ML | Why Logistic Regression in Classification ?

Using Linear Regression, all predictions >= 0.5 can be considered as 1 and rest all < 0.5 can be considered as 0. But then the question arises why classification can’t be performed using it?

Problem –

Suppose we are classifying a mail as spam or not spam and our output is y, it can be 0(spam) or 1(not spam). In case of Linear Regression, h_θ(x) can be > 1 or < 0. Although our prediction should be in between 0 and 1, the model will predict value out of the range i.e. maybe > 1 or < 0.

So, that’s why for a Classification task, Logistic/Sigmoid Regression plays its role.

$h_{\Theta} (x) = g (\Theta ^{T}x) z = \Theta ^{T}x g(z) = \frac{1}{1+e^{-z}}$

Here, we plug θ^Tx into logistic function where θ are the weights/parameters and x is the input and h_θ(x) is the hypothesis function. g() is the sigmoid function.

$h_{\Theta} (x) = P( y =1 | x ; \Theta )$

It means that y = 1 probability when x is parameterized to θ

To get the discrete values 0 or 1 for classification, discrete boundaries are defined. The hypothesis function cab be translated as

$h_{\Theta} (x) \geq 0.5 \rightarrow y = 1 h_{\Theta} (x) < 0.5 \rightarrow y = 0$

${g(z) \geq 0.5} \\ {\Rightarrow \Theta ^{T}x \geq 0.5} \\ {\Rightarrow z \geq 0.5 }$

Decision Boundary is the line that distinguishes the area where y=0 and where y=1. These decision boundaries result from the hypothesis function under consideration.

Understanding Decision Boundary with an example –
Let our hypothesis function be

$h_{\Theta}(x)= g[\Theta_{0}+ \Theta_1x_1+\Theta_2x_2+ \Theta_3x_1^2 + \Theta_4x_2^2 ]$