机器学习笔记之决策边界

Decision Boundary

In order to get our discrete 0 or 1 classification, we can translate the output of the hypothesis function as follows:

hθ(x)≥0.5→y=1hθ(x)<0.5→y=0

The way our logistic function g behaves is that when its input is greater than or equal to zero, its output is greater than or equal to 0.5:

g(z)≥0.5whenz≥0

Remember.

z=0,e0=1⇒g(z)=1/2z→∞,e−∞→0⇒g(z)=1z→−∞,e∞→∞⇒g(z)=0

So if our input to g is θTX, then that means:

hθ(x)=g(θTx)≥0.5whenθTx≥0

From these statements we can now say:

θTx≥0⇒y=1θTx<0⇒y=0

The decision boundary is the line that separates the area where y = 0 and where y = 1. It is created by our hypothesis function.

Example:

θ=⎡⎣5−10⎤⎦y=1if5+(−1)x1+0x2≥05−x1≥0−x1≥−5x1≤5

In this case, our decision boundary is a straight vertical line placed on the graph where x1=5, and everything to the left of that denotes y = 1, while everything to the right denotes y = 0.

Again, the input to the sigmoid function g(z) (e.g. θTX) doesn't need to be linear, and could be a function that describes a circle (e.g. z=θ0+θ1x21+θ2x22) or any shape to fit our data.