(x)=g(\theta^Tx)
+```
+
+### Cost Fucntion
+```
+J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}(y^{(i)}log(h_\theta(x^{(i)}))+(1-y^{(i)})log(1-h_\theta(x^{(i)}))
+```
+
+### Gradient Descent
+```
+repeat \hspace*{1mm} untill \hspace*{1mm} convergence: \{\\
+\hspace*{20mm} \theta_j:=\theta_j-\alpha\frac{1}{m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)}).x_j^{(i)} \hspace*{8mm} for \hspace*{1mm} j:=0..n
+\\\hspace*{6mm}\}
+```
+
+## Logistic Regression with multiple variables
+
+### Hypothesis
+```
+h_\theta(x)=g(\theta^Tx)
+```
+
+### Cost Fucntion
+```
+J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}[y^{(i)}log(h_\theta(x^{(i)}))+(1-y^{(i)})log(1-h_\theta(x^{(i)})]+\frac{\lambda}{2m}\sum_{j=1}^n\theta_j^2
+```
+
+### Gradient Descent
+```
+Repeat: \{
+\\
+\hspace*{20mm}\theta_0:=\theta_0-\alpha\frac{1}{m}\sum_{i=1}^m(h_\theta(x^{(i)})-y^{(i)})x_0^{(i)}
+\\
+\hspace*{20mm} \theta_j:=\theta_j-\alpha[(\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})x_j^{(i)})+\frac{\lambda}{m}\theta_j]\hspace*{8mm}j\epsilon{1,2,\dots n})
+\\
+\hspace*{6mm}\}
+
+## Nural Networks
+
+### Hypothesis
+```
+h_\theta(x)=g(\theta^Tx)
+```
+
+### Cost Fucntion
+```
+J(\Theta)=-\frac{1}{m}\sum_{i=1}^{m}\sum_{k=1}^{k}[y^{(k)}log((h_\Theta(x^{(i)}))_k)+(1-y^{(i)}_k)log(1-(h_\Theta(x^{(i)})_k)]+\frac{\lambda}{2m}\sum_{l=1}^{L-1}\sum_{i=1}^{s_l}\sum_{j=1}^{s_{l+1}}(\theta_{j,i}^{(l)})^2
+```
+
+### Gradient Descent
+```
+Repeat: \{
+\\
+\hspace*{20mm}\theta_0:=\theta_0-\alpha\frac{1}{m}\sum_{i=1}^m(h_\theta(x^{(i)})-y^{(i)})x_0^{(i)}
+\\
+\hspace*{20mm} \theta_j:=\theta_j-\alpha[(\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})x_j^{(i)})+\frac{\lambda}{m}\theta_j]\hspace*{8mm}j\epsilon{1,2,\dots n})
+\\
+\hspace*{6mm}\}