5-1 Please derive \(\frac{\partial h(\theta)}{\partial \theta}\), the first derivative of \(h(\theta)\).

\(h(\theta)=(\theta -2)^2, \frac{dh(\theta)}{d\theta}=2\theta-4\)

5-2 Show the values of \(\hat{\theta}^{(1)}\), \(\hat{\theta}^{(2)}\), and \(\hat{\theta}^{(3)}\) and give your observation under each of the following settings:

\(\hat{\theta}^{(1)}=\theta_0-\delta\frac{\partial h(\theta_0)}{\partial \theta_0}=5-1*(2*5-4)=-1\)

\(\hat{\theta}^{(2)}=\theta_1-\delta\frac{\partial h(\theta_1)}{\partial \theta_1}=-1-1*(2*(-1)-4)=5\)

\(\hat{\theta}^{(3)}=\theta_2-\delta\frac{\partial h(\theta_2)}{\partial \theta_2}=5-1*(2*5-4)=-1\)

會發現無法收斂

\(\hat{\theta}^{(1)}=\theta_0-\delta\frac{\partial h(\theta_0)}{\partial \theta_0}=5-0.5*(2*5-4)=2\)

\(\hat{\theta}^{(2)}=\theta_1-\delta\frac{\partial h(\theta_1)}{\partial \theta_1}=2-0.5*(2*2-4)=2\)

\(\hat{\theta}^{(3)}=\theta_2-\delta\frac{\partial h(\theta_2)}{\partial \theta_2}=2-0.5*(2*2-4)=2\)

第 1 次就可以收歛到 2

\(\hat{\theta}^{(1)}=\theta_0-\delta\frac{\partial h(\theta_0)}{\partial \theta_0}=5-0.2*(2*5-4)=3.8\)

\(\hat{\theta}^{(2)}=\theta_1-\delta\frac{\partial h(\theta_1)}{\partial \theta_1}=3.8-0.2*(2*3.8-4)=3.08\)

\(\hat{\theta}^{(3)}=\theta_2-\delta\frac{\partial h(\theta_2)}{\partial \theta_2}=3.08-0.2*(2*3.08-4)=2.648\)

發現收斂較慢