# Linear Decision Boundary

Assume $h_{\theta}(x)=g(\theta_0+\theta_1x_1+\theta_2x_2)$, and

$$\mathbf{\theta}= \begin{Bmatrix} \theta_0 \\ \theta_1 \\ \theta_2 \\ \end{Bmatrix}= \begin{Bmatrix} -3 \\ 1 \\ 1 \\ \end{Bmatrix}$$

where $g(x)$ is sigmoid function we had mentioned before. According to Wikipedia, the sigmoid function’s domain are all real numbers, with return value monotonically increasing most often from 0 to 1.

To predict $y=1$ (malignant tumor) ocurrancy, we set $h_{\theta}(x)=g(\theta_0+\theta_1x_1+\theta_2x_2)\ge0.5$. It means that when the occurancy of malignant tumor is greater or equal to $50\%$, we tell the patients the tumor is predict to be malignant.

For $h_{\theta}(x)\ge0.5$, we obtain $x\ge0$, in other words,
$$\theta_0+\theta_1x_1+\theta_2x_2=-3+1*\theta_1+1*\theta_2\ge0$$
Here comes
$$\theta_1+\theta_2\ge3\tag{1}$$

We can classify input data as:

$$x_1+x_2\lt3\Rightarrow h_{\theta}(x)\lt0.5\Rightarrow\text{benign}$$
x_1+x_2\ge3\Rightarrow h_{\theta}(x)\ge0.5\Rightarrow\text{malignant}

# Non-linear decision boundaries

Assume
$$h_{\theta}(x)=g(\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_1^2+\theta_4x_2^2)$$

if
$$\mathbf{\theta}= \begin{Bmatrix} -1 \\ 0 \\ 0 \\ 1 \\ 1 \\ \end{Bmatrix}$$

we’ll predict $y=1$ if $-1+x_1^2+x_2^2\ge0$, i.e. $x_1^2+x_2^2\ge1$ is a circle decision boundary.

Decision boundaries is a property of how we choose $\mathbf\theta$(i.e. hypothesis $h_{\theta}(x)$), NOT of the traning data set. The training data set is not what we use to define the decision boundary, they may be used to fit the parameters theta.

## How about the more complicate case…

$$h_{\theta}(x)=g(\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_1^2+\theta_4x_1^2x_2+\theta_5x_1^2x_2^2+\theta_6x_1^3x_2+…)$$